CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(b)

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

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Exercise 12(b)

Question 1.
Each question given below has four possible answers, out of which only one is correct. Choose the correct one.
(i) (2î – 4ĵ) . (î + ĵ + k̂) = _______.
(a) -3
(b) +2
(c) -1
(d) -2
Solution:
(d) -2

(ii) If a = î + 2ĵ – k̂, b = î + ĵ + 2k̂, c = 2î – ĵ; then
(a) \(\vec{a} \perp \vec{b}\)
(b) \(\vec{b} \perp \vec{c}\)
(c) \(\vec{a} \perp \vec{c}\)
(d) no pair of vectors are perpendicular.
Solution:
(c) \(\vec{a} \perp \vec{c}\)

(iii) (-3, λ, 1) ⊥ (1, 0, -3) ⇒ λ = _______.
(a) 0
(b) 1
(c) impossible to find
(d) any real number
Solution:
(c) impossible to find

(iv) If \(\vec{a} \cdot \vec{b}=\vec{c} \cdot \vec{a}\) for all vectors \(\vec{a}\), then
(a) \(\vec{a} \perp(\vec{b}-\vec{c})\)
(b) \(\vec{b}-\vec{c}\) = 0
(c) \(\vec{b} \neq \vec{c}\)
(d) \(\vec{b}+\vec{c}=\overrightarrow{0}\)
Solution:
(b) \(\vec{b}-\vec{c}\) = 0

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(b)

Question 2.
Find the scalar product of the following pairs of vectors and the angle between them.
(i) 3î – 4ĵ and -2î + ĵ
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(b) Q.2(1)

(ii) 2î – 3ĵ + 6k̂ and 2î – 3ĵ – 5k̂
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(b) Q.2(2)

(iii) î – ĵ and ĵ + k̂
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(b) Q.2(3)

(iv) \(\vec{a}\) = (2, -2, 1) and \(\vec{b}\) = (0, 2, 4)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(b) Q.2(4)

Question 3.
If A, B, C are the points (1, 0, 2), (0, 3, 1) and (5, 2, 0) respectively, find m∠ABC.
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(b) Q.3

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(b)

Question 4.
Find the value of λ so that the vectors \(\vec{a}\) and \(\vec{b}\) are perpendicular to each other.
(i) \(\vec{a}\) = 3î + 4ĵ, \(\vec{b}\) = -5î + λĵ
Solution:
If \(\vec{a}\) and \(\vec{b}\) are perpendicular \(\vec{a} \cdot \vec{b}\) = 0
⇒ (3î + 4ĵ) . (-5î + λĵ) = 0
⇒ -15 + 4λ = 0
⇒ λ = \(\frac{15}{4}\)

(ii) \(\vec{a}\) = î + ĵ + λk̂, \(\vec{b}\) = 4î – 3k̂
Solution:
If \(\vec{a}\) and \(\vec{b}\) are perpendicular \(\vec{a} \cdot \vec{b}\) = 0
⇒ ( î + ĵ + λk̂) . (4î – 3k̂) = 0
⇒ 4 + 0 – 3λ = 0
⇒ λ = \(\frac{4}{3}\)

(iii) \(\vec{a}\) = 2î – ĵ – k̂, \(\vec{b}\) = λî + ĵ + 5k̂
Solution:
If \(\vec{a}\) and \(\vec{b}\) are perpendicular \(\vec{a} \cdot \vec{b}\) = 0
⇒ (2î – ĵ – k̂) . (λî + ĵ + 5k̂) = 0
⇒ 2λ – 1 – 5 = 0
⇒ 2λ = 6
⇒ λ = 3

(iv) \(\vec{a}\) = (6, 2, -3), \(\vec{b}\) = (1, -4, λ)
Solution:
If \(\vec{a}\) and \(\vec{b}\) are perpendicular \(\vec{a} \cdot \vec{b}\) = 0
⇒ (6, 2, -3) . (1, -4, λ) = 0
⇒ 6 – 8 – 3λ = 0
⇒ -2 – 3λ = 0
⇒ λ = –\(\frac{2}{3}\)

Question 5.
Find the scalar and vector projections of \(\vec{a}\) on \(\vec{b}\).
(i) \(\vec{a}\) = î, \(\vec{b}\) = ĵ
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(b) Q.5(1)

(ii) \(\vec{a}\) = î + ĵ, \(\vec{b}\) = ĵ + k̂
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(b) Q.5(2)

(iii) \(\vec{a}\) = î – ĵ – k̂, \(\vec{b}\) = 3î + ĵ + 3k̂
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(b) Q.5(3)

Question 6.
In each of the problems given below, find the work done by a force \(\overrightarrow{F}\) acting on a particle, such that the particle is displaced from a point A to a point B.
(i) \(\overrightarrow{F}\) = 4î + 2ĵ + 3k̂
A (1, 2, 0), B (2, -1, 3)
Solution:
Displacement of the particle \(\overrightarrow{S}=\overrightarrow{AB}\)
= (2 – 1)î + (-1 – 2)ĵ + (3 – 0)k̂
=î – 3ĵ + 3k̂
Work done = \(\overrightarrow{F} \cdot \overrightarrow{S}\)
= (4î + 2ĵ + 3k̂) . (î – 3ĵ + 3k̂)
= 4 – 6 + 9
= 7 units.

(ii) \(\overrightarrow{F}\) = 2î + ĵ – k̂
A (0, 1, 2), B (-2, 3, 0)
Solution:
Displacement
\(\vec{S}\) = (-2 – 0)î + (3 – 1)ĵ + (0 – 2)k̂
= -2î + 2ĵ – 2k̂
Work done = \(\overrightarrow{F} \cdot \overrightarrow{S}\)
= (2î + ĵ – k̂) . (-2î + 2ĵ – 2k̂)
= -4 + 2 + 2
= 0 units.

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(b)

(iii) \(\overrightarrow{F}\) = 4î – 3k̂
A (1, 2, 0), B (0, 2, 3)
Solution:
Displacement \(\vec{S}\) = -î + 3k̂
Work done = \(\overrightarrow{F} \cdot \overrightarrow{S}\)
= (4î – 3k̂) . (-î + 3k̂)
= -4 – 9
= -13 units.

(iv) \(\overrightarrow{F}\) = 3î – ĵ – 2k̂
A (-3, -4, 1), B (-1, -1, -2)
Solution:
Displacement \(\vec{S}\) = 2î + 3ĵ – 3k̂
Work done \(\overrightarrow{F} \cdot \overrightarrow{S}\)
= (3î – ĵ – 2k̂) . (2î + 3ĵ – 3k̂)
= 6 – 3 + 6
= 9 units.

Question 7.
If \((\vec{a}+\vec{b}) \cdot(\vec{a}-\vec{b})\) = 0 show that \(|\vec{a}|=|\vec{b}|\).
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(b) Q.7

Question 8.
(i) If a and b are perpendicular vectors show that
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(b) Q.8
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(b) Q.8.1

(ii) Prove that two vectors are perpendicular iff \(|\vec{a}+\vec{b}|^2=|\vec{a}|^2+|\vec{b}|^2\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(b) Q.8.2

Question 9.
If \(\vec{a}, \vec{b}, \vec{c}\) are mutually perpendicular vectors of equal magnitude, show that \(\vec{a}+\vec{b}+\vec{c}\) is equally inclined to \(\vec{a} \cdot \vec{b} \cdot \vec{c}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(b) Q.9

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(b)

Question 10.
Prove the following by vector method.
(i) Altitudes of a triangle are concurrent;
Solution:
Let ABC be a triangle.
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(b) Q.10(1)
⇒ CF is perpendicular to AB.
Hence the altitudes of a triangle are concurrent.

(ii) Median to the base of an isosceles triangle is perpendicular to the base;
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(b) Q.10(2)
⇒ OD is perpendicular to the base AB.
Hence the median to the base of an isosceles triangle is perpendicular to the base. (Proved)

(iii) The parallelogram whose diagonals are equal is a rectangle;
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(b) Q.10(3)
⇒ m∠COA = 90°
Hence OABC is a rectangle. (Proved)

(iv) The diagonals ofa rhombus are at right angles;
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(b) Q.10(4)
Hence the diagonals of a rhombus are at right angles. (Proved)

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(b)

(v) An angle inscribed in a semi-circle is a right angle;
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(b) Q.10(5)
∴ m∠ABC = 90°
Hence the angle inscribed in a semi-circle is a right-angle. (Proved)

(vi) In any triangle ABC; a = b cos C + c cos B;
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(b) Q.10(6)

(vii) In a triangle AOB, m∠AOB = 90°. If P and Q are the points of trisection of AB, prove that OP2 + OQ2 = \(\frac{5}{9}\) AB2;
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(b) Q.10(7)

(viii) Measure of the angle between two diagonals of a cube is cos-1\(\frac{1}{3}\).
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(b) Q.10(8)

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c)

Odisha State Board CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Exercise 12(c)

Question 1.
Each question given below has four possible answers out of which only one is correct. Choose the correct one.
(i) (î + k̂) × (î + ĵ + k̂) = ______.
(a) î – k̂
(b) k̂ – î
(c) k̂ – 2î – ĵ
(d) 2
Solution:
(î + k̂) × (î + ĵ + k̂) = \(\left|\begin{array}{ccc}
\hat{i} & \hat{j} & \hat{k} \\
1 & 0 & 1 \\
1 & 1 & 1
\end{array}\right|\)
= î (0 – 1) – ĵ (1 – 1) + k̂ (1 – 0)
= -î + k̂ = k̂ – î

(ii) A vector perpendicular to the vectors î + ĵ and î + k̂ is ______.
(a) î – ĵ – k̂
(b) ĵ – k̂ + î
(c) k̂ – ĵ – î
(d) ĵ + k̂ + î
Solution:
A vector perpendicular to the vectors î + ĵ and î + k̂ is
(î + ĵ) × (î + k̂) = \(\left|\begin{array}{ccc}
\hat{i} & \hat{j} & \hat{k} \\
1 & 1 & 0 \\
1 & 0 & 1
\end{array}\right|\)
= î (1 – 0) – ĵ (1 – 0) + k̂ (0 – 1)
= î – ĵ – k̂

(iii) The area of the triangle with vertices (1, 0, 0), (0, 1, 0) and (0, 0, 1) is ______.
(a) \(\frac{1}{2}\)
(b) 1
(c) \(\frac{\sqrt{3}}{2}\)
(d) 2
Solution:

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c) Q.1

(iv) If â and b̂ are unit vectors such that â × b̂ is a unit vector, then the angle between â and b̂ is ______.
(a) of any measure
(b) \(\frac{\pi}{4}\)
(c) \(\frac{\pi}{2}\)
(d) π
Solution:
|a × b| = ab sin θ = sin θ
⇒ sin θ = 1
⇒ θ = \(\frac{\pi}{2}\)

(v) If \(\vec{a}, \vec{b} \text { and } \vec{c}\) are non-zero vectors, then \(\vec{a} \times \vec{b}=\vec{a} \times \vec{c}\) ______.
(a) \(\vec{b}=\vec{c}\)
(b) \(\vec{a} \|(\vec{b}-\vec{c})\)
(c) \(\vec{b} \| \vec{c}\)
(d) \(\vec{b} \perp \vec{c}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c) Q.1(1)

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c)

Question 2.
Let \(\vec{a}\) = 2î + ĵ, \(\vec{b}\) = -î + 3ĵ + k̂ and \(\vec{c}\) = î + 2ĵ + 5k̂ be three vectors. Find
(i) \(\vec{c} \times \vec{a}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c) Q.2(1)

(ii) \(\vec{a} \times(-\vec{b})\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c) Q.2(2)

(iii) \((\vec{a}-2 \vec{b}) \times \vec{c}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c) Q.2(3)

(iv) \((\vec{a}-\vec{c}) \times \vec{c}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c) Q.2(4)

(v) \((\vec{a}-\vec{b}) \times(\vec{c}-\vec{a})\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c) Q.2(5)

Question 3.
Find the unit vectors perpendicular to the vectors
(i) î, k̂
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c) Q.3(1)

(ii) î + ĵ, î – k̂
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c) Q.3(2)

(iii) 2î + 3k̂, î – 2ĵ
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c) Q.3(3)

(iv) 2î – 3ĵ + k̂, -î + 2ĵ – k̂
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c) Q.3(4)

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c)

Question 4.
Determine the area of parallelogram whose adjacent sides are the vectors
(i) 2î, ĵ
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c) Q.4(1)

(ii) î + ĵ, -î + 2ĵ
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c) Q.4(2)

(iii) 2î + ĵ + 3k̂, î – ĵ
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c) Q.4(3)

(iv) (1, – 3, 1), (1, 1, 1).
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c) Q.4(4)

Question 5.
Calculate the area of the traingle ABC (by vector method) where
(i) A (1, 2, 4), B (3, 1, -2), C (4, 3, 1)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c) Q.5(1)

(ii) A (1, 1, 2), B (2, 2, 3), C (3, -1, -1).
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c) Q.5(2)

Question 6.
Determine the sine of the angle between the vectors
(i) 5î – 3ĵ, 3î – 2k̂
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c) Q.6(1)

(ii) î – 3ĵ + k̂, î + ĵ + k̂
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c) Q.6(2)

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c)

Question 7.
Show that \((\vec{a} \times \vec{b})^2\) = a2b2 – \((\vec{a}, \vec{b})^2\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c) Q.7

Question 8.
If \(\vec{a} \times \vec{b}=\vec{b} \times \vec{c} \neq \overrightarrow{0}\), prove that \(\vec{a}+\vec{c}=m \vec{b}\), where m is a scalar.
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c) Q.8

Question 9.
If \(\vec{a}\) = 2î + ĵ – k̂, \(\vec{b}\) = -î + 2ĵ – 4k̂, \(\vec{c}\) = î + ĵ + k̂, find \((\vec{a} \times \vec{b}) \cdot(\vec{a} \times \vec{c})\).
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c) Q.9

Question 10.
If \(\vec{a}\) = 3î + ĵ – 2k̂, \(\vec{b}\) = 2î – 3ĵ + 4k̂ then verify that \(\vec{a} \times \vec{b}\) is perpendicular to both \(\vec{a}\) and \(\vec{b}\).
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c) Q.10

Question 11.
Find the area of the parallelogram whose diagonals are vectors 3î + ĵ – 2k̂ and î – 3ĵ + 4k̂.
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c) Q.11

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c)

Question 12.
Show that \((\vec{a}-\vec{b}) \times(\vec{a}+\vec{b})=2(\vec{a} \times \vec{b})\). Interpret this result geometrically.
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c) Q.12
= Vector area of the parallelogram ABCD.
Hence twice the vector area of a parallelogram ABCD is equal to the vector area of the parallelogram whose adjacent sides are the diagonals of the parallelogram ABCD.

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Additional Exercise

Odisha State Board Elements of Mathematics Class 12 Solutions CHSE Odisha Chapter 11 Differential Equations Additional Exercise Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Additional Exercise

(A) Multiple Choice Questions (Mcqs) With Answers

Question 1.
If f is an odd function, then write the value of \(\int_{-a}^a \frac{f(\sin x)}{f(\cos x)+f\left(\sin ^2 x\right)}\) dx
(a) 1
(b) 0
(c) -1
(d) 2
Solution:
(b) 0

Question 2.
If p and q are respectively degree and order of the differential equation y = edy/dx then write the relation between p and q.
(a) p ≠ q
(c) p ≡ q
(b) p = q
(d) None of these
Solution:
(b) p = q

Question 3.
Write the value of \(\int_0^1\){x} dx where {x} stands for fractional part of x.
(a) \(\frac{1}{2}\)
(b) \(\frac{3}{2}\)
(c) \(\frac{1}{4}\)
(d) \(\frac{2}{3}\)
Solution:
(a) \(\frac{1}{2}\)

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Additional Exercise

Question 4.
Write the value of:
\(\int_0^{\pi / 2} \frac{\sin x}{\sin x+\cos x}\) dx – \(\int_0^{\pi / 2} \frac{\cos x}{\sin x+\cos x}\) dx
(a) 1
(b) 2
(c) 0
(d) π
Solution:
(c) 0

Question 5.
Write the value of \(\int_{\frac{\pi}{4}}^{\frac{\pi}{4}}\)sin5 x cos x dx
(a) 0
(b) 1
(c) cos x
(d) sin x
Solution:
(a) 0

Question 6.
Write the particular solution of the equation \(\frac{d y}{d x}\) = sin x given that y(π) = 2
(a) y = cos x + 1
(b) y = -cos x + 1
(c) y = -cos x – 1
(d) y = -sin x + 1
Solution:
(b) y = -cos x + 1

Question 7.
Write the degree of the following differential equation:
\(\frac{\mathrm{d}^2 \mathrm{y}}{\mathrm{dx}^2}\) = \(\frac{2 y^3+\left(\frac{d y}{d x}\right)^4}{\sqrt{\frac{d^2 y}{d x^2}}}\)
(a) 0
(b) 1
(c) 2
(d) 3
Solution:
(d) 3

Question 8.
Write the order ofthe following differential equation:
\(\frac{d^2 y}{d x^2}\) = \(\frac{2 y^3+\left(\frac{d y}{d x}\right)^4}{\sqrt{\frac{d^2 y}{d x^2}}}\)
(a) 0
(b) 1
(c) 2
(d) 3
Solution:
(c) 2

Question 9.
What is F(x) if F(x) = \(\int_0^x\)e2t sin 3t dt?
(a) e2x sin 3x
(b) e2x cos 3x
(c) ex sin 3x
(d) e2x sin x
Solution:
(a) e2x sin 3x

Question 10.
\(\int \frac{d x}{\cos ^2 x \sin ^2 x}\) = ?
(a) -2 cos 2x + C
(b) -2 cot 2x + C
(c) -2 sin 2x + C
(d) 2 cot 2x + C
Solution:
(b) -2 cot 2x + C

Question 11.
If \(\int_1^2\)f(x) dx= λ, then what is the value of \(\)f(3 – x) dx?
(a) λ
(b) λ2
(c) 1λ
(d) 2λ
Solution:
(a) λ

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Additional Exercise

Question 12.
What is the value of \(\int_{-1}^1 \frac{d x}{1+x^2}\)?
(a) \(\frac{2 \pi}{2}\)
(b) 2π
(c) π
(d) \(\frac{\pi}{2}\)
Solution:
(d) \(\frac{\pi}{2}\)

Question 13.
Write the order of the following differential equation:
\(\frac{d^3 y}{d x^3}\) = \(\left(\frac{d^2 y}{d x^2}\right)^2\) + \(\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^4\) + y
(a) 1
(b) 3
(c) 2
(d) 0
Solution:
(b) 3

Question 14.
Write the degree of the following differential equation:
\(\frac{d^3 y}{d x^3}\) = \(\left(\frac{d^2 y}{d x^2}\right)^2\) + \(\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^4\) + y
(a) 1
(b) 2
(c) 3
(d) 0
Solution:
(a) 1

Question 15.
Write the particular solution of \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = (1 + x)4, y = 0 when x = -1.
(a) y = \(\frac{(1+x)^2}{5}\)
(b) y = \(\frac{(2+x)^5}{5}\)
(c) y = \(\frac{(1-x)^5}{5}\)
(d) y = \(\frac{(1+x)^5}{5}\)
Solution:
(d) y = \(\frac{(1+x)^5}{5}\)

Question 16.
Evaluate the integral ∫2x cosec2 x2 dx?
(a) cot x2 + C
(b) -cot x2 + C
(c) -cot 2x2 + C
(d) cot 2x2 + C
Solution:
(b) -cot x2 + C

Question 17.
What is the value of \(\frac{d}{d x} \int_{250}^{300}\left(x^4+5 x^3\right)^2\) dx
(a) 0
(b) 1
(c) -1
(d) 2
Solution:
(a) 0

Question 18.
Write down the integral of ∫\(e^{x^2}\) 2x dx.
(a) \(e^{2 x^2}\)
(b) 2\(e^{2 x^2}\)
(c) \(e^{x^2}\)
(d) None of the above
Solution:
(c) \(e^{x^2}\)

Question 19.
What is the integral of ∫log ex dx?
(a) \(\frac{2 x^2}{2}\) + C
(b) \(\frac{2 x^2}{3}\) + C
(c) \(\frac{x^2}{2}\) + C
(d) None of the above
Solution:
(c) \(\frac{x^2}{2}\) + C

Question 20.
What is the value of \(\int_{-2}^2\)|x| dx?
(a) 0
(b) 1
(c) 2
(d) 3
Solution:
(a) 0

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Additional Exercise

Question 21.
\(\int_{-1}^1\)|1 – x| dx = ______.
(a) 0
(b) 1
(c) 2
(d) -1
Solution:
(c) 2

Question 22.
If ∫x3\(e^{c x^4}\)dx = \(\frac{1}{20} \mathrm{e}^{\mathrm{cx}}\) then C = ______.
(a) 0
(b) 2
(c) 4
(d) 5
Solution:
(d) 5

Question 23.
\(\int_a^b\)f(x) dx = 1 ⇒ \(\int_a^b\)k f(t)dt ______.
(a) k
(b) -k
(c) 2k
(d) None of the above
Solution:
(b) -k

Question 24.
\(\int_{-1}^1\)f(x) dx = k and f is an even function then \(\int_{-1}^1\)f(x) = ______.
(a) k
(b) -k
(c) 2k
(d) None of the above
Solution:
(c) 2k

Question 25.
If ∫\(\int_0^1\)f(x) dx = 4, \(\int_0^2\)f(t) dt and \(\int_4^2\)f(u) du = 1 then \(\int_1^4\)f(x) dx = ______.
(a) 0
(b) 1
(c) 3
(d) -3
Solution:
(d) -3

Question 26.
I(f) = \(\int_a^x\)f(t) dt and Df = f'(x) then (ID – DI) f = ______.
(a) -f(a)
(b) 2f(a)
(c) f(a)
(d) None of the above
Solution:
(a) -f(a)

Question 27.
\(\int_0^\pi\)cos101 x dx = ______.
(a) 0
(b) 1
(c) -1
(d) 101
Solution:
(a) 0

Question 28.
Let f satisfies all the conditions of Rolle’s theorem in [1, 6] then \(\int_1^6\)f'(x) dx = ______.
(a) 0
(b) 1
(c) -1
(d) 6
Solution:
(a) 0

Question 29.
\(\int_{-2}^2\)|x| dx = ______.
(a) 1
(b) 2
(c) 3
(d) 4
Solution:
(d) 4

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Additional Exercise

Question 30.
Integrate ∫log x dx
(a) x. log x + x + C
(b) x. log x – x + C
(c) log x – x + C
(d) None of these
Solution:
(b) x. log x – x + C

Question 31.
Evaluate \(\int_0^2\)[x – 1] dx
(a) 0
(b) 1
(c) -1
(d) 2
Solution:
(b) 1

Question 32.
What is the value of: ∫\(\frac{f^{\prime}(x)-f(x)}{e^x}\) dx?
(a) ex f(x) + C.
(b) e2x f(x) + C.
(c) e-x f(x) + C.
(d) None of the above
Solution:
(c) e-x f(x) + C.

Question 33.
What is the value of \(\int_0^1\)x(1 – x)99 dx?
(a) \(\frac{1}{100}\)
(b) \(\frac{1}{10}\)
(c) \(\frac{1}{1010}\)
(d) \(\frac{1}{10100}\)
Solution:
(d) \(\frac{1}{10100}\)

Question 34.
Solution of \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = xy + x + y + 1 is ______.
(a) 2x + \(\frac{x^2}{2}\) + C
(b) x + \(\frac{x}{2}\) + C
(c) x + \(\frac{2 x^2}{2}\) + C
(d) x + \(\frac{x^2}{2}\) + C
Solution:
(d) x + \(\frac{x^2}{2}\) + C

Question 35.
f(x) = \(\int_0^x\)t sin t dt then f ‘(x) = ______.
(a) x cos x
(b) x sin t
(c) x sin x
(d) x tan x
Solution:
(c) x sin x

Question 36.
What is the value of the integral \(\int_a^b \frac{|x|}{x}\)dx?
(a) |b| – |a|
(b) |a| – |b|
(c) |b| + |a|
(d) |a| + |b|
Solution:
(a) |b| – |a|

Question 37.
What is the value of ∫xx (1 + ln x) dx?
(a) x2x + C
(b) xx + C
(c) 2xx + C
(d) x2 + C
Solution:
(b) xx + C

Question 38.
Evaluate: \(\int_0^{\mathrm{p} / 2}\)ln(cot x) dx.
(a) 0
(b) 1
(c) cot x
(d) sin x
Solution:
(a) 0

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Additional Exercise

Question 39.
Evaluate: \(\int_{-3}^4\)|x| dx
(a) \(\frac{2}{25}\)
(b) \(\frac{25}{2}\)
(c) \(\frac{25}{4}\)
(d) \(\frac{25}{-3}\)
Solution:
(b) \(\frac{25}{2}\)

Question 40.
Evaluate: \(\int_0^{\frac{\pi}{2}}\)(cos x – sin x) dx
(a) 0
(b) 1
(c) -1
(d) π
Solution:
(a) 0

Question 41.
Evaluate: \(\int_0^{\frac{\pi}{2}}\)log tan x dx.
(a) 1
(b) -1
(c) 0
(d) π
Solution:
(c) 0

Question 42.
Integrate: \(\frac{d x}{3 e^x-1}\)
(a) \(\ln \left(\frac{e^{3 x}-1}{e^x}\right)\) + C
(b) \(\ln \left(\frac{3 e^x+1}{e^x}\right)\) + C
(c) \(\ln \left(\frac{3 e^x-1}{e^x}\right)\) + C
(d) \(\ln \left(\frac{3 e^x+1}{e^{3 x}}\right)\) + C
Solution:
(c) \(\ln \left(\frac{3 e^x-1}{e^x}\right)\) + C

Question 43.
Evaluate: \(\int_0^1 \ln \left(\frac{1}{x}-1\right)\)dx
(a) 1
(b) 2
(c) 0
(d) -1
Solution:
(c) 0

Question 44.
Evaluate: ∫ex\(\left(\frac{1-\sin x}{1-\cos x}\right)\)dx
(a) -ex cot\(\frac{x}{2}\) + C
(b) ex tan\(\frac{x}{2}\) + C
(c) ex cot\(\frac{x}{2}\) + C
(d) -ex sin\(\frac{x}{2}\) + C
Solution:
(a) -ex cot\(\frac{x}{2}\) + C

Question 45.
Evaluate: \(\int_0^1\)x log(1 + x) dx
(a) \(\frac{1}{2}\)
(b) \(\frac{1}{4}\)
(c) \(\frac{1}{3}\)
(d) \(\frac{2}{3}\)
Solution:
(b) \(\frac{1}{4}\)

Question 46.
What is the integrating factor of the equation y’ + y cot x = cosec x?
(a) cot x
(b) sin x
(c) cos x
(d) cosec x
Solution:
(b) sin x

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Additional Exercise

(B) Very Short Type Questions With Answers

Question 1.
Write the order of the differential equation whose solution is given by
y = (c1 + c2) cos (x + c3) + c4\(e^{x+c_5}\) where c1, c2, c4 and c5 are arbitrary constants.
Solution:
y = (c1 + c2) cos (x + c3) + c4\(e^{x+c_5}\)
y = (c1 + c2) cos (x + c3) + c4\(e^{c_5}\).ex
= A cos(x + c3) + Bex
Where c1 + c2 = A, c4\(e^{c_5}\) = B
As there are 3 independent constants the order of the differential equation is 3.

Question 2.
If p and q are respectively degree and order of the differential equation y = edy/dx, then write the relation between p and q.
Solution:
Given differential equation is
y = \(e^{\frac{d y}{d x}}\) ⇒ \(\frac{d y}{d x}\) = ln y
Whose order = 1 = p
Degree = 1 = q
∴ p = q

Question 3.
Write the value of \(\int_0^1\){x} dx where {x} stands for fractional part of x.
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Additional Exercise Q.3

Question 4.
Write the order of the differential equation of the family of circles
ar2 + ay2 + 2gx + 2fy + c = 0
ax2 + ay2 + 2gx + 2fy + c = 0
Solution:
As there are 3 independent constants, the order of the differential equation is 3.

Question 5.
If p and q are the order and degree of the differential equation
y\(\left(\frac{d y}{d x}\right)^2\) + x2 \(\frac{d^2 y}{d x^2}\) + xy = sin x, then choose the correct statement out of (i) p > q, (ii) p = q, (iii) p < q.
Solution:
Order of the given differential = p = 2
Degree of the given differential equation = q = 1
∴ p > q

Question 6.
Write the order of the differential equation of the system of ellipses:
\(\frac{x^2}{a^2}\) + \(\frac{y^2}{b^2}\) = 1
Solution:
As there are two unknown constants in the system of ellipses \(\frac{x^2}{a^2}\) + \(\frac{y^2}{b^2}\) = 1 the order of the differential equation is 2.

Question 7.
What do you mean by integration? Write your answer in one sentence.
Solution:
Integration is the antiderivative of a function.

Question 8.
Write the differential equation of the family of straight lines parallel to the y-axis.
Solution:
\(\frac{d x}{d y}\) = 0 is the differential equation of family of lines parallel to y-axis.

Question 9.
Write the value of ∫\(\int_{-\pi / 4}^{\pi / 4}\)sin5 x cos x dx.
Solution:
Let f(x) = sin5 x cos x
f(-x) = sin5 (-x) cos (-x)
= -sin5 x cos x = -f(x)
i.e. f is an odd function.
Thus \(\int_{-\pi / 4}^{\pi / 4}\)sin5 x cos x dx = 0

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Additional Exercise

Question 10.
Write the degree of the differential equation ln\(\left(\frac{\mathrm{d}^2 \mathrm{y}}{\mathrm{dx}^2}\right)\) = y
Solution:
The degree of the differential equation ln\(\left(\frac{\mathrm{d}^2 \mathrm{y}}{\mathrm{dx}^2}\right)\) = y is 1.

Question 11.
What is F'(t) if F(t) = \(\int_a^t\)e3x .cos 2x dx ?
Solution:
F(t) = \(\int_a^t\)e3x .cos 2x dx
⇒ F'(t) = e3x cos 2t

Question 12.
Write the order and degree of the following differential equation:
\(\frac{d^2 y}{d x^2}\) = \(\frac{2 y^3+\left(\frac{d y}{d x}\right)^4}{\sqrt{\frac{d^2 y}{d x^2}}}\)
Solution:
Order = 2, Degree = 3

Question 13.
∫\(\frac{\cot x d x}{\ln \sin x}\) = ?
Solution:
∫\(\frac{\cot x d x}{\ln \sin x}\) = ln(ln sin x) + C

Question 14.
What is F'(x) if F(x) = \(\int_0^{\mathbf{x}}\)e2t sin 3t dt?
Solution:
If F(x) = \(\int_0^{\mathbf{x}}\)e2t sin 3t dt then F'(x) = e2x sin 3x

Question 15.
∫\(\frac{d x}{\cos ^2 x \sin ^2 x}\) = ?
Solution:
∫\(\frac{d x}{\cos ^2 x \sin ^2 x}\) = 4∫\(\frac{d x}{\sin ^2 2 x}\)
= 4∫cosec2 2x dx = -2 cot 2x + C

Question 16.
What is the value of ∫\(\frac{d}{d x}\)f(x) dx – \(\frac{d}{d x}\)(∫f(x) dx)?
Solution:
∫\(\frac{d}{d x}\)f(x) dx – \(\frac{d}{d x}\)(∫f(x) dx)
= f(x) + C – f(x) = C (constant)

Question 17.
If \(\int_1^2\)f(x) dx = λ, then what is the value \(\int_1^2\)f(3 – x) dx?
Solution:
If \(\int_1^2\)f(x) dx = λ, then \(\int_1^2\)f(3 – x) dx = λ

Question 18.
What is the value of \(\int_{-1}^1 \frac{d x}{1+x^2}\)?
Solution:
\(\int_{-1}^1 \frac{d x}{1+x^2}\) = \(\left[\tan ^{-1} x\right]_{-1}^1\)
= tan-1 1 – tan-1 (-1)
= tan-1 1 + tan-1 1
= 2tan-1 (1) = 2 . \(\frac{\pi}{4}\) = \(\frac{\pi}{2}\)

Question 19.
Write the order and the degree of the following differential equation:
\(\frac{d^3 y}{d x^3}\) = \(\left(\frac{d^2 y}{d x^2}\right)^2\) + \(\left(\frac{d y}{d x}\right)^4\) + y
Solution:
Order = 3
Degree = 1

Question 20.
Write the particular solution of \(\frac{d y}{d x}\) = (1 + x)4, y = 0 when x = -1.
Solution:
\(\frac{d y}{d x}\) = (1 + x)4 ⇒ \(\frac{(1+x)^5}{5}\) + C
Given y = 0 for x = -1
⇒ o = o + c ⇒ c = o
∴ The particular solution is y = \(\frac{(1+x)^5}{5}\)

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Additional Exercise

(C) Short Type Questions With Answers

Question 1.
Evaluate: ∫\(\frac{2 x+1}{\sqrt{x^2+10 x+29}}\)dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Additional Exercise Q.1

Question 2.
Evaluate: \(\int_0^{\pi / 2} \frac{\cos x d x}{(2-\sin x)(3+\sin x)}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Additional Exercise Q.2

Question 3.
Evaluate: ∫\(\frac{d x}{(1+x) \sqrt{1-x^2}}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Additional Exercise Q(3)

Question 4.
Solve: cosec x \(\frac{d^2 y}{d x^2}\) = x.
Solution:
cosec x \(\frac{d^2 y}{d x^2}\) = x => \(\frac{d^2 y}{d x^2}\) = x sin x
⇒ \(\frac{d y}{d x}\) = ∫x sin x dx + A
= x (-cos x) – ∫(-cos x) dx + A
= -x cos x + sin x + A
⇒ y = -∫x cos x dx + ∫sin x dx + A∫dx + B
= [x sin x – ∫sin x dx] – cos x + Ax = B
⇒ y = -x sin x – 2 cos x + Ax + B is the solution.

Question 5.
Find the particular solution of the following differential equation:
\(\frac{d y}{d x}\) = \(\frac{1+y^2}{1+x^2}\) given that y = √3 when x = 1
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Additional Exercise Q.5
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Additional Exercise

Question 6.
Evaluate: \(\int_0^a x^2\left(a^2-x^2\right)^{5 / 2}\) dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Additional Exercise Q.6

Question 7.
Evaluate: \(\int_0^a \frac{d x}{e^{4 x}-5}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Additional Exercise Q.7

Question 8.
Evaluate: ∫x2 tan-1 x dx.
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Additional Exercise Q.8

Question 9.
If f(x) = ex + \(\frac{1}{1+x^2}\) and f(0) = 1, then find f(x).
Solution:
f(x) = ex + \(\frac{1}{1+x^2}\)
⇒ f(x) = ∫\(\left(e^x+\frac{1}{1+x^2}\right)\)dx + C
= ex + tan-1 x + C
f(0) = 1
⇒ 1 = 1 + 0 + C => C = 0
Thus f(x) = ex + tan-1 x

Question 10.
Evaluate: ∫(log x)2 dx
Solution:
I = ∫(log x)2 dx
= (log x)2. x – 2∫(log x) . \(\frac{1}{x}\) . x . dx
= x (log x)2 – 2 ∫log x. dx
= x (log x)2 – 2 {(log x) x – ∫dx}
= x (log x)2 – 2x log x + 2x + C
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Additional Exercise

Question 11.
Evaluate: ∫\(\frac{2 x+9}{(x+3)^2}\)dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Additional Exercise Q.11

Question 12.
Solve: ydy + e-y x sin x dx = 0
Solution:
ydy = e-y x sin x dx = 0
⇒ y ey dy + x sin x dx = 0
⇒ ∫y ey dy + ∫x sin x dx =C
⇒ y ey – ey + (-x cos x) + sin x = C
⇒ ey (y – 1) – x cos x + sin x = C is the general solution.

Question 13.
Evaluate: ∫\(\frac{d x}{x \ln x \sqrt{(\ln x)^2-4}}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Additional Exercise Q.13

Question 14.
Find the particular solution of the differential equation \(\frac{d^2 y}{d x^2}\) = 6x given that y = 1 and \(\frac{d y}{d x}\) = 2 when x = 0.
Solution:
\(\frac{d^2 y}{d x^2}\) = 6x ⇒ \(\frac{d y}{d x}\) = 6 . \(\frac{x^2}{2}\) + A
\(\frac{d y}{d x}\) = 3x2 + A ⇒ y = x3 + Ax + B
Using the givne conditions x = 0, \(\frac{d y}{d x}\) = 2, y = 1, we get
2 = 0 + A ⇒ A = 2
and 1 = 0 + 0 + B ⇒ B = 1
The particular solution is y = x3 + 2x + 1

Question 15.
Evaluate: \(\int_0^{\frac{3}{2}}\)[x2] dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Additional Exercise Q.15
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Additional Exercise

Question 16.
Find the differential equation whose general solution is ax2 + by = 1, where a and b are arbitrary constants.
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Additional Exercise Q.16

Question 17.
Integrate: ∫\(\frac{\sin 6 x+\sin 4 x}{\cos 6 x+\cos 4 x}\) dx.
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Additional Exercise Q.17

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a)

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

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Exercise 12(a)

Question 1.
Each question given below has four possible answers out of which only one is correct. Choose the correct one.
(i) \(\vec{a}\) = î + 2ĵ + k̂, \(\vec{b}\) = 2î – 2ĵ + 2k̂ and \(\vec{c}\) = -î + 2 ĵ + k̂ then
(a) \(\vec{a}\) and \(\vec{b}\) have the same direction
(b) \(\vec{a}\) and \(\vec{c}\) have opposite directions.
(c) \(\vec{b}\) and \(\vec{c}\) have opposite directions
(d) no pair of vectors have same direction
Solution:
(d) no pair of vectors have same direction

(ii) If the vectors \(\vec{a}\) = 2î + 3ĵ – 6k̂ and \(\vec{b}\) = -α î – ĵ + 2k̂ are parallel, then α = ______.
(a) 2
(b) \(\frac{2}{3}\)
(c) –\(\frac{2}{3}\)
(d) \(\frac{1}{3}\)
Solution:
(c) –\(\frac{2}{3}\)

(iii) If the position vectors of two points A and B are 3î + k̂, and 2î + ĵ – k̂, then the vector \(\overrightarrow{BA}\) is
(a) -î + ĵ – 2k̂
(b) î + ĵ
(c) î – ĵ + 2k̂
(d) î – ĵ – 2k̂
Solution:
(c) î – ĵ + 2k̂

(iv) If \(|k \vec{a}|\) = 1, then
(a) \(\vec{a}=\frac{1}{k}\)
(b) \(\vec{a}=\frac{1}{|k|}\)
(c) \(k=\frac{1}{|\vec{a}|}\)
(d) \(k=\frac{+1}{|\vec{a}|}\)
Solution:
(d) \(k=\frac{+1}{|\vec{a}|}\)

(v) The direction cosines of the vectors \(\overrightarrow{PQ}\) where \(\overrightarrow{OP}\) = (1, 0, -2) and \(\overrightarrow{OQ}\) = (3, -2, 0) are
(a) 2, -2, 2
(b) 4, -2, -2
(c) \(\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\)
(d) \(\frac{2}{\sqrt{6}},-\frac{1}{\sqrt{6}},-\frac{1}{\sqrt{6}}\)
Solution:
(c) \(\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\)

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a)

Question 2.
Rectify the mistakes, if any
(i) \(\vec{a}-\vec{a}\) = 0
Solution:
\(\overrightarrow{0}\)

(ii) The vector \(\overrightarrow{0}\) has unique direction.
Solution:
indefinite direction

(iii) All unit vectors are equal.
Solution:
equal magnitude

(iv) \(|\vec{a}|=|\vec{b}| \Rightarrow \vec{a}=\vec{b}\)
Solution:
\(\vec{a}=\vec{b} \Rightarrow|\vec{a}|=|\vec{b}|\)

(v) Subtraction of vectors is not commutative.
Solution:
true

Question 3.
(i) If \(\vec{a}\) = (2, 1), \(\vec{b}\) = (-1, 0), find \(3 \vec{a}+2 \vec{b}\).
Solution:
\(3 \vec{a}+2 \vec{b}\) = 3 (2, 1) + 2 (-1, 0)
= (6 – 2, 3 + 0)
= (4, 3 )

(ii) If \(\vec{a}\) = (1, 1, 1) , \(\vec{b}\) = (-1, 3, 0) and \(\vec{c}\) =(2, 0, 2), find \(\vec{a}+2 \vec{b}-\frac{1}{2} \vec{c}\).
Solution:
\(\vec{a}+2 \vec{b}-\frac{1}{2} \vec{c}\)
= (1, 1, 1) + 2 (-1, 3, 0) – \(\frac{1}{2}\)(2, 0, 2)
= (1 – 2 – 1, 1 + 6 – 0, 1 + 0 – 1)
= (-2, 7, 0)

Question 4.
If A, B, C and D are the vertices of a square, find \(\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CD}+\overrightarrow{DA}\).
Solution:
Let ABCD be a square.
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.4

Question 5.
The given points A, B, C are the vertices of a triangle. Determine the vectors \(\overrightarrow{A B}, \overrightarrow{B C} \text { and } \overrightarrow{C A}\) and the lengths of these vectors in the following cases.
(i) A (4, 5, 5), B (3, 3, 3), C (1, 2, 5)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.5(1)

(ii) A (8, 6, 1), B (2, 0, 1), C (-4, 0, -5)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.5(2)

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a)

Question 6.
Find the vector from origin to the midpoint of the vector \(\overrightarrow{{P}_1 {P}_2}\) joining the points P1(4, 3) and P2(8, -5).
Solution:
P1 = (4, 3) and P2 = (8, -5)
If P is the mid-point of P1P2 then P = (6, -1).
Position vector of P = \(\overrightarrow{{OP}}\) = 6î – ĵ

Question 7.
Find the vectors from the origin to the points of trisection the vector \(\overrightarrow{{P}_1 {P}_2}\) joining P1 (-4, 3) and P2 (5, -12).
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.7

Question 8.
Find the vector from the origin to the intersection of the medians of the triangle whose vertices are A (5, 2, 1), B(-4, 7, 0) and C (5, -3, 5).
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.8

Question 9.
Prove that the sum of all the vectors drawn from the centre of a regular octagon to its vertices is the null vector.
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.9

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a)

Question 10.
Prove that the sum of the vectors represented by the sides of a closed polygon taken in order is a zero vector.
Solution:
Consider a closed polygon ABCDEFA.
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.10

Question 11.
(a) Prove that:
(i) \(|\overrightarrow{a}+\overrightarrow{{b}}| \leq|\overrightarrow{a}|+|\overrightarrow{b}|\)
State when the equality will hold;
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.11(1)

(ii) \(|\overrightarrow{a}-\overrightarrow{b}| \geq|\overrightarrow{a}|-|\overrightarrow{b}|\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.11(2)

(b) What is the geometrical significance of the relation \(|\overrightarrow{a}+\overrightarrow{b}|=|\overrightarrow{a}-\overrightarrow{b}|\)?
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.11.1

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a)

Question 12.
Find the magnitude of the vector \(\overrightarrow{PQ}\), its scalar components and the component vectors along the coordinate axes, if P and Q have the coordinates.
(i) P (-1, 3), Q (1, 2)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.12(1)

(ii) P (-1, -2), Q (-5, -6)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.12(2)

(iii) P (1, 4, -3), Q (2, -2, -1).
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.12(3)

Question 13.
In each of the following find the vector \(\overrightarrow{PQ}\), its magnitude and direction cosines, if P and Q have co-ordinates.
(i) P (2, -1, -1), Q (-1, -3, 2);
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.13(1)

(ii) P (3, -1, 7), Q (4, -3, -1).
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.13(2)

Question 14.
If \(\vec{a}\) = (2, -2, 1), \(\vec{b}\) = (2, 3, 6) and \(\vec{c}\) = (-1, 0, 2), find the magnitude and direction of
\(\vec{a}-\vec{b}+2 \vec{c}\).
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.14

Question 15.
Determine the unit vector having the direction of the given vector in each of the following problems:
(i) 5î – 12ĵ
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.15(1)

(ii) 2î + ĵ
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.15(2)

(iii) 3î + 6ĵ – k̂
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.15(3)

(iv) 3î + ĵ – 2k̂
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.15(4)

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a)

Question 16.
Find the unit vector in the direction of the vector \(\overrightarrow{r_1}-\overrightarrow{r_2}\), where \(\vec{r}_1\) = î + 2ĵ + k̂ and \(\vec{r}_2\) = 3î + ĵ – 5k̂.
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.16

Question 17.
Find the unit vector parallel to the sum of the vectors \(\vec{a}\) = 2î + 4ĵ – 5k̂ and \(\vec{b}\) = î + 2ĵ + 3k̂. Also find its direction cosines.
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.17

Question 18.
If the sum of two unit vectors is a unit vector, show that the magnitude of their difference is √3.
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.18

Question 19.
The position vectors of the points A, B, C and D are 4î + 3ĵ – k̂, 5î + 2ĵ + 2k̂, 2î – 2ĵ – 3k̂ and 4î – 4ĵ + 3k̂ respectively. Show that AB and CD are parallel.
Solution:
Given that the
position vector of A = 4î + 3ĵ – k̂
position vector of B = 5î + 2ĵ + 2k̂
position vector of C = 2î – 2ĵ – 3k̂
position vector of D = 4î – 4ĵ + 3k̂
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.19

Question 20.
In each of the following problems, show by vector method that the given points are collinear.
(i) A (2, 6, 3), B (1, 2, 7) and C (3, 10, -1)
Solution:
Given that A = (2, 6, 3), B = (1, 2, 7) and C = (3, 10, -1)
Then
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.20(1)

(ii) P (2, -1, 3), Q (3, -5, 1) and R (-1, 11, 9).
Solution:
Given that P = (2, -1, 3) Q = (3, -5, 1) and R = (-1, 11, 9)
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.20(2)
Hence the points P, Q, R are collinear. (Proved)

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a)

Question 21.
Prove that the vectors 2î – ĵ + k̂, î – 3ĵ – 5k̂, 3î – 4ĵ – 4k̂ are the sides of a right angled triangle.
Solution:
Let A, B and C be the points whose position vectors are 2î – ĵ – k̂, î – 3ĵ – 5k̂ and 3î – 4ĵ – 4k̂ respectively.
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.21

Question 22.
Prove by vector method that:
(a) the medians of a triangle are concurrent;
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.22(1)
The symmetry of the result shows that the point G also lies on the other two medians.
Hence the medians are concurrent. (Proved)

(b) the diagonals of a parallelogram bisect each other;
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.22(2)

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a)

(c) the line segment joining the midpoints of two sides of a triangle is parallel to the third and half of it;
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.22(3)

(d) the lines joining the midpoints of consecutive sides of a quadrilateral is a parallelogram;
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.22(4)
⇒ SR = PQ and SR || PQ
Hence PQRS is a parallelogram.
(Proved)

(e) in any triangle ABC, the point P being on the side \(\overrightarrow{B C} \text {; if } \overrightarrow{P Q}\) is the resultant of the vectors \(\overrightarrow{A P}, \overrightarrow{P B}\) and \(\overrightarrow{P C}\) then ABQC is a parallelogram;
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.22(5)
Hence ABQC is parallelogram. (Proved)

(f) In a parallelogram, the line joining a vertex to the midpoint of an opposite side trisects the other diagonal.
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Q.22(6)
⇒ P divides BD into the ratio 1 : 2.
Similarly we can show that Q divides BD into the ratio 2 : 1.
Hence P, Q are the points of trisection of the diagonal BD. (Proved)

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(b)

Odisha State Board Elements of Mathematics Class 12 Solutions CHSE Odisha Chapter 11 Differential Equations Ex 11(b) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Exercise 11(b)

Solve the following differential equations.
Question 1.
\(\frac{d y}{d x}\) + y = e-x
Solution:
Given equation is \(\frac{d y}{d x}\) + y = e-x … (1)
This is a linear differential equation.
Here P = 1, Q = e-x
So the integrating factor
I.F. = e∫P dx = e∫dx = ex
The solution of (1) is given by
yex = ∫e-x . ex dx = ∫dx = x + C
⇒ y – xe-x + Ce-x

Question 2.
(x2 – 1)\(\frac{d y}{d x}\) + 2xy = 1
Solution:
Given equation is (x2 – 1)\(\frac{d y}{d x}\) + 2xy = 1
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(b) Q.2

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(b)

Question 3.
(1 – x2)\(\frac{d y}{d x}\) + 2xy = x \(\sqrt{1-x^2}\)
Solution:
Given equation is
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(b) Q.3

Question 4.
x log x \(\frac{d y}{d x}\) + y = 2 log x
Solution:
Given equation is
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(b) Q.4

Question 5.
(1 + x2)\(\frac{d y}{d x}\) + 2xy = cos x
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(b) Q.5

Question 6.
\(\frac{d y}{d x}\) + y sec x = tan x
Solution:
Given equation is
\(\frac{d y}{d x}\) + y sec x = tan x
This is a linear equation where
P = sec x, Q = tan x
I.F. = e∫sec dx
= e(sec x + tan x) = sec x + tan x
The solution is y . (sec x + tan x)
= ∫(sec x + tan x) tan x dx
= ∫(sec x tan x + tan2 x) dx
= ∫(sec x . tan x + sec2 x – 1) dx
= ∫(sec x + tan x) – x + C
⇒ (y – 1) (sec x + tan x) + x = C

Question 7.
(x + tan y) dy = sin 2y dx
Given equation can be written as
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(b) Q.7

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(b)

Question 8.
(x + 2y3)\(\frac{d y}{d x}\) = y
Solution:
Given equation can be written as
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(b) Q.8

Question 9.
sin x\(\frac{d y}{d x}\)+ 3y = cos x
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(b) Q.9
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(b) Q.9.1

Question 10.
(x + y + 1)\(\frac{d y}{d x}\) = 1
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(b) Q.10

Question 11.
(1 + y2) dx + (x – \(e^{-\tan ^{-1} y}\)) dy = 0
Solution:
Given equation can be written as
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(b) Q.11

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(b)

Question 12.
x\(\frac{d y}{d x}\) + y = xy2
Solution:
Given equation can be written as
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(b) Q.12
⇒ z = -x ln x + Cx
⇒ \(\frac{1}{y}\) = -x ln x + Cx
⇒ 1 = -xy ln x + Cxy
∴ The solution is (C – ln x) xy = 1

Question 13.
\(\frac{d y}{d x}\) + y = y2 log x
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(b) Q.13

Question 14.
(1 + x2)\(\frac{d y}{d x}\) = xy – y2
Solution:
The given equation can be written as
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(b) Q.14
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(b) Q.14.1

Question 15.
\(\frac{d y}{d x}\) + \(\frac{y}{x-1}\) = \(x y^{\frac{1}{2}}\)
Solution:
The given equation can be written as
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(b) Q.15

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(b)

Question 16.
\(\frac{d y}{d x}\) + \(\frac{y}{x}\) = x2, y(1) = 1
Solution:
The given equation can be written as
\(\frac{d y}{d x}\) + \(\frac{y}{x}\) = x2, y(1) = 1 … (1)
This is a linear equation.
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(b) Q.16

Question 17.
\(\frac{d y}{d x}\) + 2y tan x = sin x, y\(\left(\frac{\pi}{3}\right)\) = 0.
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(b) Q.17

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a)

Odisha State Board Elements of Mathematics Class 12 Solutions CHSE Odisha Chapter 11 Differential Equations Ex 11(a) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Exercise 11(a)

Question 1.
Determine the order and degree of each of the following differential equations.
(i) y sec2 x dx + tan x dy = 0
Solution:
Order: 1, Degree: 1

(ii) \(\left(\frac{d y}{d x}\right)^4\) + y5 = \(\frac{d^3 y}{d x^3}\)
Solution:
Order: 3, Degree: 1

(iii) a\(\frac{d^2 y}{d x^2}\) = \(\left\{1+\left(\frac{d y}{d x}\right)^2\right\}^{\frac{3}{2}}\)
Solution:
Order: 2, Degree: 2

(iv) tan-1\(\sqrt{\frac{d y}{d x}}\) = x
Solution:
Order: 1, Degree: 1

(v) ln\(\left(\frac{d^2 y}{d x^2}\right)\) = y
Solution:
Order: 2, Degree: 1

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a)

(vi) \(\frac{\frac{d y}{d t}}{y+\frac{d y}{d t}}\) = \(\frac{y t}{d y}\)
Solution:
Order: 1, Degree: 2

(vii) \(\frac{d^2 y}{d u^2}\) = \(\frac{3 y+\frac{d y}{d u}}{\sqrt{\frac{d^2 y}{d u^2}}}\)
Solution:
Order: 2, Degree: 3

(viii) \(e^{\frac{d z}{d x}}\) = x2
Solution:
Order: 1, Degree: 1

Question 2.
Form the differential equation by eliminating the arbitrary constants in each of the following cases.
(i) y = A sec x
Solution:
y = A sec x
Then \(\frac{d y}{d x}\) = A sec x tan x = y tan x

(ii) y = C tan-1 x
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.2(2)

(iii) y = Aet + Be2t
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.2(3)

(iv) y = Ax2 + Bx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.2(4)

(v) y = -acos x + b sin x
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.2(5)

(vi) y = a sin-1 x + b cos-1 x
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.2(6)

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a)

(vii) y = at + bet
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.2(7)

(viii) y = a sin t + bet
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.2(8)

(ix) ax2 + by = 1
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.2(9)

Question 3.
Find the general solution ofthe following differential equations.
(i) \(\frac{d y}{d x}\) = \(\frac{e^{2 x}+1}{e^x}\)
Solution:
\(\frac{d y}{d x}\) = \(\frac{e^{2 x}+1}{e^x}\)
⇒ y = ∫(ex + e-x) dx = ex – e-x + C

(ii) \(\frac{d y}{d x}\) = x cos x
Solution:
\(\frac{d y}{d x}\) = x cos x
⇒ y = ∫x cos x dx
= x . sin x – ∫sin x dx – x sin x + cos x + C

(iii) \(\frac{d y}{d x}\) = t5 log t
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.3(3)

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a)

(iv) \(\frac{d y}{d x}\) = 3t2 + 4t + sec2 t
Solution:
\(\frac{d y}{d x}\) = 3t2 + 4t + sec2 t
⇒ y = t3 + 2t2 + tan t + C

(v) \(\frac{d y}{d x}\) = \(\frac{1}{x^2-7 x+12}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.3(5)

(vi) \(\frac{d y}{d u}\) = \(\frac{u+1}{\sqrt{3 u^2+6 u+5}}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.3(6)

(vii) (x2 + 3x + 2) dy – dx = 0
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.3(7)

(viii) \(\frac{d y}{d t}\) = \(\frac{\sin ^{-1} t e^{\sin ^{-1} t}}{\sqrt{1-t^2}}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.3(8)

Question 4.
Solve the following differential equations.
(i) \(\frac{d y}{d x}\) = y + 2
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.4(1)

(ii) \(\frac{d y}{d t}\) = \(\sqrt{1-y^2}\)
Solution:
\(\frac{d y}{d t}\) = \(\sqrt{1-y^2}\)
⇒ \(\frac{d y}{\sqrt{1-y^2}}\) = dt
⇒ sin-1 y = t + C

(iii) \(\frac{d y}{d z}\) = sec y
Solution:
\(\frac{d y}{d z}\) = sec y
⇒ cos y dy = dz
⇒ sin y = z + C

(iv) \(\frac{d y}{d x}\) = ey
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.4(4)

(v) \(\frac{d y}{d x}\) = y2 + 2y
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.4(5)

(vi) dy + (y2 + 1) dx = 0
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.4(6)

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a)

(vii) \(\frac{d y}{d x}\) + \(\frac{e^y}{y}\) = 0
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.4(7)

(viii) dx + cot x dt = 0
Solution:
dx + cot x dt = 0
⇒ tan x dx + dt = 0
⇒ ∫tan x dx + ∫dt = C1
⇒ In sec x + t = C1
⇒ In sec x = C1 – t
⇒ sec x = \(e^{C_1}\) . e-t
⇒ cos x = \(e^{-C_1}\) . et
⇒ cos x = Cet where C = \(e^{-C_1}\)

Question 5.
Obtain the general solution of the following differential equations.
(i) \(\frac{d y}{d x}\) = (x2 + 1) (y2 + 1)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.5(1)

(ii) \(\frac{d y}{d t}\) = e2t+3y
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.5(2)
⇒ 2e-3y + 3e2t + 6C1 = 0
⇒ 2e-3y + 3e2t = C
where C = -6C1

(iii) \(\frac{d y}{d z}\) = \(\frac{\sqrt{1-y^2}}{\sqrt{1-z^2}}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.5(3)

(iv) \(\frac{d y}{d z}\) = \(\frac{x \log x}{3 y^2+4 y}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.5(4)

(v) x2\(\sqrt{y^2+3}\) dx + y\(\sqrt{x^3+1}\) dy = 0
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.5(5)

(vi) tan y dx + cot x dy = 0
Solution:
tan y dx + cot x dy = 0
⇒ tan x . dx + cot y dy = 0
⇒ ∫tan x dx + ∫cot y dy = 0
⇒ -ln cos x + ln siny = ln C
⇒ ln\(\frac{\sin y}{\cos x}\) = ln C
⇒ \(\frac{\sin y}{\cos x}\) = C
⇒ sin y = C cos x

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a)

(vii) (x2 + 7x + 12) dy + (y2 – 6y + 5) dx = 0
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.5(7)

(viii) y dy + e-y x sin x dx = 0
Solution:
y dy + e-y x sin x dx = 0
⇒ yey dy + x sin x dx = 0
⇒ ∫yey dy + ∫x sin dx = C
[Integrating by parts.
⇒ yey – ∫ey dy + x(-cos x) – ∫(-cos x) dx = C
⇒ yey – ey – x cos x + sin x = C
⇒ (y – 1) ey – x cos x + sin x = C

Question 6.
Solve the following second order equations.
(i) \(\frac{d^2 y}{d x^2}\) = 12x2 + 2x
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.6(1)

(ii) \(\frac{d^2 y}{d t^2}\) =e2t +e-t
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.6(2)

(iii) \(\frac{d^2 y}{d \vartheta^2}\) = -sin υ + cos υ + sec2 υ
Solution:
\(\frac{d^2 y}{d \vartheta^2}\) = -sin υ + cos υ + sec2 υ
Integrating we get
\(\frac{d y}{d υ}\) = ∫sin υ dυ + ∫cos υ dυ + ∫sec2 υ dυ
= cos υ + sin υ + tan υ + A
Again integratingwe get
y = ∫(cos υ + sin υ + tan υ + A)dυ + B
where A, B are arbritrary constants.
⇒ y = sin υ – cos υ + ln |sec υ| + A.υ. + B

(iv) cosec x \(\frac{d^2 y}{d x^2}\) = x
Solution:
cosec x \(\frac{d^2 y}{d x^2}\) = x
\(\frac{d^2 y}{d x^2}\) = x sin x
Integrating we get
\(\frac{d y}{d x}\) = ∫x sin x dx + A
= x . (-cos x) – ∫(-cos x) dx + A
= -x cos x + ∫cos x dx + A
= -x cos x + sin x + A
Again integrating we get
y = -∫x cos x dx + ∫sin x + ∫A dx + B
= -{x sin x -∫1 . sin x dx} – cos x + Ax + B
= -x sin x – 2cos x + Ax + B

(v) x2\(\frac{d^2 y}{d x^2}\) + 2 = 0
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.6(5)

(vi) sec x \(\frac{d^2 y}{d x^2}\) = sec 3x
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.6(6)

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a)

(vii) \(\frac{d^2 y}{d x^2}\) = sec2 x + cos2 x
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.6(7)

(viii) e-x\(\frac{d^2 y}{d x^2}\) = x
Solution:
ex\(\frac{d^2 y}{d x^2}\) = x
⇒ \(\frac{d^2 y}{d x^2}\) = xex
Integrating we get
\(\frac{d y}{d x}\) = ∫xex dx = ∫ex dx + Ax + B
= xex – ex – ex + Ax + B
= (x – 2)ex + Ax + B

Question 7.
Find the particular solutions of the following equations subject to the given conditions.
(i) \(\frac{d y}{d x}\) = cos x, given that y = 2 when x = 0.
Solution:
\(\frac{d y}{d x}\) = cos x
Integrating we get
y = ∫cos x dx = sin x + C
Given that when x = 0, y = 2
So 2 = C
∴ The particular solution is y = sin x + 2

(ii) \(\frac{d y}{d t}\) = cos2 y subject to y = \(\frac{\pi}{4}\) when t = 0.
Solution:
\(\frac{d y}{d t}\) = cos2 y
⇒ sec2 y dy = dt
∫sec2 dy = ∫dt
⇒ tan y = t + C
When t = 0, y = \(\frac{\pi}{4}\)
So tan \(\frac{\pi}{4}\) = C ⇒ C = 1
∴ The particular solution is tan y = t + 1

(iii) \(\frac{d y}{d x}\) = \(\frac{1+y^2}{1+x^2}\) given that y = √3 when x = 1.
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.7(3)

(iv) \(\frac{d^2 y}{d x^2}\) = 6x given that y = 1 and \(\frac{d y}{d x}\) = 2 when x = 0.
Solution:
\(\frac{d^2 y}{d x^2}\) = 6x ⇒ \(\frac{d y}{d x}\) = 3x2 + 2
When x = 0, \(\frac{d y}{d x}\) = 2
So 2 = A
∴ \(\frac{d y}{d x}\) = 3x2 + 2
Again integrating we get
y = x3 + 2x + B
When x = 0, y = 1
So B = 1.
∴ The particular solution is y = x3 + 2x + 1

Question 8.
(i) Solve : \(\frac{d y}{d x}\) = sec (x + y)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.8(1)

(ii) Solve : \(\frac{d y}{d x}\) = sin(x + y) + cos(x + y)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.8(2)

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a)

(iii) Solve : \(\frac{d y}{d x}\) = cos (x + y)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.8(3)

(iv) Solve : \(\frac{d y}{d x}\) + 1 = ex+y
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(a) Q.8(4)

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(d)

Odisha State Board CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(d) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Exercise 12(d)

Question 1.
Each question given below has four possible answers out of which only one is correct. Choose the correct one.
(i) \(\vec{a} \cdot \vec{b} \times \vec{a}\) = _______.
(a) \(\overrightarrow{0}\)
(b) 0
(c) 1
(d) \(\vec{a}^2 \vec{b}\)
Solution:
\(\vec{a} \cdot(\vec{b} \times \vec{a})\) = \((\vec{b} \times \vec{a}) \cdot \vec{a}\)
= \(\vec{b} \cdot(\vec{a} \times \vec{a})\) = \(\vec{b} \cdot \overrightarrow{0}\)
= 0 [∴ Dot product is commutative and in the scalar triple product the dot and cross can be interchanged.]

(ii) \((-\vec{a}) \cdot \vec{b} \times(-\vec{c}))\) = _______.
(a) \(\vec{a} \times \vec{b} \cdot \vec{c}\)
(b) \(-\vec{a} \cdot(\vec{b} \times \vec{c})\)
(c) \(\vec{a} \times \vec{c} \cdot \vec{b}\)
(d) \(\vec{a} \cdot(\vec{c} \times \vec{b})\)
Solution:
\((-\vec{a}) \cdot \vec{b} \times(-\vec{c})\) = \(\vec{a} \cdot(\vec{b} \times \vec{c})\)

(iii) For the non-zero vectors \(\vec{a}, \vec{b}\) and \(\vec{c}, \vec{a} \cdot(\vec{b} \times \vec{c})\) = 0 if
(a) \(\vec{b} \perp \vec{c}\)
(b) \(\vec{a} \perp \vec{b}\)
(c) \(\vec{a} \| \vec{c}\)
(d) \(\vec{a} \perp \vec{c}\)
Solution:
\(\vec{a} \cdot(\vec{b} \times \vec{c})\) = \((\vec{a} \times \vec{b}) \cdot \vec{c}\)
\(\vec{c} \perp(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}})\)
but \(\vec{a} \times \vec{b}\) is perpendicular to \(\vec{a}\) and \(\vec{b}\)
∴ \(\vec{a} \| \vec{b}\)

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(d)

Question 2.
Find the scalar triple product \(\vec{b} \cdot(\vec{c} \times \vec{a})\) where \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) are respectively.
(i) î + ĵ, î – ĵ, 5î + 2ĵ + 3k̂
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(d) Q.2(1)
= 1 (0 – 3) + 1 (0 – 3) + 0 (5 – 2)
= 3 – 3 = -6

(ii) 5î – ĵ + 4k̂, 2î + 3ĵ + 5k̂, 5î – 2ĵ
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(d) Q.2(2)
= 5 (18 + 10) + 1 (12 – 25) + 4 (- 4 – 15)
= 140 – 13 – 76 = 140 – 89 = 51

Question 3.
Find the volume of the parallelopiped whose sides are given by the vectors.
(i) î + ĵ + k̂, k̂, 3î – ĵ + 2k̂
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(d) Q.3(1)
= 1 (0 + 1) – 1 (0 – 3) + 1 (0 – 0)
= 1 + 3 = 4 cube units.

(ii) (1, 0, 0), (0, 1, 0), (0, 0, 1).
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(d) Q.3(2)

Question 4.
Show that the following vector are co-planar
(i) î – 2ĵ + 2k̂, 3î + 4ĵ + 5k̂, -2î + 4ĵ – 4k̂
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(d) Q.4(1)

(ii) î + 2ĵ + 3k̂, -2î – 4ĵ + 5k̂, 3î + 6
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(d) Q.4(2)

Question 5.
Find the value of λ so that the three vectors are co-planar.
(i) î + 2ĵ + 3k̂, 4î + ĵ + λk̂ and λî – 4ĵ + k̂
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(d) Q.5(1)

(ii) (2, -1, 1), (1, 2, -3) and (3, λ, 5)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(d) Q.5(2)
⇒ 2 (10 + 3λ) + 1 (5 + 9) + 1 (λ – 6) = 0
⇒ 20 + 6λ +14 + λ – 6 = 0
⇒ 7λ + 28 = 0 ⇒ λ = -4

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(d)

Question 6.
If \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) mutually perpendiculars, show that \([\vec{a} .(\vec{b} \times \vec{c})]^2\) = a2b2c2
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(d) Q.6

Question 7.
Show that \([\vec{a}+\vec{b} \vec{b}+\vec{c} \vec{c}+\vec{a}]\) = 2\([\vec{a} \vec{b} \vec{c}]\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(d) Q.7

Question 8.
Prove that \([\vec{a} \times \vec{b} \vec{b} \times \vec{c} \vec{c} \times \vec{a}]\) = \([\vec{a} \vec{b} \vec{c}]^2\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(d) Q.8

Question 9.
For \(\vec{a}\) = î + ĵ, \(\vec{b}\) = -î + 2k̂, \(\vec{c}\) = ĵ + k̂ obtain \(\vec{a} \times(\vec{b} \times \vec{c})\) and also verify the formula \(\vec{a} \times(\vec{b} \times \vec{c})\) = \((\vec{a} \cdot \vec{c}) \vec{b}-(\vec{a} \cdot \vec{b}) \vec{c}\).
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(d) Q.9

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(d)

Question 10.
Prove that \(\vec{a} \times(\vec{b} \times \vec{c})+\vec{b} \times(\vec{c} \times \vec{a})+\vec{c} \times(\vec{a} \times \vec{b})\) and hence prove that \(\vec{a} \times(\vec{b} \times \vec{c}), \vec{b} \times(\vec{c} \times \vec{a}), \vec{c} \times(\vec{a} \times \vec{b})\) are coplanar.
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(d) Q.10

Question 11.
If \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) unit vectors and \(\hat{a} \times(\hat{b} \times \hat{c})=\frac{1}{2} \hat{b}\) find the angles that â makes with b̂ and ĉ, where b̂, ĉ are not parallel.
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(d) Q.11

CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise

Odisha State Board Elements of Mathematics Class 12 Solutions CHSE Odisha Chapter 9 Integration Additional Exercise Textbook Exercise questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise

Question 1.
∫\(\sqrt{1-\sin 2 x}\) dx
Solution:
I = ∫\(\sqrt{1-\sin 2 x}\) dx
= ∫\(\sqrt{(\cos x-\sin x)^2}\) dx
= ∫(cos x – sin x) dx
= sin x + cos x + c

Question 2.
∫\(\frac{d x}{1+\sin x}\)
Solution:
I = ∫\(\frac{d x}{1+\sin x}\)
= ∫\(\frac{1-\sin x}{\cos ^2 x}\)
= ∫sec2 x – sec x tan x dx
= tan x – sec x + c

CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise

Question 3.
∫\(\frac{\sin x}{1+\sin x}\) dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.3

Question 4.
∫\(\frac{\sec x}{\sec x+\tan x}\) dx
Solution:
I = ∫\(\frac{\sec x}{\sec x+\tan x}\) dx
= ∫\(\frac{\sec x(\sec x-\tan x)}{\sec ^2 x-\tan ^2 x}\) dx
= ∫sec2 x – sec x tan x dx
= tan x – sec x + c

Question 5.
∫\(\frac{1+\sin x}{1-\sin x}\) dx
Solution:
I = ∫\(\frac{1+\sin x}{1-\sin x}\) dx
= ∫\(\frac{(1+\sin x)^2}{\cos ^2 x}\) dx
= ∫[sec2 x+ tan2 x+ 2sec x tan x) dx
= ∫[2sec2 x – 1 + 2sec x tan x) dx
= 2tan x – x + 2sec x + c

Question 6.
∫tan-1 (sec x + tan x) dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.6

Question 7.
∫\(\frac{\cos 2 x-\cos 2 \alpha}{\cos x-\cos \alpha}\) dx
Solution:
I = ∫\(\frac{\cos 2 x-\cos 2 \alpha}{\cos x-\cos \alpha}\) dx
= ∫\(\frac{\left(2 \cos ^2 x-1\right)-\left(2 \cos ^2 \alpha-1\right)}{\cos x-\cos \alpha}\) dx
= 2 ∫(cos x + cos α) dx
= 2 sin x + 2x cos α + c

Question 8.
∫tan-1\(\sqrt{\frac{1-\cos 2 x}{1+\cos 2 x}}\) dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.8

CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise

Question 9.
∫\(\frac{d x}{\sqrt{x+1+} \sqrt{x+2}}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.9

Question 10.
∫\(\frac{2+3 x}{3-2 x}\) dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.10

Question 11.
∫\(\frac{d x}{\sqrt{x}+x}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.11

Question 12.
∫\(\frac{d x}{1+\tan x}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.12

Question 13.
∫\(\frac{x+\sqrt{x+1}}{x+2}\) dx (Hints put : \(\sqrt{x+1}\) = t)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.13

Question 14.
∫sin-1\(\sqrt{\frac{x}{a+x}}\) dx (Hints put : x = a tan2 t)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.14

CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise

Question 15.
∫ex\(\left(\frac{2+\sin 2 x}{1+\cos 2 x}\right)\) dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.15

Question 16.
∫\(\frac{\left(x^2+1\right) e^x}{(x+1)^2}\) dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.16

Question 17.
∫\(\frac{x^2-1}{x^4+x^2+1}\) dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.17

Question 18.
∫\(\frac{x^2 d x}{x^4+x^2+1}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.18

Question 19.
∫\(\sqrt{\cot x}\) dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.19
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.19.1

CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise

Question 20.
∫\((\sqrt{\tan x}+\sqrt{\cot x})\) dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.20

Question 21.
∫\(\frac{\mathrm{dx}}{x\left(x^4+1\right)}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.21

Question 22.
∫\(\frac{\mathrm{dx}}{e^x-1}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.22

Question 23.
∫\(\frac{(x-1)(x-2)(x-3)}{(x+4)(x-5)(x-6)}\) dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.23

Question 24.
∫\(\frac{d x}{\left(e^x-1\right)^2}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.24

Question 25.
∫\(\frac{d x}{\sin x \cos ^2 x}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.25

Question 26.
\(\int_2^4 \frac{\left(x^2+x\right) d x}{\sqrt{2 x+1}}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.26

CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise

Question 27.
\(\int_{-a}^a \sqrt{\frac{a-x}{a+x}}\) dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.27
Let a2 – x2 = t2
-2x dx = 2t dt
x = -a ⇒ 0 t = 0
x = a ⇒ t = 0
= 0
I = aI1 – I2 = aπ

Question 28.
\(\int_0^{\pi / 2}(\sqrt{\tan x}+\sqrt{\cot x})\) dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.28

Question 29.
\(\int_0^{\pi / 2} \frac{\cos x d x}{1+\cos x+\sin x}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.29

Question 30.
\(\int_0^1\)x (1 – x)n dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.30

Question 31.
\(\int_0^{\pi / 2}\)sin 2x log (tan x) dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.31

CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise

Question 32.
\(\int_0^{\pi / 2} \frac{\sin ^2 x d x}{\sin x+\cos x}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.32
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.32.1

Question 33.
\(\int_0^{\pi / 2} \frac{\sin ^2 x d x}{1+\sin x \cos x}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.33
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.33.1

Question 34.
\(\int_0^{\pi / 2} \frac{x d x}{\sin x+\cos x}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.34

Question 35.
Prove that \(\int_0^\pi\) x sin3 x dx = \(\frac{2 \pi}{3}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.35

CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise

Question 36.
\(\int_{\pi / 5}^{3 \pi / 10} \frac{\sin x d x}{\sin x+\cos x}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.36

Question 37.
\(\int_0^\pi\)|cos x| dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.37

Question 38.
\(\int_1^4\)(|x – 1| + |x – 2| + |x – 3|) dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.38

Question 39.
\(\int_{-\pi / 2}^{\pi / 2}\)(sin |x| + cos |x|) dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.39

Question 40.
\(\int_0^\pi\)log (1 + cos x) dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise Q.40

CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(l)

Odisha State Board Elements of Mathematics Class 12 Solutions CHSE Odisha Chapter 9 Integration Ex 9(l) Textbook Exercise questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Exercise 9(l)

Question 1.
\(\int_0^{\frac{\pi}{2}}\)sin10 θ dθ
Solution:
\(\int_0^{\frac{\pi}{2}}\)sin10 θ dθ = \(\frac{9}{10} \cdot \frac{7}{8} \cdot \frac{5}{6} \cdot \frac{3}{4} \cdot \frac{1}{2} \cdot \frac{\pi}{2}\) = \(\frac{405 \pi}{7680}\)

Question 2.
\(\int_0^{\frac{\pi}{2}}\)cos12 θ dθ
Solution:
\(\int_0^{\frac{\pi}{2}}\)cos12 θ dθ = \(\frac{11}{12} \cdot \frac{9}{10} \cdot \frac{7}{8} \cdot \frac{5}{6} \cdot \frac{3}{4} \cdot \frac{1}{2} \cdot \frac{\pi}{2}\) = \(\frac{4455 \pi}{92160}\)

Question 3.
\(\int_0^{\frac{\pi}{2}}\)sin11 θ dθ
Solution:
\(\int_0^{\frac{\pi}{2}}\)sin11 θ dθ = \(\frac{10}{11} \cdot \frac{8}{9} \cdot \frac{6}{7} \cdot \frac{4}{5} \cdot \frac{2}{3}\) = \(\frac{3840}{4455}\)

CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(l)

Question 4.
\(\int_0^{\frac{\pi}{2}}\)cos9 θ dθ
Solution:
\(\int_0^{\frac{\pi}{2}}\)cos9 θ dθ = \(\frac{8}{9} \cdot \frac{6}{7} \cdot \frac{4}{5} \cdot \frac{2}{3}\) = \(\frac{384}{405}\)

Question 5.
\(\int_0^1 \frac{x^7}{\sqrt{1-x^2}}\) dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(l) Q.5

Question 6.
\(\int_0^1 \frac{x^5\left(4-x^2\right)}{\sqrt{1-x^2}}\) dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(l) Q.6

Question 7.
\(\int_0^a x^3\left(a^2-x^2\right)^{\frac{5}{2}}\) dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(l) Q.7

Question 8.
\(\int_0^1 x^5 \sqrt{\frac{1+x^2}{1-x^2}}\) dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(l) Q.8

CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(l)

Question 9.
\(\int_0^{\infty} \frac{x^2}{\left(1+x^6\right)^n}\) dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(l) Q.9

Question 10.
\(\int_0^\pi\)sin8 θ dθ
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(l) Q.10

CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k)

Odisha State Board Elements of Mathematics Class 12 Solutions CHSE Odisha Chapter 9 Integration Ex 9(k) Textbook Exercise questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Exercise 9(k)

Evaluate the following Integrals:
Question 1.
(i) \(\int_0^{\frac{\pi}{2}} \frac{d x}{1+\tan x}\)dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.1(1)

(ii) \(\int_0^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}\)dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.1(2)

CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k)

(iii) \(\int_0^1 \frac{\ln (1+x)}{2+x^2}\)dx (x = tan θ)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.1(3)

(iv) \(\int_0^\pi \frac{x d x}{1+\sin x}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.1(4)

Question 2.
(i) \(\int_{-a}^a\)x4 dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.2(1)

(ii) \(\int_{-a}^a\)(x5 + 2x2 + x) dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.2(2)

CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k)

(iii) \(\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\)cos2 x dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.2(3)

(iv) \(\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}}\)sin5 x dx
Solution:
Let f(x) = sin5 x
Then f(-x) = sin5 (-x)
= -sin5 x = -f(x)
So f(x) is an odd function.
Thus \(\int_{-a}^a\)f(x) dx = 0
\(\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}}\)sin5 x dx = 0

Question 3.
(i) \(\int_0^\pi\)cos3 x dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.3(1)

(ii) \(\int_0^\pi\)cos2 x dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.3(2)

(iii) \(\int_0^\pi\)sin3 x cos x dx
Solution:
\(\int_0^\pi\)sin3 x cos x dx
[Put sin x = t, then cos x dx = dt
When x = 0, t = 0, when x = π, t = 0
\(\int_0^\pi\)t3 dt = 0

(iv) \(\int_0^\pi\)sin x cos2 x dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.3(4)

CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k)

Question 4.
Show that
(i) \(\int_0^1 \frac{\ln x}{\sqrt{1-x^2}}\) dx = \(\frac{\pi}{2} \ln \frac{1}{2}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.4(1)

(ii) \(\int_0^{\frac{\pi}{2}} \frac{\cos x-\sin x}{1+\sin x \cos x}\) dx = 0
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.4(2)

(iii) \(\int_0^\pi\)x ln sin x dx = \(\frac{\pi^2}{2} \ln \frac{1}{2}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.4(3)

Question 5.
(i) \(\int_0^{\pi / 2}\)ln (tan x + cot x) dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.5(1)

CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k)

(ii) \(\int_0^\pi \frac{x \tan x-\sin x}{1+\sin x \cos x}\) dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.5(2)

(iii) \(\int_1^3 \frac{\sqrt{x} d x}{\sqrt{4-x}+\sqrt{x}}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.5(3)

(iv) \(\int_0^\pi \frac{x \sin x d x}{1+\cos ^2 x}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.5(4)

(v) \(\int_0^1\)x (1 – x)100 dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.5(5)

CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k)

(vi) \(\int_{\pi / 6}^{\pi / 3} \frac{d x}{1+\sqrt{\cot x}}\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.5(6)

(vii) \(\int_0^{50}\)ex-[x] dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.5(7)

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c)

Odisha State Board Elements of Mathematics Class 12 Solutions CHSE Odisha Chapter 11 Differential Equations Ex 11(c) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Exercise 11(c)

Find the solutions of the following differential equations:
Question 1.
(x + y) dy + (x – y) dx = 0
Solution:
Given equation can be written as
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.1

Question 2.
\(\frac{d y}{d x}\) = \(\frac{1}{2}\left(\frac{y}{x}+\frac{y^2}{x^2}\right)\)
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.2

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c)

Question 3.
(x2 – y2) dx + 2xy dy = 0
Solution:
Given equation can be written as
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.3

Question 4.
x\(\frac{d y}{d x}\) + \(\sqrt{x^2+y^2}\) = y
Solution:
Given equation can be written as
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.4

Question 5.
x (x + y) dy = (x2 + y2) dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.5
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.5.1
This is the required solution.

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c)

Question 6.
y2 + x2 \(\frac{d y}{d x}\) = xy \(\frac{d y}{d x}\)
Solution:
Given equation can be written as
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.6
This is the required solution.

Question 7.
x sin\(\frac{y}{x}\) dy = \(\left(y \sin \frac{y}{x}-x\right)\)dx
Solution:
Given equation can be written as
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.7

Question 8.
x dy – y dx= \(\sqrt{x^2+y^2}\) dx
Solution:
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.8
This is the required solution.

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c)

Question 9.
\(\frac{d y}{d x}\) = \(\frac{y-x+1}{y+x+5}\)
Solution:
Given equation is
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.9
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.9.1
This is the required solution.

Question 10.
(x – y) dy = (x + y + 1) dx
Solution:
Given equation can be written as
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.10
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.10.1
This is the required solution.

Question 11.
(x – y – 2) dx + (x – 2y – 3) dy = 0
Solution:
Given equation can be written as
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.11
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.11.1
This is the required solution.

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c)

Question 12.
\(\frac{d y}{d x}\) = \(\frac{3 x-7 y+7}{3 y-7 x-3}\)
Solution:
Given equation can be written as
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.12
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.12.1
This is the required solution.

Question 13.
(2x + y + 1) dx + (4x + 2y – 1) dy = 0
Solution:
Given equation can be written as
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.13
⇒ 2z + ln (z – 1) = 3x + C
⇒ 2 (2x + y) + ln (2x + y – 1 ) = 3x + C
⇒ (x + 2y) + ln (2x + y – 1 ) = C
This is the required solution.

Question 14.
(2x + 3y – 5)\(\frac{d y}{d x}\) + 3x + 2y – 5 = 0
Solution:
Given equation can be written as
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.14
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.14.1

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c)

Question 15.
(4x + 6y + 5) dx – (2x + 3y + 4) dy = 0
Solution:
Given equation can be written as
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.15
CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.15.1
This is the required solution.