Class 12

Odisha State Board CHSE Odisha Class 12 Math Solutions Chapter 6 Probability Ex 6(b) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 6 Probability Exercise 6(b)

Question 1.
A bag contains 5 white and 3 black marbles and a second bag contains 3 white and 4 black marbles. A bag is selected at random and a marble is drawn from it. Find the probability that it is white. Assume that either bag can be chosen with the same probability.
Solution:
A bag contains 5 white and 3 black marbles and a second bag contains 3 white and 4 black marbles. A bag is selected at random and a marble is drawn from it.
Let W₁ be the event that 1st bag is choosen and a white marble is drawn and let W₂ be the event that 2nd bag is choosen and a white marble is drawn and these two events are mutually exclusive.

Question 2.
A bag contains 5 white and 3 black balls; a second bag contains 4 white and 5 black balls; a third bag contains 3 white and 6 black balls. A bag is selected at random and a ball is drawn. Find the probability that the ball is black.
(i) Do the problem assuming that the probability of choosing each bag is same.
(ii) Do the problem assuming that the probability of choosing the first bag is twice as much as choosing the second bag, which is twice as much as choosing the third bag.
Solution:
A bag contains 5 white and 3 black balls, a 2nd bag contains 4 white and 5 black balls, a 3rd bag contains 3 white and 6 black balls. A bag is selected at random and a ball is drawn.

(i) Let B₁, B₂, B₃ be the events that 1st bag is choosen and a black ball is draw. 2nd bag is choosen and a black ball is drawn, 3rd bag is drawn and a black ball is drawn. These events are mutually exclusive.
Probability of drawing a black ball
= P(B₁) + P(B₂) + P(B₃)
= \(\frac{1}{3}\) × \(\frac{3}{8}\) + \(\frac{1}{3}\) × \(\frac{5}{9}\) + \(\frac{1}{3}\) × \(\frac{6}{9}\) = \(\frac{115}{216}\)

(ii) Let the probability of choosing 1st bag be 4x. The probability of choosing the 2nd bag is 2x and that of 3rd bag is x.
It is obvious that probability of choosing 3 bags = 1
4x + 2x + x = 1 or, 7x = 1.
x = \(\frac{1}{7}\)
Probability of drawing a black ball
= \(\frac{4}{7}\) × \(\frac{3}{8}\) + \(\frac{2}{7}\) × \(\frac{5}{9}\) + \(\frac{1}{7}\) × \(\frac{6}{9}\)
= \(\frac{108+80+48}{8 \times 9 \times 7}\) = \(\frac{236}{8 \times 9 \times 7}\) = \(\frac{59}{126}\)

Question 3.
A and B play a game by alternately throwing a pair of dice. One who throws 8 wins the game. If A starts the game, find their chances of winning.
Solution:
A and B play a game by alternately throwing a pair of dice. One who throws 8 wins the game. A starts the game.
We can obtain 8 as follows:
{(6, 2), (5, 3), (4, 4) (3, 5), (6, 2)}.
∴ |S| = 6² = 36
∴ P(B) = \(\frac{5}{36}\)
⇒ P(not 8) = 1 – \(\frac{5}{36}\) = \(\frac{31}{36}\)
Since A starts the game, A can win the following situations.
(i) A throws 8
(ii) A does not throws, B does not throw 8, A throws 8,
(iii) A does not throw 8, B does not throw 8, A does not throw 8, B does not throw 8, A throws 8, etc.

Question 4.
A, B, C play a game by throwing a pair of dice in that order. One who gets 8 wins the game. If A starts the game, find their chances of winning.
Solution:
A, B, C play a game by throwing a pair of dice in that order. One who gets 8 wins the game and A starts the game.
P(B) = \(\frac{5}{36}\), P(not 8) = \(\frac{31}{36}\)

If A starts the game, then
(i) A throws 8.
(ii) A does not throw 8, B does not throw 8, C does not throw 8, A throws 8.
(iii) A does not throw 8, B does not throw 8, C does not throw 8, A does not throw 8, B does not throw 8, C does not throw 8, A throws 8, etc.

Similarly, If B wins the game, then
(i) A does not throw 8, B throw 8.
(ii) A does not throw 8, B does not throw 8, C does not throw 8, A does not throw 8, B throws 8.
(iii) A does not throw 8, B does not throw 8, C does not throw 8, A does not throw 8, B does not throw 8, C does not throw 8, A does not throw 8, B throws 8, etc.

Question 5.
There are 6 white and 4 black balls in a bag. If four are drawn successively (and not replaced), find the probability that they are alternately of different colour.
Solution:
There are 6 white and 4 black balls in a bag. Four balls are drawn without replacement. Let W and B denotes the white and black ball. There are two mutually exclusive cases WBWB and BWBW.

Question 6.
Five boys and four girls randomly stand in a line. Find the probability that no two girls come together.
Solution:
Five boys and 4 girls randomly stand in a line such that no two girls come together.
|S| = 9!

The 4 girls can stand in 6 positions in 6P4 ways. Further 5 boys can stand in 5! ways.
Probability that they will stand in a line such that no two girls come together.
= \(\frac{5 ! \times{ }^6 P_4}{9 !}\) = \(\frac{5}{42}\)

Question 7.
If you throw a pair of dice n times, find the probability of getting at least one doublet. [When you get identical members you call it a doublet. You can get a double in six ways: (1, 1), (2, 2), (3, 3), (4, 4), (5, 5) and (6, 6); thus the probability of getting a doublet is \(\frac{6}{36}\) = \(\frac{1}{6}\), so that the probability of not getting a doublet in one throw is \(\frac{5}{6}\)].
Solution:
A pair of dice is thrown n times. We get
the doublet as (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6).
Probability of getting a doublet in one throw
= \(\frac{6}{36}\) = \(\frac{1}{6}\)
Probability of not getting a doublet
= 1 – \(\frac{1}{6}\) = \(\frac{5}{6}\)
If a pair of dice is thrown n-times, the probability of not getting a doublet
= \(\left(\frac{5}{6}\right)^n\)
Probability of getting atleast one doublet
= 1 – \(\left(\frac{5}{6}\right)^n\)

Question 8.
Suppose that the probability that your alarm goes off in the morning is 0.9. If the alarm goes off, the probability is 0.8 that you attend your 8 a.m. class. If the alarm does not go off, the probability that you make your 8 a.m. class is 0.5. Find the probability that you make your 8 a.m. class.
Solution:
Let A be the event that my alarm goes off and let B be the event that I make my 8 a. m. class.
Since S = a ∪ A’, B = (B ∩ A) ∪ (B ∩ A’)
Where B ∩ A and B ∩ A’ are mutually
exclusive events.
P(B) = P (B ∩ A) + P (B ∩ A’)
= P(A). P(\(\frac{B}{A}\)) + P(A’). P(\(\frac{\mathrm{B}}{\mathrm{A}^{\prime}}\))
= 0.9 × 0.8 + 0.1 × 0.5 = 0.77

Question 9.
If a fair coin is tossed 6 times, find the probability that you get just one head.
Solution:
A fair coin is tossed 6 times.
∴ |S| = 2⁶
The six mutually exclusive events are
HTTTTT, THTTTT, TTHTTT, TTTHTT, TTTTHT, TTTTTH.
Probability of getting just one head = \(\frac{6}{2^6}\)

Question 10.
Can you generalize this situation? If a fair coin is tossed six times, find the probability of getting exactly 2 heads.
Solution:
A fair coin is tossed 6 times. Let A be the event of getting exactly 2 heads.
∴ |A| = ⁶C₂ = 15
∴ P(A) = \(\frac{15}{2^6}\)
Yes we can generalize the situation, i.e., if a fair coin is tossed n-times, then probability of getting exactly 2 heads
= \(\frac{{ }^n \mathrm{C}_2}{2^n}\) = \(\frac{{ }^6 C_2}{2^6}\)

Odisha State Board Elements of Mathematics Class 12 CHSE Odisha Solutions Chapter 7 Continuity and Differentiability Ex 7(a) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 7 Continuity and Differentiability Exercise 7(a)

Question 1.
Examine the continuity of the following functions at indicated points.
(i) f(x) = \(\left\{\begin{array}{cl}
\frac{x^2-a^2}{x-a} & \text { if } x \neq a \\
a & \text { if } x=a
\end{array}\right.\) at x = a
Solution:

(ii) f(x) = \(\left\{\begin{aligned}
\frac{\sin 2 x}{x} & \text { if } x \neq 0 \\
2 & \text { if } x=0
\end{aligned}\right.\) at x = 0
Solution:

(iii) f(x) = \(\begin{cases}(1+2 x)^{\frac{1}{x}} & \text { if } x \neq 0 \\ e^2 & \text { if } x=0\end{cases}\) at x = 0
Solution:

(iv) f(x) = \(\left\{\begin{array}{l}
x \sin \frac{1}{x} \text { if } x \neq 0 \\
0
\end{array}\right.\) at x = 0
Solution:

(v) f(x) = \(\left\{\begin{array}{l}
\frac{x^2-1}{x-1} \text { if } x \neq 1 \\
2
\end{array}\right.\) at x = 1
Solution:

(vi) f(x) = \(\begin{cases}\sin \frac{1}{x} & \text { if } x \neq a \\ 0 & \text { if } x=0\end{cases}\) at x = 0
Solution:

(vii) f(x) = [3x + 11] at x = –\(\frac{11}{3}\)
Solution:

(viii) f(x) = \(\left\{\begin{array}{l}
\frac{e^{\frac{1}{x}}-1}{e^{\frac{1}{x}}+1} \text { if } x \neq 0 \\
0
\end{array}\right.\) at x= 0
Solution:

(ix) f(x) = \(\left\{\begin{array}{l}
\frac{1}{x+[x]} \text { if } x<0 \\
-1 \quad \text { if } x \geq 0
\end{array}\right.\) at x = 0
Solution:

because [- h]is the greatest integer not exceeding – h
and so [- h ] = – 1
As L.H.L. = R.H.L. = f(0)
f(x) is cntinuous at x = 0.

(x) f(x) = \(\begin{cases}\frac{|x|}{x} & \text { if } x \neq 0 \\ 0 & \text { if } x=0\end{cases}\) at x = 0
Solution:

(xi) f(x) = \(\left\{\begin{array}{l}
2 x+1 \text { if } x \leq 0 \\
x \quad \text { if } 0<x<1 \\
2 x-1 \text { if } x \geq 1
\end{array}\right.\) at x = 0, 1
Solution:

(xii) f(x) = \(\left\{\begin{array}{l}
\frac{1}{e^{\frac{1}{x}}-1} \text { if } x>0 \\
0
\end{array} \text { if } x \leq 0\right.\) at x = 0
Solution:

(xiii) f(x) = sin\(\frac{\pi[x]}{2}\) at x = 0
Solution:

(xiv) f(x) = \(\frac{g(x)-g(1)}{x-1}\) at x = 1
Solution:
g(x) = |x – 1|
Then g(1) = |1 – 11| = 0
Now f(1) = \(\frac{g(1)-f(1)}{1-1}\) = 0/0
which we cannot determine.
Hence f(x) is discontinuous at x = 1.

Question 2.
If a function is continuous at x = a, then find
(i) \(\lim _{h \rightarrow 0}+\frac{1}{2}\{f(a+h)+f(a-h)\}\)
(ii) \(\lim _{h \rightarrow 0}+\frac{1}{2}\{f(a+h)-f(a-h)\}\)
Solution:

Question 3.
Find the value ofa such that the function f defined by \(\begin{cases}\frac{\sin a x}{\sin x} & \text { if } x \neq 0 \\ \frac{1}{a} & \text { if } x=0\end{cases}\)
is continuous at x = 0.
Solution:

Question 4.
If f(x) = \(\left\{\begin{array}{l}
a x^2+b \text { if } x<1 \\
1 \quad \text { if } x=1 \\
2 a x-b \text { if } x>1
\end{array}\right.\)
is continuous at x = 1, then find a and b.
Solution:
Let f(x) be continuous at x = 1
Then L.H.L. = R.H.L. = f(1)

Question 5.
Show that sin x is continuous for every real x.
Solution:
Let f(x) = sin x
Consider the point x = a, where ‘a’ is any real number.
Then f(a) = sin a

Thus L.H.L. = R.H.L. = f(a)
Hence f(x) = sin x is continuous for every real x.
(Proved)

Question 6.
Show that the function f defined by \(\left\{\begin{array}{l}
1 \text { if } x \text { is rational } \\
0 \text { if } x \text { is irrational }
\end{array}\right.\) is discontinuous ∀ ≠ 0 ∈ R.
Solution:
Consider any real point x = a
If a is rational then f(a) = 1.
Again lim_x→a+f(x) = lim_h→0f(a + h)
which does not exist because a + h may be rational or irrational
Similarly lim_x→a-f(x) does not exist.
Thus f(x) is discontinuous at any rational point. Similarly we can show that f(x) is discontinuous at any irrational point.
Hence f(x) is discontinuous for all x ∈ R
(Proved)

Question 7.
Show that the function f defined by f(x) = \(f(x)=\left\{\begin{array}{l}
x \text { if } x \text { is rational } \\
-x \text { if } x \text { is irrational }
\end{array}\right.\)
is continuous at x = 0 and discontinuous ∀ x ≠ 0 ∈ R.
Solution:
f(0) = 0
L.H.L. = lim_x→0f(x) = lim_h→0f(-h)
= \(\lim _{h \rightarrow 0} \begin{cases}-h & \text { if } h \text { is rational } \\ h & \text { if } h \text { is irrational }\end{cases}\) = 0
Similarly R.H.L. = 0
Thus L.H.L. = R.H.L. = f(0)
Hence f(x) is continuous at x = 0.
We can easily show that f(x) is discontinuous at all real points x ≠ 0.

Question 8.
Show that the function f defined by
f(x) = \(\left\{\begin{array}{l}
x \text { if } x \text { is rational } \\
0 \text { if } x \text { is irrational }
\end{array}\right.\)
is discontinuous everywhere except at x = 0.
Solution:
f(0) = 0
L.H.L. = lim_h→0f(-h)
= lim_h→0\(\begin{cases}-h & \text { if }-h \text { is rational } \\ 0 & \text { if }-h \text { is irrational }\end{cases}\)
Similarly R.H.L. = 0
Thus L.H.L. = R.H.L = f(0)
Hence f(x) is continuous at x = 0.
Let a be any real number except 0.
If a is rational then f(a) = a.
L.H.L. = lim_h→0f(a – h) which does not exist because a – h may be rational or may be irrational.
Similarly R.H.L. does not exist.
Thus f(x) is discontinuous at any rational point x – a ≠ 0.
Similarly f(x) is discontinuous at any irrational point.
Hence f(x) is discontinuous everywhere except at x = 0.
(Proved)

Question 9.
Show that f(x) = \(\begin{cases}x \sin \frac{1}{x}, & x \neq 0 \\ 0, & x=0\end{cases}\) is continuous at x = 0.
Solution:
Refer to No. 1(iv) of Exercise – 7(a).

Question 10.
Prove that e^x – 2 = 0 has a solution between 0 and 1. [Hints: Use continuity of e^x– 2 and fact – 2]
Solution:

∴ f(x) is continuous in [0, 1]
f(0). f(1) = (-1) (e – 2) < 0
∴ f(x) has a zero between 0 and 1
i.e. e^x – 2 = 0 has a solution between 0 and 1

Question 11.
So that x⁵ + x +1 = 0 for some value of x between -1 and 0.
Solution:
Let f(x) = x⁵ + x + 1 and any a ∈ (-1, 0)
f(a) = a⁵ + a + 1

= -1 = f(-1)
∴ f is continuous on [-1, 0]
But f(-1) f(0) = 1 × -1 < 0
∴ f has a zero between -1 and 0
⇒ x5 + x + 1 = 0 for some value of x between -1 and 0.

Odisha State Board CHSE Odisha Class 12 Math Solutions Chapter 6 Probability Additional Exercise Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 6 Probability Additional Exercise

(A) Multiple Choice Questions (Mcqs) With Answers

Question 1.
If \(\left|\begin{array}{ccc}
1+x & x & x^2 \\
x & 1+x & x^2 \\
x^2 & x & 1+x
\end{array}\right|\) = a + bx + cx² + dx³ + ex⁴ + fx⁵ then write the value of a.
(a) 0
(b) 2
(c) 1
(d) 3
Answer:
(c) 1

Question 2.
If every element of a third order determinant of value 8 is multiplied by 2, then write the value of the new determinant.
(a) 32
(b) 64
(c) 16
(d) 128
Answer:
(b) 64

Question 3.
If A is a 4 x 5 matrix and B is a matrix such that A^TB and BA^T both are defined, then write the order of B
(a) 4 x 5
(b) 1 x 5
(c) 5 x 4
(d) None of these
Answer:
(a) 4 x 5

Question 4.
If \(\left[\begin{array}{lll}
3 & 5 & 3 \\
2 & 4 & 2 \\
\lambda & 7 & 8
\end{array}\right]\) is a singular matrix, write die value of 1.
(a) λ = 2
(b) λ = 1
(c) λ = 4
(d) λ = 8
Answer:
(d) λ = 8

Question 5.
Determine the maximum value of \(\left|\begin{array}{rl}
\cos x & \sin x \\
-\sin x & \cos x-1
\end{array}\right|\)
(a) 1
(b) 2
(c) 3
(d) 0
Answer:
(b) 2

Question 6.
If \(\left[\begin{array}{cc}
x & y \\
x & \frac{x}{2}+t
\end{array}\right]\) + \(\left[\begin{array}{cc}
y & x+t \\
x+2 & \frac{x}{2}
\end{array}\right]\) = \(\left[\begin{array}{ll}
1 & 4 \\
2 & 3
\end{array}\right]\) then find x.
(a) x = 1
(b) x = 0
(c) x = 2
(d) x = -1
Answer:
(b) x = 0

Question 7.
If \(\left[\begin{array}{cc}
x & y \\
x & \frac{x}{2}+t
\end{array}\right]\) + \(\left[\begin{array}{cc}
y & x+t \\
x+2 & \frac{x}{2}
\end{array}\right]\) = \(\left[\begin{array}{ll}
1 & 4 \\
2 & 3
\end{array}\right]\) then find y.
(a) y = 1
(b) y = 3
(c) y = 2
(d) y = 0
Answer:
(a) y = 1

Question 8.
If \(\left[\begin{array}{cc}
x & y \\
x & \frac{x}{2}+t
\end{array}\right]\) + \(\left[\begin{array}{cc}
y & x+t \\
x+2 & \frac{x}{2}
\end{array}\right]\) = \(\left[\begin{array}{ll}
1 & 4 \\
2 & 3
\end{array}\right]\) then find t.
(a) t = 1
(b) t = 2
(c) t = 3
(d) t = 0
Answer:
(c) t = 3

Question 9.
Which matrix is a unit matrix?
(a) \(\left(\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right)\)
(b) \(\left(\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right)\)
(c) \(\left(\begin{array}{ll}
1 & 1 \\
0 & 1
\end{array}\right)\)
(d) \(\left(\begin{array}{ll}
1 & 1 \\
1 & 1
\end{array}\right)\)
Answer:
(b) \(\left(\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right)\)

Question 10.
If \(\left(\begin{array}{cc}
\mathbf{x}_1 & \mathbf{x}_2 \\
\mathbf{y}_1 & \mathbf{y}_2
\end{array}\right)\) – \(\left(\begin{array}{ll}
2 & 3 \\
0 & 1
\end{array}\right)\) = \(\left(\begin{array}{ll}
3 & 5 \\
1 & 2
\end{array}\right)\) then find x₁, x₂, y₁, y₂.
(a) x₁ = 8, x₂ = 5, y₁ = 3, y₂ = 1
(b) x₁ = 1, x₂ = 8, y₁ = 5, y₂ = 3
(c) x₁ = 5, x₂ = 8, y₁ = 1, y₂ = 3
(d) x₁ = 3, x₂ = 1, y₁ = 8, y₂ = 5
Answer:
(c) x₁ = 5, x₂ = 8, y₁ = 1, y₂ = 3

Question 11.
If \(\left|\begin{array}{ll}
2 & 4 \\
k & 6
\end{array}\right|\) = 0, what is the value of k?
(a) 3
(b) 4
(c) 2
(d) 6
Answer:
(a) 3

Question 12.
If \(\left|\begin{array}{ll}
\mathbf{a}_1 & \mathbf{b}_1 \\
\mathbf{c}_1 & \mathbf{d}_1
\end{array}\right|\) = k \(\left|\begin{array}{ll}
a_1 & c_1 \\
b_1 & d_1
\end{array}\right|\) hen what is the value of k?
(a) 1
(b) 2
(c) 3
(d) 4
Answer:
(d) 4

Question 13.
If A = \(\left(\begin{array}{lll}
1 & 0 & 2 \\
5 & 1 & x \\
1 & 1 & 1
\end{array}\right)\) is a singular matrix then what is the value of x?
(a) 6
(b) 7
(c) 8
(d) 9
Answer:
(d) 9

Question 14.
Evaluate \(\left|\begin{array}{ccc}
-6 & 0 & 0 \\
3 & -5 & 7 \\
2 & 8 & 11
\end{array}\right|\)
(a) 66
(b) 666
(c) 6666
(d) 6
Answer:
(b) 666

Question 15.
Evaluate \(\left|\begin{array}{lll}
1 & 1 & b+c \\
1 & b & c+a \\
1 & c & a+b
\end{array}\right|\)
(a) 0
(b) 1
(c) 11
(d) 2
Answer:
(a) 0

Question 16.
Evaluate \(\left|\begin{array}{ccc}
1^2 & 2^2 & 3^2 \\
2^2 & 3^2 & 4^2 \\
3^2 & 4^2 & 5^2
\end{array}\right|\)
(a) 54
(b) 58
(c) -54
(d) 60
Answer:
(c) -54

Question 17.
If A and B are square matrices of order 3, such that |A| = -1, |B| = 3 then |3 AB| = –
(a) 1
(b) 11
(c) 9
(d) 81
Answer:
(d) 81

Question 18.
For what k
x + 2y – 3z = 2
(k + 3)z = 3
(2k + 1)y + z = 2 is inconsistent?
(a) -3
(b) -6
(c) 3
(d) 6
Answer:
(a) -3

Question 19.
The sum of two nonintegral roots of \(\left|\begin{array}{lll}
x & 2 & 5 \\
3 & x & 3 \\
5 & 4 & x
\end{array}\right|\) = 0 is ______.
(a) 5
(b) -5
(c) 3
(d) 15
Answer:
(b) -5

Question 20.
The value of \(\left|\begin{array}{ccc}
1 & 2 & 3 \\
3 & 5 & 2 \\
8 & 14 & 20
\end{array}\right|\) is ______.
(a) 1
(b) 2
(c) 0
(d) 3
Answer:
(c) 0

Question 21.
If [x 1] \(\left[\begin{array}{cc}
1 & 0 \\
-2 & 0
\end{array}\right]\) = 0, then x equals:
(a) 0
(b) -2
(c) -1
(d) 2
Answer:
(d) 2

Question 22.
The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is:
(a) 27
(b) 18
(c) 81
(d) 512
Answer:
(d) 512

Question 23.
If A = \(\left[\begin{array}{cc}
\cos \alpha & -\sin \alpha \\
\sin \alpha & \cos \alpha
\end{array}\right]\) , and A + A’ = I, then the value of α is
(a) \(\frac{\pi}{6}\)
(b) \(\frac{\pi}{3}\)
(c) π
(d) \(\frac{3 \pi}{2}\)
Answer:
(b) \(\frac{\pi}{3}\)

Question 24.
Matrix A and B will be inverse of each other only if
(a) AB = BA
(b) AB = BA = 0
(c) AB = 0, BA = I
(d) AB = BA = I
Answer:
(d) AB = BA = I

Question 25.
The matrix P = \(\left[\begin{array}{lll}
0 & 0 & 4 \\
0 & 4 & 0 \\
4 & 0 & 0
\end{array}\right]\) is a
(a) square matrix
(b) diagonal matrix
(c) unit matrix
(d) None of these
Answer:
(a) square matrix

Question 26.
If A and B are symmetric matrices of same order, then AB – BA is a
(a) Skew-symmetric matrix
(b) Symmetric matrix
(c) Zero matrix
(d) Identity
Answer:
(a) Skew-symmetric matrix

Question 27.
If A is a square matrix of order 3, such that A(adj A) = 10I, then |adj A| is equal to
(a) 1
(b) 10
(c) 100
(d) 1000
Answer:
(c) 100

Question 28.
Let A be a square matrix of order 2 × 2, then |KA| is equal to
(a) K|A|
(b) K²|A|
(c) K³|A|
(d) 2K|A|
Answer:
(b) K²|A|

Question 29.
If A and B are invertible matrices then which of the following is not correct
(a) Adj A = |A|. A^-1
(b) det (A^-1) = (det A)^-1
(c) (AB)^-1 = B^-1A^-1
(d) (A + B)^-1 = A^-1 + B^-1
Answer:
(d) (A + B)^-1 = A^-1 + B^-1

Question 30.
If A is a skew-symmetric matrix of order 3, then the value of |A| is
(a) 3
(b) 0
(c) 9
(d) 27
Answer:
(b) 0

Question 31.
If A is a square matrix of order 3, such that A(adjA) = 10I, then ladj Al is equal to
(a) 1
(b) 10
(c) 100
(d) 1000
Answer:
(c) 100

Question 32.
Let A be a non-angular square matrix of order 3 x 3, then |A. adj Al is equal to
(a) |A|³
(b) |A|²
(c) |A|
(d) 3|A|
Answer:
(a) |A|³

Question 33.
Let A be a square matrix of order 3 × 3 and k a scalar, then |kA| is equal to
(a) k|A|
(b) |k||A|
(c) k³|A|
(d) none of these
Answer:
(c) k³|A|

Question 34.
If a, b, c are all distinct, and \(\left|\begin{array}{lll}
a & a^2 & 1+a^3 \\
b & b^2 & 1+b^3 \\
c & c^2 & 1+c^3
\end{array}\right|\) = 0 then the value of abc is
(a) 0
(b) -1
(c) 3
(d) -3
Answer:
(b) -1

Question 35.
If a, b, c are in AP, then the value of \(\left|\begin{array}{lll}
x+1 & x+2 & x+a \\
x+2 & x+3 & x+b \\
x+3 & x+4 & x+c
\end{array}\right|\) is:
(a) 4
(b) -3
(c) 0
(d) abc
Answer:
(c) 0

Question 36.
If A is a skew-symmetric matrix of order 3, then the value of |A| is
(a) 3
(b) 0
(c) 9
(d) 27
Answer:
(b) 0

Question 37.
A bag contains 3 white, 4 black and 2 red balls. If 2 balls are choosen at random (without replacement), then the probability that both the balls are white is:
(a) \(\frac{1}{18}\)
(b) \(\frac{2}{9}\)
(c) \(\frac{1}{12}\)
(d) \(\frac{1}{24}\)
Answer:
(c) \(\frac{1}{12}\)

Question 38.
Three diece are thrown simultaneously. The probability of obtaining a total score of 5 is:
(a) \(\frac{5}{216}\)
(b) \(\frac{1}{6}\)
(c) \(\frac{1}{36}\)
(d) \(\frac{1}{49}\)
Answer:
(c) \(\frac{1}{36}\)

Question 39.
An urn contains 6 balls of which two are red and four are black. Two balls are drawn at random. Probability that they are of the different colour is:
(a) \(\frac{2}{5}\)
(b) \(\frac{1}{15}\)
(c) \(\frac{8}{15}\)
(d) \(\frac{4}{15}\)
Answer:
(d) \(\frac{4}{15}\)

Question 40.
The probability of obtaining an even prime number on each die when a pair of dice is rolled is:
(a) 0
(b) \(\frac{1}{3}\)
(c) \(\frac{1}{12}\)
(d) \(\frac{1}{36}\)
Answer:
(d) \(\frac{1}{36}\)

Question 41.
Two events A and B are said to be independent if:
(a) A and B are mutually exclusive
(b) P (A’B’) = [1 – P(A)][1 – P(B)]
(c) P(A) = P(B)
(d) P(A) + P(B) = 1
Answer:
(b) P (A’B’) = [1 – P(A)][1 – P(B)]

Question 42.
A die is. thrown once, then the probability of getting number greater than 3 is:
(a) \(\frac{1}{2}\)
(b) \(\frac{2}{3}\)
(c) 6
(d) 0
Answer:
(a) \(\frac{1}{2}\)

Question 43.
If P(A) = \(\frac{6}{11}\), P(B) = \(\frac{5}{11}\) and P(A ∪ B) = \(\frac{7}{11}\), then P(A/B) is:
(a) \(\frac{2}{5}\)
(b) \(\frac{3}{5}\)
(c) \(\frac{4}{5}\)
(d) 1
Answer:
(c) \(\frac{4}{5}\)

Question 44.
Let the target be hit A and B: the target be hit by B and C: the target be hit by A and C. Then the probability that A, B and C all will hit, is:
(a) \(\frac{4}{5}\)
(b) \(\frac{3}{5}\)
(c) \(\frac{2}{5}\)
(d) \(\frac{1}{5}\)
Answer:
(c) \(\frac{2}{5}\)

Question 45.
What is the probability that ‘none of them will hit the target’?
(a) \(\frac{1}{30}\)
(b) \(\frac{1}{60}\)
(c) \(\frac{1}{15}\)
(d) \(\frac{2}{15}\)
Answer:
(b) \(\frac{1}{60}\)

(B) Very Short Type Questions With Answers

Question 1.
If \(\left|\begin{array}{ccc}
1+\mathbf{x} & \mathbf{x} & \mathbf{x}^2 \\
\mathbf{x} & 1+\mathbf{x} & \mathbf{x}^2 \\
\mathbf{x}^2 & \mathbf{x} & 1+\mathbf{x}
\end{array}\right|\) = a + bx + cx² + dx³ + ex⁴ + fx⁵ then write the value of a.
Solution:
\(\left|\begin{array}{ccc}
1+\mathbf{x} & \mathbf{x} & \mathbf{x}^2 \\
\mathbf{x} & 1+\mathbf{x} & \mathbf{x}^2 \\
\mathbf{x}^2 & \mathbf{x} & 1+\mathbf{x}
\end{array}\right|\)
= a + bx + cx² + dx³ + ex⁴ + fx⁵
which is an identity
Putting x = 0 we get
a = \(\left|\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right|\) = 1

Question 2.
If every element of a third order determinant of value 8 is multiplied by 2, then write the value of the new determinant.
Solution:
According to the question
|A| = 8
Now |KA| = Kⁿ|A|
⇒ |2A| = 2³|A| = 8 × 8 = 64
Value of the new determinant is 64.

Question 3.
If I is an identity matrix of order n, then k being a natural number, write the matrix I^k_n.
Solution:
If I is an identity matrix of order n, then I^k_n = I_n

Question 4.
If A is a 4 × 5 matrix and B is a matrix such that A^TB and BA^T both are defined, then write the order of B.
Solution:
Order of A = 4 × 5
Order of A^T = 5 × 4
Let order of B = m × n
A^TB is well defined ⇒ m = 4
BA^T is well defined ⇒ n = 5
Order of B = 4 × 5

Question 5.
Write the matrix which when added to the matrix \(\left[\begin{array}{cc}
2 & -3 \\
-4 & 7
\end{array}\right]\) gives the matrix \(\left[\begin{array}{ll}
4 & 1 \\
3 & 2
\end{array}\right]\)
Solution:
Let the required matrix is A.
\(\left(\begin{array}{cc}
2 & -3 \\
-4 & 7
\end{array}\right)\) + A = \(\left(\begin{array}{ll}
4 & 1 \\
3 & 2
\end{array}\right)\)
A = \(\left(\begin{array}{ll}
4 & 1 \\
3 & 2
\end{array}\right)\) – \(\left(\begin{array}{cc}
2 & -3 \\
-4 & 7
\end{array}\right)\) = \(\left(\begin{array}{cc}
2 & 4 \\
7 & -5
\end{array}\right)\)

Question 6.
Determine the maximum value of \(\left|\begin{array}{rl}
\cos x & \sin x \\
-\sin x & \cos x-1
\end{array}\right|\)
Solution:
Let f(x) = \(\left|\begin{array}{rl}
\cos x & \sin x \\
-\sin x & \cos x-1
\end{array}\right|\)
= cos²x – cos x + sin²x = 1 – cos x
As – 1 < cos x ≤ 1
⇒ 1 >- cos x ≥ – 1
⇒ 2 > 1 – cos x ≥ 0
The maximum value of f(x) = 2.

Question 7.
Write the value of k if:
\(\left|\begin{array}{lll}
\mathbf{a a _ { 1 }} & \mathbf{a a}_2 & \mathbf{a} \mathbf{a}_3 \\
\mathbf{a b _ { 1 }} & \mathbf{a b}_2 & \mathbf{a b} \\
\mathbf{a c _ { 2 }} & \mathbf{a c}_2 & \mathbf{a c _ { 3 }}
\end{array}\right|\) = k\(\left|\begin{array}{lll}
\mathbf{a}_1 & \mathbf{b}_1 & \mathbf{c}_1 \\
\mathbf{a}_2 & \mathbf{b}_2 & \mathbf{c}_2 \\
\mathbf{a}_3 & \mathbf{b}_3 & \mathbf{c}_3
\end{array}\right|\)
Solution:
\(\left|\begin{array}{lll}
\mathbf{a a _ { 1 }} & \mathbf{a a}_2 & \mathbf{a} \mathbf{a}_3 \\
\mathbf{a b _ { 1 }} & \mathbf{a b}_2 & \mathbf{a b} \\
\mathbf{a c _ { 2 }} & \mathbf{a c}_2 & \mathbf{a c _ { 3 }}
\end{array}\right|\) = k\(\left|\begin{array}{lll}
\mathbf{a}_1 & \mathbf{b}_1 & \mathbf{c}_1 \\
\mathbf{a}_2 & \mathbf{b}_2 & \mathbf{c}_2 \\
\mathbf{a}_3 & \mathbf{b}_3 & \mathbf{c}_3
\end{array}\right|\)
k = a³.

Question 8.
If A is a 3 × 3 matrix and |A| = 3, then write the matrix represented by A × adj A.
Solution:
|A| = 3 ⇒ A × Adj A = \(\left(\begin{array}{lll}
3 & 0 & 0 \\
0 & 3 & 0 \\
0 & 0 & 3
\end{array}\right)\)

Question 9.
If ω is a complex cube root of 1, then for what value of λ the determinant
\(\left|\begin{array}{ccc}
1 & \omega & \omega^2 \\
\omega & \lambda & 1 \\
\omega^2 & 1 & \omega
\end{array}\right|\) = 0?
Solution:

⇒ for any value of ‘A,’ the given determinant is ‘0’

Question 10.
If [1 2 3] A = [0], then what is the der of the matrix A?
Solution:
If [1 2 3] A = [0]
A is a 3 × 1 matrix

Question 11.
What is A + B if A = \(\left(\begin{array}{cc}
1 & 2 \\
3 & -1
\end{array}\right)\), B = \(\left(\begin{array}{cc}
0 & -1 \\
-2 & 1
\end{array}\right)\)?
Solution:
For A = \(\left(\begin{array}{cc}
1 & 2 \\
3 & -1
\end{array}\right)\), B = \(\left(\begin{array}{cc}
0 & -1 \\
-2 & 1
\end{array}\right)\)
A + B = \(\left(\begin{array}{ll}
1 & 1 \\
1 & 0
\end{array}\right)\)

Question 12.
Give an example of a unit matrix.
Solution:
\(\left(\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right)\) is a unit matrix of 2nd order.

Question 13.
Construct a 2 × 3 matrix having elements defined by a_ij = i – j.
Solution:
a_ij = i – j
a₁₁ =0, a₁₂ = 1 – 2 = – 1, a₁₃ = 1 – 3 =- 2
a₂₁ = 2 – 1 = 1, a₂₂ = 2 – 2 = 0, a₂₃ = 2 – 3 = -1
∴ The required matrix is \(\left(\begin{array}{ccc}
0 & -1 & -2 \\
0 & 0 & -1
\end{array}\right)\).

Question 14.
Find x, y if A = A’ where A = \(\left(\begin{array}{ll}
5 & \mathbf{x} \\
\mathbf{y} & 0
\end{array}\right)\)
Solution:
A = A’
⇒ \(\left(\begin{array}{ll}
5 & \mathbf{x} \\
\mathbf{y} & 0
\end{array}\right)\) = \(\left(\begin{array}{ll}
5 & \mathbf{y} \\
\mathbf{x} & 0
\end{array}\right)\) ⇒ x = y
∴ x and y are any real number where x = y

Question 15.
Cana matrix be constructed by taking 29 elements?
Solution:
Only two matrices can be formed by taking 29 elements. They are of order 1 × 29 and 29 × 1.

Question 16.
If \(\left|\begin{array}{ll}
2 & 4 \\
k & 6
\end{array}\right|\) = 0 , what is the value of k?
Solution:
\(\left|\begin{array}{ll}
2 & 4 \\
k & 6
\end{array}\right|\) = 0 ⇒ 12 – 4k = 0 ⇒ k = 3

Question 17.
If \(\left|\begin{array}{ll}
\mathbf{a}_1 & \mathbf{b}_1 \\
\mathbf{c}_{\mathbf{1}} & \mathbf{d}_1
\end{array}\right|\) = k = \(\left|\begin{array}{ll}
\mathbf{a}_1 & \mathbf{c}_1 \\
\mathbf{b}_1 & \mathbf{d}_1
\end{array}\right|\) then what is the value of k?
Solution:
k = 1

Question 18.
If A and B are square matrices of order 3, such that |A| = -1, |B| = 3 then |3 AB| = ______.
Solution:
|3 AB| = 27 |A| |B| = 81

Question 19.
Solve: \(\left|\begin{array}{ccc}
2 & 2 & x \\
-1 & x & 4 \\
1 & 1 & 1
\end{array}\right|\) = 0
Solution:
\(\left|\begin{array}{ccc}
2 & 2 & x \\
-1 & x & 4 \\
1 & 1 & 1
\end{array}\right|\) = 0 => \(\left|\begin{array}{ccc}
0 & 2 & x \\
-1-x & x & 4 \\
0 & 1 & 1
\end{array}\right|\) = 0
⇒ – (- 1 – x) (2 – x) = 0 ⇒ x = -1, x = 2.

(C) Short Type Questions With Answers

Question 1.
If A = \(\left[\begin{array}{ccc}
1 & 2 & 3 \\
3 & -2 & 1 \\
4 & 2 & 1
\end{array}\right]\) then show that A³ – 23A – 40I = 0
Solution:

Question 2.
Solve: \(\left|\begin{array}{ccc}
\mathbf{x + 1} & \omega & \omega \\
\omega & x+\omega^2 & 1 \\
\omega^2 & 1 & x+\omega
\end{array}\right|\) = 0
Solution:

Question 3.
If A = \(\left[\begin{array}{ccc}
1 & 2 & 0 \\
0 & 1 & 3 \\
-2 & 5 & 3
\end{array}\right]\), then verify that A + A’ is symmetric and A – A’ is skew symmetric.
Solution:

Question 4.
If A, B, C are matrices of order 2 × 2 each and 2A + B + C = \(\left[\begin{array}{ll}
1 & 2 \\
3 & 0
\end{array}\right]\), A + B + C = \(\left[\begin{array}{ll}
0 & 1 \\
2 & 1
\end{array}\right]\) and A + B – C = \(\left[\begin{array}{ll}
1 & 2 \\
1 & 0
\end{array}\right]\), then find A, B and C.
Solution:

Question 5.
Find the inverse of the following matrix: \(\left[\begin{array}{lll}
1 & 1 & 2 \\
0 & 1 & 2 \\
1 & 2 & 1
\end{array}\right]\)
Solution:
Let A = \(\left[\begin{array}{lll}
1 & 1 & 2 \\
0 & 1 & 2 \\
1 & 2 & 1
\end{array}\right]\)
Method – I
Let us find A^-1 by using elementary row transformation.
Let A = IA

Method – II
|A| = 1(1 – 4) – 1(0- 2) + 2(0- 1)
= 1(-3) – 1(-2) + 2(-1)
= -3 ≠ 0
∴ A^-1 exists.
A₁₁ = -3, A₁₂ = 2, A₁₃ = -1

Question 6.
Show that \(\left|\begin{array}{ccc}
a-b-c & 2 a & 2 a \\
2 b & b-c-a & 2 b \\
2 c & 2 c & c-a-b
\end{array}\right|\) = (a+b +c)³
Solution:

= (a + b + c)³ (1 – 0) = (a + b + c)³

Question 7.
Find the inverse of the following matrix: \(\left[\begin{array}{lll}
0 & 0 & 2 \\
0 & 2 & 0 \\
2 & 0 & 0
\end{array}\right]\)
Solution:
A = \(\left[\begin{array}{lll}
0 & 0 & 2 \\
0 & 2 & 0 \\
2 & 0 & 0
\end{array}\right]\)
|A| = 2(- 4) = – 8 ≠ 0
∴ A^-1 exists.
A₁₁ = 0, A₁₂ = 0, A₁₃ = – 4
A₂₁ = 0, A₂₂ = – 4, A₂₃ = 0
A₃₁ = 0,A₃₂ = 0, A₃₃ = – 4
∴ The matrix of cofactors

Question 8.
If the matrix A is such that \(\left[\begin{array}{cc}
1 & -1 \\
2 & 3
\end{array}\right]\)A = \(\left[\begin{array}{cc}
-4 & 1 \\
7 & 7
\end{array}\right]\), find A.
Solution:

Question 9.
Show that (a + 1) is a factor of \(\left|\begin{array}{ccc}
a+1 & 2 & 3 \\
1 & a+1 & 3 \\
3 & -6 & a+1
\end{array}\right|\).
Solution:

= (a + 1) (a² + 2a + 1 + 18) – 2(a + 1 – 9) + 3(-6 – 3a – 3)
= (a + 1)(a² + 2a + 19) – 2a + 16 – 27 – 9a
= (a + 1) (a² + 2a + 19) – 11a – 11
= (a + 1) (a² + 2a + 19) – 11(a + 1)
= (a + 1) (a² + 2a + 8)
⇒ (a + 1) is a factor of the given determinant.

Question 11.
If A = \(\left[\begin{array}{ll}
\alpha & 0 \\
1 & 1
\end{array}\right]\) and B = \(\left[\begin{array}{ll}
1 & 0 \\
5 & 1
\end{array}\right]\) show that for no values of α, A² = B.
Solution:

⇒ α2 = 1 and α + 1 = 5
⇒ α = ± 1 and α = 4
Which is not possible.
There is no α for which A² = B

Question 12.
If A = \(\left[\begin{array}{ll}
3 & -4 \\
1 & -1
\end{array}\right]\), then show that A^k = \(\left[\begin{array}{cc}
1+2 \mathrm{k} & -4 \mathrm{k} \\
\mathrm{k} & 1-2 \mathrm{k}
\end{array}\right]\), k ∈ N.
Solution:

Question 13.
If A = \(\left[\begin{array}{ccc}
1 & -2 & 2 \\
3 & 1 & -1
\end{array}\right]\), B = \(\left[\begin{array}{cc}
2 & 4 \\
1 & 2 \\
3 & -1
\end{array}\right]\), verify that (AB)^T = B^TA^T.

Question 14.
Show that for each real value of λ the system of equations
(λ + 3) + λy = 0
x + (2λ + 5)y = 0 has a unique solution.
Solution:
Given system of equations is a
homogeneous system of linear
equations.
Now
Δ = \(\left|\begin{array}{cc}
\lambda+3 & \lambda \\
1 & 2 \lambda+5
\end{array}\right|\)
= (λ + 3)(2λ + 5) – λ
= 2λ² + 11λ + 15 – λ
= 2λ² + 10λ + 15
As for 2λ² + 10A + 15, D = 100 – 120 < 0
the polynomial 2λ² + 10λ + 15 has no roots i.e. Δ ≠ 0.
Thus the system has a unique trivial solution for every real value of λ.

Question 15.
If A and B are square matrices of same order then show by means of an example that AB ≠ BA in general.
Solution:

∴ AB ≠ BA we have AB ≠ BA in general.

Question 16.
If A = \(\left|\begin{array}{cc}
0 & -\tan \frac{\theta}{2} \\
\tan \frac{\theta}{2} & 0
\end{array}\right|\), then prove that det{(I + A)(I – A)-1} = 1
Solution:

Question 17.
Solve for x: \(\left|\begin{array}{ccc}
15-2 x & 11 & 10 \\
11-3 x & 17 & 16 \\
7-x & 14 & 13
\end{array}\right|\) = 0
Solution:

Question 18.
If A = \(\left[\begin{array}{ccc}
-1 & 3 & 5 \\
1 & -3 & -5 \\
-1 & 3 & 5
\end{array}\right]\) find A³ – A².
Solution:

A² = A ⇒ A²A = A²
⇒ A³ – A² = 0

Question 19.
Prove that: A = \(\left|\begin{array}{ccc}
2 & 3 & 4 \\
1 & -2 & -3 \\
3 & 1 & -8
\end{array}\right|\) ⇒ A² – 5A + 71 = 0.
Solution:

Question 20.
Test whether the following system of equations have non-zero solution.
Write the solution set:
2x + 3y + 4z = 0,
x – 2y – 3z = 0,
3x + y – 8z = 0.
Solution:
Given equations are
2x + 3y + 4z = 0
x – 2y – 3z = 0
3x + y – 8z = 0
Now \(\left|\begin{array}{ccc}
2 & 3 & 4 \\
1 & -2 & -3 \\
3 & 1 & -8
\end{array}\right|\)
= 2(19) – 3(1) + 4(7) 0
∴ The system has no non-zero solution.
The solution set is x = 0; y = 0, z = 0.

Odisha State Board CHSE Odisha Class 12 Math Solutions Chapter 6 Probability Ex 6(a) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 6 Probability Exercise 6(a)

Question 1.
Two balls are drawn from a bag containing 5 white and 7 black balls. Find the probability of selecting 2 white balls if
Solution:
Two balls are drawn from a bag containing 5 white and 7 black balls.
∴ |S| = 12.

(i) the first ball is replaced before drawing the second.
Solution:
The 1st ball is replaced before the 2nd ball is drawn. We are to select 2 white balls. So in both the draws we will get white balls. Drawing a white ball in 1st draw and in 2nd draw are independent events.
Probability of getting 2 white balls = \(\frac{5}{12}\) × \(\frac{5}{12}\) = \(\frac{25}{144}\)

(ii) the first ball is not replaced before drawing the second.
Solution:
Here the 1st ball is not replaced before the 2nd ball is drawn. Since we are to get 2 white balls in each draw, we must get a white ball.
Now probability of getting a white ball in 1st draw = \(\frac{5}{12}\).
Probability of getting a white ball in 2nd = \(\frac{4}{11}\).
Since the two draws are independent, we have the probability of getting 2 white balls
= \(\frac{5}{12}\) × \(\frac{4}{11}\) = \(\frac{20}{132}\).

Question 2.
Two cards are drawn from a pack of 52 cards; find the probability that
(i) they are of different suits.
(ii) they are of different denominations.
Solution:
Two cards are drawn from a pack of 52 cards. The cards are drawn one after another. Each suit has 13 cards.
|S| = ⁵²C₂
(i) As the two cards are of different suits, their probability
= \(\frac{52}{52}\) × \(\frac{39}{51}\)
(ii) Each denomination contains 4 cards. As the two cards drawn are of different denominations, their probability
= \(\frac{52}{52}\) × \(\frac{48}{51}\)

Question 3.
Do both parts of problem 2 if 3 cards drawn at random.
Solution:
(i) 3 cards are drawn one after another. As they are of different suits, we have their probability
= \(\frac{52}{52}\) × \(\frac{39}{51}\) × \(\frac{26}{50}\).
(ii) As the 3 cards are of different denominations, we have their probability
= \(\frac{52}{52}\) × \(\frac{48}{51}\) × \(\frac{44}{50}\).

Question 4.
Do both parts of problem 2 if 4 cards are drawn at random.
Solution:
(i) 4 cards are drawn one after another. As they are of different suits, we have their probability
= \(\frac{52}{52}\) × \(\frac{39}{51}\) × \(\frac{26}{50}\) × \(\frac{13}{49}\).
(ii) As the cards are of different denominations, we have their probability
= \(\frac{52}{52}\) × \(\frac{48}{51}\) × \(\frac{44}{50}\) × \(\frac{40}{49}\).

Question 5.
A lot contains 15 items of which 5 are defective. If three items are drawn at random, find the probability that
(i) all three are defective
(ii) none of the three is defective.
Do this problem directly.
Solution:
(i) A lot contains 15 items of which 5 are defective. Three items are drawn at random. As the items are drawn one after another.
Their probability = \(\frac{5}{15}\) × \(\frac{4}{14}\) × \(\frac{3}{13}\)
(ii) As none of the 3 items are defective, we have to draw 3 non-defective items one after another.
Their probability = \(\frac{10}{15}\) × \(\frac{9}{14}\) × \(\frac{8}{13}\)

Question 6.
A pair of dice is thrown. Find the probability of getting a sum of at least 9 if 5 appears on at least one of the dice.
Solution:
A pair of dice is thrown. Let A be the event of getting at least 9 points and B, the event that 5 appears on at least one of the dice.
∴ B = {(1, 5), (2, 5), (3, 5), (4, 5), (5, 5), (6, 5), (5, 1), (5, 2), (5, 3), (5, 4), (5, 6)}
A = {(3, 6), (4, 5), (5, 4), (6, 3), (4, 6), (5, 5), (6, 4), (5, 6), (6, 5), (6, 6)}
∴ A ∩ B = {(4, 5), (5, 4), (5, 5), (5, 6), (6, 5)}
∴ P (A | B) = \(\frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{\mathrm{P}(\mathrm{B})}\) = \(\frac{\frac{5}{36}}{\frac{31}{36}}\) = \(\frac{5}{11}\)

Question 7.
A pair of dice is thrown. If the two numbers appearing are different, find the probability that
(i) the sum of points is 8.
(ii) the sum of points exceeds 8.
(iii) 6 appears on one die.
Solution:
A pair of dice is thrown as two numbers are different
We have |S| = 30
(i) Let A be the. event that the sum of points on the dice is 8, where the numbers on the dice are different.
A = {(2, 6), (3, 5), (5, 3), (6, 2)}
P(A) = \(\frac{|\mathrm{A}|}{|\mathrm{S}|}\) = \(\frac{4}{30}\)
(ii) Let B be the event that sum of the points exceeds 8.
B = {(3, 6), (4, 5), (5, 4), (6, 3), (5, 6), (6, 5), (4, 6), (6, 4)}
P(B) = \(\frac{|\mathrm{B}|}{|\mathrm{S}|}\) = \(\frac{8}{30}\)
(iii) Let C be the event that 6 appears on one die.
C = {(1, 6), (2, 6), (3, 6), (4, 6), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5)}
P(C) = \(\frac{|\mathrm{C}|}{|\mathrm{S}|}\) = \(\frac{10}{30}\) = \(\frac{1}{3}\)

Question 8.
In a class 30% of the students fail in Mathematics, 20% of the students fail in English and 10% fail in both. A student is selected at random.
Solution:
In a class 30% of the students fail in Mathematics, 20% of the students fail in English and 10% fail in both. Let A be the event that a student fails in Mathematics and B be the events that he fails in English.
P(A) = \(\frac{30}{100}\), P(B) = \(\frac{20}{100}\)
Where |S| = 100, P (A ∩ B) = \(\frac{10}{100}\)

(i) If he has failed in English, what is the probability that he has failed in Mathematics?
Solution:
If he has failed in English, then the probability that he has failed in Mathematics.
i.e., P\(\left(\frac{A}{B}\right)\) = \(\frac{P(A \cap B)}{P(B)}\) = \(\frac{\frac{10}{100}}{\frac{20}{100}}\) = \(\frac{1}{2}\)

(ii) If he has failed in Mathematics, what is the probability that he has failed in English?
Solution:
If he has failed in Mathematics, then the probability that he has failed in English
i.e., P\(\left(\frac{B}{A}\right)\) = \(\frac{P(A \cap B)}{P(A)}\) = \(\frac{\frac{10}{100}}{\frac{30}{100}}\) = \(\frac{1}{3}\)

(iii) What is the probability that he has failed in both?
Solution:
Probability that he has failed in both
i.e., P (A ∩ B) = \(\frac{10}{100}\) = \(\frac{1}{10}\)

Question 9.
IfA, B are two events such that
P(A) = 0.3, P(B) = 0.4, P (A ∪ B) = 0.6
Find
(i) P (A | B)
(ii) P (B | A)
(iii) P (A | B^c)
(iv) P (B | A^c)
Solution:
A and B are two set events such that
P(A) = 0.3, P(B) = 0.4, P (A ∪ B) = 0.6
We have
P (A ∪ B) = P(A) + P(B) – P (A ∩ B)
or, 0.6 = 0.3 + 0.4 – P (A ∩ B)
or, P (A ∩ B) = 0.7 – 0.6 = 0.1

Question 10.
If A, B are events such that P(A) = 0.6, P(B) = 0.4 and P (A ∩ B) = 0.2, then find
(i) P (A | B)
(ii) P (B | A)
(iii) P (A | B^c)
(iv) P (B | A^c)
Solution:
A and B are events such that
P(A) = 0.6, P(B) = 0.4, P (A ∩ B) = 0.2

Question 11.
If A and B are independent events, show that
(i) A^c and B^c are independent,
(ii) A and B^c are independent,
(iii) A^c and B are independent.
Solution:
A and B are independent events.
(i) We have P (A ∩ B) = P(A). P(B)
P (A’ ∩ B’) = P (A ∪ B)’ = 1 – P (A ∪ B)
= 1 – [P(A) + P(B) – P (A ∩ B)]
= 1 – P(A) – P(B) + P(A) P(B)
= 1 [1 – P(A)] – P(B) [1 – P(A)]
= [1 – P(A)] [1 – P(B)] = P(A’) P(B’)
∴ A’ and B’ are independent events.

(ii) P (A ∩ B^c) = P (A – B)
= P(A) – P (A ∩ B)
= P(A) – P(A) P(B)
= P(A) [1 – P(B)]
= P(A). P(B^c).
∴ A and B^c are independent events.

(iii) P (A^c ∪ B) = P (B – A)
= P(B) – P (A ∩ B)
= P(B) – P(A) P(B)
= P(B) [1 – P(A)] = P(B) P(A^c)
∴ A^c and B are independent events.

Question 12.
Two different digits are selected at random from the digits 1 through 9.
(i) If the sum is even, what is the probability that 3 is one of the digits selected?
(ii) If the sum is odd, what is the probability that 3 is one of the digits selected?
(iii) If 3 is one of the digits selected, what is the probability that the sum is odd?
(iv) If 3 is one of the digits selected, what is the probability that the sum is even?
Solution:
Two different digits are selected at random from the digits 1 through 9.
(i) Let A be the event that the sum is even and B be the event that 3 is one of the number selected.
We have to find P (B | A).
There are 4 even digits and 5 odd digits.
∴ The sum is even if both the numbers are odd or both are even.
∴ |A| = ⁴C₂ + ⁵C₂ = 6 + 10 = 16
∴ P(A) = \(\frac{16}{{ }^9 \mathrm{C}_2}\) = \(\frac{16}{36}\)
Also B = {(1, 3), (5, 3), (7, 3), (9, 3), (3, 2), (3, 4), (3, 8), (3, 6)}
∴ A ∩ B = {(1, 3), (5, 3), (7, 3), (9, 3)}
P(B) = \(\frac{8}{36}\) P (A ∩ B) = \(\frac{4}{36}\)
P(\(\frac{B}{A}\)) = \(\frac{P(A \cap B)}{P(A)}\) = \(\frac{\frac{4}{36}}{\frac{16}{36}}\) = \(\frac{1}{4}\)

(ii) Let A be the event that the sum is odd. The sum is odd if one of the numbers selected is odd and other is even.
∴ P(A) = \(\frac{5}{9}\) × \(\frac{4}{8}\) + \(\frac{4}{9}\) × \(\frac{5}{8}\) = \(\frac{20}{36}\)
Let B be the event that one of the numbers selected is 3.
∴ B = {(1, 3), (2, 3), (4, 3), (5, 3), (6, 3), (7, 3), (8, 3), (9, 3)}
∴ A ∩ B = {(2, 3), (4, 3), (6, 3), (8, 3)}

Question 13.
If P(A) = 0.4, P (B | A) = 0.3 and P (B^c | A^c) = 0.2. find
(i) P (A | B)
(ii) P (B | A^c)
(iii) P(B)
(iv) P(A^c)
(v) P (A ∪ B)
Solution:
P(A) = 0.4, P (B | A) = 0.3 and P (B^c | A^c) = 0.2.


(iii) P(B) = 0.6
= \( \frac{6}{10} \)
= \( \frac{3}{5} \)

(iv) P(A^c) = 1 – P(A)
= 1 – 0.4 = 0.6
= \( \frac{6}{10} \) = \( \frac{3}{5} \)

(v) P (A ∪ B) = P(A) + P(B) – P (A ∩ B)
= 0.4 + 0.6 – 0.12
= 1.0 – 0.12 = 0.88

Question 14.
If P(A) = 0.6, P (B | A) = 0.5, find P (A ∪ B) if A, B are independent.
Solution:
P(A) = 0.6, P (B | A) = 0.5
We have P (B | A) = \(\frac{\mathrm{P}(\mathrm{B} \cap \mathrm{A})}{\mathrm{P}(\mathrm{A})}\) = 0.5
or, P (B ∩ A) = 0.5 × P(A)
= 0.5 × 0.6 = 0.3
As A and B are independent events, we have
P (B ∩ A) = P(B) P(A) = 0.3
or, P(B) = \(\frac{0.3}{\mathrm{P}(\mathrm{A})}\)
= \(\frac{0.3}{0.6}\)
= \(\frac{1}{2}\) = 0.5
P (A ∪ B) = P(A) + P(B) – P(A ∩ B)
= 0.6 + 0.5 – 0.3 = 0.8

Question 15.
Two cards are drawn in succession from a deck of 52 cards. What is the probability that both cards are of denomination greater than 2 and less than 9?
Solution:
Two cards are drawn in succession from a deck of 52 cards.
There are 6 denominations which are greater than 2 and less than 9. So there are 24 cards whose denominations are greater than 2 and less than 9.
∴ Their probability = \(\frac{24}{52}\) × \(\frac{23}{51}\).

Question 16.
From a bag containing 5 black and 7 white balls, 3 balls are drawn in succession. Find the probability that
(i) all three are of the same colour.
(ii) each colour is represented.
Solution:
From a bag containing 5 black and 7 white balls, 3 balls are drawn in succession.
(i) The 3 balls drawn are of same colour.
∴ Probability of drawing 3 balls of black colour
= \(\frac{5}{12}\) × \(\frac{4}{11}\) × \(\frac{3}{10}\) = \(\frac{1}{22}\)
Probability of drawing 3 white balls
= \(\frac{7}{12}\) × \(\frac{6}{11}\) × \(\frac{5}{10}\) = \(\frac{7}{44}\)
∴ Probability of drawing 3 balls of same colour
= \(\frac{5}{12}\) × \(\frac{4}{11}\) × \(\frac{3}{10}\) + \(\frac{7}{12}\) × \(\frac{6}{11}\) × \(\frac{5}{10}\) = \(\frac{9}{44}\)

(ii) Balls of both colour will be drawn. If B represents black ball and W represents the white ball.
∴ The possible draws are WWB, WBW, BWW.

Question 17.
A die is rolled until a 6 is obtained. What is the probability that
(i) you end up in the second roll
(ii) you end up in the third roll.
Solution:
A die is rolled until a 6 is obtained
(i) We are to end up in the 2nd roll i.e., we get 6 in the 2nd roll. Let A be the event of getting a 6 in one roll of a die.
∴ P(A) = \(\frac{1}{6}\) ⇒ P(A’) = 1 – \(\frac{1}{6}\) = \(\frac{5}{6}\)
∴ Probability of getting a 6 in the 2nd roll
= \(\frac{5}{6}\) × \(\frac{1}{6}\) = \(\frac{5}{36}\)
(ii) Probability of getting a 6 in the 3rd roll
= \(\frac{5}{6}\) × \(\frac{5}{6}\) × \(\frac{1}{6}\) = \(\frac{25}{216}\)

Question 18.
A person takes 3 tests in succession. The probability of his (her) passing the first test is 0.8. The probability of passing each successive test is 0.8 or 0.5 according as he passes or fails the preceding test. Find the probability of his (her) passing at least 2 tests.
Solution:
A person takes 3 tests in succession. The probability of his passing the 1st test is 0.8. The probability of passing each successive test is 0.8 or 0.5 according as he passes or fails the preceding test.
Let S denotes the success (passing) in a test and F denotes the failure in a test.
∴ P(S) = 0.8
∴ P(F) = 1 – P(S) = 1 – 0.8 = 0.2
We have the following mutually exclusive cases:

Event			Probability
S	S	S	0.8 × 0.8 × 0.8 = 0.512
S	S	F	0.8 × 0.8 × 0.2 = 0.128
S	F	S	0.8 × 0.2 × 0.5 = 0.080
F	S	S	0.2 × 0.5 x 0.8 = 0.080

∴ Probability of atleast 2 successes
= 0.512 + 0.128 + 0.080 + 0.080
= 0.8 = \(\frac{4}{5}\)

Question 19.
A person takes 4 tests in succession. The probability of his passing the first test is p, that of his passing each succeeding test is p or y depending on his passing or failing the preceding test. Find the probability of his passing
(i) at least three test
(ii) just three tests.
Solution:
A person takes 4 tests in succession. The probability of his passing the 1st test is P, that of his passing each succeeding test is P or P/2 depending bn his passing or failing the preceding test. Let S and F denotes the success and failure in the test.
∴ P(S) = P, P(F) = 1 – P
We have the following mutually exclusive tests:

Question 20.
Given that all three faces are different in a throw of three dice, find the probability that
(i) at least one is a six
(ii) the sum is 9.
Solution:
Three dice are thrown once showing different faces in a throw.
|S| = 6³ = 216
Let A be the event that atleast one is a six.
Let B be the event that all three faces are different.
|B| = ⁶6₃
(i) Now A^c is the event that there is no six. A^c ∩ B is the event that all 3 faces are different and 6 does not occur.
|A^c ∩ B| = ⁵C₃
P (A^c | B) = \(\frac{P\left(A^C \cap B\right)}{P(B)}\)
= \(\frac{{ }^5 \mathrm{C}_3 / 216}{{ }^6 \mathrm{C}_3 / 216}\) = \(\frac{1}{2}\)

(ii) Let A be the event that the sum is 9.
A ∩ B = {(1,3, 5), (1,5, 3), (3, 5,1), (3, 1, 5), (5, 1, 3), (5, 3, 1), (1, 2, 6), (1, 6, 2), (2, 1, 6), (2, 6, 1), (6, 2, 1), (6, 1, 2), (2, 3, 4), (2, 4, 3), (3, 2, 4), (2, 3, 4), (3, 4, 2), (4, 3, 2)}
|A ∩ B| = 18
P (A | B) = \(\frac{P(A \cap B)}{P(B)}\)
= \(\frac{18 / 216}{20 / 216}\) = \(\frac{9}{10}\)

Question 21.
From the set of all families having three children, a family is picked at random.
(i) If the eldest child happens to be a girl, find the probability that she has two brothers.
(ii) If one child of the family is a son, find the probability that he has two sisters.
Solution:
A family is picked up at random from a set of families having 3 children.
(i) The eldest child happens to be a girl. We have to find the probability that she has two brothers. Let G denotes a girl and B denotes a boy.
∴ P(B) = \(\frac{1}{2}\), P(G) = \(\frac{1}{2}\)
P(BB | G) = P(B) × P(G) = \(\frac{1}{2}\) × \(\frac{1}{2}\) = \(\frac{1}{4}\)

(ii) The one child of the family is a son. We have to find the probability that he has two sisters. We have the following mutually exclusive events:
BGG, GBG, GGB.
∴ The required probability
= P(B) × P(G) × P(G) + P(G) × P(B) + P(G) + P(G) × P(G) × P(B)
= \(\frac{1}{2}\) × \(\frac{1}{2}\) × \(\frac{1}{2}\) + \(\frac{1}{2}\) × \(\frac{1}{2}\) × \(\frac{1}{2}\) + \(\frac{1}{2}\) × \(\frac{1}{2}\) × \(\frac{1}{2}\) = \(\frac{3}{8}\)

Question 22.
Three persons hit a target with probability \(\frac{1}{2}\), \(\frac{1}{3}\) and \(\frac{1}{4}\) respectively. If each one shoots at the target once,
(i) find the probability that exactly one of them hits the target
(ii) if only one of them hits the target what is the probability that it was the first person?
Solution:
Three persons hit a target with probability
\(\frac{1}{2}\), \(\frac{1}{3}\) and \(\frac{1}{4}\)
Let A, B, C be the events that the 1st person, 2nd person, 3rd person hit the target respectively.

(i) As the events are independent, the probability that exactly one of them hit the target
= P(AB’C’) + P(A’BC’) + P(A’B’C)
= P(A) P(B’) P(C’) + P(A’) P(B) P(C’) + P(A’) P(B’) P(C)

(ii) Let E₁ be the event that exactly one person hits the target.
∴ P(E₁) = \(\frac{11}{24}\)
Let E₂ be the event that 1st person hits the target
∴ P(E₂) = P(A) = \(\frac{1}{2}\)
∴ E₁ ∩ E₂ = AB’C’
⇒ P(E₁ ∩ E₂)
= P(A) × P(B’) P(C’) = \(\frac{6}{24}\)
∴ P(E₂ | E₁) = \(\frac{6 / 24}{11 / 24}\) = \(\frac{6}{11}\)

Odisha State Board CHSE Odisha Class 12 Math Solutions Chapter 3 Linear Programming Additional Exercise Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 3 Linear Programming Additional Exercise

(A) Multiple Choice Questions (Mcqs) With Answers

Question 1.
Write the value of cos^-1 cos(3π/2).
(a) π
(b) \(\frac{\pi}{2}\)
(c) \(\frac{\pi}{4}\)
(d) 2π
Solution:
(b) \(\frac{\pi}{2}\)

Question 2.
Sets A and B have respectively m and n elements. The total number of relations from A to B is 64. If m < n and m ≠ 1, write the values of m and n respectively.
(a) m = 3, n = 2
(b) m = 2, n = 2
(c) m = 2, n = 3
(d) m = 3, n = 3
Solution:
(c) m = 2, n = 3

Question 3.
Write the principal value of
sin^-1 (\(-\frac{1}{2}\)) + cos^-1 cos(\( -\frac{\pi}{2}\))
(a) \(\frac{\pi}{2}\)
(b) \(\frac{\pi}{3}\)
(c) \(\frac{\pi}{4}\)
(d) π
Solution:
(b) \(\frac{\pi}{3}\)

Question 4.
Write the maximum value of x + y subject to: 2x + 3y < 6, x > 0, y > 0.
(a) 3
(b) 1
(c) 2
(d) 0
Solution:
(a) 3

Question 5.
Let A has 3 elements and B has m elements. Number of relations from A to B = 4096. Find the value of m.
(a) 2
(b) 4
(c) 1
(d) 3
Solution:
(b) 4

Question 6.
Let A is any non-empty set. Number of binary operations on A is 16. Find |A|.
(a) 2
(b) 1
(c) 3
(d) 4
Solution:
(a) 2

Question 7.
Give an example of a relation which is reflexive, transitive but not symmetric.
(a) x < y on Z
(b) x = y on Z
(c) x > y on Z
(d) None of the above
Solution:
(a) x < y on Z

Question 8.
Find the least positive integer r such that – 375 ∈ [r]₁₁
(a) r = 5
(b) r = 6
(c) r = 3
(d) r = 10
Solution:
(d) r = 10

Question 9.
Find three positive integers x_i, i = 1, 2, 3 satisfying 3x ≡ 2 (mod 7)
(a) x = 1, 3, 9…
(b) x = 2, 4, 6…
(c) x = 3, 10, 17…
(d) x = 2, 10, 18…
Solution:
(c) x = 3, 10, 17…

Question 10.
If the inversible function f is defined as f(x) = \(\frac{3 x-4}{5}\) write f^-1(x)
(a) \(\frac{5 x+4}{3}\)
(b) \(\frac{4 x+5}{3}\)
(c) \(\frac{5 x-4}{3}\)
(d) \(\frac{5 x+4}{2}\)
Solution:
(a) \(\frac{5 x+4}{3}\)

Question 11.
Let f : R → R and g : R → R defined as f(x) = |x|, g(x) = |5x – 2| then find fog.
(a) |5x + 2|
(b) |5x – 2|
(c) |2x – 2|
(d) |2x – 5|
Solution:
(b) |5x – 2|

Question 12.
Let ∗ is a binary operation defined by a ∗ b = 3a + 4b – 2, find 4 ∗ 5.
(a) 20
(b) 12
(c) 30
(d) 36
Solution:
(c) 30

Question 13.
Let the binary operation on Q defined as a ∗ b = 2a + b – ab, find 3 ∗ 4.
(a) -2
(b) -1
(c) 2
(d) 1
Solution:
(a) -2

Question 14.
Let ∗ is a binary operation on Z defined as a ∗ b = a + b – 5 find the identity element for ∗ on Z.
(a) e = 1
(b) e = 5
(c) e = -5
(d) e = -1
Solution:
(b) e = 5

Question 15.
Find the number of binary, operations on the set {a, b}.
(a) 12
(b) 14
(c) 15
(d) 16
Solution:
(d) 16

Question 16.
Let ∗ is a binary operation on [0, ¥) defined as a ∗ b = \(\sqrt{a^2+b^2}\) find the identity element.
(a) e = 0
(b) e = 2
(c) e = 1
(d) e = 3
Solution:
(a) e = 0

Question 17.
Find least non-negative integer r such that 7 × 13 × 23 × 413 ≡ r (mod 11).
(a) r = 13
(b) r = 49
(c) r = 7
(d) r = 23
Solution:
(c) r = 7

Question 18.
Find least non-negative integer r such that 1237(mod 4) + 985 (mod 4) ≡ r (mod 4).
(a) r = 1
(b) r = 2
(c) r = -2
(d) r = -1
Solution:
(b) r = 2

Question 19.
Let ∗ is a binary operation on R – {0} defined as a ∗ b = \(\frac{a b}{5}\). If 2 ∗ (x ∗ 5) = 10, then find x:
(a) x = 25
(b) x = -5
(c) x = 5
(d) x = 1
Solution:
(a) x = 25

Question 20.
Find the principal value of cos^-1 (\( -\frac{1}{2}\)) + 2sin^-1 (\( \frac{1}{2}\)).
(a) \(\frac{5 \pi}{6}\)
(b) \(\frac{5 \pi}{2}\)
(c) \(\frac{5 \pi}{4}\)
(d) \(\frac{\pi}{6}\)
Solution:
(a) \(\frac{5 \pi}{6}\)

Question 21.
Evaluate sin^-1 (\(\frac{1}{\sqrt{5}}\)) + cos^-1 (\(\frac{3}{\sqrt{10}}\))
(a) \(\frac{\pi}{2}\)
(b) \(\frac{\pi}{5}\)
(c) \(\frac{\pi}{4}\)
(d) π
Solution:
(c) \(\frac{\pi}{4}\)

Question 22.
Evaluate cos^-1 (\(\frac{1}{2}\)) + 2sin^-1 (\(\frac{1}{2}\)).
(a) \(\frac{2 \pi}{5}\)
(b) \(\frac{2 \pi}{3}\)
(c) π
(d) \(\frac{\pi}{3}\)
Solution:
(b) \(\frac{2 \pi}{3}\)

Question 23.
Find the value of tan^-1 √3 – sec^-1 (-2).
(a) –\(\frac{\pi}{3}\)
(b) \(\frac{2 \pi}{3}\)
(c) \(\frac{\pi}{3}\)
(d) \(\frac{3 \pi}{2}\)
Solution:
(a) –\(\frac{\pi}{3}\)

Question 24.
Evaluate tan (2 tan^-1 \(\frac{1}{3}\))
(a) 2
(b) 1
(c) 0
(d) -1
Solution:
(a) 2

Question 25.
Evaluate : sin^-1 (sin \(\frac{3 \pi}{5}\)).
(a) \(-\frac{2 \pi}{3}\)
(b) \(\frac{\pi}{3}\)
(c) \(\frac{2 \pi}{5}\)
(d) \(\frac{\pi}{5}\)
Solution:
(c) \(\frac{2 \pi}{5}\)

Question 26.
tan^-1 (2cos\(\frac{\pi}{3}\)) is ________.
(a) \(\frac{\pi}{2}\)
(b) \(\frac{\pi}{4}\)
(c) \(\frac{\pi}{5}\)
(d) \(\frac{\pi}{3}\)
Solution:
(b) \(\frac{\pi}{4}\)

Question 27.
Evaluate : sin^-1 (sin \(\frac{2 \pi}{3}\)) is ________.
(a) \(\frac{\pi}{2}\)
(b) \(\frac{\pi}{4}\)
(c) \(\frac{\pi}{5}\)
(d) \(\frac{\pi}{3}\)
Solution:
(d) \(\frac{\pi}{3}\)

Question 28.
The value of sin(tan^-1 x + tan^-1 \( \frac{1}{x}\)), x > 0 = ________.
(a) -1
(b) 1
(c) 0
(d) 2
Solution:
(b) 1

Question 29.
2sin^-1 \( \frac{4}{5}\) + sin^-1 \( \frac{24}{25}\) = ________.
(a) 12
(b) 15
(c) 16
(d) 20
Solution:
(b) 15

Question 30.
Evaluate: tan^-1 1 = (2cos\(\frac{\pi}{3}\))
(a) \(\frac{\pi}{4}\)
(b) \(\frac{\pi}{2}\)
(c) \(\frac{\pi}{3}\)
(d) \(\frac{2 \pi}{3}\)
Solution:
(a) \(\frac{\pi}{4}\)

Question 31.
If sin^-1 (\( \frac{\pi}{5}\)) + cosec^-1 (\(\frac{5}{4}\)) = \(\frac{5}{2}\) then find the value of x.
(a) 1
(b) 2
(c) 3
(d) 4
Solution:
(c) 3

Question 32.
Evaluate:
tan^-1 (\( \frac{-1}{\sqrt{3}}\)) + cot^-1 (\( \frac{1}{\sqrt{3}}\)) + tan^-1(sin(\( -\frac{\pi}{2}\))).
(a) \(\frac{- \pi}{12}\)
(b) \(\frac{2 \pi}{5}\)
(c) \(\frac{\pi}{12}\)
(d) \(\frac{\pi}{6}\)
Solution:
(a) \(\frac{- \pi}{12}\)

Question 33.
Evaluate sin^-1 (cos(\( \frac{33 \pi}{5}\)))
(a) \(\frac{\pi}{10}\)
(b) \(\frac{- \pi}{10}\)
(c) \(\frac{\pi}{5}\)
(d) \(\frac{\pi}{2}\)
Solution:
(b) \(\frac{- \pi}{10}\)

Question 34.
Express the value of the following in simplest form. tan (\( \frac{\pi}{4}\) + 2cot^-1 3)
(a) 7
(b) 12
(c) 3
(d) 6
Solution:
(a) 7

Question 35.
Express the value of the following in simplest form sin cos^-1 tan sec^-1 √2
(a) cos 0
(b) cot 0
(c) tan 0
(d) sin 0
Solution:
(d) sin 0

Question 36.
tan \(\left\{\frac{1}{2} \sin ^{-1} \frac{2 x}{1+x^2}+\frac{1}{2} \cos ^{-1} \frac{1-y^2}{1+y^2}\right\}\)
(a) \(\frac{x-y}{1+x y}\)
(b) \(\frac{x+y}{1-x y}\)
(c) \(\frac{x-y}{1+y}\)
(d) \(\frac{x+y}{x y}\)
Solution:
(b) \(\frac{x+y}{1-x y}\)

Question 37.
The relation R on the set A = [1, 2, 3] given by R = {(1, 1), (1, 2), (2, 2), (2, 3), (3, 3)} is:
(a) Reflexive
(b) Symmetric
(c) Transitive
(d) Equivalence
Solution:
(a) Reflexive

Question 38.
Let f : R → R be defined as f(x) = 3x – 2. Choose the correct answer.
(a) f is one-one onto
(b) f is many-one onto
(c) f is one-one but not onto
(d) f is neither one-one nor onto
Solution:
(a) f is one-one onto

Question 39.
Let R be a relation defined on Z as R = {(a, b) ; a² + b² = 25 }, the domain of R is:
(a) {3, 4, 5}
(b) {0, 3, 4, 5}
(c) {0, 3, 4, 5, -3, -4, -5}
(d) None of the above
Solution:
(c) {0, 3, 4, 5, -3, -4, -5}

Question 40.
let R be the relation in the set N given by R={(a, b) : a = b – 2, b > 6}. Choose the correct answer.
(a) (2, 4) • R
(b) (3, 8) • R
(c) (6, 8) • R
(d) (8, 10) • R
Solution:
(d) (8, 10) • R

Question 41.
Set A has 3 elements and set B has 4 elements. Then the number of injective functions that can be defined from set A to set B is
(a) 144
(b) 12
(c) 24
(d) 64
Solution:
(c) 24

Question 42.
Let R be a relation on set of lines as L₁ R L₂ if L₁ is perpendicular to L₂. Then
(a) R is Reflexive
(b) R is transitive
(c) R is symmetric
(d) R is an equivalence relation
Solution:
(c) R is symmetric

Question 43.
A Relation from A to B is an arbitrary subset of:
(a) A × B
(b) B × B
(c) A × A
(d) B × B
Solution:
(a) A × B

Question 44.
Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as aRb if a is congruent to b ∀ a, b ∈ T. Then R is
(a) reflexive but not transitive
(b) transitive but not symmetric
(c) equivalence
(d) None of these
Solution:
(c) equivalence

Question 45.
The maximum number of equivalence relations on the set A = {1, 2, 3} are
(a) 1
(b) 2
(c) 3
(d) 5
Solution:
(d) 5

Question 46.
Let A = {1, 2, 3} and consider the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}. Then R is
(a) reflexive but not symmetric
(b) reflexive but not transitive
(c) symmetric and transitive
(d) neither symmetric nor transitive
Solution:
(a) reflexive but not symmetric

Question 47.
Which of the following functions from Z into Z are bijective?
f(x) = x³
f(x) = x + 2
f(x) = 2x + 1
f(x) = x² + 1
Solution:
f(x) = x + 2

Question 48.
Let R be a relation on the set N of natural numbers denoted by nRm <=> n is a factor of m (i.e. n | m). Then, R is
(a) Reflexive and symmetric
(b) Transitive and symmetric
(c) Equivalence
(d) Reflexive, transitive but not symmetric
Solution:
(c) Equivalence

Question 49.
Let S = {1, 2, 3, 4, 5} and let A = S × S. Define the relation R on A as follows: (a, b) R (c, d) iff ad = cb. Then, R is
(a) reflexive only
(b) Symmetric only
(c) Transitive only
(d) Equivalence relation
Solution:
(d) Equivalence relation

Question 50.
Let X = {-1, 0, 1}, Y = {0, 2} and a function f : X → Y defined by y = 2x⁴, is
(a) one-one onto
(b) one-one into
(c) many-one onto
(d)many-one into
Solution:
(c) many-one onto

Question 51.
Let A = R – {3}, B = R – {1}. Let f : A → B be defined by f(x) = (x-2)/(x-3). Then,
(a) f is bijective
(b) f is one-one but not onto
(c) f is onto but not one-one
(d) None of these
Solution:
(a) f is bijective

Question 52.
The function f : R → R given by f(x) = x³ – 1 is
(a) a one-one function
(b) an onto function
(c) a bijection
(d) neither one-one nor onto
Solution:
(c) a bijection

Question 53.
Let f : [0, ∞) → [0, 2] be defined by f(x) = 2x/1+x, then f is
(a) one-one but not onto
(b) onto but not one-one
(c) both one-one and onto
(d) neither one-one nor onto
Solution:
(a) one-one but not onto

Question 54.
If N be the set of all natural numbers, consider f : N → N such that f(x) = 2x, ∀ × ∈ N, then f is
(a) one-one onto
(b) one-one into
(c) many-one onto
(d) None of these
Solution:
(b) one-one into

Question 55.
Let f : R → R be a function defined by f(x) = x³ + 4, then f is
(a) injective
(b) surjective
(c) bijective
(d) none of these
Solution:
(c) bijective

Question 56.
Given set A = {a, b, c}. An identity relation in set A is
(a) R = {(a, b), (a, c)}
(b) R = {(a, a), (b, b), (c, c)}
(c) R = {(a, a), (b, b), (c, c), (a, c)}
(d) R= {(c, a), (b, a), (a, a)}
Solution:
(b) R = {(a, a), (b, b), (c, c)}

Question 57.
Set A has 3 elements and the set B has 4 elements. Then the number of injective functions that can be defined from set A to set B is
(a) 144
(b) 12
(c) 24
(d) 64
Solution:
(c) 24

Question 58.
sin(sec^-1 x + cosec^-1 x) =
(a) 1
(b) -1
(c) π/2
(d) π/3
Solution:
(a) 1

Question 59.
The principle value of sin^-1 (√3/2) is:
(a) 2π/3
(b) π/6
(C) π/4
(d) π/3
Solution:
(d) π/3

Question 60.
Simplified form of cos^-1 (4x³ – 3x)
(a) 3 sin^-1 x
(b) 3 cos^-1 x
(c) π – 3 sin^-1 x
(d) None of these
Solution:
(b) 3 cos^-1 x

Question 61.
tan^-1 √3 – sec^-1 (-2) is equal to
(a) π
(b) -π/3
(c) π/3,
(d) 2π/3
Solution:
(b) -π/3

Question 62.
If y = sec^-1 x then
(a) 0 ≤ y ≤ π
(b) 0 ≤ y ≤ π/2
(c) -π/2 ≤ y ≤ π/2
(d) None of these
Solution:
(d) None of these

Question 63.
If x + (1/x) = 2 then the principal value of sin^-1 x is
(a) π/4
(b) π/2
(c) π
(d) 3π/2
Solution:
(b) π/2

Question 64.
The principle value of sin^-1 (sin 2π/3) is :
(a) 2π/3
(b) π/3
(c) -π/6
(d) π/6
Solution:
(b) π/3

Question 65.
The value of cos^-1 (1/2) + 2sin^-1 (1/2) is equal to
(a) π/4
(b) π/6
(c) 2π/3
(d) 5π/6
Solution:
(b) π/6

Question 66.
Principal value of tan^-1 (-1) is
(a) π/4
(b) -π/2
(c) 5π/4
(d) -π/4
Solution:
(d) -π/4

Question 67.
A Linear function, which is minimized or maximized is called
(a) an objective function
(b) an optimal function
(c) A feasible function
(d) None of these
Solution:
(a) an objective function

Question 68.
The maximum value of Z = 3x + 4y subject to the constraints : x+ y ≤ 4, x ≥ 0, y ≥ 0 is:
(a) 0
(b) 12
(c) 16
(d) 18
Solution:
(c) 16

Question 69.
The maximum value of Z = 2x +3y subject to the constraints : x + y ≤ 1, 3x + y ≤ 4, x, y ≥ 0 is
(a) 2
(b) 4
(c) 5
(d) 3
Solution:
(c) 5

Question 70.
The point in the half plane 2x + 3y – 12 ≥ 0 is:
(a) (-7,8)
(c) (-7,-8)
(b) (7, -8)
(d) (7, 8)
Solution:
(d) (7, 8)

Question 71.
Any feasible solution which maximizes or minimizes the objective function is Called:
(a) A regional feasible solution
(b) An optimal feasible solution
(c) An objective feasible solution
(d) None of these
Solution:
(b) An optimal feasible solution

Question 72.
Objective function of a LPP is
(a) a constraint
(b) a function to be optimized
(c) a relation between the variables
(d) none of these
Solution:
(b) a function to be optimized

(B) Very Short Type Questions With Answers

Question 1.
If R is a relation on A such that R = R^-1, then write the type of the relation R.
Solution:
We know that (a, b) ∈ R ⇒ (b, a) ∈ R^-1
As R = R^-1, so R is symmetric. [2019(A)

Question 2.
Write the value of cos^-1 cos (\(\frac{3 \pi}{2}\)). [2019(A)
Solution:
cos^-1 cos (\(\frac{3 \pi}{2}\)) = cos^-1 (0) = \(\frac{\pi}{2}\)

Question 3.
Sets A and B have respectively m and n elements. The total number of relations from A to B is 64. If m < n and m ≠ 1, write the values of m and n respectively. [2018(A)
Solution:
|A| = m and |B| = n
Number of relations from A to B = 2^mn.
A.T.Q. 2^mn = 64 = 2⁶.
⇒ mn = 6, m < n with m ≠ 1.
∴m = 2, n = 3

Question 4.
Write the principal value of sin^-1 (\(\frac{- 1}{2}\)) + cos^-1 cos(\(\frac{- \pi}{2}\)) [2018(A)
Solution:
sin^-1 (\(\frac{- 1}{2}\)) + cos^-1 (cos\(\frac{- \pi}{2}\)) = \(\frac{- \pi}{6}\) + \(\frac{\pi}{2}\) = \(-\frac{\pi}{3}\)

Question 5.
Write the maximum value of x + y subject to: 2x + 3y ≤ 6, x ≥ 0, y ≥ 0. [2011(A)
Solution:
2x + 3y = 6 intersects the axes at (3, 0) and (0, 2)
∴ The maximum value of x + y = 3.

Question 6.
Define ‘feasible’ solution of an LPP. [2009(A)
Solution:
The solutions of LPP which satisfy all the constraints and non-negative restrictions are called feasible solutions.

Question 7.
Mention the quadrant in which the solution of an LPP with two decision variables lies when the graphical method is adopted. [2008(A)
Solution:
The solution lies in XOY or 1st quadrant.

Question 8.
Write the smallest equivalence relation on A = {1, 2, 3}.
Solution:
The relation R = {(1, 1), (2, 2), (3, 3)} is the smallest equivalence relation on set A.

Question 9.
Congruence modulo 3 relation partitions the set Z into how many equivalence classes?
Solution:
The relation congruence modulo 3 on the set Z partitions Z into three equivalence classes.

Question 10.
Give an example of a relation which is reflexive, symmetric but not transitive.
Solution:
The relation R = {(a, b), (b, a), (a, c), (c, a), (a, a), (b, b), (c, c)} defined on the set {a, b, c} is reflexive, symmetric but not transitive.

Question 11.
Give an example of a relation which is reflexive, transitive but not symmetric.
Solution:
‘‘The relation x ≤ y on Z” is reflexive, transitive but not symmetric.

Question 12.
Give an example of a relation which is reflexive but neither symmetric nor transitive.
Solution:
The relation R = {(a, a), (b, b), (c, c), (a, b), (b, c)} defined on the set A = {a, b, c} is reflexive but neither symmetric nor transitive.

Question 13.
Find three positive integers x_i, i =1, 2, 3 satisfying 3x ≡ 2 (mod 7)
Solution:
3x ≡ 2 mod 7
Least positive value of x = 3
Each member of [3] is a solution
∴ x = 3, 10, 17…

Question 14.
State the reason for relation R on {1, 2, 3} defined as {(1, 2), (2, 1)} is not transitive.
Solution:
(1, 2), (2, 1) ∈ R but (1, 1) ∉ R
∴ R is not transitive.

Question 15.
Give an example of a function which is injective but not surjective.
Solution:
f(x) = \(\frac{x}{2}\) from Z → R is injective but not surjective.

Question 16.
Let X = {1, 2, 3, 4}. Determine whether
f : X → X defined as given below have inverses. Find f^-1 if it exists:
f = {(1, 2), (2, 2), (3, 2), (4, 2)}
Solution:
f is not injective hence not invertible.

Question 17.
Let ∗ is a binary operation defined by a ∗ b = 3a + 4b – 2, find 4 ∗ 5.
Solution:
4 ∗ 5 = 3 × 4 + 4 × 5 – 2
= 12 + 20 – 2
= 30

Question 18.
Let the binary operation on Q defined as a ∗ b = 2a + b – ab, find 3 ∗ 4.
Solution:
3 ∗ 4 = 6 + 4 – 12 = -2

Question 19.
Let ∗ is a binary operation on Z defined as a ∗ b = a + b – 5 find the identity element for ∗ on Z.
Solution:
Let e is the identity element.
⇒ a ∗ e = e ∗ a = a
⇒ a + e – 5 = a
⇒ e = 5

Question 20.
Find the number of binary operations on the set {a, b}.
Solution:
Number of binary operations on
{a, b} = 2²² = e⁴ =16.

Question 21.
Let * is a binary operation on [0, ¥) defined as a * b = \(\sqrt{\mathbf{a}^2+\mathbf{b}^2}\) find the identity element.
Solution:
Let e is the identity element
⇒ a * e = \(\sqrt{\mathbf{a}^2+\mathbf{e}^2}\) = a
⇒ a² + e² = a²
⇒ e = 0

Question 22.
Evaluate cos^-1 (\(\frac{1}{2}\)) + 2 sin^-1 (\(\frac{1}{2}\)).
Solution:
cos^-1 (\(\frac{1}{2}\)) + 2 sin^-1 (\(\frac{1}{2}\))
= \(\frac{\pi}{3}\) + 2 × \(\frac{\pi}{6}\) = \(\frac{2 \pi}{3}\)

Question 23.
Find the value of tan^-1 √3 – sec^-1 (-2)
Solution:
tan^-1 √3 – sec^-1 (-2)
= \(\frac{\pi}{3}\) – \(\frac{2 \pi}{3}\) = – \(\frac{\pi}{3}\).

Question 24.
Evaluate tan (2  tan^-1 \(\frac{1}{3}\))
Solution:
tan (2  tan^-1 \(\frac{1}{3}\)) = tan tan^-1 \(\left(\frac{\frac{2}{3}}{1-\frac{2}{3}}\right)\)
= tan tan^-1 (2) = 2

Question 25.
Evaluate: sin^-1 (sin \(\frac{3 \pi}{5}\)).
Solution:
sin^-1 (sin \(\frac{3 \pi}{5}\)) = sin^-1 sin (\(\pi \frac{-2 \pi}{5}\))
= sin^-1 sin \(\frac{2 \pi}{5}\) = \(\frac{2 \pi}{5}\)

Question 26.
Evaluate tan^-1 1 = (2 cos \(\frac{\pi}{3}\))
Solution:
tan^-1 (2 cos \(\frac{\pi}{3}\))
= tan^-1 (2 × 1/2) = tan^-1 1 = \(\frac{\pi}{4}\)

Question 27.
Define the objective function.
Solution:
If C₁, C₂, C₃ …. Cn are constants and x₁, x₂, …… x_n are variables then the linear function z = C₁x₁ + C₂x₂ +…… C_nx_n which is to be optimized is called an objective function.

Question 28.
Define feasible solution.
Solution:
A set of values of the variables x₁, x₂, …… x_n is called a feasible solution of LPP if it satisfies the constraints and non-negative restrictions of the problem.

Question 29.
Define a convex set.
Solution:
A set is convex if every point on the line segment joining any two points lies on it.

Question 30.
State extreme point theorem.
Solution:
Let S is a convex polygon bounded by the straight lines. The linear function z = Ax + By attains its optimum value at the vertices of S.

(C) Short Type Questions With Answers

Question 1.
Construct the multiplication table X₇ on the set {1, 2, 3, 4, 5, 6}. Also find the inverse element of 4 if it exists. [2019(A)
Solution:
Given set A = { 1, 2, 3, 4, 5, 6} Binary operation ∗ defined on A is X₇.
i.e. a ∗ b = a × b mod 7
= The remainder on dividing a × b by 7
The composition table for this operation is:

∗	1	2	3	4	5	6
1	1	2	3	4	5	6
2	2	4	6	1	3	5
3	3	6	2	5	1	4
4	4	1	5	2	6	3
5	5	3	1	6	4	2
6	6	5	4	3	2	1

As the second row is identical to first row, we have ‘1’ as the identity element.
As 4 ∗ 2 = 2 ∗ 4 = 1 we have 4^-1 = 2

Question 2.
Let R be a relation on the set R of real numbers such that aRb iff a – b is an integer. Test whether R is an equivalence relation. If so, find the equivalence class of 1 and \(\frac{1}{2}\). [2019(A)
Solution:
The relation R on the set of real numbers is defined as
R = { (a, b) : a – b ∈ Z}
Reflexive:
∀ a ∈ R (set of real numbers)
a – a = 0 ∈ Z
⇒ (a, a) ∈ R
⇒ R is reflexive
Symmetric:
Let (a, b) ∈ R
⇒ a – b ∈ Z
⇒ b – a ∈ Z
⇒ (b, a) ∈ R
⇒ R is symmetric.
Transitive:
Let (a, b), (b, c) ∈ R
⇒ a – b and b – c ∈ Z
⇒ a – b + b – c ∈ Z
⇒ a – c ∈ Z
⇒ (a, c) ∈ R
⇒ R is transitive
Thus R is an equivalence relation
[1] = { x ∈ R : x – 1 ∈ Z} = Z
\(\frac{1}{2}\) = { x ∈ R : x – \(\frac{1}{2}\) ∈ Z}
= {x ∈ R : x = \(\frac{2 k+1}{2}\), k ∈ Z}

Question 3.
Two types of food X and Y are mixed to prepare a mixture in such a way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. These vitamins are available in 1 kg of food as per the table given below: [2019(A)

Vitamin
Food	A	B	C
X	1	2	3
Y	2	2	1

1 kg of food X costs ₹16 and 1 kg of food Y costs ₹20. Formulate the LPP so as to determine the least cost of the mixture containing the required amount of vitamins.
Solution:
Let x kg of food X and Y kg of food y are to be mixed to prepare the mixture.
Total cost = 16x + 20y to be minimum.
According to the question
Total vitamin A = x + 2y ≥ 10 units
Total vitamin B = 2x + 2y ≥ 12 units
Total vitamin C = 3x + y ≥ 8 units.
∴ The required LPP is minimize
Z= 16x + 20y
Subject to : x + 2y ≥ 10
x + y ≥ 6
3x + y ≥ 8
x, y ≥ 0

Question 4.
Let ~ be defined by (m, n) ~ (p, q) if mq = np, where m, n, p, q e Z – {0}. Show that it is an equivalence relation. [2018(A)
Solution:
Let A = z – {0}
~ is a relation on A x A defined as (m, n) ~ (p, q) ⇔ mq = np
Reflexive : For all (m, n) ∈ A × A
We have mn = nm
⇒ (m, n) ~ (m, n)
∴ ~ is reflexive.
Symmetric: Let {m, n), (p, q) ∈ A × A and (m, n) ~ (p, q)
⇒ mq = np
⇒ pn = qm
(p, q) ~ (m, n)
∴ ~ is symmetric.
Transitive: Let (m, n), (p, q), (x, y) ∈ A × A
and (m, n) ~ (p, q), (p, q) ~ (x, y)
⇒ mq = np and py = qx
⇒ mqpy = npqx
⇒ my = nx
⇒ (m, n) ~ (x, y)
∴ ~ is transitive.
Thus ~ is an equivalence relation.

Question 5.
Solve the following LPP graphically:
Minimize Z = 4x + 3y
subject to 2x + 5y ≥ 10
x, y ≥ 0. [2018(A)
Solution:
Given LPP is:
Minimize: Z = 4x + 3y
Subject to: 2x + 5y ≥ 10
x, y ≥ 0
Step – 1 Considering the constraints as equations we have 2x + 5y = 10

x	5	0
y	0	2

Let us draw the graph.

Step – 2 As 0(0, 0) does not satisfy 2x + 5y > 0 and x, y > 0 is the first quadrant, we have the shaded region is the feasible region whose vertices are A(5, 0) B(0, 2).
Step – 3 Z (5, 0) = 20
Z (0, 2) = 6 … Minimum
As the feasible region is unbounded. Let us draw the half plane.
4x + 3y < 6

x	0	\(\frac{3}{2}\)
y	2	0

Step – 4 As there is no point common to the feasible region and the half plane 4x + 3y < 6, we have Z is minimum for x = 0, y = 2 and Z(min) = 6

Question 6.
Find the feasible region of the system 2y – x > 0, 6y – 3x < 21, x > 0, y > 0. [2017 (A)
Solution:
Step – 1: Treating the constraints as equations we have
2y – x = 0
6y – 3x = 21
⇒ 2y – x = 0
2y – x = 7
Step – 2: Let us draw the lines.
Table – 1

x	0	2
y	0	1

x	1	37
y	4	5

Step – 3: Clearly A(1, 3) Satisfies both the constraints, x > 0, y > 0 is the first quadrant. Thus the shaded region is the feasible region.

Question 7.
Solve the following LPP graphically:
Maximize: Z = 20x + 30y
Subject to: 3x + 5y ≤ 15
x, y ≥ 0. [2014 (A), 2016 (A), 2017 (A)
Solution:
Given LPP is
Maximize: Z = 20x + 30y
Subject to: 3x + 5y ≤ 15
x, y ≥ 0
Step – 1 Treating the constraints as equations we get 3x + 5y = 15.
Step- 2 Let us draw the graph

x	5	0
y	0	3

Step – 3
As 0(0, 0) satisfies 3x + 5y ≤ 15 the shaded region is the feasible region.
Step – 4
The vertices ofthe feasible region are 0(0, 0), A(5, 0) and B(0, 3).
Z(0) = 0, Z(A) = 100, Z(B) = 90
Z attains maximum at A for x = 5 and y = 0.
The given LPP has a solution, x = 5, y = 0 and Z(max) = 100.

Question 8.
Find the feasible region of the following system:
2x + y ≥ 6, x – y ≤ 3, x ≥ 0, y ≥ 0. [2016 (A)
Solution:
Given system of inequations are
2x + y ≥ 6, x – y ≤ 3, x ≥ 0, y ≥ 0
Step- 1: Consider 2x + y = 6
x – y = 3
Step – 2: Let us draw the graph
Table- 1

x	3	0
y	0	6

Table- 2

x	3	0
y	0	-3

Step – 3: 0(0, 0) satisfies x – y < 3, does not satisfy 2x + y > 6 and x > 0, y > 0 is the first quadrant. Thus the shaded region is the feasible region.

Question 9.
Solve the following LPP graphically:
Minimize: Z = 6x₁ + 7x₂
Subject to: x₁ + 2x₂ ≥ 1
x₁, x₂ ≥ 0. [2015 (A)
Solution:
Given LPP is
Minimize: Z = 6x₁+ 7x₂
Subject to: x₁ + 2x₂ ≥ 2
x₁, x₂ ≥ 0
Let us draw the line x₁ + 2x₂ = 2

x1	0	2
x2	1	0

Clearly (0, 0) does not satisfy x₁ + 2x₂ ≥ 2 and x₁, x₂ ≥ 0 is the first quadrant.
The shaded region is the feasible region.
The coordinates of vertices are A(2, 0) and B(0, 1).

Point	Z = 6x1 + 7x2
A (2, 0)	12
B (0,1)	7 → Minimum

As there is no point common to the half plane 6x₁ + 7x₂ < 7 and the feasible region.
Z is minimum when x₁ = 0, y₁ =1 and the minimum value of z = 7

Question 10.
Find the feasible region of the following system:
2y – x ≥ 0, 6y – 3x ≤ 21, x ≥ 0, y ≥ 0. [2015 (A)
Solution:
Given system is
2y – x ≥ 0
6y – 3x ≤ 21
x, y ≥ 0.
Considering the constraints as equations we have
2y – x = 0
and 6y – 3x = 21

x	0	2
x	0	1

⇒ 3y – x = 7

x	-7	2
x	0	3

Let us draw the lines

Clearly (2, 0) does not satisfy 2y – x ≥ 0 and satisfies 6y – 3x ≤ 21
∴ The shaded region is the feasible region.

Question 11.
Find the maximum value of z = 50x₁ + 60x₂
subject to 2x₁ + 3x₂ ≤ 6
x₁, x₂ ≥ 0. [2013 (A)
Solution:
Let us consider the constraints as equations.
2x₁ + 3x₂ = 6 … (1)
The table of some points on (1) is

x1	0	3
x2	2	0

Let us draw the line 2x₁ + 3x₂ = 6

As (0, 0) satisfies the inequality 2x₁ + 3x₂ ≤ 6 and x₁, x₂ ≥ 0 is the first quadrant, the shaded region is the feasible region with corner points O(0, 0), A(3, 0) and B(0, 2).

Corner point	z = 50x1 + 60x2
O(0, 0)	0
A(3, 0)	150
B(0, 2)	120

Thus Z_(max) = 150 for x₁ = 3, x₂ = 0

Question 12.
Shade the feasible region satisfying the inequations 2x + 3y ≤ 6, x ≥ 0, y ≥ 0 in a rough sketch. [2011(A)
Solution:
Let us consider the line 2x + 3y = 6

x1	0	3
x2	2	0

Let us draw the line on the graph

The feasible region is shaded in the figure.

Question 13.
Show the feasible region for the following constraints in a graph:
2x + y ≤ 4, x ≥ 0, y ≥ 0. [2010(A)
Solution:
Let us draw the graph of 2x + y = 4.

The shaded region shows the feasible region.

Odisha State Board Elements of Mathematics Class 12 Solutions CHSE Odisha Chapter 5 Determinants Ex 5(a) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 5 Determinants Exercise 5(a)

Question 1.
Evaluate the following determinants.
(i) \(\left|\begin{array}{ll}
1 & 1 \\
2 & 3
\end{array}\right|\)
Solution:
\(\left|\begin{array}{ll}
1 & 1 \\
2 & 3
\end{array}\right|\) = 3 – 2 = 1

(ii) \(\left|\begin{array}{ll}
2 & -3 \\
1 & -4
\end{array}\right|\)
Solution:
\(\left|\begin{array}{ll}
2 & -3 \\
1 & -4
\end{array}\right|\) = -8 + 3 = -5

(iii) \(\left|\begin{array}{ll}
\sec \theta & \tan \theta \\
\tan \theta & \sec \theta
\end{array}\right|\)
Solution:
\(\left|\begin{array}{ll}
\sec \theta & \tan \theta \\
\tan \theta & \sec \theta
\end{array}\right|\) = sec² θ – tan² θ = 1

(iv) \(\left|\begin{array}{ll}
0 & x \\
2 & 0
\end{array}\right|\)
Solution:
\(\left|\begin{array}{ll}
0 & x \\
2 & 0
\end{array}\right|\) = 0 – 2x = -2x

(v) \(\left|\begin{array}{cc}
1 & \omega \\
-\omega & \omega
\end{array}\right|\)
Solution:
\(\left|\begin{array}{cc}
1 & \omega \\
-\omega & \omega
\end{array}\right|\) = ω + ω² = -1

(vi) \(\left|\begin{array}{cc}
4 & -1 \\
3 & 2
\end{array}\right|\)
Solution:
\(\left|\begin{array}{cc}
4 & -1 \\
3 & 2
\end{array}\right|\) = 8 + 3 = 11

(vii) \(\left|\begin{array}{ll}
\cos \theta & \sin \theta \\
\sin \theta & \cos \theta
\end{array}\right|\)
Solution:
\(\left|\begin{array}{ll}
\cos \theta & \sin \theta \\
\sin \theta & \cos \theta
\end{array}\right|\) = cos² θ – sin² θ = cos 2θ

(viii) \(\left|\begin{array}{lll}
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1
\end{array}\right|\)
Solution:
\(\left|\begin{array}{lll}
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1
\end{array}\right|\) = 0
as the rows are identical.

(ix) \(\left|\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right|\)
Solution:
\(\left|\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right|\) = 1\(\left|\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right|\) = 1 – 0 = 1

(x) \(\left|\begin{array}{ccc}
2 & 3 & 1 \\
0 & 0 & 0 \\
-1 & 2 & 0
\end{array}\right|\)
Solution:
\(\left|\begin{array}{ccc}
2 & 3 & 1 \\
0 & 0 & 0 \\
-1 & 2 & 0
\end{array}\right|\) = 0
as all the entries in the 2nd row are zero.

(xi) \(\left|\begin{array}{ccc}
1 & x & y \\
0 & \sin x & \sin y \\
0 & \cos x & \cos y
\end{array}\right|\)
Solution:
\(\left|\begin{array}{ccc}
1 & x & y \\
0 & \sin x & \sin y \\
0 & \cos x & \cos y
\end{array}\right|\) = 1\(\left|\begin{array}{cc}
\sin x & \sin y \\
\cos x & \cos y
\end{array}\right|\)
= sin x cos y – cos x sin y = sin (x – y)

(xii) \(\left|\begin{array}{lll}
1 & 2 & 3 \\
1 & 2 & 3 \\
3 & 4 & 5
\end{array}\right|\)
Solution:
\(\left|\begin{array}{lll}
1 & 2 & 3 \\
1 & 2 & 3 \\
3 & 4 & 5
\end{array}\right|\) = 0 (∵ R₁ = R₂)

(xiii) \(\left|\begin{array}{lll}
0.2 & 0.1 & 3 \\
0.4 & 0.2 & 7 \\
0.6 & 0.3 & 2
\end{array}\right|\)
Solution:
\(\left|\begin{array}{lll}
0.2 & 0.1 & 3 \\
0.4 & 0.2 & 7 \\
0.6 & 0.3 & 2
\end{array}\right|\)
= 2\(\left|\begin{array}{lll}
0.2 & 0.1 & 3 \\
0.4 & 0.2 & 7 \\
0.6 & 0.3 & 2
\end{array}\right|\) = 0 (∵ C₁ = C₂)

(xiv) \(\left|\begin{array}{ccc}
1 & \omega & \omega^2 \\
\omega & \omega^2 & 1 \\
\omega^2 & 1 & \omega
\end{array}\right|\)
Solution:
\(\left|\begin{array}{ccc}
1 & \omega & \omega^2 \\
\omega & \omega^2 & 1 \\
\omega^2 & 1 & \omega
\end{array}\right|\)

(xv) \(\left|\begin{array}{lll}
1 & 1 & 1 \\
2 & 2 & 2 \\
3 & 3 & 3
\end{array}\right|\)
Solution:
\(\left|\begin{array}{lll}
1 & 1 & 1 \\
2 & 2 & 2 \\
3 & 3 & 3
\end{array}\right|\) = 0 (∵ C₁ = C₂)

(xvi) \(\left|\begin{array}{ccc}
-6 & 0 & 0 \\
3 & -5 & 7 \\
2 & 8 & 11
\end{array}\right|\)
Solution:
\(\left|\begin{array}{ccc}
-6 & 0 & 0 \\
3 & -5 & 7 \\
2 & 8 & 11
\end{array}\right|\)
= (-6) \(\left|\begin{array}{cc}
-5 & 7 \\
8 & 11
\end{array}\right|\) = = (-6) (- 55 – 56)
= (-6) (-111) = 666

(xvii) \(\left|\begin{array}{lll}
1 & 0 & 0 \\
2 & 3 & 5 \\
4 & 1 & 3
\end{array}\right|\)
Solution:
\(\left|\begin{array}{lll}
1 & 0 & 0 \\
2 & 3 & 5 \\
4 & 1 & 3
\end{array}\right|\)
= 1 \(\left|\begin{array}{ll}
3 & 5 \\
1 & 3
\end{array}\right|\) = 9 – 5 = 4

(xviii) \(\left|\begin{array}{ccc}
-18 & 17 & 19 \\
3 & 0 & 0 \\
-14 & 5 & 2
\end{array}\right|\)
Solution:
\(\left|\begin{array}{ccc}
-18 & 17 & 19 \\
3 & 0 & 0 \\
-14 & 5 & 2
\end{array}\right|\)
= -3 \(\left|\begin{array}{cc}
17 & 19 \\
5 & 2
\end{array}\right|\)
(Expanding along 2nd row)
= – 3 (34 – 95)
= (-3) (-61) = 183

Question 2.
State true or false.
(i) If the first and second rows of a determinant be interchanged then the sign of the determinant is changed.
Solution:
True

(ii) If first and third rows of a determinant be interchanged then the sign of the determinent does not change.
Solution:
False

(iii) If in a third order determinant first row be changed to second column. Second row to 1st column and third row to third column, then the value of the determinant does not change.
Solution:
False

(iv) A row and a column of a determinant can have two or more common elements.
Solution:
False

(v) The minor and the co-factor of the element a₃₂ of a determinant of third order are equal.
Solution:
False

(vi) \(\left|\begin{array}{lll}
3 & 1 & 3 \\
0 & 4 & 0 \\
1 & 3 & 1
\end{array}\right|\) = 0
Solution:
True

(vii) \(\left|\begin{array}{lll}
6 & 4 & 2 \\
4 & 0 & 7 \\
5 & 3 & 4
\end{array}\right|\) = \(\left|\begin{array}{lll}
6 & 4 & 5 \\
4 & 0 & 3 \\
2 & 7 & 3
\end{array}\right|\)
Solution:
True

(viii) \(\left|\begin{array}{lll}
2 & 3 & 4 \\
5 & 6 & 7 \\
1 & 2 & 3
\end{array}\right|\) = \(\left|\begin{array}{lll}
4 & 2 & 3 \\
7 & 5 & 6 \\
3 & 1 & 2
\end{array}\right|\)
Solution:
True

Question 3.
Fill in the blanks with appropriate answer from the brackes.
(i) The value of \(\left|\begin{array}{ccc}
0 & 8 & 0 \\
25 & 520 & 25 \\
1 & 410 & 0
\end{array}\right|\) = _______. (0, 25, 200, -250)
Solution:
200

(ii) If ω is the cube root of unity, then \(\left|\begin{array}{ccc}
1 & \omega & \omega^2 \\
\omega & \omega^2 & 1 \\
\omega^2 & 1 & \omega
\end{array}\right|\) = _______. (1, 0, ω, ω²)
Solution:
0

(iii) The value of the determinant \(\left|\begin{array}{lll}
1 & a & b+c \\
1 & b & c+a \\
1 & c & a+b
\end{array}\right|\) = _______. (a + b – c, (a + b + c)², 0, 1 + a + b + c)
Solution:
0

(iv) If \(\left|\begin{array}{lll}
a & b & c \\
b & a & b \\
x & b & c
\end{array}\right|\) = 0, then x = _______. (a, b, c, a + b + c)
Solution:
a

(v) \(\left|\begin{array}{lll}
a_1+a_2 & a_3+a_4 & a_5 \\
b_1+b_2 & b_3+b_4 & b_5 \\
c_1+c_2 & c_3+c_4 & c_5
\end{array}\right|\) can be expressed at the most as _______, different 3rd order determinants. (1, 2, 3, 4)
Solution:
4

(vi) Minimum value of \(\left|\begin{array}{cc}
\sin x & \cos x \\
-\cos x & 1+\sin x
\end{array}\right|\) is _______. (-1, 0, 1, 2)
Solution:
0

(vii) The determinant \(\left|\begin{array}{lll}
1 & 1 & 1 \\
1 & 2 & 3 \\
1 & 3 & 6
\end{array}\right|\) is not equal to _______. \(\left(\left|\begin{array}{lll}
2 & 1 & 1 \\
2 & 2 & 3 \\
2 & 3 & 6
\end{array}\right|,\left|\begin{array}{lll}
2 & 1 & 1 \\
3 & 2 & 3 \\
4 & 3 & 6
\end{array}\right|,\left|\begin{array}{lll}
1 & 2 & 1 \\
1 & 5 & 3 \\
1 & 9 & 6
\end{array}\right|,\left|\begin{array}{ccc}
3 & 1 & 1 \\
6 & 2 & 3 \\
10 & 3 & 6
\end{array}\right|\right)\)
Solution:
\(\left|\begin{array}{lll}
2 & 1 & 1 \\
2 & 2 & 3 \\
2 & 3 & 6
\end{array}\right|\)

(viii) With 4 different elements we can construct _______ number of different determinants of order 2. (1, 6, 8, 24)
Solution:
6

Question 4.
Solve the following:
(i) \(\left|\begin{array}{cc}
4 & x+1 \\
3 & x
\end{array}\right|\)
Solution:
\(\left|\begin{array}{cc}
4 & x+1 \\
3 & x
\end{array}\right|\) = 5
or, 4x – 3x – 3 = 5 or, x = 8

(ii) \(\left|\begin{array}{ccc}
\boldsymbol{x} & a & a \\
m & m & m \\
b & x & b
\end{array}\right|\) = 0
Solution:

(Replacing C₁ and C₂ by C₁ – C₃ and C₂ – C₃ respectively)
⇒ m |(x – a) (-x + b)| = 0
⇒ m (x – a) (b – x) – 0 ⇒ x = a, b

(iii) \(\left|\begin{array}{lll}
7 & 6 & x \\
2 & x & 2 \\
x & 3 & 7
\end{array}\right|\) = 0
Solution:

or, (x – 7) (7x + x² – 1 8) = 0
or, (x – 7) (x² + 7x – 18) = 0
or, (x – 7) (x + 9) (x – 2) = 0
∴ x = -9, 2, 7

(iv) \(\left|\begin{array}{ccc}
0 & x-a & x-b \\
x+a & 0 & x-c \\
x+b & x+c & 0
\end{array}\right|\) = 0
Solution:

or, – (x – a) {0 – (x + b) (x – c)} + (x – b) (x + a) (x + c) = 0
or, (x – a) (x + b) (x – c) + (x – b) (x + a) (x + c) = 0
or, (x² + bx – ax – ab) (x – c) + (x² + ax – bx – ab) (x + c) = 0
or, x³ – cx² + bx² – bcx – ax² + acx – abx + abc + x³ + cx² + ax² + acx- bx² – bcx – abx – abc = 0
or, 2x³ – 2abx – 2bcx + 2acx = 0
or, 2x (x² – ab – bc + ac) = 0
x = 0, x² = ab + bc – ca
∴ x = 0, x = \(\sqrt{a b+b c-c a}\)

(v) \(\left|\begin{array}{ccc}
\boldsymbol{x}+\boldsymbol{a} & \boldsymbol{b} & \boldsymbol{c} \\
\boldsymbol{b} & \boldsymbol{x}+\boldsymbol{c} & \boldsymbol{a} \\
\boldsymbol{c} & \boldsymbol{a} & \boldsymbol{x}+\boldsymbol{b}
\end{array}\right|\) = 0
Solution:

⇒ (x + a + b + c) {x² + bx + cx + bc – a² – bx – b² + ca + ab – cx – c² = 0}
⇒ (x + a + b + c) {x² – a² – b² – c² + ab + bc + ca} = 0
⇒ x + a + b + c = 0
or, x² – a² – b² – c² + ab + bc + ca = 0
⇒ x = – (a + b + c)
∴ or x = \(\sqrt{a^2+b^2+c^2-a b-b c-c a}\)

(vi) \(\left|\begin{array}{ccc}
1+x & 1 & 1 \\
1 & 1+x & 1 \\
1 & 1 & 1+x
\end{array}\right|\) = 0
Solution:

(vii) \(\left|\begin{array}{ccc}
1 & 4 & 20 \\
1 & -2 & 5 \\
1 & 2 x & 5 x^2
\end{array}\right|\) = 0
Solution:

⇒ -30x² + 30 + 30x + 30 = 0
⇒ -30x² + 30x + 60 = 0
⇒ x² – x – 2 = 0
⇒ x² – 2x + x + 2 = 0
⇒ (x – 2) (x + 1) = 0
⇒ x = 2, -1

(viii) \(\left|\begin{array}{ccc}
x+1 & \omega & \omega^2 \\
\omega & x+\omega^2 & 1 \\
\omega^2 & 1 & x+\omega
\end{array}\right|\) = 0
Solution:

⇒ x(x² + xω – xω² + xω² + ω³ – ω⁴ – xω – ω² + ω³ – 1 + ω² + ω – ω³) = 0
⇒ x(x² + ω³ – ω⁴ – ω² + ω³ – 1 + ω² + ω – ω³) = 0
⇒ x(x² + ω³ – ω + ω – 1) = 0
⇒ x(x² + 1 – ω + ω – 1) = 0 (∵ ω³ = 1)
⇒ x³ = 0
⇒ x = 0

(ix) \(\left|\begin{array}{ccc}
2 & 2 & x \\
-1 & x & 4 \\
1 & 1 & 1
\end{array}\right|\) = 0
Solution:

or, 10 + 10x – x² – 8x – x – 8 = 0
or, x² – 2x + x – 2 = 0
or, (x – 2) (x + 1) = 0
x = 2, x = -1

(x) \(\left|\begin{array}{lll}
x & 1 & 3 \\
1 & x & 1 \\
3 & 6 & 3
\end{array}\right|\) = 0
Solution:

or, x(3x – 6) – 0 + 3(6 – 3x) = 0
or, 3x² – 6x + 18 – 9x = 0
or, 3x² – 15x + 18 = 0
or, x² – 5x + 6 = 0
or, (x – 3) (x – 2) = 0
x = 3 or, x = 2

Question 5.
Evaluate the following
(i) \(\left|\begin{array}{ccc}
2 & 3 & 4 \\
1 & -1 & 3 \\
4 & 1 & 10
\end{array}\right|\)
Solution:

(ii) \(\left|\begin{array}{lll}
\boldsymbol{x} & \mathbf{1} & 2 \\
\boldsymbol{y} & \mathbf{3} & 1 \\
z & 2 & 2
\end{array}\right|\)
Solution:

(iii) \(\left|\begin{array}{ccc}
x & 1 & -1 \\
2 & y & 1 \\
3 & -1 & z
\end{array}\right|\)
Solution:

= x (yz + z – z + 1) – (2z – 2 – 3y – 3)
= xyz + x – 2z + 3y + 5
= xyz + x + 3y – 2z + 5

(iv) \(\left|\begin{array}{lll}
a & h & g \\
h & b & f \\
g & f & c
\end{array}\right|\)
Solution:

= a(bc – f²) – h (ch – fg) + g (hf – bg)
= abc – af² – ch² + fgh + fgh – bg²
= abc + 2fgh – af² – bg² – ch²

(v) \(\left|\begin{array}{lll}
a & h & g \\
h & b & f \\
g & f & c
\end{array}\right|\)
Solution:

(vi) \(\left|\begin{array}{ccc}
\sin ^2 \theta & \cos ^2 \theta & 1 \\
\cos ^2 \theta & \sin ^2 \theta & 1 \\
-10 & 12 & 2
\end{array}\right|\)
Solution:

(vii) \(\left|\begin{array}{ccc}
-1 & 3 & 2 \\
1 & 3 & 2 \\
1 & -3 & -1
\end{array}\right|\)
Solution:

(viii) \(\left|\begin{array}{ccc}
11 & 23 & 31 \\
12 & 19 & 14 \\
6 & 9 & 7
\end{array}\right|\)
Solution:

(ix) \(\left|\begin{array}{ccc}
37 & -3 & 11 \\
16 & 2 & 3 \\
5 & 3 & -2
\end{array}\right|\)
Solution:

(x) \(\left|\begin{array}{ccc}
2 & -3 & 4 \\
-4 & 2 & -3 \\
11 & -15 & 20
\end{array}\right|\)
Solution:

= 2(40 – 45) + 3(-80 + 33) + 4(60 – 22)
= -10 – 141 + 152 = -151 + 152 = 1

Question 6.
Show that x = 1 is a solution of \(\left|\begin{array}{ccc}
x+1 & 3 & 5 \\
2 & x+2 & 5 \\
2 & 3 & x+4
\end{array}\right|\) = 0
Solution:

or, (x + 9) {(x – 1)²} – 0
or, x = -9, 1
∴ x = 1 is a solution of the given equation.

Question 7.
Show that (a + 1) is a factor of \(\left|\begin{array}{ccc}
a+1 & 2 & 3 \\
1 & a+1 & 3 \\
3 & -6 & a+1
\end{array}\right|\) = 0
Solution:

= (a+ 1) {(a + 1)² + 18} – 2(a + 1 – 9) + 3(- 6 – 3a – 3)
= (a + 1) (a² + 2a + 1 + 18) – 2(a – 8) + 3(- 9 – 3a)
= (a + 1) (a² + 2a + 19) – 2a + 16 – 27 – 9a
= (a + 1) (a² + 2a + 19) – 11a – 11
= (a + 1) (a² + 2a + 19) – 11(a + 1)
= (a + 1) (a² + 2a + 19 – 11)
= (a + 1) (a² + 2a + 8)
∴ (a + 1) is a factor of the above determinant.

Question 8.
Show that \(\left|\begin{array}{ccc}
a_1 & b_1 & -c_1 \\
-a_2 & b_2 & c_2 \\
a_3 & b_3 & -c_3
\end{array}\right|=\left|\begin{array}{lll}
a_1 & b_1 & c_1 \\
a_2 & b_2 & c_2 \\
a_3 & b_3 & c_3
\end{array}\right|\)
Solution:

Question 9.
Prove the following
(i) \(\left|\begin{array}{lll}
a & b & c \\
\boldsymbol{x} & y & z \\
\boldsymbol{p} & q & r
\end{array}\right|=\left|\begin{array}{lll}
\boldsymbol{y} & \boldsymbol{b} & \boldsymbol{q} \\
\boldsymbol{x} & \boldsymbol{a} & p \\
z & c & r
\end{array}\right|=\left|\begin{array}{lll}
\boldsymbol{x} & \boldsymbol{y} & z \\
\boldsymbol{p} & \boldsymbol{q} & r \\
a & b & c
\end{array}\right|\)
Solution:

(ii) \(\left|\begin{array}{ccc}
1+a & 1 & 1 \\
1 & 1+b & 1 \\
1 & 1 & 1+c
\end{array}\right|\) = abc \(\left(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\)
Solution:

(iii) \(\left|\begin{array}{lll}
b+c & c+a & a+b \\
q+r & r+p & p+q \\
y+z & z+x & x+y
\end{array}\right|=2\left|\begin{array}{lll}
a & b & c \\
p & q & r \\
x & y & z
\end{array}\right|\)
Solution:

(iv) \(\left|\begin{array}{lll}
(a+1)(a+2) & a+2 & 1 \\
(a+2)(a+3) & a+3 & 1 \\
(a+3)(a+4) & a+4 & 1
\end{array}\right|\) = -2
Solution:

(v) \(\left|\begin{array}{ccc}
a+d & a+d+k & a+d+c \\
c & c+b & c \\
d & d+k & d+c
\end{array}\right|\) = abc
Solution:

(vi) \(\left|\begin{array}{ccc}
1 & 1 & 1 \\
b+c & c+a & c+a \\
b^2+c^2 & c^2+a^2 & a^2+b^2
\end{array}\right|\) = (b – c) (c – a) (a – b)
Solution:

(vii) \(\left|\begin{array}{lll}
a & a^2 & a^3 \\
b & b^2 & b^3 \\
c & c^2 & c^3
\end{array}\right|\) = abc (a – b) (b – c) (c – a)
Solution:

(viii) \(\left|\begin{array}{ccc}
\boldsymbol{b}+\boldsymbol{c} & \boldsymbol{a} & \boldsymbol{a} \\
\boldsymbol{b} & \boldsymbol{c}+\boldsymbol{a} & \boldsymbol{b} \\
\boldsymbol{c} & \boldsymbol{c} & \boldsymbol{a}+\boldsymbol{b}
\end{array}\right|\) = 4abc
Solution:

= (b + c – a) {(a + b) (c + a – b) – b (c – a – b)} + a (b – c – a) (c – a – b)
= (b + c – a)(ca + a² – ab + bc + ab – b² – bc + ab + b²) + a(bc – ab – b² – c² + ca + bc – ac + a² + ab)
= (b + c – a) (a² + ab + ca) + a (a² – b² – c² + 2bc)
= a²b + ab² + abc + ca² + abc + c²a – a³ – a²b – ca² + a³ – b²a – c²a + 2abc = 4abc

(ix) \(\left|\begin{array}{ccc}
b^2+c^2 & a b & a c \\
a b & c^2+a^2 & b c \\
c a & c b & a^2+b^2
\end{array}\right|\) = 4a²b²c²
Solution:

(x) \(\left|\begin{array}{ccc}
a & b & c \\
a^2 & b^2 & c^2 \\
b c & c a & a b
\end{array}\right|\) = (b – c) (c – a) (a – b) (bc + ca + ab)
Solution:

= (a – b) (b – c) (c – a) – (- ab + c²) + c (a + b + c)
= (a – b) (b – c) (c – a) (ab – c² + ca + bc + c²)
= (a – b) (b – c) (c – a) (ab + bc + ca)

(xi) \(\left|\begin{array}{ccc}
a-b-c & 2 a & 2 a \\
2 b & b-c-a & 2 b \\
2 c & 2 c & c-a-b
\end{array}\right|\) = (a + b+ c)³
Solution:

(xii) \(\left|\begin{array}{ccc}
(v+w)^2 & u^2 & u^2 \\
v^2 & (w+u)^2 & v^2 \\
w^2 & w^2 & (u+v)^2
\end{array}\right|\) = 2uvw (u + v + w)³
Solution:

Question 10.
Factorize the following
(i) \(\left|\begin{array}{ccc}
x+a & b & c \\
b & x+c & a \\
c & a & x+b
\end{array}\right|\)
Solution:

= (x + a + b + c) [(x + c – b) (x + b – a) – (a – c) (a – x – c)]
= (x + a + b + c) (x² + xb – ax + cx +bc – ca – bx – b² + ab – a² + ax + ac + ac – cx – c²)
= (x + a + b + c) (x² – a² – b² – c² + ab + bc + ca)

(ii) \(\left|\begin{array}{ccc}
a & b & c \\
b+c & c+a & a+b \\
a^2 & b^2 & c^2
\end{array}\right|\)
Solution:

(iii) \(\left|\begin{array}{ccc}
x & 2 & 3 \\
1 & x+1 & 3 \\
1 & 4 & x
\end{array}\right|\)
Solution:

Question 11.
Show that by eliminating α and from the equations.
a_i α + b_i β + c_i = 0, i = 1, 2, 3 we get \(\left|\begin{array}{lll}
a_1 & b_1 & c_1 \\
a_2 & b_2 & c_2 \\
a_2 & b_3 & c_3
\end{array}\right|\) = 0
Solution:
We have
a₁ α + b₁ β + c₁ = 0    …..(1)
a₂ α + b₂ β + c₂ = 0    …..(2)
a₃ α + b₃ β + c₃ = 0    …..(3)
Solving (2) and (3) by cross-multiplication method we have

Question 12.
Prove the following:
(i) \(\left|\begin{array}{lll}
1 & b c & a(b+c) \\
1 & c a & b(c+a) \\
1 & a b & c(a+b)
\end{array}\right|\) = 0
Solution:

(ii) \(\left|\begin{array}{ccc}
x+4 & 2 x & 2 x \\
2 x & x+4 & 2 x \\
2 x & 2 x & x+4
\end{array}\right|\) = (5x + 4) (4- x)²
Solution:

(iii) \(\left|\begin{array}{l}
\sin \alpha \cos \alpha \cos (\alpha+\delta) \\
\sin \beta \cos \beta \cos (\beta+\delta) \\
\sin \alpha \cos \gamma \cos (\gamma+\delta)
\end{array}\right|\) = 0
Solution:

(iv) \(\left|\begin{array}{ccc}
1 & x & x^2 \\
x^2 & 1 & x \\
x & x^2 & 1
\end{array}\right|\) = (1 -x³)²
Solution:

Question 13.
Prove that the points (x₁, y₁), (x₂, y₂), (x₃, y₃) are collinear if \(\left|\begin{array}{lll}
x_1 & y_1 & 1 \\
x_2 & y_2 & 1 \\
x_3 & y_3 & 1
\end{array}\right|\) = 0
Solution:
From geometry, we know that, if the points A, B, C, are collinear, then the area of the triangle ABC with vertices (x₁, y₁), (x₂, y₂) and (x₃, y₃) is zero.
\(\left|\begin{array}{lll}
x_1 & y_1 & 1 \\
x_2 & y_2 & 1 \\
x_3 & y_3 & 1
\end{array}\right|\) = 0

Question 14.
If A + B + C = π, prove that \(\left|\begin{array}{lll}
\sin ^2 A & \cot A & 1 \\
\sin ^2 B & \cot B & 1 \\
\sin ^2 C & \cot C & 1
\end{array}\right|\) = 0
Solution:

Question 15.
Eliminate x, y, z from a = \(\frac{x}{y-z}\), b = \(\frac{y}{z-x}\), c = \(\frac{z}{x-y}\)
Solution:
We have
a = \(\frac{x}{y-z}\), b = \(\frac{y}{z-x}\), c = \(\frac{z}{x-y}\)
ay – az – x = 0, bz – bx – y = 0, cx – cy – z = 0
x – ay + az = 0
bx + y – bz = 0
cx – cy – z = 0
Now eliminating x, y, z from the above equations we have,

or, – 1 – bc + a(-b + bc) + a(-bc – c) = 0
or, – 1 – bc – ab + abc – abc – ac = 0
or, ab + bc + ca + 1 = 0

Question 16.
Given the equations
x = cy + bz, y = az + ex and z = bx + ay where x, y and z are not all zero, prove that a² + b² + c² + 2abc = 1 by determinant method.
Solution:
x = cy + bz, y = az + cx and z = bx + ay

or, 1 – a² + c(-c – ab) – b(ca + b) = 0
or, 1 – a² – c² – abc – abc – b² = 0
or, a² + b² + c² + 2abc = 1

Question 17.
If ax + hy + g = 0, hx + by +f = 0 and gx + fy + c = λ, find the value of λ, in the form of a determinant.
Solution:

Odisha State Board CHSE Odisha Class 12 Sociology Solutions Unit 5 Change and Development in India Long Answer Questions.

CHSE Odisha 12th Class Sociology Unit 5 Change and Development in India Long Answer Questions

Long Questions With Answers

Question 1.
What is Globalisation ? Discuss the different impacts of globalisation on society?
Answer:
Globalisation is a vast, complex and multi- faceted term, hence it is difficult to give a comprehensive definition A prominent development is marked in the international marketing environment. Today is the trend towards increasing economic interdependence and globalisation of markets. Besides, as we know globalisation is not a new term. Though the term was not used there has always been a trend for business transcending national boundaries.

Globalisation refers to a trend towards international business which gives stresses international competition. It refers to the greatest use of markets and the forces of competition to coordinate economic activities. It also means opening up the economy to foreign competition. Globalisation means being able to manufacture in the most cost-effective way possible anywhere in the world.

At the same time, it also refers to being able to prepare raw materials and drawing management resources from the cheapest source anywhere in the world. It considered the entire world as its market. Hence, globalisation refers to a process of increasing economic integration and growing economic inter-dependence between countries in the world economy.

It is associated not only with increasing cross-border movements of goods and services, capital technology, information and people but also with an organisation of economic activities which cross national boundaries. Thus, globalisation is a kind of new world order and reduction of states or demise of the state system. In short, globalisation means thinking globally, producing and making globally.

Impact of Globalisation on Indian Society:
Before stepping to analyse the impact of globalisation on Indian society it will be pertinent to know when India, adopts the principle of globalisation. Under the pressure from International Monetary Fund and the World Bank and due to the increasing realisation of Indian planners, leaders and administrators that globalisation is a panacea for Indian poverty.

the Indian economy has been opening up to globalisation since the 1980s. Restricting the policy framework and industrial production, inflow of capital goods and technology, and growing foreign collaboration and foreign credit have to a great extent turned the economy of global developments. However, the following are the impacts of globalization.

Free market economy:
One of the immediate impacts of globalisation is that market became free and open to competition to all. There is an increasing realisation that a free market is better for the growth of the economy.

Encourages foreign investment:
Globalisation encourages foreign investment in different sectors of the Indian economy. Different sectors of the Indian economy are made open to different multinational or foreign companies. These companies enter India and invest a number of foreign capital because of which the Indian economy gets a boost.

More employment opportunities:
Because of globalisation a large number of foreign and multi-national companies have entered India and settled in different industries within India. This resulted in the creation of large-scale- employment in Indian society. Both direct and indirect employment is created.

Privatisation :
Globalisation also encourages privatisation in India. Because of globalisation disinvestment process set in. Privatisation refers to process whereby public operations are transferred to the private sector. Privatisation as a tool of public policy and as a concept has emerged only in recent times.

Liberalisation:
Another impact of globalisation is liberalisation. It aims to free the Indian industrial economy from the cobwebs of unnecessary bureaucratic control. It was introduced in Indian society to integrate the Indian economy with the world economy. It also aims at to remove restrictions on direct foreign investment as also to free domestic entrepreneurs from the restrictions of MRTP.

The decline of small and cottage industries:
Another impact of globalisation is the fall or decline of small and cottage industries. Being unable to face the competitions posed by the large scale and multinational companies the small and cottage industries wither away. They cannot insist on facing cutthroat competition from these industries.

Development of global culture:
Another important impact of globalisation is the development of a global culture. The whole world is a village in miniature.

The demise of the nation-state:
Globalisation resulted in the Denise of nation-states or states. It creates a new world order in which the state has little role to play. Thus, these are different impacts of globalisation.

Question 2.
Write a short note on urbanisation?
Answer:
Urbanisation refers to the process of growth in cities it terms of their social structure, population, physical outlay and cultural organisations. The physical and social structure of society to a large extent governs the nature of urbanisation. No doubt the nature of urbanism differs from security to society depending upon its cultural historicity and transition.

In the abstract, urbanisation is universally associated with a wide living that farmers privacy, anonymity with physic or unity to quickly adapt to new ideas or innovations and greater industrialism or sense of identity. It promotes plurality of the styles of a high degree of elitism in cultural life and dominates literally traditional learnings and skills in economic and cultural domains.

Socially it is not characterised by a predominance of conjugal families, or a faster pace of work pattern. Urbanism promotes the emergence of overlapping cultural and social enclaves based on principles of kinship, religion, language and religion etc. In which people interact different levels of social and cultural contexts.

The problem of studying neighbourhoods in cities and towns is a part of the tradition of urban community studies which is relatively new in India. While some socialists have studied small towns as communities, others have studied words or neighbourhoods in parts traditional cities revealing homogeneity in terms of casts and religious- groups. The community organisation in such neighbourhoods differed from that in neighbourhoods or in namely established housing estates.

A large percentage of the sector’s population felt that they are not bound by common interests and problems. This suggests that planned neighbourhoods need necessarily be communities. More intensive studies of both traditional neighbourhoods and new housing estates would be essential to understand the processes of continuity and change in traditional urbanism.

Question 3.
Explain globalization and discuss its merits and demerits?
Answer:
Some of the positive impacts, advantages or merits of the process of globalization are discussed below:
Improves efficiency:
Globalization brings efficiency in production and increases the efficiency of labourers. Free trade and the opening up of the economy are the main basis of globalization. This leads to specialization of production which is possible only due to the increase of efficiency of technology, labourers and management production of specialised products leads to export.

Eliminates poverty:
Globalization eliminates poverty and a higher growth rate. It gives a boost to the stagnant economy and eliminates poverty. Globalization creates more employment opportunities which means less poverty.

Promotes healthy competition:
Globalization creates or promotes healthy competition
among producers. Because it has given birth to the world market and a producer has to produce qualitative products or goods for the global market one could not produce qualitative products of the world-class standard has existence will be at stake, its motto is to compete perish. All this promotes healthy competition among producers.

Creates global village:
Globalization helps in the development of a global village. It increases interdependence among nation-states by breaking up national boundaries. It also aims at the establishment of one world and one government.

Improves financial situation:
Adequate finance is a precondition for development. A poor or developing country needs more finances to establish industrial ventures under globalization, and more financial help or assistance is available from different financial institutions like the IMF world bank. Bank Insurance and multinational corporations.

Multinational Corporations make direct investments and provide technical know-how, market management skill and many other associated benefits. All this helps to improve the financial situation of a developing country at the initial stage.

Encourages migration:
Globalization encourages cross-border migration of workers which makes up the deficiency of workers in developed countries. Knowledge, workers IT and computer engineers have a chance to move freely searching for good salary and better service conditions. migration reduces pressure on land and brought more foreign currencies to the country. At the same time, it also solves the problem of unemployment. This globalization by encouraging migration creates many benefits.

Strengthens democracy:
Globalization provides economic freedom to many. Because of better economic freedom more and more people actively, participate in the democratic process of the country. Thus, globalization has strengthened democracy.

Encourages international cooperation:
Encouraging cross-border migration and breaking up national boundaries and creating world market globalization increases international cooperation in different spheres which works towards world peace. Globalization has many benefits for its credit. But it is not an unmixed blessing.

Cities have criticised globalization due to its following disadvantages.
Increases inequality:
Globalization increases inequality both between rich and poor people as well as between developed and undeveloped nations. Under the process of globalization, the rich become richer and the poor become poorer. Similarly developed or rich countries enjoy all the benefits from the process of globalization and become richer or developed day by day whereas developing or poor countries suffer from misery and poverty They can’t compete with them in the market and become losers.

Closer of Industries:
Globalization encourages free trade which may lead to the closure of many domestic or small-scale industries. These industries fail to compete with the multinationals and become sick. Due to the process of globalization a large number of small-scale industries have been closed down. This leads to a decrease in production and creates unemployment.

Divides the world:
As a divisive process globalization divides the world into rich and poor nations or into underdeveloped, developing and developed nations. This division creates many problems and intensifies conflicts and tensions.

Creates uncertainties:
Globalization creates many uncertainties among workers industrialists among financial institutions. Workers fear retrenchment, industrialists fear the closure of their industries, and financial institutions fear a recession. All these uncertainties affect production and upset the economy of underdeveloped or developing countries.

Degenerates Human values:
Globalization degenerates human values, and progress or development is always viewed in terms of economic growth. Achievement of high economic growth is the only. Human values have little importance.

Exploitation:
It seems as if exploitation is the main objective of globalization. Under the process of globalization, multinational companies exploit poor workers as well as poor underdeveloped and developing nations. They take advantage of cheap labour and resources. Maternity of the poor lost their occupation.

Negative impacts on agriculture:
Globalization has several negative impacts on agriculture. Increasing emphasis on intensive irrigation more use of chemical fertilisers and pesticides abandoning traditional practices and increasing productivity have proved to be dangerous. Too much stress on the modernization of agriculture strongly affects agriculture as well as the environment.

Cultural erosion:
Globalisation led to the erosion of culture. Due to the impact of western culture, people become alien to their own culture. People become stronger in their own land.

Weakening of states:
Under the process of globalization power of state weakened state act as an agent of multinational companies.lt protects their interest and neglects the weaker sections. Multinational companies interfere with policies and their course.

Globalization created lot of economic insecurities like cutthroat competition, retrenchment, unemployment etc. Globalization led to an increase of crimes which threatens the existence of mankind. Control of the state on the domestic economy diminishes. Globalization causes Brain Drain which harms poor nations. Frequent and unnecessary interference multinational companies in the domestic affairs of developing countries acts as a threat to the unity and sovereignty of these countries.

Question 4.
What do you mean by liberalization? discuss its merits and demerits?
Answer:
Liberalization is another process of social change in India. lt is considered one of constituent parts of economic reforms. As an important economic concept liberalization is becoming more popular day by day. Liberalization is mainly a western economic theory. This process has entered India due to the process of modernization and western impact on Indian society.

However, the process of Liberalization began in India during the mid-seventies due to the crisis in the Indian economy. In order to save India from the acute financial crises the then prime minister Narsimha Rao and his finance minister Dr Manmohan Singh introduced liberalization in India, by accepting liberalization as the economic policy of the government of India.

As a result, liberalization became one of the aspects of the new economic policy gives stressing on reduction of governmental control on trade, business and industry. It abolished industrial licensing for all projects except a few like security strategic concerns. Liberalization refers to the reduction of governmental control to the minimum in matters of trade, business, investment and industry.

It aims at the abolition of licenses and permits raj and opening up of the national economy to the world economy. It means the government must shame private entrepreneurs while taking economic decisions.lt aims to set trade, business and industry free and to enable it to run on commercial lines.

The main idea behind the process of liberalization is that as trade and commerce are global subject hence it should not be confined to a particular boundary. Hence governmental restrictions over economic and commercial activities should be minimised to the maximum.

Merits of Liberalizations:

• Liberalization provides better opportunities for competition
• Liberalization helps to increase the export of the country.
• It helps in the free movement of goods and services.
• It has led to the production of Eqailitative products.
• It has led to rapid industrialization.
• It has provided maximum liberty to private enterprises.
• It helps to reduce unnecessary governmental control.

Demerits of Liberalization:

• Liberalization has negative impacts on small-scale industries.
• It has seriously damaged the power of the state.
• It has seriously affected our agriculture and environment.
• Under liberalization, the rich become richer and the poor become poorer which is not a good bend.
• It also creates unemployment and poverty.
• Conditions of unskilled labour is very pitiable under liberalization.

Thus, from the above, it is concluded that liberalization itself is neither good nor bad. It is a double-edged weapon. It can provide many benefits to mankind and can also be harmful and can spell disaster. Hence, much depends on its use and its own attitude towards it. But we should be conscious while following this economic principle.

Question 5.
Explain urbanization and discuss its causes and consequences?
Answer:
Urbanization is one of the most important processes of social change in India Because of the tremendous increase in urban population all over the world including India the importance of the process of urbanization has increased manifold. The term urbanization perhaps comes from the urban. The term urban is very ancient in nature.

Ordinarily, by urban area, we mean an area with a high-density of population. It also refers to a way of life. According to the 1981 census, an urban area refers to all places municipality corporation, cantoment board etc or an area which has a minimum population of5000 and at least 75 per cent of the male working population is engaged in non-agricultural activities and a density of population at least 400 persons per sq. KM. Urban centuries or cities are very ancient in nature.

There were cities of urban centres in ancient civilization. 5000 years ago there was a city civilization in India. There was the existence of chief cities like Harappa and Mohenjodaro, Vatsayana, Meghasthenese and Kautilya in their books mention the existence of cities during ancient times. The Muslim rulers built great cities like Agra and Delhi.

Then Britishers built many cities, but the exact origin of the city is last in the obscurity of the past. However, the first cities seem to have appeared in between 6000 and 5000 B.C. these cities were small and hard to distinguish from towns. But the city in its real sense came into existence by 3000 B.C. After that, there was a fall for more than 2000 years.

Then cities came into existence in Greece, Rome, India, Egypt etc. but in spite of the growth of cities, most of the population of India live in villages which is true even today. Though India has been a land of villages but has also had an urban tradition since time immemorial. Though. there were cities in ancient civilization as well as in Indian society, it is only in the last two centuries that urbanization has become a characteristic form of human life.

Causes of Urbanization :
Urbanization is a worldwide phenomenon. The percentage of urban population and growth of urban centres has increased rapidly At about 30 per cent of India’s total population lives in urban areas. Thus rapid growth of Urbanization is caused by several factors. Some of the factors which cause urbanization are as follows:

The national increase in population:
The population of the world increases naturally This provides employment to the increasing population and meets the increasing demand of products of this population. Industries are set up and urban centres grow revolving around these industries. Besides more and more people migrate from rural areas to urban areas.

In search of employment, better health facilities and better living a result of urbanization spreads. Besides now- a- days there is a growing trend to live in urban areas which resulted in the growth of new urban centres and the spread of urbanization.

Migration:
Migration is another important cause of urbanization. Migration means the movement of people from one place to another. It refers to a kind of geographical mobility. normally from rural areas to better opportunities. Sometimes urban people also migrate to rural areas to live in a natural and pollution-free environment. Migration helps in the spread of urbanization.

Expansion of urban areas:
The expansion of urban areas resulted in urbanization. Due to the expansion of urban areas the outlying rural areas become urban areas and the process of urbanization spreads over.

Industrialization:
Rapid industrialization is also another important cause of urbanization, Urban areas develop around industrial centres. Due to the installation of more industries, new urban centres grow which resulted in the spread of urbanization. Besides people migrate from villages to industrial towns to work there which helps in the spread of urbanization.

Impact or consequences of urbanization :
Urbanization is not an unmixed blessing. It has many negative impacts on human living and social relationships. It has resulted in the breakdown of traditional social institutions and brought a number of changes in society. However, some of the impacts or consequences of urbanization are discussed below.

Impact on family:
Urbanization has a number of impacts on families. It leads to a decline in family size. It leads to the breaking up of a joint family and the creation of a nuclear family. Similarly, it also affects family lies and led to the decline of family control. Urban family loses their control over children It also weakens family bonds.

Impact on marriage:
Urbanization greatly affects our marriage system. Parental control over marriages gradually declines. Marriage ceases to be religious and becomes secular. Rites and rituals in marriage decline day by day. Due to the free mixing of boys and girls the number of love marriages increases. It also affects mortal bonds and marriage ceases to be permanent. The number of divorces is increased. The age of marriage also increases, and many people even remain unmarried.

The decline in fellow and sympathy:
Urbanization leads to a decline in fellow feelings and sympathy. Due to rapid population growth and overcrowding nature, fellow feeling and sympathy sharply is declined among urban people. Urban people remain so busy that they have little time to take part in others’ jobs and sorrows. Even urban people do not know their next-door neighbours. Everyone is concerned only about himself and has little concern for others.

The decline in family control:
Urbanization leads to a decline in family control. In an urban area, we found a nuclear family and it has little control over its members Besides urban people have no time to spend with their family and to know what their children are doing. Loss of family ties resulted in the decline of family control.

The decline in the influence of Religion:
Religion has lost its control over the minds of urban people. Urban people are more materialistic in nature and is self- centred. Different religious rites, rituals and practices lose their importance in urban areas.

Impact on the role and status of women:
Urbanization considerably affects the role and status of women. It has led to the increasing role of women in different spehers of society. They are now enjoying economic freedom and are at par with their male counterpart. A large number of women are working in industries, offices and business houses. All this has led to a change in the status of women. The increasing role and status of women considerably affect family life and husband-wife relations.

Impact on caste :
Urbanization deeply affects our traditional caste system. Many caste rules are under change and the caste system has lost its earlier rigidities and become more flexible. People are no more following their caste occupations and not obeying caste rules even during marriage. More and more intercaste marriages are taking place and some caste associations are emerging caste is now playing a major role in politics.

Development of slums:
One of the important consequences of urbanization is the development of slums. Due to the rapid growth of population and shortage of land area in urban areas most of people are living in slums. Their slums are the breeding ground for criminal activities and the spread of diseases.

The decline in moral values:
Another evil impact of urbanization is the degeneration of the moral values of urban dwellers. Due to the spread of education, economic independence, the decline of religion, and the growth of materialism, there is a great deal of change in the moral values of people which causes many social problems.

Question 6.
Define globalization and discuss its aims and features?
Answer:
The term globalization comes from the word global means covering or relating to the whole world. In other words, global means looking at everything from the whole world’s point of view not an individual point of view. It means borderless development internationalization of all aspects of human life. It also means thinking, doing and producing globally. Globalization is mainly economic in character.

now- a – days globalization is active in the economic field means economic globalization. It refers process of increasing economic integration and growing economic interdependence among countries. Because of globalization whole world inter linked and inter-connected through economic social, political and cultural relations.

Globalization means manufacturing things or products in the most effective way anywhere in the world it aims at procuring raw materials and management personnel from anywhere in the world. Globalization considers the entire world as a market. Globalization also refers to the process of opening up national markets to the global market.

Definitions:
According to Anita, “Globalization is a process through which an increasingly free flow of ideas, people, goods and capital leads to the integration of economies and societies. According to D.N. Dhanagare “globalization refers to the growing economic integration international level based significantly or activities of multinational corporations”.

According to the European Commission, “ globalization is the process by which markets the productions in different countries are becoming increasingly interdependent dynamic of trade in goods and services and flows of capital technology”. Anthony Gidden, Globalization can be defined as the intensification of worldwide social relations, which link distant localities.

such a way that local happenings are shaped by events occurring many miles away and vice-versa”. According to MC Grew, “Globalization refers to those processes operating at a global scale which cut across national boundaries integrating of connecting communities and organizations in space-time combinations making the world in reality and in experience more interconnected” Aims of Globalization:

• Opening up national economics and developing a single economic system.
• Reduction of trade barriers and free movement of products.
• The disintegration of geographic boundaries.
• Free flow of international trade.
• Integration of local economy with the worked economy.

Features of Globalization:
Globalization has the following features.

A complex process:
Globalization is a complex process. Increasing interdependence among nations, free flow of products, labour and trade and increasing socio-cultural contacts among nations makes it more complex and complicated.

A composite process:
Globalization is a composite process. Because a combination of a series of developments in the world led to its emergence. Development, in science and technology, development in die field of communication, increased social mobility and a number of other developments led to the development of globalization. A single cause factor or development is not responsible for globalization. Hence, it is a composite process.

A historical process :
Globalization is a historical process because erosion of the process goes to the period of the industrial revolution of the 16th century, but the trend for business transcending national boundaries is very old Hence, globalization is not a new concept but rather very ancient in nature.

An integrating process:
Globalization is a process of increasing economic integration. In this process markets, finance and technology are well integrated.

A multi-dimensional process:
Globalization is a multi-dimensional process because it has many faces. It can be understood from different angles. From an economic angle, it refers to the integration of the national economy with the world economy.

From a political angle, globalization refers to the emergence of a world state with the erosion of the sovereignty of the state. From a cultural angle, it refers to increased socio-cultural contact among nations all over the world. From an ideological angle, it refers to the victory of liberalization and capitalism over socialism. Globalization is associated with new technology like computers, the internet, electronic media, television and many others.

Globalization envisions the development of the world community. Globalization is also characterized by the development of multinational business corporations. Globalization is a self-contradictory process as. it contains the existence of contradictory forces like integration versus fragmentation, universalization versus particularization, and homogeneity versus heterogeneity.

Globalization is a dynamic process The process of globalization started in India in 1990. India opened its economy to the world economy then, but in the beginning, it follows a protective policy to safeguard its own industries. But now things have changed.

Question 7.
What is industrialization changing life and its positive effects on the Industrial Revolution?
Answer:
The Industrial Revolution affected every part of life in Great Britain but proved to be a mixed blessing. Eventually, industrialization led to a better quality of life for most people. the change in machine production initially caused human suffering Rapid industrialization brought plentiful Jobs but out also caused unhealthy working conditions air and water pollution and the illness of child labour.

It also led to rising class tensions, especially between the working class and the middle class. The pace of industrialization accelerated rapidly in Britain. By the 1800 people could earn higher wages in factors than a form. With this money, more people could offer to heat their homes with coal from walls and dine on Scottish beef. They were better clothing too, woven on power looms on England’s Industrial cities swelled with waves of job seekers.

Positive Effects of the Industrial Revolution Despite the problems followed industrialization the industrial Revolution had a number of positive effects. It created jobs for workers. It contributed to the wealth of the nation. It fastened technological progress and invention. It greatly increased the production of goods and raised the standard of living, perhaps.

most important it provided the hope of improvement in people’s lives. industrial Revolution produced a number of other benefits as well. These included healthier diets, better housing the cheaper, mass-produced clothing. Because the Industrial Revolution created a demand for engineers as well as clerical and professional workers, it expanded educational opportunities.

The middle and upper classes prospered immediately from the Industrial Revolution for the workers it took longer but their lives gradually improved during the 1800s Labourers eventually won higher wages shorter horns, and better working conditions after they joined together to form labour unions. The long-term effects of the Industrial Revolution are still evident most people today in industrialized countries can be offered consumer goods that would have been considered luxuries 60 or 60 years ago.

In addition, their living and working conditions are much improved over those of workers in the 19th century. Also, profits derived from industrialization produced tax revenues. These funds have allowed local state and federal governments to invest in urban improvements and the standard of living of most city dwellers.

Question 8.
Defining Modernization and politicians’ modernization?
Answer:
Modernization originally referred to the contrast and transition between a traditional agrarian society and the kind of modem society that is based on trade and industry, For example, traditional and modern would describe the difference between medieval England and late-Victorian Britain. A traditional society is vertically organized by hierarchical division by class or caste – a specialization of prestige.

But a modem society is horizontally organized by function, and the major social systems include the political system public administration (social service) the armed forces the legal system the economy religion, education the health service and the mass media. while a traditional society is like a pyramid of top-down authority a modem society is more like a mosaic held together the cement of mutual interdependence.

A further contrast is that traditional societies consist of a single, unified system with a single centre of power, while a modem society is composed of a plurality of autonomous systems each other do not absorb each other. Modem societies are fundamentally hetero- generous with multiple centres of power and thus is no accident but intrinsic to their nature.

Indeed the continued process of modernization tends to break down any remaining vestiges of hierarchy and centralized domination of social functions. Modernization is a product of the selection process. This means that not all political initiatives that are self-described as modernization can be considered genuine modernization.

Many such modernizing reforms actually diminish the selection processes that tend to be generally complex functionally. Thus mismatch between theoric and reality arises from a terminological ambiguity which modernization means different things different contexts. In this book, we follow humans in arguing that true modernization is the increase in the functional specialization of societies.

that the functionality of a social system is defined by its having prevailed over other social system variants during a historical competition. In other words, functionality is relative and the most functional system is one that has displaced other system variants a competitive, Selection processes are therefore intrinsic to modernization.

But another use of modernization is as a synonym for rationalization. Rationalization usually entails the reform of a social system by central government along the lines of making out more of a rational bureaucracy involving standardization of exploit procedures hierarchical command system The confusion is across from the fact that (as weber famously noted).

the emergence of rational bureaucracies characterized many modem states such as the nineteenth century. Germany later thus ideal of rational bureaucracy as being the most efficient mode of the organization was to dominate the social system of the USSR and outs satellites.

Modernisation and Its Impact on Indian society:
The term’Modemisation’ is a broader and more complex term. According to S.H. Alatas, “Modernisation is a process by which modem scientific knowledge is introduced in the society with the ultimate purpose of achieving a better and more satisfactory life in the broadest sense of the term accepted by the society concerned”.

Prof Yogendra Singh says, “Modernisation symbolizes a rational attitude towards issues and their evaluation but not from a particularistic point of view. He also says modernization is rooted in scientific knowledge, technological skill. Prof S.Ci Dube says “Modernisation refers to a common behavioural pattern characterised by A rational and scientific worldview.

Growth and ever-increasing application of science and technology. Adaptation of new institutions emerged in society to cope with the new situation dominated by science and technology. C.E. Black in his writing, “Dynamics Modernisation” modernisation as “Modernisation is a process by which historically evolved institutions are adopted.

the rapidly changing functions that reflect the unprecedented increase in man’s knowledge permitting control over environment, accompany the scientific revolution”. Here, Black has given prime importance to the institutions and their roles the process of modernisation. W.E. Moore (1961) suggested that a modem society has specific economic, political and cultural characteristics.

In the economic sphere, modern society is characterised by:
Development in technology. Specialization the economic role. Scope for saving and investment. Expansion of market(from local international).

In the political sphere modernization of society expects:
Declining of traditional rulers. Formulation of ideology for the rulers to handle the power. Decentralization of power among the members of the society. The scope must be provided to all to participate in the decision-making process.

In the cultural sphere, a modernizing society is characterised by:
Growing differential among major elements of culture like religion, philosophy and science. Spread of literacy, secular education. Introduction of a complex institutional system for the advancement of specialized roles. Expansion of media communication.

Development of new cultural elements based on:
Progress and improvement Expression of ability Emphasis on the dignity of the individual and his efficiency, Modernisation is a process of adaptation of new values, cultural elements and technology in the various fields of life. It is indeed the ability of a society of confronting, overcome and prepare itself to meet new challenges.

While doing so society adopts two methods:
By rearranging its social structure. By modifying the traditional norms and values. The learner emphasized mobility high-level participation. A modem man is more mobile in the sense that he can more frequently move from one place to another and from one occupational another, from one status to another. A high degree of participation indicates a strong sense of participation in common affairs of the state and community.

Characteristics of Modernisation:

• It is a revolutionary process.
• It is a multidimensional process.
• It is a universal process.
• It is a complex process.
• It is a global process.
• It is an irreversible process.
• It is a continuous and lengthy process.
• It is a systematic process
• It indicates scientific temper, rationality and secular attitude.
• It is a phased process.
• Modernized society is an open society
• It is a progressive society.
• It is a critical process because it requires not only a relatively stable new structure but is also capable of adopting continuously changing conditions and problems.
• It is a centralized process.

Eisenstadt (1965) in his article, Transformation of Social, Political and Cultural Orders in Modernisation” has given his opinion modernisation requires three structural characteristics of a society. Firstly, a high level of structural differentiation. Secondly, high level of social mobilization and thirdly relatively centralized and autonomous institutional frameworks.

Modernisation is critical in the sense that it requires not only a relatively stable new structure in society but also expects that the society acquires the capability to adapt to continuously changing conditions and problems. Its success depends on the ability of society to respond to the elements. But all societies don’t respond to modernisation uniformly.

Herbert Blunter in his writing. Industrialisation and the Traditional Order” has mentioned five different ways through which a traditional society can respond to the process of modernisation.

Rejective response:
A traditional society may not like the elements of modernisation and the society may reject it. Mainly two factors come to the forefront to reject modernization. Human factors included powerful groups, zamindars/ landlords, middlemen etc. protect their vested interests. Values system of the society which includes traditional values, customs, belief systems etc. Both factors try to maintain traditional order and reject the process of modernisation.

Disjunctive Response:
In this type of response, modernisation as a process operates as a detached development. The old elements and new elements co-exist but without any interference. People do not face any type of conflicting situation due to modernisation. They could lead their traditional life.

Assimilative Response:
Society, in this case, accepts elements of modernisation without affecting it. organisation and way of life. It assimilates the elements within its system without disruption. For example, in Indian rural society, the farmers use fertilizer and other modem machinery like tractors without affecting their pattern of life.

Supportive Response :
In a supportive response, society accepts modem elements to strengthen the conditional order. Traditional groups and institutions want to take advantage of the use of modem elements. Here modernisation acts as the supportive source of the traditional pattern. For example, the introduction of science and technology in the educational system.

Disruptive Response:
This type of response takes place when the traditional order is underestimated at many points. It occurs when society tries to accommodate modem elements in the traditional order. For example, the situation of the Odia language in Odia. Considering these five responses two types of situations may occur in society.

In one situation society may respond to all these at different points or periods and in another situation, society may express all these responses with different combinations. In India, response to modernisation depends on three factors as it constitutes a multi-dimensional process. Firstly, the nature of the choice that our society has made on the preference of the people in accepting modem elements.

Secondly, the interest of the people in using modem elements also counts much for that expresses the nature of our response to the changes due to modernisation. Thirdly, the role of the cultural tradition based on history is important as a value system controls our behaviour in using and interpreting modem elements.

Modernsation in India:
Due to modernisation, so many changes are founded in India:

• Introduction of new institutions like banking, mass media communication etc.
• Introduction of new value systems such as equality, justice, individualism, secularism etc.
• Acceptance of scientific innovation.
• Increase in the standard of living.
• Introduction of large-scale industries.
• Restructuring of the political system, i.e. introduction of democracy.
• Introduction of structural changes in social institutions like marriage, family, caste, etc.
• The emergence of the middle class.
• There are some eliminative changes like the disappearance of cultural traits, behaviour patterns, values etc. For example, the abolition of feudal power.
• There is shifting of attitude from sacred to secular.
• The emergence of new forms is because of the synthesis of old and new elements. For example,- the nuclear family in structured but functioning as a joint.
• Adoption of new cultural traits as a new election system.

Question 9.
What is Industrialization? Discuss its Impact?
Answer:
Industrialization is the process by which an economy is transformed from primarily agricultural to one based on the manufacturing of goods. Individual manual labour as often. replaced by mechanized mass production and craftsmen are replaced by assembly lines characteristic of industrialization include economic growth, more efficient division of labour and the use of technological innovation to solve problems as opposed to dependency on conditions outside human control.

Industrialization is most commonly associated with the European Industrial Revolution of the 18th and early centuries. The inset of the second world war also led to a great deal of industrialization which resulted in the growth and development of large urban centres and submits outs effects on society are still undetermined to some extent, however, it has resulted in a lower birthrate and a higher average income.

Impact on Indian Society: The Industrial Revolution traces its roots to the late 19th century in Britain. The growth of the metals and textiles industries allowed for the mass production of basic personal and commercial goods. As manufacturing activities grew transportation, finance and communications industries expanded to support the new production capacities.

The Industrial Revolution led to improved ented expansion in wealth and financial well-being for some. It also led to increased labour specialization and allowed cities to support a larger population motivating a rapid demographic shift, people left rural areas in large numbers seeking potential fortunes in budding industries.

The revolution quickly spread beyond Britain with manufacturing centres being established in continental Europe and the United States. World War II created unprecedented demand for certain manufactured goods, leading to the building of production capacity. After the war reconstruction in Europe occurred alongside a massive population expansion in North America.

There provided further catalysts that kept capacity utilization high and stimulated future growth of industrial activity. Innovation specialization and wealth creations were the causes and effects of industrialization in this period. The late 20th century was noteworthy for rigid industrialization in other parts of the worked notably East Asia. The Asian Tigers of their own industrial revolution after moving towards a merely mixed economy and away from heavy central planning.

Odisha State Board CHSE Odisha Class 12 Sociology Solutions Unit 5 Change and Development in India Objective & Short Answer Type Questions.

CHSE Odisha 12th Class Sociology Unit 5 Change and Development in India Objective & Short Answer Type Questions

Multiple Choice Questions With Answers

Question 1.
Who of the following used the term globalization first?
(a) Ronald Robertson
(b) G J. Holyoake
(c) M. N. Srinivas
Answer:
(a) Ronald Robertson.

Question 2.
Globalisation is?
(a) The increasing integration of the national economy into the world economy through the removal of barriers to international trade and capital movements.
(b) Tariff and non-tariff barriers to imports and exports and restrictions on the inflow and outflow of capital cease to exist in a fully globalized world economy.
(c) Free market economy of internationalization of the economy.
Answer:
(c) Free market economy of internationalization of the economy

Question 3.
Liberalization is?
(a) the removal of unnecessary control in laws and procedures.
(b) the opening of the economy to the world by removing barriers against free trade.
(c) the economy seems to be taking place mainly in the industrial areas.
Answer:
(a) the removal of unnecessary control in laws and procedures.

One Word Answers

Question 1.
The reduction of governmental control to the minimum in matters of trade, business, investment, and industry is called?
Answer:
Liberalization.

Question 2.
When did the process of liberalization start in India?
Answer:
1991

Question 3.
The process by which rural areas transformed into urban areas is known as?
Answer:
Urbanization

Question 4.
The book modernization of Indian tradition was written by whom?
Answer:
Y.Singh

Question 5.
Which term was Srinivas first used before Sanskritization to explain the socio-cultural change in Indian society?
Answer:
Religion and society among changes in South India.

Question 6.
Who first used the term globalization?
Answer:
Ronal Robertson.

Question 7.
The process of integrating the local economy with the world economy by reducing the barriers of trade and investment is called is?
Answer:
Liberalization.

Question 8.
When India accepts the economic policy of Liberalization?
Answer:
1991

Question 9.
A process of increasing economical integration- and growing economic interdependence between countries in the world economy?
Answer:
Globalisation.

Question 10.
Opening up the economy to foreign competition?
Answer:
Globalisation

Correct The Sentences

Question 1.
M.N. Srinivas write the book modernization of Indian traditions”?
Answer:
Y. Sing wrote the book modernization of Indian traditions”.

Question 2.
M.N. Srinivas first used the term globalization?
Answer:
Ronald Robertson first used the term globalization.

Question 3.
The process of integrating the local economy with the world economy by reducing trade and investment barriers is called secularization?
Answer:
The process of integrating the local economy with the world economy by reducing trade and investment barriers is called Liberalization.

Question 4.
The process of reduction of Government control is caused by Westernization.
Answer:
The process of reduction of Government control is called Liberalization.

Question 5.
Does Prof. Y. Singh write the book Religion and Society among the Coorgs of South India?
Answer:
Prof Y. Singh writes the book “Modernization of Indian Traditions.

Question 6.
Does liberalization refer to the greatest use of markets and the forces of competition to co- ordiante economic activities?
Answer:
Globalisation refers to the greatest use of markets and the forces of competition to co-ordinate economic activities.

Question 7.
Liberalization is a free market economy?
Answer:
Liberalization is a free market economy.

Fill In The Blanks

Question 1.
The term Sanskritization was used by ________.
Answer:
M.N. Srinivas

Question 2.
first used the term globalization _______.
Answer:
Ronald Robertson

Question 3.
Write the book Modernization of Indian Traditions _______.
Answer:
Y.Singh

Question 4.
India accepts the economic policy of liberalization since _______.
Answer:
1991

Question 5.
As now- a- days globalization is active in the economic field it means globalization _______.
Answer:
economic

Question 6.
Globalization means manufacturing things or products in the most effective way _______ anywhere in the
Answer:
World

Question 7.
Globalization is not a phenomenon _______.
Answer:
New

Question 8.
Globalization considers the entire as a market ________.
Answer:
World

Question 9.
The International Monetary Fund and the World Bank, the Indian economy has _______ been opening up to global capital since
Answer:
1980.

Question 10.
New industrial policy enumerated in industrial licensing and opened up _______ the economy considerably.
Answer:
1991

Question 11.
Globalization is a free market ______.
Answer:
economy

Short Type Questions With Answers

Question 1.
Globalisation?
Answer:
Globalisation refers to a process of increasing economic integration and growing economic inter-dependence between countries in the world economy. It is associated not only with an increasing cross-border movement of goods, services, capital technology information, and people but also with an organization of economic activities which straddles national boundaries opening up the economy to foreign competitions.

Question 2.
Urbanisation and Secularisation?
Answer:
Urbanization refers to the cultural values and patterns that dominate the life of a city. Whereas urbanization refers to the process of growth in cities both in terms of their social structure, population, physical outlay, and cultural organizations. The physical and social structure of society to a large extent governs the nature of an organization.

No doubt die nature of urbanization differs from society to society depending upon its cultural constraints and transition. One of the greatest changes in Indian society has been the change from a sacred society to a secular society. The nation of purity of society recurred serve below at the hands of the process of secularisation.

Question 3.
Define modernization?
Answer:
Modernization originally referred to the contrast and transition between a traditional agrarian society and the kind of modem society that is based on trade and industry. For example, traditional and modem would describe the difference between medieval. England and victorian Britain.

Question 4.
What is Modernisation?
Answer:
The term modernization is a broader and more complex term. According to S.H. Atatas modernization is a process by which modem scientific knowledge as introduced in society with the ultimate purpose of achieving a better and more satisfying life on the broadest sense of the term accepted by the society concerned.

Question 5.
Mention any two characteristics of Modernisation?
Answer:

• Development in technology
• Specialization in the economic role.

Question 6.
Mention any two characteristics of industrialization?
Answer:

• Industrial Revolution
• Later periods of Industrialization

Question 7.
What is Industrialization?
Answer:
Industrialization in the process by which an economy in transformed from primarily agricultural to one based on the manufacturing of goods Individual manual labor is often replaced by mechanized mass production and craftsman are replaced by assembly lines. Characteristics growth more efficient division of labor, and the use of technological innovation to solve problems as opposed to dependency on conditions outside human control.

Question 8.
Discuss the impact of Globalisation?
Answer:
Free market economy:
One of the immediate impacts of globalization is that market became free and open to competition to all. There is an increasing realization that a free market is better for the growth of the economy.

Encourages foreign investment:
globalization encourages foreign investment in different sectors of the Indian economy. Different sectors of the Indian economy are made open to different multinational or foreign companies. Those companies enter into India and on rest amount of the foreign capital because of which the Indian economy gets a boost.

More employment opportunities:
Because of globalization a large number of foreign and multi-national companies have entered India and settled in different industries India.

Privatization:
Globalization also encourages presentations on India.

Liberalization:
Another impact of globalization. The decline of small and cottage industries.

Question 9.
Write a short note on urbanization?
Answer:
Urbanization refers to the process of growth in cities in terms of their social structure population, physical outlay, and cultural organizations. The physical and social structure of society to a large extent governs the nature of urbanization and differs from security society depending upon it.

Urbanization is universally associated with a widely living physic or unity to quickly adapt to new ideas or innovation greater industrialization or sense of identity. It promotes plurality the styles of a high degree of editions as cultural life dominates literally traditional learnings and skills on economic and cultural domains socially at is not characterized predominance of conjugal families faster pace of work pattern.

Question 10.
Discuss the merits of globalization?
Answer:
Improves Efficiency:
Globalization brings efficiency in production and increases the efficiency of laborers. Eliminates Poverty: Globalization eliminates poverty through a higher growth rate.

Promotes healthy competition:
Globalization creates or promotes healthy Competition among producers.

Creates global village:
Globalization helps in the development of a global village.

Improves financial situation:
Adequate finance is a pre-condition for development.

Encourages migration:
Globalization encourages cross-border migration of workers which makes up for the deficiency of workers in developed countries.

Question 11.
Discuss the demerits of Globalization?
Answer:
Increases inequality:
Globalization increases inequality both between rich and poor people as well as between developed and underdeveloped nations.

Closer of Industries:
Globalization encourages free trade which may lend to the closure of many domestic or small-scale industries.

Divides the world:
As a divisive process globalization divides the world into rich and poor nations or into underdeveloped, developing and developed nations.

Creates uncertainties:
Globalization creates many uncertainties among works industrialists and among financial institutions.

Degenerates Human values:
Globalization human values progress or development is always viewed in terms of economic growth.

Exploitation:
It seems as if exploitation is the many objectives of globalization.

Odisha State Board CHSE Odisha Class 12 Psychology Solutions Unit 4 Long Answer Questions Part-3.

CHSE Odisha 12th Class Psychology Unit 4 Long Answer Questions Part-3

Long Questions With Answers

Question 1.
What are the different types of psychotherapy? On what basis are they classified?
Answer:
Different types of psychotherapy are:

• Psychodynamic therapy
• Behaviour therapy
• Humanistic-existential therapy
• Biomedical therapy

Also, there are many alternative therapies such as yoga, meditation, acupuncture, herbal remedies etc.
Basis of classification of psychotherapy:

On the cause which has led to the problem:
Psychodynamic therapy is of the view that intrapsychic conflicts, i.e. the conflicts that are present within the psyche of the person, are the source of psychological problems.

On how did the cause come into existence:
The psychodynamic therapy, unfulfilled desires of childhood and unresolved childhood fears lead to intrapsychic conflicts.

What is the chief method of treatment?
Psychodynamic therapy uses the methods of free association and reporting of dreams to elicit the thoughts and feelings of the client.

What is the nature of the therapeutic relationship between the client and the therapist?
Psychodynamic therapy assumes that the therapist understands the client’s intrapsychic conflicts better than the client and hence it is the therapist who interprets the thoughts and feelings of the client to her/him so that s/he gains an understanding of the same.

What is the chief benefit to the client?
Psychodynamic therapy values emotional insight as the important benefit that the client derives from the treatment. Emotional insight is present when the client understands her/his conflicts intellectually; is able to accept the same emotionally, and is able to change her/his emotions towards the conflicts.

On the duration of treatment:
Hie duration of classical psychoanalysis may continue for several years. However, several recent versions of psychodynamic therapies are completed in 10—15 sessions.

Question 2.
A therapist asks the client to reveal all her/his thoughts including early childhood experiences. Describe the technique and type of therapy being used.
Answer:
In this case psychodynamic, therapy is used in the treatment of the client. Since the psychoanalytic approach views intrapsychic conflicts to be the cause of the psychological disorder. The first step in the treatment is to elicit this intrapsychic conflict. Psychoanalysis has invented free association and dream interpretation as two important methods for eliciting intrapsychic conflicts.

The free association method is the main method for understanding the client’s problems. Once a therapeutic relationship is established, and the client feels comfortable, the therapist makes her/him lie down on the couch, close her/his eyes and asks her/him to speak whatever comes to mind without censoring it in any way. The client is encouraged to freely associate one thought with another, and this method is called the method of free association.

The censoring superego and the watchful ego are kept in abeyance as the client speaks whatever comes to mind in an atmosphere that is relaxed and trusting. As the therapist does not interrupt, the free flow of ideas, desires and conflicts of the unconscious, which had been suppressed by the ego, emerges into the conscious mind. This free uncensored verbal narrative of the client is a window into the client’s unconscious to which the therapist gains access.

Along with this technique, the client is asked to write down her/his dreams upon waking up. Psychoanalysts look upon dreams as symbols of the unfulfilled desires present in the unconscious. The images of dead dreams are symbols which signify intrapsychic forces. Dreams use symbols because they are indirect expressions and hence would not alert the ego.

If the unfulfilled desires are expressed directly, the ever-vigilant ego would suppress them and that would leads to anxiety. These symbols are interpreted according to an accepted convention of translation as indicators of unfulfilled desires and conflicts.

Question 3.
Discuss the various techniques used in behaviour therapy.
Answer:
Various techniques used in behaviour therapy:
A range of techniques is available for changing behaviour. The principles of these techniques are to reduce the arousal level of the client, alter behaviour through classical conditioning or operant conditioning with different contingencies of reinforcements, as well as to use vicarious learning procedures, if necessary. Negative reinforcement and aversive conditioning are the two major techniques of behaviour modification.

Negative reinforcement refers to following an undesired response with an outcome that is gainful or not liked. For example, one learns to put on woollen clothes, bum firewood or use electric heaters to avoid the unpleasant cold weather. One learns to move away from dangerous stimuli because they provide negative reinforcement.

Aversive conditioning refers to the repeated association of an undesired response with an aversive consequence. For example, an alcoholic is given a mild electric shock and asked to smell the alcohol. With repeated pairings, the smell of alcohol is aversive as the pain of the shock is associated with it and the person will give up alcohol.

Positive reinforcement is given to increase the deficit if adaptive behaviour occurs rarely. For example, if a child does not do homework regularly, positive reinforcement may be used by the child’s mother by preparing the child’s favourite dish whenever s/he does homework at the appointed time. The positive reinforcement of food will increase the behaviour of doing homework at the appointed time.

The token economy in which persons with behavioural problems can be given a token as a reward every time a wanted behaviour occurs. The tokens are collected and exchanged for a reward such as an outing for the patient or a treat for the child. Unwanted behaviour can be reduced and waited behaviour can be increased simultaneously through differential reinforcement.

Positive reinforcement for the wanted behaviour and negative reinforcement for the unwanted behaviour attempted together may be one such method. The other method is to positively reinforce the wanted behaviour and ignore the unwanted behaviour. The latter method is less painful and equally effective. For example, let us consider the case of a girl who sulks and cries when she is not taken to the cinema when she asks.

The parent is instructed to take her to the cinema if she does not cry and sulk but not to take her if she does. Further, the parent is instructed to ignore the girl when she cries and sulks. The wanted behaviour of politely asking to be taken to the cinema increases and the unwanted behaviour of crying and sulking decreases.

Question 4.
Explain with the help of an example how cognitive distortions take place.
Answer:
Cognitive distortions are ways of thinking which are general in nature but which distort reality in a negative manner. These patterns of thought are called dysfunctional cognitive structures. They lead to errors of cognition about social reality. Aaron Beck’s theory of psychological distress states that childhood experiences provided by the family and society develop core, schemas or systems, which include beliefs and action patterns in the individual.

Thus, a client, who was neglected by the parents as a child, develops the core schema of “I am not wanted”. During the course of their life, a critical incident occurs in her/his life. S/he is publicly ridiculed by the teacher in school. This critical incident triggers the core schema of “I am not wanted” leading to the development of negative automatic thoughts. Negative thoughts are persistent irrational thoughts such as “nobody loves me”, “I am ugly”, “l am stupid”, “I will not succeed”, etc.

Such negative automatic thoughts are characterised by cognitive distortions. Repeated occurrence of these thoughts leads to the development of feelings of anxiety and depression. The therapist uses questioning, which is a gentle, non-threatening disputation of the client’s beliefs and thoughts. Examples of such questions would be, “Why should everyone love you?”, “What does it mean to you to succeed?” etc.

Question 5.
Which therapy encourages the client to seek personal growth and actualise their potential? Write about the therapies which are based on this principle.
Answer:
Humanistic-existential therapy encourages the client to seek personal growth and actualise their potential. It states that psychological distress arises from feelings of loneliness, alienation, and an inability to find meaning and genuine fulfilment in life.
The therapies which are based on this principle are:

Existential therapy:
There is a spiritual unconscious, which is the storehouse of love, aesthetic awareness, and values of life. Neurotic anxieties arise when the problems of life are attached t6 the physical, psychological or spiritual aspects of one’s existence. Frankl emphasised the role of spiritual anxieties in leading to meaninglessness and hence it may be called existential anxiety, i.e. neurotic anxiety of spiritual origin.

Client-centred therapy:
Client-centred therapy was given by Carl Rogers. He combined scientific rigour with the individualised practice of client-centred psychotherapy. Rogers brought into psychotherapy the concept of self, with freedom and choice as the core of one’s being. The therapy provides a warm relationship in which the client can reconnect with her/his disintegrated feelings. The therapist shows empathy, i.e. understanding the client’s experience as if it were her/his own, is warm and has unconditional positive regard, i.e. total acceptance of the client as s/he is. Empathy sets up an emotional resonance between the therapist and the client.

Gestalt therapy:
The German word gestalt means ‘whole’. This therapy was given by Frederick (Fritz) Peris together with his wife Laura Peris. The goal of gestalt therapy is to increase an individual’s self-awareness and self-acceptance. The client is taught to recognise the bodily processes and die emotions that are being blocked out from awareness. The therapist does this by encouraging the client to act out fantasies about feelings and conflicts. This therapy can also be used in group settings.

Question 6.
What are the factors that contribute to healing in psychotherapy? Enumerate some of the alternative therapies.
Answer:
Factors Contributing to Healing in Psychotherapy are:

A major factor in healing is the techniques adopted by the therapist and the implementation of the same with the patient/client. If the behavioural system and the CBT school are adopted to heal an anxious client, the relaxation procedures and the cognitive restructuring largely contribute to the healing.

The therapeutic alliance, which is formed between the therapist and the patient/ client, has healing properties, because of the regular availability of the therapist and the warmth and empathy provided by the therapist.

At the outset of therapy, while the patient/client is being interviewed in the initial sessions to understand the nature of the problem, s/he unburdens the emotional problems being faced. This process of emotional unburdening is known as catharsis and it has healing properties.

There are several non-specific factors associated with psychotherapy. Some of these factors are attributed to the patient/client and some to the therapist. These factors are called non-specific because they occur across different systems of psychotherapy and across .different clients/patients and different therapists. Non-specific factors attributable to the client/patient are the motivation for change, the expectation of improvement due to the treatment, etc.

These are called patient variables. Non-specific factors attributable to the therapist are positive nature, absence of unresolved emotional conflicts, presence of good mental health, etc. These are called therapist variables. Some of the alternative therapies are Yoga, meditation, acupuncture, herbal remedies etc.

Question 7.
What are the techniques used in the rehabilitation of the mentally ill?
Answer:
The treatment of psychological disorders has two components, i.e. reduction of symptoms, and improving the level of functioning or quality of life. In the case of milder disorders such as generalised anxiety, reactive depression or phobia, reduction of symptoms is associated with an improvement in the quality of life. However, in the case of severe mental disorders such as schizophrenia, reduction of symptoms may not be associated with an improvement in the quality of life.

Many patients suffer from negative symptoms such as disinterest and lack of motivation to do work or interact with people. The aim of rehabilitation is to empower the patient to become a productive member of society to the extent possible. In rehabilitation, the patients are given occupational therapy, social skills training, and vocational therapy. In occupational therapy, the patients are taught skills such as candle making, paper bag making and weaving to help them to form a work discipline.

Social skills. training helps the patients to develop interpersonal skills through role play, imitation and instruction. The objective is to teach the patient to function in a Social group. Cognitive retraining is given to improve the basic cognitive functions of attention, memory and executive functions. After the patient improves sufficiently, vocational training is given wherein the patient is helped to gain the skills necessary to undertake productive employment.

Question 8.
How would a social learning theorist account for a phobic fear of lizards/ cockroaches? How would a psychoanalyst account for the same phobia?
Answer:
Systematic desensitisation is a technique introduced by Wolpe for treating phobias or irrational fears. The client is interviewed to elicit fear-provoking situations and together with the client, the therapist prepares a hierarchy of anxiety-provoking stimuli with the least anxiety-provoking stimuli at the bottom of the hierarchy. The therapist relaxes the client and asks the client to think about the least anxiety-provoking situation.

The client is asked to stop thinking of the fearful situation if the slightest tension is felt. Over sessions, the client is able to imagine more severe fear-provoking situations while maintaining relaxation. The client gets systematically desensitised to the fear.

Question 9.
Should Electroconvulsive Therapy (ECT) be used in the treatment of mental disorders?
Answer:
Yes, Electro-convulsive Therapy (ECT) can be used in the treatment of mental disorders. Electroconvulsive Therapy (ECT) is another form of biomedical therapy. Mild electric shock is given via electrodes to the brain of the patient to induce convulsions. The shock is given by the psychiatrist only when it is necessary for the improvement of the patient. ECT is not a routine treatment and is given only when drugs are not effective in controlling, the symptoms of the patient.

Question 10.
What kind of problems is cognitive behaviour therapy best suited for?
Answer:
Cognitive behaviour treatment best suited for a wide range of psychological disorders such as anxiety, depression, panic attacks, borderline personality, etc. adopts a bio-CBT psychosocial approach to the delineation of psychopathology. It combines cognitive therapy with behavioural techniques.

Question 11.
What is the nature and process of therapeutic approaches?
Answer:
Psychotherapy is a voluntary relationship between the one seeking treatment or the client and the one who treats the therapist. The purpose of the relationship is to help the client to solve the psychological problems faces by her or him. The relationship is conducive to building the trust of the client so that problems may be freely discussed.

Psychotherapies aim at changing maladaptive behaviours, decreasing the sense of personal distress and helping the client to adapt better to her/his environment. The inadequate marital, occupational and social adjustment also requires that major changes be made in an individual’s personal environment. All psychotherapeutic approaches have the following characteristics:

• there is the systematic application of principles underlying the different theories of therapy.
• persons who have received practical training under expert supervision can practice psychotherapy and not everybody. An untrained person may unintentionally cause more harm than good.
• the therapeutic situation involves a therapist and a client who seeks and receives help for her/his emotional problems (this person is the focus of attention in the therapeutic process).
• the interaction of these two persons — the therapist and the client— results in the consolidation/formation of the therapeutic relationship. This is a confidential, interpersonal and dynamic relationship.

This human relationship is central to any sort of psychological therapy and is the vehicle for change. All psychotherapies aim at a few or all of the following goals :

• Reinforcing the client’s resolve for betterment.
• Lessening emotional pressure.
• Unfolding the potential for positive growth.
• Modifying habits.
• Changing thinking patterns
• Increasing self-awareness.
• Improving interpersonal relations and communication.
• Facilitating decision-making.
• Becoming aware of one’s choices in life.
• Relating to one’s social environment in a more creative and self-aware manner.

Question 12.
What is the relationship between the client and therapist?
Answer:
Therapeutic Relationship :
The special relationship between the client and the therapist is known as the therapeutic relationship or alliance. It is neither a passing acquaintance nor a permanent and lasting relationship. There are two major components of a therapeutic alliance. The first component is the contractual nature Of the relationship in which two willing individuals, the client and the therapist, enter into a partnership which aims at helping the client overcome her/his problems.

The second component of the therapeutic alliance is the limited duration of the therapy. This alliance lasts until the client becomes able to deal with her/his problems and take control of her/ his life. This relationship has several unique properties. It is a trusting and confiding relationship. The high level of trust enables the client to unburden herself/himself to the therapist and confide her/his psychological and personal problems to the latter.

The therapist encourages this by being accepting, empathic, genuine and warm to the client. The therapist conveys by her/his words and behaviours that s/he is not judging the client and will continue to show the same positive feelings towards the client even if the client is rude or confides in all the ‘wrong’ things that s/he may have done or thought about. This is the unconditional positive regard that the therapist has for the client. The therapist has empathy for the client.

Empathy:
Empathy is different from sympathy and intellectual understanding of another person’s situation. Iii sympathy, one has compassion and pity towards, the .suffering of another but is not able to feel like the other person. Intellectual understanding is cold in the sense that the person is unable to feel like the other person and does not feel sympathy either. On the other hand, empathy is present when one is able to understand the plight of another person and feel like the other person.

It means understanding things from the other person’s perspective, i.e. putting oneself in the other person’s shoes. Empathy enriches the therapeutic relationship and transforms it into a healing relationship. The therapeutic alliance also requires that the therapist must keep strict confidentiality of the experiences, events, feelings or thoughts disclosed by the client. The therapist must not exploit the trust and confidence of the client in any way. Finally, it is a professional relationship and must remain so.

Question 13.
Write the types of therapies.
Answer:
Though all psychotherapies aim at removing human distress and fostering effective behaviour, they differ greatly in concepts, methods, and techniques. Psychotherapies may be classified into three broad groups, viz. the psychodynamic, behaviour sad existential psychotherapies. In terms of chronological order, psychodynamic therapy emerged first followed by behaviour therapy while existential therapies which are also called the third force, emerged last. The classification of psychotherapies is based on the following parameters:

What is the cause, which has led to the problem?
Psychodynamic therapy is of the view that intrapsychic conflicts, i.e. the conflicts that are present within the psyche of the person, are the source of psychological problems. According to behaviour therapies, psychological problems arise due to faulty learning of behaviours and cognitions. Existential therapies postulate that questions about the meaning of one’s life and existence are the cause of psychological problems.

How did the cause come into existence?
In psychodynamic therapy, unfulfilled desires of childhood and unresolved childhood fears lead to intrapsychic conflicts. Behaviour therapy postulates that faulty conditioning patterns, faulty learning, and faulty thinking and beliefs lead to maladaptive behaviours that, in turn, lead to psychological problems. Existential therapy places importance on the present. It is the current feelings of loneliness, alienation, a sense of the futility of one’s existence, etc., which cause psychological problems.

What is the chief method of treatment?
Psychodynamic therapy uses the methods of free association and reporting of dreams to elicit the thoughts and feelings of the client. This material is interpreted to the client to help her/him to confront and resolve the conflicts and thus overcome problems. Behaviour therapy identifies faulty conditioning patterns and sets up alternate behavioural contingencies to improve behaviour.

The cognitive methods employed in this type of therapy challenge the faulty thinking patterns of the client to help her/him overcome psychological distress. Existential therapy provides a therapeutic environment which is positive, accepting and non-judgmental. The client is able to talk about the problems and the therapist acts as a facilitator. The client arrives at the solutions through a process of personal growth.

What is the nature of the therapeutic relationship between the client and the therapist?
Psychodynamic therapy assumes that the therapist understands the client’s intrapsychic conflicts better than the client and hence it is the therapist who interprets the. thoughts and feelings of the client to her/him so that s/he gains an understanding of the same. Behaviour therapy assumes that the therapist is able to discern the faulty behaviour and thought patterns of the client.

It further assumes that the therapist is capable of finding out the correct behaviour and thought patterns, which would be adaptive for the client. Both psychodynamic and behaviour therapies assume that the therapist is capable of arriving at solutions to the client’s problems. In contrast to these therapies, existential therapies emphasise that the therapist merely provides a warm, empathic relationship in . which the client feels secure to explore the nature and causes of her/his problems by herself/ himself.

What is the chief benefit to the client?
Psychodynamic therapy values emotional insight as the important benefit that the client derives from the treatment. Emotional insight is present when the client understands her/his conflicts intellectually; is able to accept the same emotionally and is able to change her/his emotions towards the conflicts. The client’s symptoms and distresses reduce as a consequence of this emotional insight.

Behaviour therapy considers changing faulty behaviour and thought patterns to adaptive ones as the chief benefit of the treatment. Instituting adaptive or healthy behaviour and thought patterns ensures the reduction of distress and the removal of symptoms. Humanistic therapy values personal growth as the chief benefit. Personal growth is the process of gaining an increasing understanding of oneself and one’s aspirations, emotions and motives.

What is the duration of treatment?
The duration of classical psychoanalysis may continue for several years. However, several recent versions of psychodynamic therapies are completed in 10-15 sessions. Behaviour and cognitive behaviour therapies as well as existential therapies are shorter and are completed in a few months. Thus, different types of psychotherapies differ on multiple parameters.

However, they all share the common method of providing treatment for psychological distress’ through psychological means. The therapist, the therapeutic relationship, and the process of therapy become the agents of change in the client leading to the alleviation of psychological distress. The process of psychotherapy begins by formulating the client’s problem.

Odisha State Board CHSE Odisha Class 12 Psychology Solutions Unit 4 Long Answer Questions Part-2.

CHSE Odisha 12th Class Psychology Unit 4 Long Answer Questions Part-2

Long Questions With Answers

Question 1.
Write the classification of biological disorders.
Answer:
In order to understand psychological disorders, we need to begin by classifying them. A classification of such disorders consists of a list of categories of specific psychological disorders grouped into various classes on the basis of some shared characteristics. Classifications are useful because they enable users like psychologists, psychiatrists and social workers to communicate with each other about the disorder and help in understanding the causes of psychological disorders and the processes involved in their development and maintenance.

The American Psychiatric Association (APA) has published an official manual describing and classifying various kinds of psychological disorders. The current version of it, the Diagnostic and Statistical Manual of Mental Disorders, IV Edition (DSM-IV), evaluates the patient on five axes or dimensions rather than just one broad aspect of ‘mental disorder’. These dimensions relate to biological, psychological, social and other aspects.

The classification scheme officially used in India and elsewhere is the tenth revision of the International Classification of Diseases (ICD-10), which is known as the ICD-10 Classification of Behavioural and Mental Disorders. It was prepared by the World Health Organisation (WHO). For each disorder, a description of the main clinical features or symptoms and of other associated features including diagnostic guidelines is provided in this scheme.

Question 2.
What are the approaches to understanding abnormal behaviour?
Answer:
In order to understand something as complex as abnormal behaviour, psychologists use different approaches. Each approach in use today emphasises a different aspect of human behaviour and explains and treats abnormality in line with that aspect. These approaches also emphasise the role of different factors such as biological, psychological and interpersonal and socio-cultUral factors.

We will examine some of the approaches which are currently being used to explain abnormal behaviour. Biological factors influence all aspects of our behaviour. A wide range of biological factors such as faulty genes, endocrine imbalances, malnutrition, injuries and other conditions may interfere with the normal development and functioning of the human body. These factors may be potential causes of abnormal behaviour. We have already come across the biological model.

According to this model, abnormal behaviour has a biochemical or physiological basis. Biological researchers have found that psychological disorders are often related to problems in the transmission of messages from one neuron to another. You have studied in Class XI, that a tiny space called a synapse separates one neuron from the next and the message must move across that space.

When an electrical impulse reaches a neuron’s ending, the nerve ending is stimulated to release a chemical, called a neurotransmitter. Studies indicate that abnormal activity by certain neurotransmitters can lead to specific psychological disorders. Anxiety disorders have been linked to low activity of the neurotransmitter gamma-aminobutyric acid (GABA) schizophrenia to the excess activity of dopamine, and depression to low activity of serotonin.

Genetic factors have been linked to mood disorders, schizophrenia, mental retardation and other psychological disorders. Researchers have not, however, been able to identify the specific genes that are the culprits. It appears that in most cases, no single gene is responsible for a particular behaviour or a psychological disorder. In fact, many genes combine to help bring about our various behaviours and emotional reactions, both functional and dysfunctional.

Although there is sound evidence to believe that genetic/ biochemical factors are involved in mental disorders as diverse as schizophrenia, depression, anxiety, etc. and biology alone cannot account for most mental disorders. There are several psychological models which provide a psychological explanation of mental disorders. These models maintain that psychological and interpersonal factors have a significant role to play in abnormal behaviour.

These factors include maternal deprivation (separation from the mother, or lack of warmth and stimulation during early years of life), faulty parent-child relationships (rejection, overprotection, over permissiveness, faulty discipline, etc.), maladaptive family structures (inadequate or disturbed family) and severe stress. The psychological models include the psychodynamic, behavioural, cognitive and humanistic-existential models.

The psychodynamic model is the oldest and most famous of the modern psychological models. You have already read about this model in Chapter 2 on Self and Personality. Psychodynamic theorists believe that behaviour, whether normal or abnormal, is determined by psychological forces within the person of which s/he is not consciously aware. These internal forces are considered dynamic, i.e. they interact with one another and their interaction gives shape to behaviour, thoughts and emotions.

Abnormal symptoms are viewed as the result of conflicts between these forces. This model was first formulated by Freud who believed that three central forces shape personality — instinctual needs, drives and impulses (id), rational thinking (ego), and moral standards (superego). Freud stated that abnormal behaviour is a symbolic expression of unconscious mental conflicts that can be generally traced to early childhood or infancy.

Another model that emphasises the role of psychological factors is the behavioural model. This model states that both normal and abnormal behaviours are learned and psychological disorders are the result of learning maladaptive ways of behaving. The model concentrates on behaviours that are learned through conditioning and propose that what has been learned can be unlearned.

Learning can take place by classical conditioning (temporal association in which two events repeatedly occur close together in time), operant conditioning (behaviour is followed by a reward), and social learning (learning by imitating others’ behaviour). These three types of conditioning account for behaviour, whether adaptive or maladaptive. Psychological factors are also emphasised by the cognitive model. This model states that abnormal functioning can result from cognitive problems.

People may hold assumptions and attitudes about themselves that are irrational and inaccurate. People may also repeatedly think in illogical ways and makeover generalisations, that is, – they may draw broad, negative conclusions on the basis of a single insignificant event. Another psychological model is the humanistic-existential model which focuses on broader aspects of human existence.

Humanists believe that human beings are born with a natural tendency to be friendly, cooperative and constructive, and are driven to self-actualise, i.e. to fulfil this potential for goodness and growth. Existentialists believe that from birth we have total freedom to give meaning to our existence or to avoid that responsibility. Those who shirk from this responsibility would live empty, inauthentic and dysfunctional lives.

In addition to the biological and psychosocial factors, socio-cultural factors such as war and violence, group prejudice and discrimination, economic and employment problems and rapid social change, put stress on most of us and cafes also lead to psychological problems in some individuals. According to the sociocultural model, abnormal behaviour is best understood in light of the social and cultural forces that influence an individual.

As behaviour is shaped by societal forces, factors such as family structure and communication, social networks, societal conditions and societal labels and roles become more important. It has been found that certain family systems are likely to produce abnormal functioning in individual members. Some families have an enmeshed structure in which the members are over involved in each other’s activities, thoughts and feelings.

Children from this kind of family may have difficulty in becoming independent in life. The broader social networks in which people operate include their social and professional relationships. Studies have shown that people who are isolated and lack social support, i.e. strong and fulfilling interpersonal relationships in their lives are likely to become more depressed and remain depressed longer than those who have good friendships.

Socio-cultural theorists also believe that abnormal functioning is influenced by the societal labels and roles assigned to troubled people. When people break the norms of their society, they are called deviant and ‘mentally ill’. Such labels tend to stick so that the person may be viewed as ‘crazy’ and encouraged to act sick. The person gradually learns to accept and play the sick role, and functions in a disturbed manner.

In addition to these models, one of the most widely accepted explanations of abnormal behaviour has been provided by the diathesis-stress model. This model states that psychological disorders develop when a diathesis (biological predisposition to the disorder) is set off a stressful situation. This model has three components. The first is the diathesis or the presence of some biological aberration which may be inherited.

The second component is that the diathesis may carry a vulnerability to developing a psychological disorder. This means that the person is ‘at risk’ or ‘predisposed’ to develop the disorder. The third component is the presence of pathogenic stressors, i.e. factors/stressors that may lead to psychopathology. If such “at risk” persons are exposed to these stressors, their predisposition may actually evolve into a disorder. This model has been applied to several disorders including anxiety, depression, and schizophrenia.

Question 3.
What are the major psychological disorders?
Answer:
Anxiety Disorders:
One day while driving home, Deb felt his heart beating rapidly, he started sweating profusely and even felt short of breath. He was so scared that he stopped the car and stepped out. In the next few months, these attacks increased and now he was hesitant to drive for fear of being caught in traffic during an attack. Deb started feeling that he had gone crazy and would die. Soon he remained indoors and refused to move out of the house.

We experience anxiety when we are waiting to take an examination or visit a dentist, or even give a solo performance. This is normal and expected and even motivates us to do our tasks well. On the other hand, high levels of anxiety that are distressing and interfere with effective functioning indicate the presence of an anxiety disorder— the most common category of psychological disorders. Everyone has worries and fears.

The term anxiety is usually defined as a diffuse, vague, very unpleasant feeling of fear and apprehension. The anxious individual also shows combinations of the following symptoms: rapid heart rate, shortness of breath, diarrhoea, loss of appetite, fainting, dizziness, sweating, sleeplessness, frequent urination and tremors. There are many types of anxiety disorders (see Table 4.2).

They include generalised anxiety disorder, which consists of prolonged, vague, unexplained and intense fears that are not attached to any particular object. The symptoms include worry and apprehensive feelings about the future; hypervigilance, which involves constantly scanning the environment for dangers. It is marked by motor tension, as a result of which the person is unable to relax, is restless and visibly shaky and tense.

Another type of anxiety disorder is panic disorder, which consists of recurrent anxiety attacks in which the person experiences intense terror. A panic attack denotes an abrupt surge of intense anxiety rising to a peak when thoughts of particular stimuli are present. Such thoughts occur in an unpredictable manner. The clinical features include shortness of breath, dizziness, trembling, palpitations, choking, nausea, chest pain or discomfort, fear of going crazy, losing control or dying.

You might have met of heard of someone who was afraid to travel in a lift or climb to the tenth floor of a building or refused to enter a room if s/he saw a lizard. You may have also felt it yourself or seen a friend unable to speak a word of a well-memorised and rehearsed speech before an audience. These kinds of fears are termed as phobias. People who have phobias have irrational fears related to specific objects, people, or situations. Phobias often develop gradually or begin with a generalised anxiety disorder. Phobias can be grouped into three main types, i.e. specific phobias, social phobias and agoraphobia.

Specific phobias:
Specific phobias are the most commonly occurring type of phobia. This group includes irrational fears such as intense fear of a certain type of animal, or of being in an enclosed space. Intense and incapacitating fear and embarrassment when dealing with others characterises social phobias.

Agoraphobia:
Agoraphobia is the term used when people develop a fear of entering unfamiliar situations. Many agoraphobics are afraid of leaving their homes. So their ability to carry out normal life activities is severely limited. Have you ever noticed someone washing their hands every time they touch something, or washing even things like coins, or stepping only within the patterns on the floor or road while walking.

People affected by the obsessive-compulsive disorder are unable to control their preoccupation with specific ideas or are unable to prevent themselves from repeatedly carrying out a particular act or series of acts that affect their ability to carry out normal activities.

Obsessive behaviour:
Obsessive behaviour is the inability to stop thinking about a particular idea or topic. The person involved/often finds these thoughts to be unpleasant and shameful.

Compulsive behaviour:
Compulsive behaviour is the need to perform certain behaviours over and over again. Many compulsions deal with counting, ordering, checking, touching and washing. Very often people who have been caught in a natural disaster (such as a tsunami) or have been victims of bomb blasts by terrorists, or been in a serious accident or in a war-related situation, experience posttraumatic stress disorder (PTSD). PTSD symptoms vary widely but may include recurrent dreams, flashbacks, impaired concentration and emotional numbing.

Somatoform Disorders:
These are conditions in which there are physical symptoms in the absence of physical disease. In somatoform disorders, the individual has psychological difficulties and complains of physical symptoms, for which there is no biological cause. Somatoform disorders include pain disorders, somatisation disorders, conversion disorders, and hypochondriasis.

Pain disorders:
Pain disorders involve reports of extreme and incapacitating pain, either without any identifiable biological symptoms or greatly in excess of what might be expected to accompany biological symptoms. How people interpret pain influences their overall adjustment. Some pain sufferers can learn to use active coping, i.e. remaining active and ignoring the pain. Others engage in passive coping, which leads to reduced activity and social withdrawal.

Patients with somatisation disorders have multiple recurrent or chronic bodily complaints. These complaints are likely to be presented in a dramatic and exaggerated way. Common complaints are headaches, fatigue, heart palpitations, fainting spells, vomiting, and allergies. Patients with this disorder believe that they are sick, provide long and detailed histories of their illness and take large quantities of medicine.

The symptoms of conversion disorders are the reported loss of part or all of some basic body functions. Paralysis, blindness, deafness and difficulty in walking are generally among the symptoms reported. These symptoms often occur after a stressful experience and may be quite sudden.

Hypochondriasis:
Hypochondriasis is diagnosed if a person has a persistent belief that s/he has a serious illness, despite medical reassurance, lack of physical findings, and failure to develop the disease. Hypochondriacs have an obsessive preoccupation and concern with the condition of their bodily organs, and they continually worry about, their health.

Question 4.
Write the major anxiety disorders.
Answer:
Generalised Anxiety Disorder:
prolonged, vague, unexplained and intense fears that have no object, accompanied by hypervigilance and motor tension.

Panic Disorder:
frequent anxiety attacks characterised by feelings of intense terror arid dread; unpredictable ‘panic attacks’ along with physiological symptoms like breathlessness, palpitations, trembling, dizziness, and a sense of losing control or even dying.

Phobias :
irrational fears related to specific objects, interactions with others, and unfamiliar situations.

Obsessive-compulsive Disorder :
being preoccupied with certain thoughts that are viewed by the person to be embarrassing or shameful, and being unable to check the impulse to repeatedly carry out certain acts like checking, washing, counting, etc.

Post-traumatic Stress Disorder (PTSD) :
recurrent dreams, flashbacks, impaired concentration and emotional numbing followed by a traumatic or stressful event like a natural disaster, serious accident, etc.

Question 5.
What is dissociative disorders?
Answer:
Dissociative Disorders: Dissociation can be viewed as a severance of the connections between ideas and emotions. Dissociation involves feelings of unreality, estrangement, depersonalisation, and sometimes a loss or shift of identity. Sudden temporary alterations of consciousness that blot out painful experiences are a defining characteristic of dissociative disorders.

Four conditions are included in this group: dissociative amnesia, dissociative fugue, dissociative identity disorder, and depersonalisation. Salient features of somatoform and dissociative disorders are given.

Salient Features of Somatoform and Dissociative Disorders
Dissociative Disorders

Dissociative amnesia:
The person is unable to recall important, personal information often related to a stressful and traumatic report. The extent of forgetting is beyond normal.

Dissociative fugue:
The person suffers from a rare disorder that combines amnesia with travelling away from a stressful environment.

Dissociative identity (multiple personalities) :
The person exhibits two or more separate and contrasting personalities associated with a history of physical abuse.

Somatoform Disorders
Hypochondriasis:
A person interprets insignificant symptoms as signs of a serious illness despite repeated medical evaluations that point to no pathology disease.

Somatisation :
A person exhibits vague and recurring physical/bodily symptoms such as pain, acidity, etc., without any organic cause.

Conversion :
The person suffers from a loss or impairment of motor or sensory function (e.g., paralysis, blindness, etc.) that has no physical cause but may be a response to stress and psychological problems.

Dissociative amnesia:
Dissociative amnesia is characterised by extensive but selective memory loss that has no known organic cause (e.g., head injury). Some people cannot remember anything about their past. Others can no longer recall specific events, people, places, Or objects, while their memory for other events remains intact. This disorder is often associated with overwhelming stress.

Dissociative fugue:
Dissociative fugue has, as its essential feature, an unexpected travel away from home and the workplace, the assumption of a new identity, and the inability to recall the previous identity. The fugue usually ends when the person suddenly ‘wakes up’ with no memory of the events that occurred during the fugue.

Dissociative identity disorder:
Dissociative identity disorder often referred to as multiple personalities, is the most dramatic of the dissociative disorders. It is often associated with traumatic experiences in childhood. In this disorder, the person assumes alternate personalities that may or may not be aware of each other.

Depersonalisation:
Depersonalisation involves a dreamlike state in which the person has a sense of being separated both from self and from reality. In depersonalisation, there is a change of self-perception, and the person’s sense of reality is temporarily lost or changed.

Question 6.
What is mood disorders?
Answer:
Mood disorders are characterised by disturbances in mood or prolonged emotional state. The most common mood disorder is depression, which covers a variety of negative moods and behavioural changes. Depression can refer to a symptom Oi a disorder. In day-to-day life, we often use the term depression to refer to normal feelings after a significant loss, such as the break-up of a relationship, or the failure to attain a significant goal. The main types of mood disorders include depressive, manic dead bipolar disorders.

Major depressive disorder:
Major depressive disorder is defined as a period of depressed mood and/or loss of interest or pleasure in most activities, together with other symptoms which may include a change in body weight, constant sleep problems, tiredness, inability to think clearly, agitation, greatly slowed behaviour and thoughts of death and suicide. Other symptoms include excessive guilt or feelings of worthlessness.

Factors Predisposing towards Depression :
Genetic makeup or heredity is an important risk factor for major depression and bipolar disorders. Age is also a risk factor. For instance, women are particularly at risk during young adulthood, while for men the risk is highest in early middle age. Similarly, gender also plays a great role in this differential risk addition. For example, women in comparison to men are more likely to report a depressive disorder.

Other risk factors are experiencing negative life events and a lack of social support. Another less common mood disorder is mania. People suffering from mania become euphoric (‘high’), extremely active, excessively talkative, and easily distractible. Manic episodes rarely appear by themselves; they usually alternate with depression. Such a mood disorder, in which both mania and depression are alternately present, is sometimes interrupted by periods of normal mood.

This is known as a bipolar mood disorder. Bipolar mood disorders were earlier referred to as manic-depressive disorders. Among the mood disorders, the lifetime risk of a suicide attempt is highest in case of bipolar mood disorders. Several risk factors in addition to the mental health status of a person predict the likelihood of suicide. These include age, gender, ethnicity, or race and recent occurrence of serious life events. Teenagers and young adults are as much at high risk for suicide, as those who are over 70 years.

Gender is also an influencing factor, i.e. men have a higher rate of contemplated suicide than women. Other factors that affect suicide rates are cultural attitudes toward suicide. In Japan, for instance, suicide is the culturally appropriate way to deal with feelings of shame and disgrace. Negative expectations, hopelessness, setting unrealistically high standards and being over-critical in self-evaluation are important themes for those who have suicidal, preoccupations.

Suicide can be prevented by being alert to some of the symptoms which include :

• changes in eating and sleeping habits
• withdrawal from friends, family and regular activities
• violent actions, rebellious behaviour, running away
• drug and alcohol abuse.
• marked personality change
• persistent boredom
• difficulty in concentration.
• complaints about physical symptoms, and
• loss of interest in pleasurable activities.
  However, seeking timely help from a professional counsellor/psychologist can help to prevent the likelihood of suicide.

Question 7.
What is Schizophrenic Disorders and state its symptoms?
Answer:
Schizophrenia is the descriptive term for a group of psychotic disorders in which personal, social and occupational functioning deteriorate as a result of disturbed thought processes, strange perceptions, unusual emotional states, and motor abnormalities. It is a debilitating disorder. The social and psychological costs of schizophrenia are tremendous, both to patients as well as to their families and society.

Symptoms of Schizophrenia:
The symptoms of schizophrenia can be grouped into three categories, viz. positive symptoms (i.e. excesses of thought, emotion and behaviour), negative symptoms (i.e. deficits of thought, emotion, and behaviour) and psychomotor symptoms.

Positive symptoms:
Positive symptoms are ‘pathological excesses’ or ‘bizarre addition?’ to a person’s behaviour. Delusions, disorganised thinking and speech, heightened perception and hallucinations, and inappropriate effects are the ones most often found in schizophrenia. Many people with schizophrenia develop delusions. A delusion is a false belief that is firmly held on inadequate grounds. It is not affected by rational argument and has no basis in reality.

Delusions of persecution:
Delusions of persecution are the most common in schizophrenia. People with this delusion believe that they are being plotted against, spied on, slandered, threatened, attacked Or deliberately victimised. People with schizophrenia may also experience delusions of reference in which they attach special and personal meaning to the actions of others or to objects and events.

Delusions of grandeur:
In delusions of grandeur, people believe themselves to be specially empowered persons and in delusions of control, they believe that their feelings, thoughts and actions are controlled by others. People with schizophrenia may not be able to think logically and may speak in peculiar ways. These formal thought disorders can make communication extremely difficult.

These include rapidly shifting from one topic to another so that the normal structure of thinking is muddled and becomes illogical (loosening of associations, derailment), inventing new words or phrases (neologisms), and persistent aid inappropriate repetition of the same thoughts (perseveration). Schizophrenics may have hallucinations, i. e. perceptions that occur in the absence of external stimuli.

Auditory hallucinations:
Auditory hallucinations are most common in schizophrenia. Patients hear sounds or voices that speak words, phrases and sentences directly to the patient (second-person hallucination) or talk to one another referring/to the patient as s/he (third-person hallucination). Hallucinations can also involve the other senses.

These include tactile hallucinations (i.e. forms of tingling, burning), somatic hallucinations (i.e. something happening inside the body such as a snake crawling inside one’s stomach), visual hallucinations (i.e. vague perceptions of colour or distinct visions of people or objects), gustatory hallucinations (i.e. food or drink taste strange), and olfactory hallucinations (i.e. smell of poison or smoke). People with schizophrenia also show inappropriate effects, i.e’. emotions that are unsuited to the situation.

Negative symptoms:
Negative symptoms are ‘pathological deficits’ and include poverty of speech, blunted and flat affect, loss of volition, and social withdrawal. People with schizophrenia show alogia or poverty of speech, i.e. a reduction in speech and speech content, felony people with schizophrenia show less anger, sadness, joy, and other feelings than most people do. Thus they have blunted effect Some show no emotions at all, a condition is known as flat affect. Also, patients with schizophrenia experience avolition or apathy and an inability to start or complete a course of action.

People with this disorder may withdraw socially and become totally focused on their own ideas and fantasies. People with schizophrenia also show psychomotor Symptoms. They move less spontaneously or make odd grimaces and gestures. These symptoms may take extreme forms known as catatonia. People in a catatonic stupor remain motionless and silent for long stretches of time. Some show catatonic rigidity, i.e. maintaining a rigid, upright posture for hours. Others exhibit catatonic posturing, i.e. assuming awkward, bizarre positions for long periods.

Question 8.
Write the: Sub-types of Schizophrenia.
Answer:
According to DSM-IV-TR, the sub-types of schizophrenia and their characteristics are:

• Paranoid type :
  Preoccupation with delusions or auditory hallucinations; no disorganised speech or behaviour or inappropriate affect.
• Disorganised type:
  Disorganised speech and behaviour; inappropriate or flat affect; no catatonic symptoms.
• Catatonic type :
  Extreme motor immobility; excessive motor inactivity; extreme negativism (i.e. resistance to instructions) or mutism (i.e. refusing to speak).
• Undifferentiated type :
  Does not fit any of the sub-types but meets symptom criteria.
• Residua] type:
  Has experienced at least one episode of schizophrenia; no positive symptoms but shows negative symptoms.

Question 9.
What is Behavioural and Developmental Disorders?
Answer:
There are certain disorders that are specific to children and if neglected can lead to serious consequences later in life. Children have less self-understanding and they have not yet developed a stable sense of identity nor do they have an adequate frame of reference regarding reality, possibility and value. As a result, they are unable to cope with stressful events which might be reflected in behavioural and emotional problems.

On the other hand, although their inexperience and lack of self-sufficiency make them easily upset by problems that seem minor to an adult, children typically bounce back more quickly. We will now discuss several disorders of childhood like Attention-deficit Hyperactivity Disorder (ADHD), Conduct Disorder, and Separation Anxiety Disorder. These disorders, if not attended to, can lead to more serious and chronic disorders as the child moves into adulthood.

Classification of children’s disorders has followed a different path than that of adult disorders. Achenbach has identified two factors, i.e. extermination and internalisation, which include the majority of childhood behaviour problems. The externalising disorders, or under-controlled problems, include behaviours that are disruptive and often aggressive and aversive to others in the child’s environment.

Internalising disorders, or overcontrolled problems, are those conditions where the child experiences depression, anxiety, and discomfort that may not be evident to others. There are several disorders in which children display disruptive or externalising behaviours. We will now focus on three prominent disorders, viz. Attention-deficit Hyperactivity Disorder (ADHD), Oppositional Defiant Disorder (ODD), and Conduct Disorder.

The two main features of (ADHD) are inattention and hyperactivity-impulsivity. Children who are inattentive find it difficult to sustain mental effort during work or play. They have a hard time keeping their minds on any one thing or in following instructions. Common complaints are that the child does not listen, cannot concentrate, does not follow instructions, is disorganised, easily distracted, forgetful, does not finish assignments and is quick, to lose interest in boring activities.

Children who are impulsive seem unable to control their immediate reactions or to think before they act. They find it difficult to wait or take turns and have difficulty resisting immediate temptations or delaying gratification. Minor mishaps such as knocking things over are common whereas more serious accidents and injuries can also occur. Hyperactivity also takes many forms. Children with (ADHD) are in constant motion. Sitting still through a lesson is impossible for them.

The child may fidget, squirm, climb and run around the room aimlessly. Parents and teachers describe them as ‘driven by a motor’, always on the go, and talking incessantly. Boys are four times more likely to be given this diagnosis than girls. Children with Oppositional Defiant Disorder (ODD) display age-inappropriate amounts of stubbornness, are irritable, defiant, disobedient, and behave in a hostile manner. Unlike ADHD, the rates of ODD in boys and girls are not very different.

The terms Conduct Disorder and Antisocial Behaviour refer to age-inappropriate actions and attitudes that violate family expectations, societal norms, and the personal or property rights of others. The behaviours typical of conduct disorder include aggressive actions that cause or threaten harm to people or animals, non-aggressive conduct that causes property damage, major deceitfulness or theft, and serious rule violations.

Children show many different types of aggressive behaviour, such as verbal aggression (i.e. name-calling, swearing), physical aggression (i.e. hitting, fighting), hostile aggression (i.e. directed at inflicting injury to others) and proactive aggression (i.e. dominating and bullying others without provocation). Internalising disorders include Separation Anxiety Disorder (SAD) and Depression. Separation anxiety disorder is an internalising disorder unique to children.

Its most prominent symptom is excessive anxiety or even panic experienced by children at being separated from their parents. Children with SAD may have difficulty being in a room by themselves, going to school alone, are fearful of entering hew situations, and cling to and shadow their parents’ every move. To avoid separation, children with SAD may fuss, scream, throw severe tantrums, or make suicidal gestures.

The ways in which children express and experience depression are related to their level of physical, emotional, and cognitive development. An infant may show sadness by being passive and unresponsive; a pre¬schooler may appear withdrawn and inhibited; a school-age child may be argumentative and combative, and a teenager may express feelings of guilt and hopelessness. Children may also have more serious disorders called Pervasive Developmental Disorders.

These disorders are characterised by severe and widespread impairments in social interaction and communication skills, and stereotyped patterns of behaviours, interests and activities. Autistic disorder or autism is one of the most common of these disorders. Children with autistic disorder have marked difficulties in social interaction and communication a restricted range of interests, and a strong desire for routine.

About 70 per cent of children with autism are also mentally retarded. Children with autism experience profound difficulties in relating to other people. They are unable to initiate social behaviour and seem unresponsive to other people’s feelings. They are unable to share experiences or emotions with others. They also show serious abnormalities in communication and language that persist over time.

Many autistic children never develop speech and those who do, have repetitive and deviant speech patterns. Children with autism often show narrow patterns of interest and repetitive behaviours such as lining up objects or stereotyped body movements such as rocking. These motor movements may be self-stimulatory such as hand flapping or self-injurious such as banging their head against the wall.

Question 10.
What is Substance-use Disorders?
Answer:
Addictive behaviour, whether it involves excessive intake of high-calorie food resulting in extreme obesity or involving the abuse of substances such as alcohol or cocaine, is one of the most severe problems being faced by society today. Disorders relating to maladaptive behaviours resulting from regular and consistent use of the substance involved are called substance abuse disorders.

These disorders include problems associated with using and abusing Such drugs as alcohol, cocaine and heroin, which alter the way people think, feel and behave. There are two sub-groups of substance-use disorders, i.e. those related to substance dependence and those related to substance abuse.

Insubstance dependence:
In substance dependence, there is an intense craving for the substance to which the person is addicted, and the person shows tolerance, withdrawal symptoms and compulsive drug-taking. Tolerance means that the person has to use more and more of a substance to get the same effect. Withdrawal refers to physical symptoms that occur when a person stops or cuts down on the use of a psychoactive substance, i.e. a substance that has the ability to change an individual’s consciousness, mood and thinking processes.

Insubstance abuse:
In substance abuse, there are recurrent and significant adverse consequences related to the use of substances. People who regularly ingest drugs damage their family and social relationships, perform poorly at work and create physical hazards. We will now focus on the three most common forms of substance abuse, viz. alcohol abuse and dependence, heroin abuse and dependence and cocaine abuse and dependence.

Alcohol Abuse and Dependence People who abuse alcohol drink large amounts regularly and rely on it to help Heroin Abuse and Dependence Heroin intake significantly interferes with social and occupational functioning. Most abusers further develop a dependence on heroin, revolving their lives around the substance, building up a tolerance for it and experiencing a withdrawal reaction when they stop taking it.

The most direct and stopping it results in feelings of depression, fatigue, sleep problems, irritability and anxiety. Cocaine poses serious dangers. It has dangerous effects on psychological functioning and physical well-being.

Question 11.
Describe the nature and scope of psychotherapy. Highlight the importance of therapeutic relationships in psychotherapy.
Answer:
Nature and scope of psychotherapy: Psychotherapy is a voluntary relationship between the one seeking treatment or the client and the one who treats or the therapist. The purpose of the relationship is to help the client to solve the psychological problems faces by her or him. The relationship is conducive for building the trust of the client so that problems may be freely discussed.

Psychotherapies aim at changing maladaptive behaviours, decreasing the sense of personal distress and helping the client to adapt better to her/his environment. The inadequate marital, occupational and social adjustment also requires that major changes be made in an individual’s personal environment.

AH, psychotherapies aim at a few or all of the following goals :

• Reinforcing the client’s resolve for betterment.
• Lessening emotional pressure.
• Unfolding the potential for positive growth.
• Modifying habits,
• Changing thinking patterns.
• Increasing self-awareness.
• Improving interpersonal relations and communication.
• Facilitating decision-making.
• Becoming aware of one’s choices in life.

Relating to one’s social environment in a more creative and self-aware manner. The special relationship between the client and the therapist is known as the
therapeutic relationship or alliance.

There are two major components of a therapeutic alliance:

• The first component is the contractual nature of the relationship in which two willing individuals, the client and the therapist, enter into a partnership that aims at helping the client overcome her/his problems.
• The second component of the therapeutic alliance is the limited duration of the therapy. This alliance lasts until the client becomes able to deal with her/his problems and take control of her/his life.

This relationship has several unique properties. It is a trusting and confiding relationship. The high level of trust enables the client to unburden herself/himself to the therapist and confide her/his psychological and personal problems to the latter. The therapist encourages this by being accepting, empathic, genuine, and warm to the client.

The therapist conveys by her/his words and behaviours that she is not judging the client and will continue to show the same positive feelings towards the client even if the client is rude or confides in all the wrong things that she may have done or thought about. The therapeutic alliance also requires that the therapist must keep strict confidentiality of the experiences, events, feelings, or thoughts disclosed by the client. The therapist must not exploit the trust and confidence of the Client in any way.

Odisha State Board CHSE Odisha Class 12 Psychology Solutions Unit 4 Long Answer Questions Part-1.

CHSE Odisha 12th Class Psychology Unit 4 Long Answer Questions Part-1

Long Questions With Answers

Question 1.
Identify the symptoms associated with depression and mania.
Answer:
Symptoms associated with depression change in body weight, constant sleep problems, tiredness, inability to think clearly, agitation, greatly slowed behaviour, and thoughts of death and suicide. Other symptoms include excessive guilt or feelings of worthlessness. Symptoms associated with mania are people become euphoric (‘high’), extremely active, excessively talkative and easily distractible.

Question 2.
Describe the characteristics of hyperactive children.
Answer:
Hyperactive children are suffering from Attention-deficit Hyperactivity Disorder (ADHD) which can lead to more serious and chronic disorders as the child moves into adulthood if not attended. Children display disruptive or externalising behaviours. The two main features of ADHD are inattention and hyperactivity-impulsivity. Children who are inattentive find it difficult to sustain mental effort during work or play.

They have a hard time keeping their minds on any one thing or following instructions. Common complaints are that the child does not listen, cannot concentrate, does not follow instructions, is disorganised, easily distracted, forgetful, does not finish assignments and is quick to lose interest in boring activities. Children who are impulsive seem unable to control their immediate reactions or to think before they act. They find it difficult to wait or take turns, and have difficulty resisting immediate temptations or delaying gratification.

Minor mishaps such as knocking things over are common whereas more serious accidents and injuries can also occur. Hyperactivity also takes many forms. Children with ADHD are in constant motion. Sitting still through a lesson is impossible for them. The child may fidget, squirm, climb and run around the room aimlessly. Parents and teachers describe them as ‘driven by a motor’, always on the go and talk incessantly. Boys are four times more likely to be given this diagnosis than girls.

Question 3.
What do you understand by substance abuse and dependence?
Answer:
In substance abuse, there are recurrent and significant adverse consequences related to the use of substances. People who regularly ingest drugs damage their family and social relationships, perform poorly at work, and create physical hazards. In substance dependence, there is an intense craving for the substance to which the person is addicted, and the person shows tolerance, withdrawal symptoms and compulsive drug-taking.

Tolerance means that the person has to use more and more of a substance to get the same effect. Withdrawal refers to physical symptoms that occur when a person stops or cuts down bn the use of a psychoactive substance, i.e. a substance that has the ability to change an individual’s consciousness, mood and thinking processes.

Question 4.
Can a distorted body image lead to eating disorders? Classify the various forms of it.
Answer:
Yes, distorted body image can lead to eating disorders. The various forms of eating disorders are anorexia nervosa, bulimia nervosa, and binge eating.
Anorexia nervosa:
In this eating disorder, the individual has a distorted body image that leads her/him to see herself/himself as overweight. Often refusing to’ eat, exercising compulsively and developing unusual habits such as refusing to eat in front of others, the anorexic may lose large amounts of weight and even starve herself/himself to death.

Bulimia nervosa:
In this disorder, the individual may eat excessive amounts of food, then purge her/his body of food by using medicines such as laxatives or diuretics or by vomiting. The person often feels disgusted and ashamed when s/he binges and is relieved of tension and negative emotions after purging.

Binge eating:
In this disorder, there are frequent episodes of out-of-control eating.

Question 5.
“Physicians make diagnosis looking at a person’s physical symptoms”. How are psychological disorders diagnosed?
Answer:
Psychological disorders can be diagnosed by observations, interviews, counselling etc. In ancient days, abnormal behaviour can be explained by the operation of supernatural and magical forces such as evil spirits (bhoot-pret) or the devil (shaitan). In many Societies, the shaman, or medicine man (Ojha) is a person who is believed to have contact with supernatural forces and is the medium through which spirits communicate with human beings.

Through the shaman, an afflicted person can learn which spirits are responsible for her/his problems and what needs to be done to appease them. A recurring theme in the history of abnormal psychology is the belief that individuals behave strangely because their bodies and their brains are not working properly. This is the biological or organic approach. In the modem era, there is evidence that body and brain processes have been linked to many types of maladaptive behaviour. For certain types of disorders, correcting these defective biological processes results in improved functioning. Another approach is the psychological approach.

According to this point of view, psychological problems are caused by inadequacies in the way an individual thinks, feels, or perceives the world. The American Psychiatric Association (APA) has published an official manual describing and classifying various kinds of psychological disorders. The current version of it, the Diagnostic and Statistical Manual of Mental Disorders, TV Edition (DSM-IV), evaluates the patient on five axes or dimensions rather than just one broad aspect of ‘mental disorder’.

These dimensions relate to biological, psychological, social and other aspects. The classification scheme officially used in India and elsewhere is the tenth revision of the International Classification of Diseases (ICD-10), which is known as the ICD-10 Classification of Behavioural and Mental Disorders. It was prepared by the World Health Organisation (WHO). For each disorder, a description of the main clinical features or symptoms, and of other associated features including diagnostic guidelines is provided in this scheme.

Question 6.
Distinguish between obsessions and compulsions.
Answer:
Obsessions is the inability to stop thinking about a particular idea or topic. The person involved, often finds these thoughts to be unpleasant and shameful while Compulsions is the need to perform certain behaviours Over and over again. Many compulsions deal with counting, ordering, checking, touching and washing.

Question 7.
Can a long-standing pattern of deviant behaviour be considered abnormal? Elaborate.
Answer:
The first approach views abnormal behaviour as a deviation from social norms. Many psychologists have stated that ‘abnormal’ is simply a label that is given to a behaviour which is deviant from social expectations. Abnormal behaviour, thoughts and emotions are those that differ markedly from a society’s ideas of proper functioning. Each society has norms, which are stated or unstated rules for proper conduct.

Behaviours, thoughts and emotions that break societal norms are called abnormal. A society’s norms grow from its particular cultural history, values, institutions, habits, skills, technology and arts. Thus, a society whose culture values competition and assertiveness may accept aggressive behaviour, whereas one that emphasises cooperation and family values (such as in India) may consider aggressive behaviour as unacceptable Or even abnormal.

A society’s values may change over time, causing its views of what is psychologically abnormal to change as well. Serious questions have been raised about this definition. It is based on the assumption that socially accepted behaviour is not abnormal, and that normality is nothing more than conformity to social norms. The second approach views abnormal behaviour as maladaptive.

Many psychologists believe that the best criterion for determining the normality of behaviour is not whether society accepts it but whether it fosters the well-being of the individual and eventually of the group to which s/he belongs. Well-being is not simply maintenance and survival but also includes growth and fulfilment, i.e. the actualisation of potential, which you must have studied in Maslow’s need hierarchy theory.

According to this criterion, conforming behaviour can be seen as abnormal if it is maladaptive, i.e. if it interferes with optimal functioning and growth. For example, a student in the class prefers to remain silent even when s/he has questions in her/his mind. Describing behaviour as maladaptive implies that a problem exists; it also suggests that vulnerability in the individual, inability to cope, or exceptional stress in the environment have led to problems in life.

Question 8.
While speaking in public the patient changes topics frequently, is this a positive or a negative symptom of schizophrenia? Describe the other symptoms and sub-types of schizophrenia.
Answer:
Positive symptoms:
These are ‘pathological excesses’ or ‘bizarre additions’ to a person’s behaviour. Delusions, disorganised thinking and speech, heightened perception and hallucinations, and inappropriate effects are the ones most often found in schizophrenia.

Negative symptoms:
These are ‘pathological deficits’ and include poverty of speech, blunted and flat affect, loss of volition, and social withdrawal. People with schizophrenia show alogia or poverty of speech, i.e. a reduction in speech and speech content. Many people with schizophrenia show less anger, sadness, joy and other feelings than most people do. Thus they have blunted effect.

Some show no emotions at all, a condition is known as flat affect. Also patients with schizophrenia experience avolition, apathy and an inability to start or complete a course of action. People with this disorder may withdraw socially and become totally focused on their own ideas and fantasies.

Sub-types of Schizophrenia: According to DSM-IV-TR, the sub-types of schizophrenia and their characteristics are:

Paranoid type:
Preoccupation with delusions or auditory hallucinations; no disorganised speech or behaviour or inappropriate affect.

Disorganised type:
Disorganised speech and behaviour; inappropriate or flat affect; no catatonic symptoms.

Catatonic type:
Extreme motor immobility; excessive motor inactivity; extreme negativism (i.e. resistance to instructions) or mutism (i.e. refusing to speak).

Undifferentiated type:
Does not fit any of the sub-types but meets symptom criteria.

Residual type:
Has experienced at least one episode of schizophrenia; no positive symptoms but shows negative symptoms.

Question 9.
What do you understand by the term ‘dissociation’? Discuss its various forms.
Answer:
Dissociation can be viewed as a severance of the connections between ideas and emotions. Dissociation involves feelings of unreality, estrangement, depersonalisation, and sometimes a loss or shift of identity. Sudden temporary alterations of consciousness that blot out painful experiences are a defining characteristic of dissociative disorders. Four conditions are included in this group: dissociative amnesia, dissociative fugue, dissociative identity disorder, and depersonalisation.

Various forms of dissociation are as follows:

Dissociative amnesia:
It is characterised by extensive but selective memory loss that has no known organic cause (e.g. head injury). Some people cannot remember anything about their past. Others can no longer recall specific events, people, places, or objects, while their memory for other events remains intact. This disorder is often associated with overwhelming stress.

Dissociative fugue:
It has, as its essential feature, an unexpected travel away from home and workplace, the assumption of a new identity, and the inability to recall the previous identity. The fugue usually ends when the person suddenly ‘wakes up’ with no memory of the events that occurred during the fugue.

Dissociative identity disorder:
It is often referred to as multiple personalities, is the most dramatic of the dissociative disorders. It is often associated with traumatic experiences in childhood. In this disorder, the person assumes alternate personalities that may or may not be aware of each other.

Depersonalisation:
It involves a dreamlike state in which the person His a sense of being separated both from self and from reality. In depersonalisation, there is a change of self-perception, and the person’s sense of reality is temporarily lost or changed.

Question 10.
What are phobias? If someone had an intense fear of snakes, could this simple phobia be a result of faulty learning? Analyse how this phobia could have developed.
Answer:
Phobias are irrational fears related to specific objects, interactions with others, and unfamiliar situations. If someone had an intense fear of snakes, this simple phobia cannot be a result of faulty learning. It is a. specific phobia which is most common. This group includes irrational fears such as intense fear of a certain type of animal, or of being in an enclosed space. This phobia often develops gradually or begins with generalised anxiety disorders.

Question 11.
Anxiety has been called the “butterflies in the stomach feeling”. At what stage does anxiety become a disorder? Discuss its types.
Answer:
Everyone has worries and fears. The term anxiety is usually defined as a diffuse, vague, very unpleasant feeling of fear and apprehension. The anxious individual also shows combinations of the following symptoms: rapid heart rate, shortness of breath, diarrhoea, loss of appetite, fainting, dizziness, sweating, sleeplessness, frequent urination and tremors.

Different types of anxiety disorders and their symptoms are as follows:

Generalised anxiety disorder:
This disorder consists of prolonged, vague, unexplained and intense fears that are not attached to any particular object. The symptoms include worry and apprehensive feelings about the future; hypervigilance, which involves constantly scanning the environment for dangers.

Panic disorder:
This disorder consists of recurrent anxiety attacks in which the person experiences intense terror. The clinical symptoms include shortness of breath, dizziness, trembling, palpitations, choking, nausea, chest pain or discomfort, fear of going crazy, losing control or dying.

Obsessive-compulsive disorder:
People are unable to control their preoccupation with specific ideas or are unable to prevent themselves from repeatedly carrying out a particular actor series of acts that affect their ability to carry out normal activities. Obsessive behaviour is the inability to stop thinking about a particular idea or topic. The person involved, often finds these thoughts to be unpleasant and shameful. Compulsive behaviour is the need to perform certain behaviours over and over again. Many compulsions deal with counting, ordering, checking, touching and washing.

Phobias:
These are irrational fears related to specific objects, interactions with others, and unfamiliar situations.

Question 12.
What is the concept of abnormality and psychological disorders?
Answer:
Although many definitions of abnormality have been used over the years, none has won universal acceptance. Still, most definitions have certain common features, often called the ‘four Ds’: deviance, distress, dysfunction and danger. That is, psychological disorders are deviant (different, extreme, unusual, even bizarre), distressing (unpleasant and upsetting to the person and to others), dysfunctional (interfering with the person’s ability to carry out daily activities in a constructive You must have come across people who are unhappy, troubled and dissatisfied.

Their minds and hearts are filled with Sorrow, unrest and tension and they feel that they are unable to move ahead in their lives; they feel life is a painful, uphill struggle, sometimes, not worth living. Famous analytical psychologist Carl Jung has quite remarkably said, “How can I be substantial without casting a shadow? I must have a dark side, too, if I am to be whole and by becoming conscious of my shadow, I remember once more that I am a human being like any other”.

At times, some of you may have felt nervous before an important examination, tense and concerned about your future career or anxious when someone close to you was unwell. All of us face major problems at some point in our lives. However, some people have an extreme reaction to the problems and stresses of life. In this chapter, we will try to understand what goes wrong when people develop psychological problems, what are the causes and factors which lead to abnormal behaviour, and the various signs and symptoms associated with different types of psychological disorders.

The study of psychological disorders has intrigued and mystified all cultures for more than 2,500 years. Psychological disorders or mental disorders (as they are commonly referred to), like anything unusual, may make us uncomfortable and even a little frightened. Unhappiness, discomfort, anxiety and unrealised potential are seen all over the world. These failures in living are due mainly to failures in adaptation to life challenges.

As you must have studied in the previous chapters, adaptation refers to the person’s ability to modify her/his behaviour in response to changing environmental requirements. When the behaviour cannot be modified according to the needs of the situation, it is said to be maladaptive. Abnormal Psychology is the area within psychology that is focused on maladaptive behaviour – its causes, consequences, and treatment way), and possibly dangerous (to the person or to others).

This definition is a useful starting point from which we can explore psychological abnormality. Since the word ‘abnormal’ literally means “away from the normal”, it implies deviation from some clearly defined norms or standards. In psychology, we have no ‘ideal model’ or even ‘normal model’ of human behaviour to use as a base for comparison. Various approaches have been used in distinguishing between normal and abnormal behaviours.

From these approaches, there emerge two basic and conflicting views: The first approach views abnormal behaviour as a deviation from social norms. Many psychologists have stated that ‘abnormal’ is simply a label that is given to a behaviour which is deviant from social expectations. Abnormal behaviour, thoughts and emotions are those that differ markedly from a society’s ideas of proper functioning. Each society has norms, which are stated or unstated rules for proper conduct.

Behaviours, thoughts and emotions that break societal norms are called abnormal. A society’s norms grow from its particular culture — its history, values, institutions, habits, skills, technology and arts. Thus, a society whose culture values competition and assertiveness may accept aggressive behaviour, whereas one that emphasises cooperation and family values (such as in India) may consider aggressive behaviour as unacceptable or even abnormal.

A society’s values may change over time, causing its views of what is psychologically abnormal to change as well. Serious questions have been raised about this definition. It is based on the assumption that socially accepted behaviour is not abnormal and that normality is nothing more than conformity to social norms. The second approach views abnormal behaviour as maladaptive.

Many psychologists believe that the best criterion for determining the normality of behaviour is not whether society accepts it but whether it fosters the well-being of the individual and eventually of the group to Which s/he belongs. Well-being is not simply maintenance and survival but also includes growth and fulfilment, i.e. the actualisation of potential, which you must have studied in Maslow’s need hierarchy theory.

According to this criterion, conforming behaviour can be seen as abnormal if it is maladaptive, i.e. if it interferes with optimal functioning and growth. For example, a student in the class prefers to remain silent even when s/he has questions in her/his mind. Describing behaviour as maladaptive implies that a problem exists; it also suggests that vulnerability in the individual, inability to cope, or exceptional stress in the environment have led to problems in life.

If you talk to people around, you will see that they have vague ideas about psychological disorders that are characterised by superstition, ignorance and fear. Again it is commonly believed that psychological disorder is something to be ashamed of. the stigma attached to mental illness means that people are hesitant to consult a doctor or psychologist because they are ashamed of their problems. Actually, a psychological disorder which indicates a failure in adaptation should be viewed as any other illness.

Question 13.
Write the historical view of psychological disorders.
Answer:
Historical Background :
To understand psychological disorders, we would require a brief historical account of how these disorders have been viewed over the ages. When we study the history of abnormal psychology, we find that certain theories have occurred over and over again. One ancient theory that is still encountered today holds that abnormal behaviour can be explained by the operation of supernatural and magical forces such as evil spirits (bhoot-pret), or the devil (shaitan).

Exorcism, i.e. removing the evil that resides in the individual through countermagic and prayer, is still commonly used. In many societies, the shaman, or medicine man (Ojha) is a person who is believed to have contact with supernatural forces and is the medium through which spirits communicate with human beings. Through the shaman, an afflicted person can learn which spirits are responsible for her/his problems and what needs to be done to appease them.

A recurring theme in the history of abnormal psychology is the belief that individuals behave strangely because their bodies and their brains are not working properly. This is the biological or organic approach. In the modem era, there is evidence that body and brain processes have been linked to many types of maladaptive behaviour. For certain types of disorders, correcting these defective biological processes results in improved functioning.

Another approach is the psychological approach. According to this point of view, psychological problems are caused by inadequacies in the way an individual thinks, feels, or perceives the world. All three of these perspectives—supernatural, biological or organic, and psychological — have recurred throughout the history of Western civilisation.

In the ancient Western world, it was philosopher physicians of ancient Greece such as Hippocrates, Socrates, and in particular Plato who developed the organismic approach and viewed disturbed behaviour as arising out of conflicts between emotion and reason. Galen elaborated on the role of the four senses of humour in personal character and temperament. According to him, the material world was made up of four elements, viz. earth, air, fire and water which combined to form four essential body fluids, viz. blood, black bile, yellow bile and phlegm.

Each of these fluids was seen to be responsible for a different temperament. Imbalances among the humour were believed to cause various disorders. This is similar to the Indian notion of the three doshas of Vata, Pitta, and Kapha which were mentioned in the Atharva Veda and Ayurvedic texts. You have already read about it in Chapter 2. In the Middle Ages, demonology and superstition gained renewed importance in the explanation of abnormal behaviour.

Demonology related to a belief that people with mental problems were evil and there are numerous instances of ‘witch-hunts’ during this period. During the early Middle Ages, the Christian spirit of charity prevailed and St. Augustine wrote extensively about feelings, mental anguish, and conflict. This laid the groundwork for modem psychodynamic theories of abnormal behaviour. The Renaissance Period was marked by increased humanism and curiosity about behaviour.

Johann Weyer emphasized psychological conflict and disturbed interpersonal relationships as causes of psychological disorders. He also insisted that ‘ witches’ were mentally disturbed and required medical, not theological; treatment. The seventeenth and eighteenth centuries were known as the Age of Reason and Enlightenment, as the scientific method replaced faith and dogma as ways of understanding abnormal behaviour.

The growth of a scientific attitude towards psychological disorders in the eighteenth century contributed to the Reform Movement and to increased compassion for people who suffered from these disorders. Reforms of asylums were initiated in both Europe and America. One aspect of the reform movement was the new inclination for deinstitutionalization which placed emphasis on providing community Care for recovered mentally ill individuals.

In recent years, there has been a convergence of these approaches, which has resulted in an interactional, or biopsychosocial approach. From this perspective, all three factors, i.e. biological, psychological and social play important roles in influencing the expression and outcome of psychological disorders.