Class 12

Odisha State Board CHSE Odisha Class 12 Math Notes Chapter 9 Integration will enable students to study smartly.

CHSE Odisha 12th Class Math Notes Chapter 9 Integration

Indefinite integral:
If \(\frac{d}{d x}\)F(x) = f(x) then the indefinite integral of f(x) w.r.t x is
∫f(x)dx = F(x) + C
which represents the entire class of anti-derivatives.

(a) Algebra of integrals:
(i) ∫[f(x) ± g(x)] dx = ∫f(x) dx ± ∫g(x) dx
(ii) ∫af(x) dx = α ∫f(x) dx
(iii) ∫f(x) g(x) dx = f(x) ∫g(x) dx – ∫\(\left[(\frac{d}{d x} f(x)\right) \cdot \int g(x) d x]\) dx (Integration by parts)

(b) Some standard indefinite integrations:
(1) ∫x_n dx = \(\frac{x^{n+1}}{n+1}\) + C, n ≠ (-1)
(2) ∫\(\frac{d x}{x}\) = log_e|x| + C.
(3) ∫sin x dx = -cos x + C
(4) ∫cos x dx = sin x + C
(5) ∫tan x dx = -log |cos x| + C or log |(sec x)| + C
(6) ∫cot x dx = log |(sin x)| + C or -log |(cosec x)| + C
(7) ∫sec x dx = log |sec x + tan x| + C
(8) ∫cosec x dx = log |cosec x – cot x| + C
(9) ∫sec² x dx = tan x +C
(10) ∫cosec² x dx = -cot x + C
(11) ∫sec x tan x dx = sec x + C
(12) ∫cosec x cot x dx = -cosec x + C

[Note: To integrate by parts choose 1st function according to I LATE]
Where I → Inverse trigonometric functions.
L → Logarithmic function
A → Algebraic function
T → Trigonometric function
E → Exponential function

(c) Techniques of integration:

Definite integration:

Odisha State Board CHSE Odisha Class 12 Math Notes Chapter 13 Three Dimensional Geometry will enable students to study smartly.

CHSE Odisha 12th Class Math Notes Chapter 13 Three Dimensional Geometry

Important Formulae:
Distance Formula:
The distance between two points (x₁, y₁, z₁) and (x₂, y₂, z₂)
= \(\sqrt{\left(x_1-x_1\right)^2+\left(y_2-y_1\right)^2+\left(z_2-z_1\right)^2}\)

Division Formula:
(i) Internal division:
If P (x, y, z) divides the line segment joining, A (x₁, y₁, z₁) and B (x₂, y₂, z₂) into the ratio m : n internally then

Remark:
(i) If P (x, y, z) divides the line segment joining the points A (x₁, y₁, z₁) and B (x₂, y₂, z₂) into the ratio λ : 1 then

(ii) Co-ordinates of the mid-point of the line segment joining the points (x₁, y₁, z₁) and (x₂, y₂, z₂) are
\(\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}\right)\)

Direction Cosines:
Suppose that a straight line makes angles α, β, γ with the positive directions of x-axis, y-axis and z-axis respectively.
Then direction cosines of the line are < cos α, cos β, cos γ >
We denote l = cos α, m = cos β and n = cos γ
Then l² + m² + n² = 1

Direction Ratios:
The direction ratios of a straight line are proportional to direction cosines.
If < a, b, c > are d. rs. and < l, m, n > are d.cs then

Direction ratios of a line segment joining the points (x₁, y₁, z₁) and (x₂, y₂, z₂) are
< x₂ – x₁, y₂ – y₁, z₂ – z₁ >

The projection of a line segment joining the points A (x₁, y₁, z₁) and B (x₂, y₂, z₂) onto the line ‘L’ with d.cs. < l, m, n >
= l (x₂ – x₁) + m (y₂ – y₁) + n (z₂ – z₁)

Angle between two lines:
Angle between two lines with d.cs.
< l₁, m₁, n₁ > and < l₂, m₂, n₂ > is given by cos θ = l₁l₂ + m₁m₂ + n₁n₂
(i) Two lines are parallel if their d.cs. are equal or d.r.s. are proportional.
(ii) Two lines are perpendicular if l₁l₂ + m₁m₂ + n₁n₂ = 0

Plane
Important formulae:
1. The general equation of the plane is ax + by + cz + d = 0
2. Equation of the plane passing through a poing (x₁, y₁, z₁) and having l, m, n direction cosines of the normal to the plane is l (x – x₁) + m (y – y₁) + n (z – z₁) = 0
3. Equation of the plane in intercept form is \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\) = 1
where a, b, c are the intercepts from the axes.
4. Equation of the plane in normal form is lx + my + nz = p
where < l, m, n > are d.cs of the normal and p is the length of the normal.
5. Equation of the plane passing through three points.
(x₁, y₁, z₁), (x₂, y₂, z₂) and (x₃, y₃, z₃) is
\(\left|\begin{array}{rrr}
x-x_1 & y-y_1 & z-z_1 \\
x_2-x_1 & y_2-y_1 & z_2-z_1 \\
x_3-x_1 & y_3-y_1 & z_3-z_1
\end{array}\right|\) = 0

6. (i) Angle between two planes is the angle between their normals.
(ii) If two planes are
a₁x + b₁y + c₁z + d₁ = 0 and
a₂x + b₂y + c₂z + d₂ = 0
then the direction ratios of their normal are < a₁, b₁, c₁ > and < a₂, b₂, c₂ >
(iii) If θ is the angle between two planes then
cos θ = \(\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \cdot \sqrt{a_2^2+b_2^2+c_2^2}}\)
(iv) Two planes a₁x + b₁y + c₁z + d₁ = 0 and a₂x + b₂y + c₂z + d₂ = 0 are parallel if \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\).
(v) The above two planes are perpendicular if a₁a₂ + b₁b₂ + c₁c₂ = 0.

7. The distance of a point (x₁, y₁, z₁) from a plane ax + by + cz + d = 0 is
\(\left|\frac{a x_1+b y_1+c z_1+d}{\sqrt{a^2+b^2+c^2}}\right|\)

8. Equations of the planes bisecting the angle between two planes
a₁x + b₁y + c₁z + d₁ = 0 and
a₂x + b₂y + c₂z + d₂ = 0 are
\(\frac{a_1 x+b_1 y+c_1 z+d_1}{\sqrt{a_1^2+b_1^2+c_1^2}}=\pm \frac{a_2 x+b_2 y+c_2 z+d_2}{\sqrt{a_2^2+b_2^2+c_2^2}}\)

The Straight Line
Important formulae:
1. Unsymmetrical From:
The joint equation of two planes represent a stright line. Thus the equation of a straight line in unsymmetrical form a₁x + b₁y + c₁z + d₁ = 0 and a₂x + b₂y + c₂z + d₂ = 0

2. Symmetrical Form:
Equation of a straight line through a point (x₀, y₀, z₀) and having d.c. < l, m, n > is
\(\frac{x-x_0}{l}=\frac{y-y_0}{m}=\frac{z-z_0}{n}\)

3. Two-point Form:
The equation of a straight line passing through two points (x₁, y₁, z₁) and (x₂, y₂, z₂) is
\(\frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1}=\frac{z-z_1}{z_2-z_1}\)

4. Condition that a line will lie on a Plane:
The straight line
\(\frac{x-x_0}{l}=\frac{y-y_0}{m}=\frac{z-z_0}{n}\) lie in a plane ax + by + cz + d = 0
if (i) al + bm + cn = 0
and (ii) ax₀ + by₀ + cz₀ + d = 0

5. Condition for Two Lines to be Coplanar:

6. Angle between a line and a plane:
The angle between the line

7. Distance of a Point from a Line:
The distance of a point (x₁, y₁, z₁) from a line

Odisha State Board CHSE Odisha Class 12 Math Notes Chapter 4 Matrices will enable students to study smartly.

CHSE Odisha 12th Class Math Notes Chapter 4 Matrices

It is a system of mn numbers arranged in a rectangular system of m rows and n columns.
Example : \(\left(\begin{array}{lll}
a_{11} & a_{12} & a_{1 n} \\
a_{21} & a_{22} & a_{2 n} \\
a_{m_1} & a_{m_2} & a_{m_n}
\end{array}\right)\) is a m × n matrix.

Note : We write the above matrix in short as [a_ij]_m×n or [aij]_m×n or ||a_ij||_m×n.

Important types matrices:
(a) Square matrix:
It is a matrix where the number of rows are equal to the number of columns.

(b) Null (zero) matrix:
If all the entries of a matrix are zero, then the matrix is a zero matrix denoted by 0_m×n.

(c) Diagonal matrix:
It is a square matrix in which all the elements except those in main (or leading) diagonal are zero.

(d) Unit matrix:
It is a diagonal matrix where all the elements in the leading (main) diagonal are one.

(e) Scalar matrix:
It is a diagonal matrix with all the elements in the leading (main) diagonal are α (α ≠ 0 or 1).

(f) Singular matix:
A square matrix ‘A’ is singular iff |A| = 0, otherwise it is a non singular matrix.

(g) Symmetric matrix:
A square matrix is symmetric if A = A’ and skew-symmetric if A = -A’.

Matrix algebra:
(a) Addition and subtraction:
If A = [a_ij]_m×n and B = [b_ij]_m×n then A ± B = [a_ij ± b_ij]_m×n

(b) Scalar multiplication:
If A = [a_ij]_m×n then for any scalar ‘k’
kA = [ka_ij]_m×n

(c) Matrix multiplication:

Properties:
1. Matrix addition is commutative as well as associative.
2. 0_m×n is the additive identity.
3. A is the zero additive inverse of A.
4. We can add or subtract matrices if they are of same order.
5. We can multiply two matrices if the number of columns of 1st is equal to the number of rows of 2nd.
6. Matrix multiplication is non commutative but associative.
7. Matrix multiplication is distributive over addition.
8. AB = 0 ≠ A = 0 or B = 0 for two matrices A and B. Also AB = AC ≠ B = C.
9. If A is a square matrix of order n then
I_n =  is the multiplicative identity.

Transpose and adjoint of a matrix:
(a) The transpose of a matrix A = [a_ij]_m×n is A^T or A’ = [a_ji]_n×m.
Properties:
(i) (A’)’ = A
(ii) (A ± B)’ = A’ ± B’
(iii) (AB)’ = B’A’
(iv) (KA)’ = KA’
(v) Any matrix A can be expressed as sum of a symmetric and a skew-symmetric matrix as

(b) Adjoint of a matrix:
If A is a square matrix then Adj A = The transpose of the matrix of co-factors

Properties:
1. (Adj A) A = A (Adj A) = |A|I_n
∴ A^-1 = \(\frac{{Adj} A}{|~A|}\)
2. |Adj A| = A^n-1
3. Adj (AdjA) = |A|^n-2A
4. (AdjA)’ = Adj (A’)
5. (AB)^-1 = B^-1A^-1

Solution of system of linear equations (matrix method):
Let the system of linear equations is
a₁x + b₁y + c₁z = d₁
a₂x + b₂y + c₂z = d₂
a₃x + b₃y + c₃z = d₃
Let

Note: If the system is homogeneous i.e. d₁ = d₂ = d₃ = 0, then the system has a trivial solution x = 0, y = 0, z = 0 for |A| ≠ 0. In case |A| = 0 then the system has infinitely many solutions.

Odisha State Board CHSE Odisha Class 12 Math Notes Chapter 3 Linear Programming will enable students to study smartly.

CHSE Odisha 12th Class Math Notes Chapter 3 Linear Programming

Definition:
A general linear programming problem LPP is to obtain x₁, x₂, x₃ …… , x_n so as to

Optimize:
Z = c₁x₁ + c₂x₂ + c₃x₃ + ……. + c_nx_n … (A)
subject to
a₁₁x₁ + a₁₂x₂ + ……. + a_1nx_n  ≤ (or ≥) b₁
a₂₁x₁ + a₂₂x₂ + ……. + a_2nx_n  ≤ (or ≥) b₂ … (B)
where x₁, x₂, …… , x_n ≥ … 0 … (C)
and a_ij, b_i, c_j with i = 1, 2, … , m; j = 1, 2, … ,n are real constants.

In the LPP given above, the function Z in (A) is called the objective function. The variables x₁ x₃, ……, x_n are called decision variables. The constants c₁ c₂, ……, c_n are called cost coefficients. The inequalities in (B) are called constraints. The restrictions in (C) are called non-negative restrictions. The solutions which satisfy all the constraints in (B) and the non-negative restrictions in (C) are called feasible solutions.

The LPP involves three basic elements:

1. Decision variables whose values we seek to determine,
2. Objective (goal) that we aim to optimize,
3. Constraints and non-negative restrictions that the variables need to satisfy.

Types of Linear Programming Problems

As we have already discussed we come across different types of problems which we need solve depending on the objective functions and the constraints. Here we discuss a few important types of LPPs. before learning how to formulate them.

(i) Manufacturing Problem
A manufacturer produces different items so as to maximise his profit. He has to determine the number of units of products he must produce while satisfying a number of constraints because each unit of product requires availability of some amount of raw material, certain manpower, certain machine hours etc.

(ii) Diet Problem
Suppose a person is advised to take vitamins/nutrients of two or more types. The vitamins/nutrients are available in different proportions in different types of foods. If the person has to take a minimum amount of the vitamins/nutrients then the problem is to determine appropriate quantity of food of each type so that cost of food is kept at the minimum.

(iii) Allocation Problem
In this type of problem one has to allocate different resources/tasks to different units/persons depending on the nature of the gain or outcome.

(iv) Transportation Problem
These problems involve transporting materials from sources to destinations for sale or distribution of products or collection of raw materials etc. Here the aim is to use various options for transportation such as distance, time etc so as to keep the cost of transportation to a minimum.

Working procedure to solve LPP graphically
Step-1. Taking all inequations of the constraints as equations, draw lines represented by each equation and considering the inequalities of the constraint inequations complete the feasible region.
Step-2. Determine the vertices of the feasible region either by inspection or by and solving the two equations of the intersecting lines.
Step-3. Evaluate the objective function Z = ax + by at each vertex.
Case (i) F.R. is bounded: The vertex which gives the optional value (maximum or minimum) of Z gives the desired optional solution to the LPP.
Case (ii) F.R. is unbounded: When M is the maximum value of Z at a vertex Vmax, determine the open half plane corresponding to the inequation ax + by > M. If this open half plane has no points in common with the F.R. then M is the maximum value of Z and the point Vmax gives the desired solution. Otherwise, Z has no maximum value.

Similarly consider the open half plane ax+by < m when m is the minimum value of Z at the vertex Vmin. If this half-plane has no point common with the F.R. then m is the minimum value of Z and Vmin gives the desired solution. Otherwise Z has no minimum value.

Odisha State Board CHSE Odisha Class 12 Math Notes Chapter 10 Area Under Plane Curves will enable students to study smartly.

CHSE Odisha 12th Class Math Notes Chapter 10 Area Under Plane Curves

Area of the portion bounded by x-axis the curve
y = f(x) and two ordinates at x = a and x = b.

= Area of the portion bounded by y-axis,
the curve x = f(y) and two abscissa at y = c and y = d.

Area between two curves y = f(x), y g(x) with g(x) < f(x) in [a, b] and between two ordinates
x = a and x = b is given by \(\int_a^b\){f(x) – g(x)}dx

Odisha State Board CHSE Odisha Class 12 Math Notes Chapter 7 Continuity and Differentiability will enable students to study smartly.

CHSE Odisha 12th Class Math Notes Chapter 7 Continuity and Differentiability

Definition:
A function f is said to be continuous at a point a D_f if
(i) f(x) has definite value f(a) at x = a,
(ii) lim_x->a f(x) exists,
(iii) lim_x->a f(x) = f(a).
If one or more of the above conditions fail, the function f is said to be discontinuous at x = a. The  above definition of continuity of a function at a point can also be formulated as follows:
A function f is said to be continuous at x = a if
(i) holds and for a given ∈ > 0, there exists a δ > 0 depending on ∈ such that
|x – a| < 8 ⇒ |f(x) – f(a)| < ∈.
A function f is continuous on an interval if it is continuous at every point of the interval.
If the interval is a closed interval [a, b] the function f is continuous on [a, b] if it is continuous on (a, b),
lim_x->a+ f(x) = f(a) and lim_x->b- f(x) = f(b).

Differentiation of a function:
(a) Differential coefficient (or derivative) of a function y = f(x) with respect to x is

Fundamental theorems:

Then to get \(\frac{d y}{d x}\) it is convenient to take log of both sides before differentiation.

Derivative of some functions:

Higher order derivative:

Leibnitz Theorem:
If ‘u’ and ‘v’ are differentiable functions having ‘n’th derivative then
\(\frac{d^n}{d x^n}\)(u.v) = C₀u_nv + C₁u_n-1v₁ + C₂u_n-2v₂ + ….. + C_nuv_n

Partial derivatives and Homogeneous functions:
(a) If z = f(x, y) is any function of two variables then the partial derivative of z w.r.t. x and y are given below.

(b) Homogeneous function:
z = f(x > y) is a homogeneous function of degree ‘n’ if f(tx, ty) = tⁿ f(x, y).

Euler’s Theorem:
If z = f(x, y) is a homogeneous function of degree ‘n’ then \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}\) = nf(x, y).

Odisha State Board CHSE Odisha Class 12 Math Notes Chapter 6 Probability will enable students to study smartly.

CHSE Odisha 12th Class Math Notes Chapter 6 Probability

Important terms:
(a) Random experiment: It is an experiment whose results are unpredictable.
(b) Sample space: It is the set of all possible outcomes of an experiment. We denote the sample space by ‘S’.
(c) Sample point: Each element of a sample space is a sample point.
(d) Event: Any subset of a sample space is an event.
(e) Simple event: It is an event with a single sample point.
(f) Compound event: Compound events are the events containing more than one sample point.
(g) Mutually exclusive events: Two events A and B are mutually exclusive if A ∩ B = φ (i.e. occurrence of one excludes the occurrence of other)
(h) Mutually exhaustive events: The events A₁, A₂, A₃ ……. A_n are mutually exhaustive if A₁ ∪ A₂ ∪ A₃ ∪ A_n = S.
(i) Mutually exclusive and exhaustive events.
The events A₁, A₂, A₃ ……. A_n are mutually exclusive and exhaustive if
(i) A₁ ∩ A₂ = φ for i ≠ j
(ii) A₁ ∪ A₂ ∪ …… A_n = S.
(j) Equally likely events: Two events are equally likely if they have equal chance of occurrence.
(k) Impossible and certain events: φ is the impossible and S is the sure or certain event.
(i) Independent events: The events are said to be independent if the occurrence or non-occurrence of one does not affect the occurrence of non-occurrence of other.

Probability of an event:
Let S be the sample space and A is an event then the probability of A is
\(P(A)=\frac{|A|}{|S|}=\frac{\text { No.of out comes favourable to } A}{\text { Total number of possible outcomes. }}\)
Note:
1. P(φ) = 0
2. P(S) = 1
3. P(A’) = P (not A) = 1 – P(A)
4. P(A) + P(A’) = 1

Odds in favour and odds against an event:
Let in an experiment is the number of cases favourable to A and ‘n’ is the number of cases not in favour of A then

Addition theorem:
If A and B are any two events then
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
Note:
If A and B are mutually exclusive then
P(A ∪ B) = P(A) + P(B)
(∴ P(A ∩ B) = P(φ) – 0)

Conditional probability:
Let A and B are any two events and P(B) ≠ 0 then the conditional probability of A when B has already happened
P(A/B) = \( \frac{P(A \cap B)}{P(B)} \)
Note:
1. P(A ∩ B) = P(B) . P(A/B)
2. If A and B are mutually independent events then P(A/B) = P(A).
∴ P(A ∩ B) = P(A) . P(B)
3. If A and B are independent events then (i) A’ and B’ (ii) A’ and B (iii) A and B’ are also independent.
4. P(A₁ ∩ A₂ ∩ A₃ ….. ∩ A_n) = P(A₁) . P(A₂/A₁) . P(A₃/A₂ ∩ A₁) ….. P(A_n/A₁ ∩ A₂ ∩ …. ∩ A_n-1)
5. Let A₁, A₂ ….. A_n are mutually exhaustive and exclusive events and A is any event which occurs with A₁ or A₂ or A₃ … or A_n then
P(A) = P(A₁) . P(A/A₁) + P(A₂) . P(A/ A₂) + …….. + P(A_n) . P(A/A_n).
This is called the total conditional probability theorem.

Baye’s theorem:
If A₁, A₂ …… A_n are mutually exclusive and exhaustive events and A is any event which occurs with A₁ or A₂ or A₃ or …. A_n then
\( P\left(A_i / B\right)=\frac{P\left(A_i\right) \cdot P\left(A / A_i\right)}{\sum_{i=1}^n P\left(A_i\right) P\left(A / A_i\right)} \)

Odisha State Board CHSE Odisha Class 12 Math Notes Chapter 2 Inverse Trigonometric Functions will enable students to study smartly.

CHSE Odisha 12th Class Math Notes Chapter 2 Inverse Trigonometric Functions

Definitions:
Let us first consider the function
Sin : R → [-1, 1]
Let y = Sin x, x ∈ R. Look at the graph of sin x. For y ∈ [-1, 1], there is a unique number x in each of the intervals…, \(\left[-\frac{3 \pi}{2},-\frac{\pi}{2}\right],\left[-\frac{\pi}{2}, \frac{\pi}{2}\right],\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right], \ldots\) such that y = sin x.

Hence any one of these intervals can be chosen to make sine function bijective.

We usually choose \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) as the domain of sine function. Thus sin: \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) → [-1, 1] is bijective and hence admits of an inverse function with range \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) denoted by sin^-1 or arcsin (see footnote).

Each of the above-mentioned intervals as range gives rise to different branches of sin^-1 function. The function sin^-1 with the range \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) is called the principal branch which is defined below.

sin^-1 : [-1, 1] → \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) defined by y = sin^-1 x ⇔ x = sin y.

The values of y (=sin^-1 x) in \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) are called principal values of sin^-1.

Similar considerations for other trigonometric functions give rise to respective inverse functions. We define below the principal branches of cos^-1, tan^-1 and cot^-1.

cos^-1 : [-1, 1] → [0, π] defined by
y = cos^-1 x ⇔ x = cos y
tan^-1 : R → \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) defined by
y = tan^-1 x ⇔ x = tan y
cot^-1 : R → (0, π) defined by
y = cot^-1 x ⇔ x = cot y

Important Properties

Property – I
We know that when
f : X → Y is invertible then fof^-1 = I_y and f^-1of = I_x.
Applying this we have.
(i) sin (sin^-1 x) = x, x ∈ [-1, 1]
cos (cos^-1 x) = x, x ∈ [-1, 1]
tan (tan^-1 x) = x, x ∈ R
cot (cot^-1 x) = x, x ∈ R
sec (sec^-1 x) = x, x ∈ R -(-1, 1)
cosec (cosec^-1 x) = x, x ∈ R -(-1, 1)

Property – II
(i) sin^-1(-x) = -sin^-1 x, x ∈ [-1, 1]
(ii) cosec^-1(-x) = -cosec^-1 x , |x| ≥ 1
(iii) tan^-1(-x) = -tan^-1 x, x ∈ R

Odisha State Board CHSE Odisha Class 12 Math Notes Chapter 1 Relation and Function will enable students to study smartly.

CHSE Odisha 12th Class Math Notes Chapter 1 Relation and Function

Ordered Pair
It is a pair of numbers of functions listed in a specific order, eg. (a, b) is an ordered pair.

Cartesian Product:
Let A and B are two non empty sets. The cartesian product of A and B = A × B = {(a, b) : a ∈ A, b ∈ B}.

Relation
A relation R from A to B is a subset of A × B i.e. R ⊆ A × B.

Relation on a set:
R is a relation on A if R ⊆ A × A.

Domain, Range and Co-domain of a relation:
Let R is a relation from A to B.
Dom R = {x ∈ A: (x, y) ∈ R for y ∈ B }
Rng R = {y ∈ B : (x, y) ∈ R for x ∈ A}
Co-domain of R = B

Types of Relation
(a) Empty or void relation:
As Φ ⊂ A × B we have Φ is a relation known as empty or void relation.

(b) Universal Relation:
As A × B ⊆ A × B, we have A × B is a relation, known as universal relation.

(c) Identity Relation:
A relation R on A is an identity relation of (a, a) ∈ R, For a ∈ A.

(d) Reflexive Relation:
A relation R on A is reflexive if (a, a) ∈ R for all a ∈ A.

(e) Symmetric Relation:
A relation R on A is symmetric if (a, b) ∈ R ⇒ (b, a) ∈ R, where a, b ∈ A.

(f) Transitive Relation:
A relation R on A is transitive if (a, b), (b, c) ∈ R ⇒ (a, c) ∈ R where a, b, c ∈ A.

(g) Anti Symmetric Relation:
A relation R on A is anti-symmetric if (a, b), (b, a) ∈ R ⇒ a = b.

(h) Equivalence Relation:
A relation R on A is an equivalence relation if it is reflexive, symmetric as well as Transitive.

(i) Partial ordering:
A relation R on A is a partial ordering if it is reflexive, transitive and antisymmetric.

(j) Total ordering:
A relation R on A is a total ordering. If it is a partial ordering and either (a, b) or (b, a) ∈ R for a, b ∈ A.

Equivalence Class:
Let R is an equivalence relation on X for any x ∈ X the equivalence class of x.
= [x] = {y ∈ X : (x, y) ∈ R}
(a) For all x ∈ X, [x] ≠ Φ
(b) If (x, y) ∈ R, then [x] = [y]
(c) If (x, y) ∉ R then [x] ∩ [y] = Φ
(d) An equivalence relation partitions a set into disjoint equivalence classes.

Function
Function as a Rule:
Let A and B are two non empty sets. If each element of A is mapped to a unique element of B by rule ‘f’ then f is a function.
Function as a relation
A relation f from A to B is a function if
(i) Dom f = A
(ii) (x, y), (x, z) ∈ f ⇒ y = z

Domain, Co-domain and Range of a function

Domain of a Real Function:
Let f : A → B, defined as y = f(x)
Dom f = {x ∈ A: = f(x) for y ∈ B}

Range of a function:
Let f : A → B, defined as y = f(x)
Rng f = {y ∈ B : y = f(x) for all x ∈ A}

Co-Domain of a function:
Let f : A → B defined as y = f(x)
Co dom f = B
Note: Rng f ⊆ Co dom f

Types of Functions

Injective (one-one) function:
f : A→ B is one one if f(x₁) = f(x₂) ⇒ x₁ = x₂
Step-1
Methods of check Injective function:
Consider any two arbitrary x₁, x₂ ∈ A.
Step-2
Put f(x₁) = f(x₂) and simplify.
Step-3
If we get x₁ = x₂ then f is one-one, otherwise f is many one.

Surjective (onto) function:
A function f : A → B is onto if Rng f = Co-domain f = B.

Methods of Check for onto
Method-1:
Find Range of f
If Rng f = B then f is onto.

Method-2:
Step-1
Consider any arbitrary y ∈ B
Step-2
Write y = f(x) and simplify to express x in terms of y.
Step-3
If x ∈ A, then find f(x)
Step-4
If f(x) = y, then f is onto
In case x ∉ A or f(x) ≠ y then f is not onto, it is into function.

Bijective function (one-one and onto):
f : A → B is a bijective function if its both one-one and onto.

Composition of functions:
Let f : X → Y and g : Y → Z, the composition of f and g denoted by gof, is defined as
gof : X → Z defined by
gof(x) = g(f (x)) for all x ∈ X.

Note:

1. gof is defined when Rngf ⊆ Dom g.
2. Dom gof = Dom f.
3. As gof(x) = g(f(x)), first f rule is applied then g rule is applied.
4. gof ≠ fog
5. If f : R → R and g : R→ R then gof and fog exist.
6. ho(gof) = (hog)of
7. If f : X → Y and g : Y → Z are one-one, then gof : X → Z is also one-one.
8. If f : X → Y and g : Y → Z are onto then go f : X → Z is also onto.
9. Let f : X → Y and g : Y → Z then
   (i) gof : X → Z is onto ⇒ g is onto.
   (ii) gof is one-one ⇒ f is one-one.
   (iii) gof is on to and g is one-one then f is onto.
   (iv) gof is one-one and f is onto ⇒ g is one-one.

Invertible functions:
A function f : X → Y is invertible if there exists a function g : Y → X, such that gof = idx and fog = idy.
The function g is called the inverse of f denoted by g = f^-1.

Note:

1. Not all functions are invertible.
2. Only bijective functions are invertible.
3. Inverse of a function may not be a function.
4. Dom f^-1 = Y
5. \(\left(f^{-1}\right)^{-1}\)
6. (gof)^-1 = f^-1og^-1

Methods to check invertibility of a function and to find f^-1:

Method-1:
Step-1
Write f : X→ Y, defined as y = f(x) = an expression in x.
Step-2
Take any arbitrary y ∈ Y. and write y = f(x).
Step-3
Express x in terms of y and check that x ∈ X.
Step-4
Define g : Y → X as x = g(y) = Value of x in terms of y.
Step-5
find gof (x) and fog of (y).
Step-6
If gof (x) = x and fog (y) = y then f is invertible with f^-1.

Method-2:
Step-1
Show that ‘f’ is one-one.
Step-2
Show that ‘f’ is on to.
Step-3
Either from step-2 or otherwise, write x in terms of y and write f^-1 : Y → X defined as f^-1(y) = x (in terms of y).

Even and odd function:
A function f is even if f(-x) and odd if f(-x) = -f(x).

Note:

1. fi(x) + f(-x) is always even.
2. f(x) – f(-x) is always odd.
3. Every function can be expressed as the sum of one even and one odd function as
   f(x) = \(\frac{f(x)+f(-x)}{2}+\frac{f(x)-f(-x)}{2}\)

Binary operations:
Let A is a nonempty set. Then a binary operation * on a set A is a function *:
A × A → A

Note :

1. Closure property: An operation * on a non-empty set A is a said to satisfy closure property if for every a, b ∈ A
   ⇒ a * b ∈ A.
2. If * is a binary operation on a non-empty set A, then must satisfy the closure property.
3. Number of binary operations on A where |A| = n is \(n^{n^2}\).

Properties of a Binary operation
Let * is a binary operation on A.
1. Commutative Law:
* is commutative if for all a, b ∈ A. a * b = b * a

2. Associative Law:
* is a associative if for all a, b, c ∈ A.
(a * b) * c = a * (b * c)

3. Distributive Law:
Let ‘*’ and ‘a’ are two binary operations on A.
‘*’ is said to be distributive over ‘o’ if for all a, b, ∈ A.
a * (b o c) = (a * b) o (a * c)

4. Existence of identity element:
e ∈ A is said to be the identity element for the binary operation if for all a ∈ A.
a * e = e * a = a

5. Existence of inverse of an element:
a^-1 ∈ A is inverse of a ∈ A if a * a^-1 = a^-1 * a = e, where e is the identity.

Operation or composition table for a binary operation.
Let * is any operation on a finite set A = {a₁, a₂ …… a_n}
The table containing the results of the operation * is known as the operation table.
We can study different properties of binary operation * from the table.

Operation table for * on A:

Study of properties from operation table:
1 . If all the entries of the table belong to A, then A is a binary operation.
2. If each row coincides with the corresponding column, then * is commutative.
3. If elements of a row are identical to the top row, then the leftmost element of that row is the identity element.
4. Mark the position of identity elements in the table. The leading elements in the corresponding row and column of that cell are inverse of each other.

Odisha State Board Elements of Mathematics Class 12 CHSE Odisha Solutions Chapter 13 Three Dimensional Geometry Additional Exercise Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 13 Three Dimensional Geometry Additional Exercise

Question 1.
Find the equation in vector and Cartesian form of the plane passing through the point (3, -3, 1) and normal to the line joining the points (3, 4, -1) and (2, -1, 5)
Solution:

Changing the vector equation to cartesian form we have
(xi + yj + zk) . (-i – 5j + 6k) = 18
⇒ -x – 5y + 6z = 18
⇒ x + 5y – 6z + 18 = 0

Question 2.
Find the vector equation of the plane whose Cartesian form of equation is 3x – 4y + 2z = 5
Solution:
The cartesian equation is 3x – 4y + 2z = 5
The vector equation is \(\vec{r}\). (3i – 4j + 2k) = 5

Question 3.
Show that the normals to the planes \(\vec{r}\) . (î – ĵ + k̂) = 3 and \(\vec{r}\) . (3î + 2ĵ – k̂) = 0 are perpendicular to each other.
Solution:

Question 4.
Find the angle between the planes \(\vec{r} \cdot(2 \hat{i}-\hat{j}+2 \hat{k})=6 \text { and } \vec{r} \cdot(3 \hat{i}+6 \hat{j}-2 \hat{k})=9\).
Solution:

Question 5.
Find the angle between the line \(\vec{r}=(\hat{i}+2 \hat{j}-\hat{k})+\lambda(\hat{i}-\hat{j}+\hat{k})\) and the \(\vec{r} \cdot(2 \hat{i}-\hat{j}+\hat{k})\) = 4.
Solution:

Question 6.
Prove that the acute angle between the lines whose direction cosines are given by the relations l + m + n = 0 and l² + m² – n² = 0 is \(\frac{\pi}{3}\).
Solution:
l + m + n = 0 ⇒ n = -(l + m)
l² + m² – n² = 0
⇒ l² + m² – (l + m)² = 0
⇒ 2lm = 0
Now l = 0 ⇒ m + n = 0 ⇒ m = -n

Question 7.
Prove that the three lines drawn from origin with direction cosines l₁, m₁, n₁ ; l₂, m₂, n₂ ; l₃, m₃, n₃ are coplanar if \(\left|\begin{array}{lll}
l_1 & m_1 & n_1 \\
l_2 & m_2 & n_2 \\
l_3 & m_3 & n_3
\end{array}\right|\) = 0.
Solution:

Question 8.
Prove that three lines drawn from origin with direction cosines proportional to (1, -1, 1), (2, -3, 0), (1, 0, 3) lie on one plane.
Solution:

= 1 (-9) + 1 (6) + 1(3) = 0
∴ The lines are co-planar.

Question 9.
Determine k so that the lines joining the points P₁ (k, 1, -1) and P₂ (2k, 0, 2) shall be perpendicular to the line from P₂ to P₃ (2 + 2k, k, 1).
Solution:

Question 10.
Find the angle between the lines whose direction ratios are proportional to a, b, c and b-c, c-a, a-b.
Solution:

Question 11.
O is the origin and A is the point (a, b, c). Find the equation of the plane through A at right angles to \(\overrightarrow{OA}\).
Solution:

Question 12.
Find the equation of the plane through (6, 3, 1) and (8, -5, 3) parallel to x-axis.
Solution:
Given points are A (6, 3, 1) and B (8, -5, 3) D.C.S of x-axis are < 1, 0, 0 >
Let P (x, y, z) is any point on the plane.

⇒ 2 (y – 3) + 8 (z – 1) = 0
⇒ 2y – 6 + 8z – 8 = 0
⇒ y + 4z – 7 = 0

(A) Multiple Choice Questions (Mcqs) With Answers

Question 1.
Write the value of k such that the line \(\frac{x-4}{1}=\frac{y-2}{1}=\frac{z-k}{2}\) lies on the plane 2x – 4y + z = 7.
(a) k = 7
(b) k = 2
(c) k = 5
(d) k = 3
Solution:
(a) k = 7

Question 2.
If \((\vec{a} \times \vec{b})^2+(\vec{a} \cdot \vec{b})^2\) = 144, write the value of ab.
(a) 4
(b) 12
(c) 3
(d) 24
Solution:
(b) 12

Question 3.
If the vectors \(\vec{a}, \vec{b}\text { and }\vec{c}\) from the sides \(\overline{BC}, \overline{CA} \text { and } \overline{AB}\) respectively of a triangle ABC, then write the value of \(\vec{a} \times \vec{c}+\vec{b} \times \vec{c}\).
(a) 1
(b) -1
(c) 2
(d) 0
Solution:
(d) 0

Question 4.
If \(|\vec{a}|=3|\vec{b}|=2 \text { and } \vec{a} \cdot \vec{b}=0\), then write the value of \(|\vec{a} \times \vec{b}|\).
(a) 6
(b) 0
(c) 2
(d) 3
Solution:
(a) 6

Question 5.
If \(|\vec{a} \times \vec{b}|^2+|\vec{a} \cdot \vec{b}|^2\) = 144 and \(|\vec{a}|\) = 4 then \(|\vec{b}|\) is equal to:
(a) 3
(b) 8
(c) 12
(d) 16
Solution:
(a) 3

Question 6.
Write down the equation to the plane perpendicular to the y-axis at the point (0, -2, 0).
(a) y – 2 = 0
(b) y + 2 = 0
(c) y = 0
(d) 2y + 1 = 0
Solution:
(b) y + 2 = 0

Question 7.
How many straight lines in space through the origin are equally inclined to the coordinate axes?
(a) 2
(b) 1
(c) 0
(d) -1
Solution:
(a) 2

Question 8.
Write the value of a if the vectors \(\vec{a}=2 \hat{i}+3 \hat{j}-6 \hat{k} \text { and } \vec{b}=\alpha \hat{i}-\hat{j}+2 \hat{k}\) are parallel.
(a) \(\frac{2}{3}\)
(b) \(\frac{3}{2}\)
(c) \(\frac{-3}{2}\)
(d) \(\frac{-2}{3}\)
Solution:
(d) \(\frac{-2}{3}\)

Question 9.
Write the equation of the line passing through the point (4, -6, 1) and parallel to the line \(\frac{x-1}{1}=\frac{y+2}{3}=\frac{z-1}{-1}\).
(a) \(\frac{x+4}{1}=\frac{y+6}{3}=\frac{z+1}{-1}\)
(b) \(\frac{x-4}{1}=\frac{y+6}{3}=\frac{z-1}{-1}\)
(c) \(\frac{x-4}{1}=\frac{y-6}{3}=\frac{z-1}{-2}\)
(d) \(\frac{x+4}{-1}=\frac{y+6}{3}=\frac{z-1}{-1}\)
Solution:
(b) \(\frac{x-4}{1}=\frac{y+6}{3}=\frac{z-1}{-1}\)

Question 10.
What is the image of the point (-2, 3, -5) with respect to the zx-plane?
(a) (-2, -3, -5)
(b) (2, -3, -5)
(c) (2, 3, 5)
(d) (2, 3, -5)
Solution:
(a) (-2, -3, -5)

Question 11.
If \(|\vec{a} \times \vec{b}|^2=k|\vec{a} \cdot \vec{b}|^2+|\vec{a}|^2|\vec{b}|^2\), then what is the value of k?
(a) 1
(b) 2
(c) 0
(d) -1
Solution:
(d) -1

Question 12.
Write the value of m and n for which the vectors (m – 1) î + (n + 2) ĵ + 4k̂ and (m + 1) î + (n – 2) ĵ + 8k̂ will be parallel.
(a) m = 3 and n = 6
(b) m = -3 and n = 6
(c) m = -3 and n = -6
(d) m = 3 and n = -6
Solution:
(d) m = 3 and n = -6

Question 13.
For what value of λ the vectors λî + 3ĵ + λk̂ and λî – 2ĵ +k̂ are perpendicular to each other.
(a) (-2, -3)
(b) (2, 3)
(c) (2, -3)
(d) (-2, 3)
Solution:
(c) (2, -3)

Question 14.
If |x| = 1, |y| = 2 and |z| = 3, then how many points in R³ are there having coordinates (x, y, z)?
(a) 8
(b) 2
(c) 6
(d) 4
Solution:
(a) 8

Question 15.
Write the equation of the plane passing through the point (1, -2, 3) and perpendicular to the y-axis.
(a) y – 2 = 0
(b) y = -2
(c) y + 2 = 0
(d) 2y – 2 = 0
Solution:
(c) y + 2 = 0

Question 16.
What is the projection of î + ĵ – k̂ upon the vector î?
(a) 0
(b) 1
(c) 2
(d) None of the above
Solution:
(b) 1

Question 17.
If \(\vec{a}=2 \hat{i}+\hat{j}, \vec{b}=\hat{k}\) what is \(\vec{a} \cdot \vec{b}\)?
(a) 0
(b) 1
(c) 2
(d) None of the above
Solution:
(a) 0

Question 18.
What is the work done by a force \(\overrightarrow{\mathrm{F}}\) = 4i + 2j + 3k in displacing a particle from A (1, 2, 0) to B (2, -1, 3)?
(a) 5
(b) 6
(c) 7
(d) 8
Solution:
(c) 7

Question 19.
If \(\vec{a} \times \vec{b}=\hat{n}\) then what is the angle between \(\vec{a} \text { and } \vec{b}\)?
(a) \(\frac{\pi}{4}\)
(b) \(\frac{\pi}{2}\)
(c) \(\frac{\pi}{3}\)
(d) None of the above
Solution:
(b) \(\frac{\pi}{2}\)

Question 20.
The plane 2x + 3z = 5 is parallel to:
(a) x-axis
(b) y-axis
(c) z-axis
(d) line x = y = z
Solution:
(b) y-axis

Question 21.
The equation of the plane containing the points (1, 0, 0), (0, 2, 0) and (0, 0, 3) is given by:
(a) x + 2y + 3z = 1
(b) 3x + 2y + z = 2
(c) 6x + 3y + 2z = 6
(d) 6x + 3y + 2z = 8
Solution:
(c) 6x + 3y + 2z = 6

Question 22.
If on action of force f = 2i + j – k, a particle displaced from A (0, 1, 2) to B (-2, 3, 0) then what is the work done by the force?
(a) 1
(b) 2
(c) 0
(d) None of the above
Solution:
(c) 0

Question 23.
If \(\vec{a}, \vec{b}, \vec{c}\) are unit vectors and \(\vec{a}+\vec{b}+\vec{c}=0\) then evaluate \(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}\).
(a) \(\frac{3}{2}\)
(b) –\(\frac{3}{2}\)
(c) \(\frac{2}{3}\)
(d) –\(\frac{2}{3}\)
Solution:
(b) –\(\frac{3}{2}\)

Question 24.
What is the value of (î + ĵ) × (ĵ + k̂) × (k̂ + î)?
(a) 0
(b) 1
(c) 2
(d) 3
Solution:
(c) 2

Question 25.
Write the distance of the point of intersection of the plane, ax + by + cz + d = 0 and the z-axis from the origin.
(a) \(\left|\frac{d}{c}\right|\)
(b) \(\left|\frac{d}{a}\right|\)
(c) \(\left|\frac{a}{c}\right|\)
(d) \(\left|\frac{a}{d}\right|\)
Solution:
(a) \(\left|\frac{d}{c}\right|\)

Question 26.
Write down the equation of the plane through (0, 0, 0) perpendicular to the line joining (0, 0, 1) and (0, 0, -1).
(a) x = 0
(b) z = 0
(c) y = 0
(d) None of the above
Solution:
(b) z = 0

Question 27.
What is the distance of the point (1, 1, 1) from the plane y = x?
(a) 0
(b) 1
(c) -1
(d) None of the above
Solution:
(a) 0

Question 28.
What is the angle between the planes y + x = 0 and z = 0?
(a) 45°
(b) 60°
(c) 75°
(d) 90°
Solution:
(d) 90°

Question 29.
For what ‘k’ the line \(\frac{x-3}{2}=\frac{y+k}{-1}=\frac{z+1}{-5}\) lies on the plane 2x – y + z – 7 = 0.
(a) 0
(b) 1
(c) -1
(d) 2
Solution:
(d) 2

Question 30.
Projection of the line segment joining (1, 3, -1) and (3, 2, 4) on z-axis is
(a) 4
(b) 5
(c) 3
(d) 2
Solution:
(b) 5

Question 31.
The image of the point (6, 3, -4) with respect to yz-plane is ______.
(a) (-6, 3, 4)
(b) (6, 3, -4)
(c) (-6, 3, -4)
(d) (-6, -3, -4)
Solution:
(c) (-6, 3, -4)

Question 32.
Find the equation of a plane through (1, 1, 2) and parallel to x + y + z – 1 = 0
(a) x + y – z + 4 = 0
(b) x + y – z – 4 = 0
(c) x + y + z – 4 = 0
(d) x + y + z + 4 = 0
Solution:
(c) x + y + z – 4 = 0

Question 33.
The distance between the parallel planes 2x – 3y + 6z + 1 = 0 and 4x – 6y + 12z – 5 = 0 is ______.
(a) \(\frac{1}{4}\)
(b) \(\frac{2}{3}\)
(c) \(\frac{1}{2}\)
(d) None
Solution:
(c) \(\frac{1}{2}\)

Question 34.
Find k if the normal to the plane parallel to the line joining (-1, 1, -4) and (5, 6, -2) has d.rs (3, -2, k).
(a) 1
(b) -1
(c) 2
(d) -2
Solution:
(b) -1

Question 35.
If a line makes angles 35° and 55° with x-axis and y-axis respectively, then the angle which this line subtends with z-axis is:-
(a) 35°
(b) 45°
(c) 55°
(d) 90°
Solution:
(d) 90°

Question 36.
Write the equation of the plane passing through (3, -6, -9) and parallel to xz-plane.
(a) z = -5
(b) z = -9
(c) z = 9
(d) z = -7
Solution:
(b) z = -9

Question 37.
In which condition x + y + z = α + β + γ will contain the line \(\frac{x-\alpha}{l}=\frac{y-\beta}{m}=\frac{z-\gamma}{n}\).
(a) l + m – n = 0
(b) l – m – n = 0
(c) l – m + n = 0
(d) l + m + n = 0
Solution:
(d) l + m + n = 0

Question 38.
The angle between the planes x + y + 1 = 0 and y + z + 1 = 0 is ______.
(a) 30°
(b) 45°
(c) 60°
(d) 75°
Solution:
(c) 60°

Question 39.
Find the number of points (x, y, z) in space other than the point (1, -2, 3) such that |x| = 1, |y|= 2, |z| = 3.
(a) 2
(b) 3
(c) 5
(d) 7
Solution:
(d) 7

Question 40.
Write the ratio in which the line segment joining the points (1, 2, -2) and (4, 3, 4) is divided by the xy-plane.
(a) 1:2
(b) 3:4
(c) 2:3
(d) 2:5
Solution:
(a) 1:2

(B) Very Short Type Questions With Answers

Question 1.
Write the value of k such that the line \(\frac{x-4}{1}=\frac{y-2}{1}=\frac{z-k}{2}\) lies on the plane 2x – 4y + z = 7.
Solution:
The line \(\frac{x-4}{1}=\frac{y-2}{1}=\frac{z-k}{2}\) lies on the plane 2x – 4y + z = 7.

Question 2.
Write the equations of the line 2x + z – 4 = 0 = 2y + z in the symmetrical form.
Solution:
Given line is 2x + z – 4 = 0 = 2y + z
⇒ z = -(2x – 4) = -2(x – 2) and z = -2y
∴ -2(x – 2) = -2y = z
∴ The equation of the line in symmetrical form is \(\frac{x-2}{1}=\frac{y}{1}=\frac{z}{-2}\).

Question 3.
Write the distance between parallel planes 2x – y + 3z = 4 and 2x – y + 3z = 18
Solution:
Distance between the given parallel planes
= \(\frac{|18-4|}{\sqrt{4+1+9}}=\frac{14}{\sqrt{14}}=\sqrt{14}\)

Question 4.
Write down the equation to the plane perpendicular to the y-axis at the point (0, -2, 0).
Solution:
The equation of the plane perpendicular to y-axis at (0, -2, 0) is
(x – 0) . 0 + (y + 2) . 1 + (z – 0) . 0 = 0
⇒ y + 2 = 0

Question 5.
Under which conditions the straight line \(\frac{\mathrm{x}-\mathrm{a}}{l}=\frac{\mathrm{y}-\mathrm{b}}{\mathrm{m}}=\frac{\mathrm{z}-\mathrm{c}}{\mathrm{n}}\) intersects the plane Ax + By + Cz = 0 at a point other than (a, b, c)?
Solution:
The line \(\frac{\mathrm{x}-\mathrm{a}}{l}=\frac{\mathrm{y}-\mathrm{b}}{\mathrm{m}}=\frac{\mathrm{z}-\mathrm{c}}{\mathrm{n}}\) will intersect the plane Ax + By + Cz + D = 0 at a point other than (a, b, c) if Al + Bm + Cn ≠ 0 and Aa + Bb + Cc + D ≠ 0.

Question 6.
How many straight lines in space through the origin are equally inclined to the coordinate axes?
Solution:
There are two lines in space through origin which are equally inclined to coordinate axes.

Question 7.
Write the value of a if the vectors \(\vec{a}=2 \hat{i}+3 \hat{j}-6 \hat{k} \text { and } \vec{b}=\alpha \hat{i}-\hat{j}+2 \hat{k}\) are parallel.
Solution:

Question 8.
Write the equation of the line passing through the point (4, -6, 1) and parallel to the line \(\frac{x-1}{1}=\frac{y+2}{3}=\frac{z-1}{-1}\).
Solution:
Equation of the line passing through (4, -6, 1) and parallel to
\(\frac{x-1}{1}=\frac{y+2}{3}=\frac{z-1}{-1} \text { is } \frac{x-4}{1}=\frac{y+6}{3}=\frac{z-1}{-1}\).

Question 9.
What is the image of the point (-2, 3, -5) with respect to the zx-plane?
Solution:
Image of the point (-2, 3, -5) w.r.t. zx-plane is (-2, -3, -5)

Question 10.
What is the point of intersection of the line x = y = z with the plane x + 2y + 3z = 6?
Solution:
Given line is x = y = z = λ (say)
Any point on this line has coordinates (λ, λ, λ).
Putting x = y = z = λ in x + 2y + 3z = 6
We get 6λ = 6 ⇒ λ = 1
∴ The point of interesetion is (1, 1, 1).

Question 11.
How many directions a null vector has?
Solution:
A null vector has infinitely many directions (arbitrary direction).

Question 12.
For what value of λ the vectors λî + 3ĵ + λk̂ and λî – 2ĵ + k̂ are perpendicular to each other?
Solution:
Given vectors are perpendicular to each other iff λ² – 6 + λ = 0.
⇒ λ² + 3λ – 2λ- 6 = 0
⇒ λ (λ + 3) – 2 (λ + 3) = 0
⇒ (λ – 2) (λ + 3) = 0
⇒ λ = 2, λ = (-3)

Question 13.
If |x| = 1, |y| = 2 and |z| = 3, then how many points in R³ are there having coordinates (x, y, z)?
Solution:
Required number of points are ‘8’.

Question 14.
Write the equation of the plane passing through the point (1, -2, 3) and perpendicular to the y-axis.
Solution:
D.rs. of any line parallel to y-axis are < 0, 1, 0 >.
∴ The equation of required plane is
(x – 1) 0 + (y + 2) . 1 + (z – 3) . 0 = 0
⇒ y + 2 = 0

(C) Short Type Questions With Answers

Question 1.
Find the point where the line \(\frac{x-2}{1}=\frac{y}{-1}=\frac{z-1}{2}\) meets the plane 2x + y + z = 2.
Solution:
Equation of the line is \(\frac{x-2}{1}=\frac{y}{-1}=\frac{z-1}{2}\)
Co-ordinates of any point on the line are (k + 2, -k, 2k + 1)
This point lies on the plane
⇒ 2 (k + 2) + (-k) + (2k + 1) = 2
⇒ 3k + 5 = 2 ⇒ k = -1
⇒ The required point of intersection is (1, 2, -1)

Question 2.
If the sum of two unit vectors is a unit vector, show that the magnitude of their difference is √3.
Solution:

Question 3.
The position vectors of two points A and B are 3î + ĵ + 2k̂ and î – 2ĵ – 4k̂ respectively. Find the equation of the plane passing through B and perpendicular to \(\overrightarrow{AB}\).
Solution:

Question 4.
Find the equation of the plane through the point (2, 1, 0) and passing through the intersection of the planes 3x – 2y + z – 1 = 0 and x – 2y + 3z = 1.
Solution:
Equation of any plane passing through the intersection of given two planes is:
(3x – y + z – 1) + λ (x – 2y + 3z – 1) = 0
⇒ x (3 + 2λ) + y (-1 – 2λ) + z (1 + 3λ) – 1 – λ = 0
This plane passes through A (2, 1, 0)
⇒ 2 (3 + λ) – 1 – 2λ – 1 – λ = 0
⇒ 6 + 2λ – 2 – 3λ = 0
⇒ λ = 4
∴ Equation of the required plane is 7x – 9y + 13z – 5 = 0

Question 5.
Prove that the vectors 2î – ĵ + k̂, î – 3ĵ – 5k̂, 3î – 4ĵ – 4k̂ are the sides of a right angled triangle.
Solution:

Question 6.
Prove that \(|\overrightarrow{a}+\overrightarrow{b}| \leq|\overrightarrow{a}|+|\overrightarrow{b}|\). Write when equality will hold
Solution:

Question 7.
The projections of a line segment \(\overline{OP}\), through the origin O on the coordinate axes are 6, 2, 3. Find the length of the line segment \(\overline{OP}\) and its direction cosines.
Solution:

Question 8.
Prove that the lines \(\frac{x+4}{3}=\frac{y+6}{5}=\frac{z-1}{-2}\) and 3x – 2y + z + 5 = 0 = 2x + 3y + 4z – 4 are coplanar.
Solution:
Given lines are
\(\frac{x+4}{3}=\frac{y+6}{5}=\frac{z-1}{-2}\) = r (say) … (1)
and 3x – 2y + z + 5 = 0
2x + 3y + 4z – 4 = 0 … (2)
Two lines are coplanar if either they are parallel or intersecting coordinates of any point on the line (1) are P (3r – 4, -5r – 6, -2r + 1) putting in the equations (2) we get.
3 (3r – 4) – 2 (5r – 6) + (-2r + 1) + 5 = 0
and 2 (3r – 4) + 3 (5r – 6) + 4 (-2r + 1) -4 = 0
⇒ -3r + 6 = 0 and 13r – 26 = 0
⇒ r = 2 and r = 2
Thus two lines are intersecting.
⇒ The lines are co-planar.

Question 9.
If the sun of two unit vectors is a unit vector then find the magnitude of the difference.
Solution:

Question 10.
Find the equation of a plane parallel to the plane 2x – y + 3z + 1 = 0 and at a distance of 3 units away from it.
Solution:
Any plane parallel to 2x – y + 3z + 1 = 0 … (1)
has equation 2x – y + 3z + λ = 0.
Distance between two parallel planes (1) and (2) is 3 units.
Equation of required plane is.

Question 11.
Show that the poiints (3, -2, 4), (1, 1, 1) and (-1, 4, -2) are collinear.
Solution:
Let the given points are A (3, -2, 4), B (1, 1, 1) and C (-1, 4, -2).

The points A, B and C are collinear.

Question 12.
Determine the value of m for which the following vectors are orthogonal:
(m + 1) j + m²ĵ – mk̂, (m² – m + 1) î – mĵ + k̂
Solution:

Question 13.
Find the equation of the plane passing through the line x = y = z and the point (3, 2, 1).
Solution:
Equation of any plane through the line is (x – y) + λ (y – z) = 0 … (1)
Since the point (3, 2, 1) is on the plane, so it satisfies (1)
i.e., ( 3 – 2) + λ ( 2 – 1) = 0 ⇒ 1 + λ = 0
⇒ λ = -1
Using the value of λ in (1) we get
x – y + (-y) – z = 0 ⇒ x – 2y + z = 0 is the required equation of the plane.

Question 14.
Find the scalar projection of the vector \(\overrightarrow{a}=3 \hat{i}+6 \hat{j}+9 \hat{k} \text { on } \overrightarrow{b}=2 \hat{i}+2 \hat{j}-\hat{k}\).
Solution:

Question 15.
Prove that the straight line \(\frac{x-1}{2}=\frac{y+2}{-3}=\frac{z-3}{1}\) lies on the plane 7x + 5y + z = 0.
Solution:
Given line is \(\frac{x-1}{2}=\frac{y+2}{-3}=\frac{z-3}{1}\) lies on the plane 7x + 5y + z = 0.
As 7 × 1 + 5 × (-2) + 3 = 0 the point (1,-2, 3) lies on the plane.
D.rs. of the given line = < 2, -3, 1 > and
d.rs. of the normal to the plane are < 7, 5, 1 >
As 2 × 7 + (-3) × 5 + 1 × 1 = 0
The given line is parallel to the given plane.
Hence the given line lies in the plane.

Odisha State Board Elements of Mathematics Class 12 CHSE Odisha Solutions Chapter 13 Three Dimensional Geometry Ex 13(b) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 13 Three Dimensional Geometry Exercise 13(b)

Question 1.
State, which of the following statements are true (T) or false (F).
(a) Through any four points one and only one plane can pass.
Solution:
False

(b) The equation of xy-plane is x + y = 0.
Solution:
False

(c) The plane ax + by + c = 0 is perpendicular to z-axis.
Solution:
False

(d) The equation of the plane parallel to xz-plane and passing through (2, -4, 0) is y + 4 = 0.
Solution:
True

(e) The planes 2x – y + z – 1 = 0 and 6x – 3y + 3z = 1 are coincident.
Solution:
False

(f) The planes 2x + 4y – z + 1 = 0 and x – 2y – 6z + 3 = 0 are perpendicular to each other.
Solution:
True

(g) The distance of a point from a plane is same as the distance of the point from any line lying in that plane.
Solution:
False

Question 2.
Fill in the blanks by choosing the appropriate answer from the given ones:
(a) The equation of a plane passing through (1, 1, 2) and parallel to x + y + z – 1 = 0 is ______. [x + y + z = 0, x + y + 2z – 1 = 0, x + y + z = 0, x + y + z = 4]
Solution:
x + y + z = 4

(b) The equation of plane perpendicular to z-axis and passing through (1, -2, 4) is ______. [x = 1, y + 2 = 0, z – 4 = 0, x + y + z – 3 = 0]
Solution:
z – 4 = 0

(c) The distance between the parallel planes 2x – 3y + 6z + 1 = 0 and 4x – 6y + 12z – 5 = 0 is ______. \(\left[\frac{1}{2}, \frac{1}{7}, \frac{4}{7}, \frac{6}{7}\right]\)
Solution:
\(\frac{1}{2}\)

(d) The plane y – z + 1 = 0 is ______. [parallel to x-axis, perpendicular to x-axis, parallel xy-plane, perpendicular to yz-plane].
Solution:
parallel to x-axis

(e) A plane whose normal has direction ratios < 3, -2, k > is parallel to the line joining (-1, 1, -4) and (5, 6, -2). Then the value of k = ______. [6, -4, -1, 0]
Solution:
k = -4

Question 3.
Find the equation of planes passing through the points.
(a) (6, -1, 1), (5, 1, 2) and (1, -5, -4)
Solution:
Equation of the plane through the points (6, -1, 1), (5, 1, 2) and (1, -5, -4) is

⇒ (x – 6) (-10 + 4) – (y + 1) (5 + 5) + (z – 1) (4 + 10) = 0
⇒ -6 (x – 6) – 10 (y + 1) + 14 (z – 1) = 0
⇒ -6x + 36 – 10y – 10 + 14z – 14 = 0
⇒ -6x – 10y + 14z + 12 = 0
⇒ 3x + 5y – 7z – 6 = 0

(b) (2, 1, 3), (3, 2, 1) and (1, 0, -1)
Solution:
Equation of the plane though the given points is

⇒ (x – 2) (-4 – 2) – (y – 1) (-4 – 2) + (z – 3) (-1 + 1) = 0
⇒ (x – 2) (-6) – (y – 1) (-6) = 0
⇒ x – y – 1 = 0

(c) (-1, 0, 1), (-1, 4, 2) and (2, 4, 1)
Solution:
Equation of the plane through the given points is

⇒ (x + 1) (0 – 4) – y ( 0 – 3) + (z – 1) (0 – 12) = 0
⇒ -4x – 4 + 3y – 12z + 12 = 0
⇒ -4x + 3y – 12z + 8 = 0
⇒ 4x – 3y + 1 2z – 8 = 0

(d) (-1, 5, 4), (2, 3, 4) and (2, 3, -1)
Solution:
Equation of the plane through the given points is

⇒ (x + 1) (10 – 0) – (y – 5) (-15 – 0) + (z – 4) (-6 + 6) = 0
⇒ 10x + 10 + 15y – 75 = 0
⇒ 2x + 3y – 13 = 0

(e) (1, 2, 3), (1, -4, 3) and (-1, 3, 2)
Solution:
Equation of the plane through the given points is

⇒ (x – 1) (6 – 0) – (y – 2) (0 – 0) + ( z – 3) ( 0 – 12) = 0
⇒ (x – 1) 6 + (z – 3) (-12) = 0
x – 1 – 2z + 6 = 0
⇒ x – 2z + 5 = 0

Question 4.
Find the equation of plane in each of the following cases:
(a) Passing through the point (2, 3, -1) and parallel to the plane 3x – 4y + 7z = 0.
Solution:
Given plane is 3x – 4y + 7z = 0 … (1)
Any plane parallel to the plane (1) is given by 3x – 4y + 7z + d = 0.
If it passes through the point (2, 3, -1) then 6 – 12 – 7 + d = 0 or, d = 13
Equation of the plane is 3x – 4y + 7z + 13 = 0

(b) Passing through the point (2, -3, 1) and (-1, 1, -7) and perpendicular to the plane x – 2y + 5z + 1 = 0.
Solution:
Equation of the plane passing through the point (2, -3, 1) is
a (x – 2)+ b (y + 3) + c (z – 1) = 0 … (1)
If it passes through the point (-1, 1, -7)
then a (-1 – 2) + b (1 + 3) + c (-7 – 1) = 0
or 3a – 4b + 8c = 0 … (2)
Again given that plane (1) is perpendicular to the plane
x – 2y + 5z + 1 = 0
so a – 2b + 5c = 0 … (3)
Solving (2) and (3) we get
\(\frac{a}{-20+16}=\frac{b}{8-15}=\frac{c}{-6+4}\)
or, \(\frac{a}{-4}=\frac{b}{-7}=\frac{c}{-2} \text { or, } \frac{a}{4}=\frac{b}{7}=\frac{c}{2}\)
Equation of the required plane is
4 (x – 2) + 7 (y + 3) + 2 (z – 1) = 0
or, 4x + 7y + 2z + 11 = 0

(c) Passing through the foot of the perpendiculars drawn from P (a, b, c) on the coordinate planes.
Solution:
Let A, B, C be the feet of the perpendiculars drawn from point P(a, b, c) on to the coordinate axes. Then
A = (a, 0, 0), B = (0, b, 0), C = (0, 0, c)
Equation of the plane through A, B, C is
\(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\) = 1

(d) Passing through the point (-1, 3, 2) perpendicular to the planes x + 2y + 2z = 5 and 3x + 3y + 2z = 8.
Solution:
Equation of the plane through the point (-1, 3, 2) is
a (x + 1) + b (y – 3) + c (z – 2) = 0
If this plane is perpendicular to the plane
x + 2y + 2z = 5 and 3x + 3y + 2z = 8
then a + 2b +2c = 0
3a + 3b + 2c = 0
Solving we get \(\frac{a}{4-6}=\frac{b}{6-2}=\frac{c}{3-6}\)
or \(\frac{a}{-2}=\frac{b}{4}=\frac{c}{-3} \text { or, } \frac{a}{2}=\frac{b}{-4}=\frac{c}{3}\)
Equation of the required plane is
2 (x + 1) – 4 (y – 3) + 3 (z – 2) = 0
or, 2x – b – 3z – 8 = 0

(e) Bisecting the line segment joining (-1, 4, 3) and (5, -2, -1) at right angles.
Solution:

Let A = (-1, 4, 3) and B (5, -2, -1).
Suppose that a plane bisects AB at right angle. Let C be the mid-point of AB. Then the plane passes through C and the line AB is perpendicular to the plane
C = \(\left(\frac{-1+5}{2}, \frac{4-2}{2}, \frac{3-1}{2}\right)\) = (2, 1, 1)
D.rs. of AB are < 6, -6, -4 >
Equation of the plane is
6 (x – 2) – 6 (y – 1) – 4 (z – 1) = 0
or, 3x – 6 – 3y + 3 – 2z + 2 = 0
or, 3x – 3y – 2z – 1 = 0

(f) Parallel to the plane 2x – y + 3z + 1 = 0 and at a distance 3 units away from it.
Solution:
Given plane is 2x – y + 3z + 1 = 0 … (1)
Equation of the plane parallel to the plane (1) is 2x – y + 3z +k = 0 … (2)
Given that the distance between two planes (1) and (2) is 3 units.
We see that (0, 1, 0) is a point on (1).
The length of the perpendicular from (0, 1, 0) onto the plane (2).

Question 5.
(a) Write the equation of the plane 3x – 4y + 6z – 12 = 0 in intercept form and hence obtain the coordinates of the point where it meets the co-ordinate axes.
Solution:
Given plane is 3x – 4y + 6z – 12 = 0
⇒ 3x – 4y + 6z = 12
⇒ \(\frac{x}{4}+\frac{y}{-3}+\frac{z}{2}\) = 1
This is in intercept form. The coordinates of the points where the plane meets the coordinate axes are (4, 0, 0), (0, -3, 0), (0, 0, 2).

(b) Write the equation of the plane 2x – 3y + 5z + 1 = 0 in normal form and find its distance from the origin. Find also the distance from the point (3, 1, 2).
Solution:
Given plane is 2x – 3y + 5z + 1 = 0
2x – 3y + 5z = -1
-2x + 3y – 5z = 1
Its normal form is

(c) Find the distance between the parallel planes 2x – 2y + z + 1 = 0 and 4x – 4y + 2z + 3 = 0.
Solution:
The parallel planes are
2x – 2y + z + 1 = 0 … (1)
and 4x – 4y + 2z + 3 = 0 … (2)
We see that (0, 0, -1) lies on the plane (1).
Distance between two planes = Length of the perpendicular from (0, 0, -1) onto the plane (2)
\(=\left|\frac{-2+3}{\sqrt{16+16+4}}\right|=\frac{1}{6}\)

Question 6.
In each of the following cases, verify whether the four given points are coplanar or not.
(a) (1, 2, 3), (-1, 1, 0), (2, 1, 3), (1, 1, 2)
Solution:
As

The given points are coplanar.

(b) (1, 1, 1), (3, 1, 2), (1, 4, 0), (-1, 1, 0)
Solution:
As

The given points are coplanar.

(c) (0, -1, -1), (4, 5, 1), (3, 9, 4), (-4, 4, 4)
Solution:
As

= 100 – 210 + 110
= 0
The given points are coplanar.

(d) (-6, 3, 2), (3, -2, 4), (5, 7, 3) and (-13, 17, -1)
Solution:
As

= 14 × (-26) + 2 × 182 = 0
The given points are coplanar.

Question 7.
Find the equation of plane in each of the following cases:
(a) Passing through the intersection of planes 2x + 3y – 4z + 1 = 0, 2x – y + z + 2 = 0 and passing through the point (3, 2, 1).
Solution:
Given planes are
2x + 3y – 4z + 1 = 0 … (1)
and 3x – y + z + 2 = 0 … (2)
Any plane through the line of intersection of the planes (1) and (2) is
(2x + 3y – 4z + 1) + k (3x – y + z + 2) = 0
or (2 + 3k) x + (3 – k) y + (k – 4) z + (2k + 1) = 0 … (3)
If the plane (3) passes through the point (3, 2, 1)
then (2 + 3k) 3 + (3 – k) 2 + (k – 4) + 2k + 1 = 0
⇒ 6 + 9k + 6 – 2k + k – 4 + 2k + 1 = 0

or, -7x + 39y – 49z – 8 = 0
or, 7x – 39y + 49z + 8 = 0

(b) Which contains the line of intersection of the planes x + 2y + 3z – 4 = 0 and 2x + y – z + 5 = 0 and perpendicular to the plane 5x + 3y + 6z + 8 = 0.
Solution:
Given planes are
x + 2y + 3z – 4 = 0 … (1)
and 2x + y – z + 5 = 0 … (2)
Any plane containing the line of intersection of the planes (1) and (2) is
x + 2y + 3z – 4 + k (2x + y – z + 5) = 0
⇒ (2k + 1) x + (k + 2) y + (3 – k) z + (5k – 4) = 0 … (3)
Given that the plane (3) is perpendicular to the plane
5x + 3y + 6z + 8 = 0 … (4)
Two planes are perpendicular if their normals are perpendicular.
D.rs. of the normal of the plane (3) are < 2k + 1, k + 2, 3 – k >.
D.rs. of the normal of the plane (4) are < 5, 3, 6 >.
If the normal are perpendicular then
5 (2k + 1) + 3 (k + 2) + 6 (3 – k) = 0
⇒ 10k + 3k – 6k + 5 + 6 + 18 = 0
⇒ 7k = -29

or, -51x – 15y + 50z – 173 = 0
or, 51x + 15y – 50z + 173 = 0

(c) Passing through the intersection of ax + by + cz + d = 0 and a₁x + b₁y + c₁z + d₁ = 0 and perpendicular to xy-plane.
Solution:
Given planes are
ax + by + cz + d = 0 … (1)
and a₁x + b₁y + c₁z + d₁ = 0 … (2)
Any plane passing through the line of intersection of the planes (1) and (2) is given by:
ax + by + cz + d + k (a₁x + b₁y + c₁z + d₁) = 0
or, (a + ka₁) x + (b + kb₁) y + (c + kc₁) z + (d + kd₁) = 0 … (3)
D.rs. of the normal of the plane (3) are < a + ka₁, b + kb₁, c + kc₁).
D.rs. of the normal of the xy-plane i.e., z = 0 are < 0, 0, 1 >
If the plane (3) is perpendicular to the xy-plane then
(a + ka₁) × 0 + (b + kb₁) × 0 + (c + kc₁) × 1 = 0

(d) Passing through the intersection of the planes x + 3y – z + 1 = 0 and 3x – y+ 5z + 3 = 0 and is at a distance 2/3 units from origin.
Solution:
Given planes are
x + 3y – z + 1 = 0 … (1)
and 3x – y + 5z + 3 = 0 … (2)
Any plane through the line of intersection of the planes (1 ) and (2) is
(x + 3y – z + 1)+ k (3x – y + 5z + 3) = 0
⇒ (3k + 1) x + (3 – k) y + (5k – 1) z + (3k + 1) = 0
The length of the perpendicular from origin onto the plane (3)
\(\frac{3 k+1}{\sqrt{(3 k+1)^2(3-k)^2+(5 k-1)^2}}=\frac{2}{3}\) (Given)
⇒ 9 (3k + 1)²
⇒ 4 {(3k + 1)² + (3 – k)² + (5k – 1)²}
⇒ 81k² + 54k + 9
= 4 {9k² + 6k + 1 + 9 + k² – 6k + 25k² + 1 – 10k}
= 4 {35k² – 10k + 11}
= 140k² – 40k + 44
⇒ 59k² – 94k + 35 = 0
⇒ 59k² – 59k – 35k + 35 = 0
⇒ 59k (k – 1) – 35 (k – 1) = 0
⇒ (k – 1) (59k – 35) = 0
⇒ k – 1 = 0 or 59k – 35 = 0
⇒ k = 1 or, k = \(\frac{35}{59}\)
∴ The planes are 4x + 2y + 4z + 4 = 0 and 82x + 71y + 58z + 82 = 0

Question 8.
Find the angle between the following pairs of planes.
(a) x + 3y – 5z + 1 = 0 and 2x + y – z+ 3 = 0
Solution:
Given planes are
x + 3y – 5z + 1 = 0 … (1)
and 2x + y – z + 3 = 0 … (2)
Angle between two planes is equal to the angle between their normals.
D.rs. of the normal to the plane (1) are < 1, 3, -5 >
D.rs. of the normal to the plane (2) are < 2, 1, -1 >.
If θ is the angle between the planes then

(b) x + 2y + 2z – 3 = 0 and 3x + 4y + 5z + 1 = 0
Solution:
Given planes are
x + 2y + 2z – 3 = 0 and 3x + 4y + 5z + 1 = 0
D.rs. of normals of two planes are < 1, 2, 2 > and < 3, 4, 5 >.
If θ is the angle between the planes then

(c) x + 2y + 2z – 7 = 0 and 2x – y +z = 6
Solution:
Two planes are
x + 2y + 2z – 7 = 0 and 2x – y + z = 6
The d.rs. of the normals are < 1, 2, 2 > and <2, -1, 1 >.
If θ is the angle between two planes then

Question 9.
(a) Find the equation of the bisector of the angles between the following pairs of planes and specify the ones which bisects the acute angles;
(i) 3x – 6y + 2z + 5 = 0 and 4x – 12y + 3z – 3 = 0
Solution:
Given planes are
3x – 6y + 2z + 5 = 0 … (1)
and 4x – 12y + 3z – 3 = 0 … (2)
The equations of the bisectors are given by

So the plane (3) is a bisector for obtuse angle and the plane (4) is the bisector for acute angle.

(ii) 2x + y – 2z – 1 = 0 and 4x – 12y + 3z + 3 = 0
Solution:
Given planes are
2x + y – 2z – 1 = 0 … (1)
and 4x – 12y + 3z + 3 = 0 … (2)
The equation of bisector planes are

∴ θ > 45°
∴ Equation (3) is the bisector of the use angle and equation (4) is the bisector of acute angle.

(b) Show that the origin lies in the interior of the acute angle between planes.
x + 2y + 2z + 9 and 4x – 3y + 12z + 13 = 0;
Find the equation of bisector of the acute angle.
Solution:
Given planes are x + 2y + 2z – 9 = 0
and 4x – 3y + 12z + 13 = 0
These equations can be written as
-x – 2y – 2z+ 9 = 0 … (1)
4x – 3y+ 12z + 13 = 0 … (2)
Here a₁a₂ + b₁b₂ + c₁c₂
= (-1) 4 + (-2) (-3) + (-2) . 12
= -4 + 6 – 24 = -22 < 0
So the origin lies in the acute angle between the given planes.
The equations of the planes bisecting the angle between the given planes are

⇒ tan θ < 1 ⇒ θ < 45°
Hence the plane (3) bisects the acute angle between the given planes.

Question 10.
(a) Prove that the line joining (1, 2, 3), (2, 1, -1) intersects the line joining (-1, 3, 1) and (3, 1, 5)
Solution:
Let A = (1, 2, 3), B = (2, 1, -1), C = (-1, 3, 1), D = (3, 1, 5)
The line AB intersects CD if A, B, C, D are coplanar.
Equation of the plane

or, (x – 1) (2 + 4) – (y – 2) (-2 – 8) + (z – 3) (1 – 2) = 0
or, 6 (x – 1) + 10 (y – 2) – (z – 3) = 0
or, 6x+ 10y – z – 23 = 0
The point (3, 1, 5) lies on the plane (1) because
6x + 10y – z – 23 = 18 + 10 – 5 – 23 = 0 … (1)
As (1) is satisfied for the point (3, 1, 5), the point lies on the plane (1).
Hence the four points are coplanar. So the line AB is either parallel to the line CD.
or, AB is intersecting to CD.
D.rs. of AB are < 1, -1, -4 >.
D.rs. of CD are < 4, -2, 4 >.
So d.rs. of AB are not proportional to d.rs. of CD. So they are not parallel. Hence they are intersecting.

(b) Show that the point (-\(\frac{1}{2}\), 2, 0) is the circumcentre of the triangle formed by the points (1, 1, 0), (1, 2, 1) and (-2, 2, -1).
Solution:

Thus PA = PB = PC
Hence P is the circumcentre of the ΔABC.

Question 11.
Show that plane ax + by + cz + d = 0 divides the line segment joining
(x₁, y₁, z₁) and (x₂, y₂, z₂) in a ratio – \(\frac{a x_1+b y_1+c z_1+d}{a x_2+b y_2+c z_2+d}\).
Solution:
Let A = (x₁, y₁, z₁) , B = (x₂, y₂, z₂)
Suppose that the line AB intersects the plane
ax + by + cz + d = 0 at P. … (1)
Suppose the P divides AB into the ratio m:n.
Then the coordinates of P are

Question 12.
A variable plane is at a constant distance p from the origin and meets the axes at A, B, C. Through A, B, C planes are drawn parallel to the co-ordinate planes. Show that the locus of their points of intersection is \(\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{1}{p^2}\).
Solution:

The planes through the points A, B, C parallel to the coordinate planes are
x = a, y = b and z = c respectively.
Let P be the point of intersection of these planes.
Then p = (a, b, c)
Hence the locus of P is obtained by putting a = x, b = y, c = z in (2).
Hence the required locus is
\(\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{1}{p^2}\). (Proved)

Question 13.
A variable plane passes through a fixed point (a, b, c) and meets the co-ordinate axes at A, B, C. Show that the locus of the point common to the planes drawn through A, B and C parallel to the coordinate planes is \(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}\) = 1.
Solution:
Let the variable plane be
\(\frac{a}{\alpha}+\frac{b}{\beta}+\frac{c}{\gamma}\) = 1 … (1)
It meets the axes at A (α, 0, 0), B (0, β, 0) and C (0, 0, γ).
If P is the point of intersection of the planes drawn through A, B and C parallel to the coordinate planes then P = (α, β, γ).
Again the plane (1) passes through a fixed point (a, b, c).
So \(\frac{a}{\alpha}+\frac{b}{\beta}+\frac{c}{\gamma}\) = 1
∴ The locus of P is \(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}\) = 1. (Proved)

Question 14.
The plane 4x + 7y + 4z + 81 = 0 is rotated through a right angle about its line of intersection with the plane 5x + 3y + 10z – 25 = 0. Find the equation of the plane in new position.
Solution:
Given that the plane
4x + 7y + 4z + 81 = 0 … (1)
is rotated through a right angle about the line of intersection with the plane
5x + 3y + 10z – 25 = 0 … (2)
Let the equation of the plane in new position be
4x + 7y + 4z + 81 + λ (5x + 3y -10z – 25) = 0
or, (4 + 5λ) x + (7 + 3λ) y + (4 + 10λ) z + (81 – 25λ) = 0
Then the angle between the planes (1 ) and (3) is 90° i.e., they are perpendicular.
So 4 (4 + 5λ) + 7 (7 + 3λ) + 4 (4 + 10λ) = 0
16 + 20λ + 49 + 21λ + 16 + 40λ = 0
81λ = -81
⇒ λ = -1
∴ The equation of plane in new position is
-x + 4y – 6z + 106 = 0
or, x – 4y + 6z – 106 = 0

Question 15.
The plane lx + my = 0 is rotated about its line of intersection with the plane z = 0 through angle measure α. Prove that the equation of the plane in new position is lx + my ± z \(\sqrt{l^2+m^2}\) tan α = 0.
Solution:
Given plane is lx + my = 0 … (1)
The plane (1) is rotated through an angle α about line of intersection of the plane (1) and the plane z = 0 … (2)
Let the equation of the plane in it’s new position be lx + my + kz = 0. … (3)
D.rs. of the normal to the plane (1) are < l, m, 0 >.
D.rs. of the normal to the plane (3) are < l, m, k >.
If α is the angle between the planes (1) and (3) then