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KENDRIYA VIDYALAYA SANGATHAN, RAIPUR REGION (2022-23)
CLASS XII – MATHEMATICS
CHAPTER 1 : RELATIONS AND FUNCTIONS
Ordered Pair:
A pair of elements listed in a specific order separated by comma and enclosing the pair in
parenthesis is called an ordered pair.
For example, (a, b) is an ordered pair with a as the first element and b as the second
element.
Cartesian Product or Cross Product of sets A and B:
The set of ordered pairs (a, b) such that a∈ A , b∈ B is called the cartesian product of A to B.
The set of ordered pairs (b, a) such that a∈ A, b∈ B is called the cartesian product of B to A.
It is written as:
A x B = {(a, b): a∈ A , b∈ B }
A x B = {(b, a): a∈ A , b∈ B }
Number of elements in A x B:
If n(A) = p and n(B) = q then n( A x B ) = pq
Relation from Set A to set B:
Let A and B be two non-empty sets, then a relation R from set A to set B is a subset of
cartesian product A x B.
Relation on a Set:
Let A be a non-empty set. Then, a relation from A to A is called a relation on set A.
Domain, Range and Codomain of Relation:
Let R be a relation from set A to set B, then the set of all the first elements of the ordered
pairs in R is called the domain and the set of all the second elements of the ordered pairs in R is
called the range of R, i.e., Domain of R = {a: (a, b) ∈ R} and Range of R = {a: (a, b) ∈ R}. The set B is
called the codomain of relation R.
Empty Relation:
A relation from set A to set B is said to be empty if no element of A is related to any element
of B, and is denoted by ∅. An empty relation is a subset of A x B.
Universal Relation:
A relation from set A to set B is said to be universal if each element of A is related to every
element of B. Universal relation U = A x B.
NOTE: Empty relation and Universal relation are said to be trivial relations.
Identity Relation:
A relation R on the set A is an identity relation if and only if R = {(a, a) for each a ∈ A}.
Types of Relations:
A relation on a non-empty set A is said to be
(i) Reflexive, if (a, a) ∈ R for all a ∈ A.
(ii) Symmetric, if (a, b) ∈ R implies (b, a) ∈ R, for all a, b∈ A.
(iii) Transitive, if (a, b) ∈ R and (b, c) ∈ R implies (a, c) ∈ R, for all a, b, c ∈ A.
Equivalence Relation:
A relation R on a set A is said to be an equivalence relation if R is reflexive, symmetric and
transitive.
Equivalence Classes:
Let R be an equivalence relation on a set A. The set of all those elements of A, which are
related to a, where a ∈ A, is said equivalence class determined by a and is denoted by [a].
Given an arbitrary relation R on an arbitrary set A, R divides A into mutually disjoint subsets
𝐴𝑖 , called partitions or subdivisions of A, satisfying the conditions:
(i) All elements of 𝐴𝑖 are related to each other, for each 𝑖.
(ii) No element of 𝐴𝑗 is related to any element of 𝐴𝑖 , for all 𝑖 ≠ 𝑗.
(iii) 𝐴𝑖 ∩ 𝐴𝑗 = ∅, for all 𝑖, 𝑗.
Function (Mapping):
For any two non-empty sets A and B, a function f from A to B is a rule or mapping which
associates each element of set A to a unique element in set B. It is denoted by f : A→B.
Domain, Codomain and Range of a Function:
Let f : A→B then elements of set A are called domain of f and the elements of set B
are called codomain of f. The set of all the images obtained in set B corresponding to each element
belongs to A under f is called range.
Types of Functions:
One-one (or injective function): A function f : A→B is called a one-one or injective function, if
distinct elements of A have distinct images in B.
i.e., for every x1 , x 2 ∈ A, f(x 1 ) = f(x 2 ) implies x 1 = x 2
Number of Functions:
If a set A has m elements and set B has n elements, then
The total number of functions from A to B = n
m
Number of Surjective Functions (Onto Functions):
If a set A has m elements and set B has n elements, then
The number of onto functions from A to B = n
m
n C 1 (n-1)
m
n C 2 (n-2)
m
n C 3 (n-3)
m +…. -
n Cn- 1
Number of Injective Functions (One to One Functions):
If set A has m elements and set B has n elements, n≥m, then the number of injective functions or
one to one function is given by n!/(n - m)!.
Number of Bijective functions:
If there is bijection between two sets A and B, then both sets will have the same number of
elements. If n(A) = n(B) = m, then number of bijective functions = m!.
MULTIPLE CHOICE QUESTIONS
1. If A1,2,3 and let R1,1) , (2,2) , (3,3) , (1,2) , (2,1) , (2,3) , (3,2) , then R is:
(a) Reflexive, symmetric but not transitive (b) symmetric, transitive but not reflexive
(c) Reflexive and transitive but not symmetric (d) an equivalence relation
2. Let R be a relation defined on Z by a Rb a b, then R is:
(a) symmetric, transitive but not reflexive (b) Reflexive, symmetric but not transitive
(c) Reflexive and transitive but not symmetric (d) an equivalence relation
3. Let R be a relation defined on Z as follows: (a, b) ∈ R a
2
2 = 25, then domain of R is:
(a) 3,4,5 (b) 0,3,4,5 (c) 0, 3, 4, 5 (d) none of these
4. The relation R defined on the set A1,2,3,4,5 by R = (a, b): |a
2 − 𝑏
2 | (^) < 16 is given by:
(a) 1,1) , (2,1) , (3,1) , (4,1) , (2,3) (b) 2,2) , (3,2) , (4,2) , (2,4)
(c) 3,3) , (4,3) , (5,4) , (3,4) (d) none of these
5. Let R be a relation defined on Z as R = (x, y): |𝑥 − y | ≤ 1 Then R is:
(a) Reflexive and transitive (b) Reflexive and symmetric
(c) Symmetric and transitive (d) an equivalence relation
6. Let A= 1,2,3, B= 1,4,6,9 and R is a relation from A to B define by ‘ x is greater than y ’. Then
range of R is given by:
(a) 1,4,6,9 (b) 4,6,9 (c) 1 (d) none of these
7. A relation R is defined from 2,3,4,5 to 3,6,7,10 by x R y x is relatively prime to y. Then
the domain of R is given by:
(a) 2,3,5 (b) 3,5 (c) 2,3,4 (d) 2,3,4,5
8. Let the function f: R -b R - 1 be defined by 𝑓(𝑥) =
𝑥+𝑎
𝑥+𝑏
; 𝑎 ≠ 𝑏, then:
(a) f is one-one but not onto (b) f is onto but not one-one
(c) f is both one-one and onto (d) none of these
9. The function f: 0, ) R given by 𝑓(𝑥) =
𝑥
𝑥+ 1
is:
(a) f is both one-one and onto (b) f is one-one but not onto
(c) f is onto but not one-one (d) neither one-one nor onto
10. Which of the following functions from Z to itself is bijection?
(a) 𝑓(𝑥) = 𝑥
3 (b) 𝑓(𝑥) = 𝑥 + 2 (c) 𝑓(𝑥) = 2 𝑥 + 1 (d) 𝑓(𝑥) = 𝑥
2
11. If the function f : 2, ) B, defined by fx =𝑥
2 − 4 𝑥 + 5 is a bijection, then B is:
(a) R (b) 1, (c) 4, (d) 5,
ASSERTION-REASON TYPE QUESTIONS
Each of the following questions contains two statements: Assertion (A) and Reason (R). Each of the
questions has four alternative choices, only one of which is the correct statement.
(a) Both 'A' and 'R' are true and 'R' is the correct explanation of 'A'.
(b) Both 'A' and 'R' are true but 'R' is not thecorrect explanation of 'A'.
(c) 'A' is true but 'R' is false.
(d) 'A' is false but 'R' is true.
12. Assertion (A) : f(x) = |x - 2| + |x - 3| + |x - 5| is an odd function for all values of x between
3 and 5.
Reason (R) : f(-x) = - f (x) for odd function.
13. Let f(x) = 1 / 3 ∗ (
|𝑠𝑖𝑛 𝑥|
𝐶𝑂𝑆 𝑥
𝑠𝑖𝑛 𝑥
|𝑐𝑜𝑠 𝑥|
Assertion (A): The period of f (x) is 2π.
Reason (R): The period of sin x and cos x is 2π.
14. Assertion (A): Every even function y = f(x) is not one-one for all in x ∈ Df.
Reason (R): Even function is symmetrical about y – axis.
Case Study
15. Sherlin and Danju are playing Ludo at home during Covid-19. While rolling the dice, Sherlin's sister
Raji observed and noted the possible outcomes of the throw every time belongs to set {1, 2, 3, 4,
5, 6}. Let A be the set of players while B be the set of all possible outcomes.
A=(S,D), B={1,2,3,4,5,6}
21. Show that the relation R in the set R of real numbers, defined as R = {(a, b) : a ≤ b
2 } is neither
reflexive nor symmetric nor transitive
22. Consider f : R− {−
4
3
} → R− {
4
3
} given by f(𝑥) =
4 𝑥+ 3
3 𝑥+ 4
. Show that f is bijective. 23. Show that the relation R defined in the set A of all polygons as R = {(P 1 , P 2 ) : P 1 and P 2 have same
number of sides}, is an equivalence relation.
24. Let f : N → N be defined by
f(n)^ = {
n + 1 , if n is odd
n − 1 , if n is even
, show that f is bijective.
LONG ANSWER QUESTIONS ( 5 marks each)
25. Let A = { 1 , 2 , 3 , … , 9 }^ and R be the relation in A defined by ( a , b) R (c , d) iff 𝑎 + d = b + c for all
a , b, c ,d ∈ A .Prove that R is an equivalence relation. Also obtain the equivalence class [(2,5)].
26. Show that the relation R in the set A = { 1 , 2 , 3 , 4 , 5 }^ given by R = {(𝑎, 𝑏)^ ∶ |𝑎 −
𝑏|^ is divisible by 2 }^ is an equivalence relation. Show that all the elements of { 1 , 3 , 5 }^ are related
to each other and all the elements of { 2 , 4 } are related to each other, but no element of { 1 , 3 , 5 }
is related to any element of { 2 , 4 }.
27. Let A= {𝑥 ∈ 𝑍: 0 ≤ 𝑥 ≤ 12 }. Show that R = {(𝑎, b): 𝑎, b ∈ A, |𝑎 − b|is divisible by 4 } is an
equivalence relation. Find the set of all elements related to 1. Also write the equivalence class [ 2 ].
28. Let R be a relation on the set A = N × N ,where N is the set of natural numbers, defined by
(𝑥, 𝑦) (^) R (u, v) (^) if and only if 𝑥v = 𝑦u. Show that R is an equivalence relation.
29. Show that the relation R in the set N × N, defined by (𝑎, b)^ R (c, d)^ iff 𝑎
2
2 = b
2
c
2 ∀ 𝑎, b, c, d ∈ N, is an equivalence relation.
30. Let N denote the set of all natural numbers and R be the relation on N × N defined by
(𝑎, b) R (c, d) iff 𝑎d(b + c) = b c (𝑎 + d). Show that R is an equivalence relation.
31. Show that the function f : R → {x ∈ R : – 1 < x < 1} defined by f(x) =
𝑥
1 +|𝑥|^
, x ∈ R is one one and
onto function.
ANSWERS
a 2. c 3. c 4. d 5. b 6. c 7. d 8. c 9. b 10. b 11. b
(a) 13. (b) 14. (d) 15. (i) 6
2 (ii) 2
12 (iii) reflexive and transitive
- (i) ) injective (ii) {1,4,9, 16,...} (iii) neither injective nor surjective
- There is a maximum of 5 equivalence relations on the set A = {1,2,3}.
They are
R 1 = {(1,1),(2,2),(3,3)}
R 2 = {(1,1),(2,2),(3,3),(1,2),(2,1)}
R 3 = {(1,1),(2,2),(3,3),(1,3),(3,1)}
R 4 = {(1,1),(2,2),(3,3),(2,3),(3,2)}
R 5 = {(1,1),(2,2),(3,3),(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)}
25. {(1,4), (2,5), (3,6), (4,7), (5,8), (6,9)} 27. { 1 , 3 , 5 }; [ 2 ]^ = { 2 , 6 , 10 }
KENDRIYA VIDYALAYA SANGATHAN, RAIPUR REGION (2022-23)
CLASS XII – MATHEMATICS
CHAPTER 2 : INVERSE TRIGONOMETRIC FUNCTIONS
CBSE SYLLABUS: - Definition, range, domain, principal value branch. Graphs of inverse trigonometric
function.
Gist of topic: The domain of sine function is the set of all real numbers and range is the closed
interval [–1, 1].
If we restrict its domain to
, then it becomes one-one and onto with range [– 1, 1].
Actually, sine function restricted to any of the intervals .............
2
or
etc., is one-one and its range is [–1, 1].
therefore, define the inverse of sine function in each of these intervals.
We denote the inverse of sine function by sin
Thus, sin
- 1 is a function whose domain is [– 1, 1] and range could be any of the intervals
or , and so on. Corresponding to each such interval,
we get a branch of the function sin
- 1 . The branch with range ,
is called the principal value
branch, whereas other intervals as range give different branches of sin
function sin
- 1 , we take it as the function whose domain is [–1, 1] and range is
. We write
sin
We can, therefore, define the inverse of cosine function in each of these intervals. We denote the
inverse of the cosine function by cos
Thus, cos
- 1 is a function whose domain is [–1, 1] and range could be any of the intervals [–π, 0], [0,
π], [π, 2π] etc. Corresponding to each such interval,
we get a branch of the function cos
- 1 . The branch with range [0, π] is called the principal value
branch of the function cos
We write cos
Domain and Range of inverse-trigonometric function
Competency based question: Type 1-MCQ
Q1 One branch of cos
- 1 other than the principal value branch corresponds to
(A)
(B)
C) (0, π) (D) [2π, 3π]
Q2The domain of y = cos
2
(A) [3, 5] (B) [0, π] (C) 5 , 3 5 , 3
, (D) 5 , 3 3 , 5
Q3If sin
- 1 x + sin - 1 y = π/2 , then value of cos - 1 x + cos - 1 y is
(A)
(B) π (C) 0 (D) 2
Q4 The domain of the function cos
- 1 (2 x – 1) is ..........
(A) [0, 1] (B) [–1, 1] (C) [–2, 1] (D) [0, π]
Q5 the value of
sin cos
is
(A)
(B)
5
(C)
(D)
Q6The domain of the function y = sin
2 ) is
(A) [0, 1] (B) (0, 1) (C) [–1, 1] (D) φ
Q7 The domain of the function defined by f (x) = sin
(A) [–1, 1] (B) [–1, π + 1] (C) ( – ∞ , ∞) (D) φ
Q8 Which of the following corresponds to the principal value branch of sin
Q2 Find the value of : cos^ tan^3 3
sin 2 tan
1 1
Q3 Find the value of
sin sin
1 .
Q4 Write the principal value ofcos
− 1 (cos
7 𝜋
6
Q5Evaluate:tan ( 2 tan
− 1 1
5
Q6.Find the value of : cos
− 1 (cos
13 𝜋
6
) + tan
− 1 (tan
3 𝜋
4
Q7 Evaluate : sin [
π
6
− sin
− 1 (−
√ 3
2
)]
Q8 Find the value of sin (sin
− 1 1
2
− 1 3
5
Q9. Evaluate : sin ( 2 cos
− 1 (−
3
5
Q10. The value of cos
− 1 (−
1
2
) − 2 sin
− 1 (
1
2
Answer: (1). 5
𝟓𝝅
𝟔
𝟓
𝟏𝟐
𝝅
𝟏𝟐
𝟑+𝟒√𝟑
𝟏𝟎
𝟐𝟒
𝟐𝟓
KENDRIYA VIDYALAYA SANGATHAN RAIPUR REGION
CHAPTER 3 &4 : MATRICES & DETERMINANTS
SCHEMATIC DIAGRAM
MATRIX : If mn elements can be arranged in the form of m row and n column in a rectangular array then this
arrangement is called a matrix.
Order of a matrix: A matrix having m row and n column is called a matrix of 𝑚 × 𝑛 order.
Types of Matrices
(i) Column matrix: A matrix is said to be a column matrix if it has only one column e.g[
2
1
3
]
(ii) Row matrix: A matrix is said to be a row matrix if it has only one row e.g..[1 2 3]
(iii) Square matrix : A matrix in which the number of rows are equal to the number of columns, is said to be a
square matrix. Thus an m × n matrix is said to be a square matrix if m = n and is known as a square matrix of
order ‘n’ e.g.[
𝑎 𝑏 𝑐
𝑑 𝑒 𝑓
𝑔 ℎ 𝑖
]
CHAPTER2 INVERSE TRIGONOMETRIC FUNCTION.docx
(iv) Diagonal matrix: A square matrix B = [bij] (^) m × m is said to be a diagonal matrix if all its non diagonal
elements are zero, that is a matrix B = [bij] m × m is said to be a diagonal matrix if bij = 0, when i ≠ j
e.g. [
𝑎 0 0
0 𝑒 0
0 0 𝑖
]
(v) Scalar matrix : A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal, that is, a
square matrix B = [bij] n × n is said to be a scalar matrix if bij = 0, when i ≠ j bij = k, when i = j, for some
constant k.
e.g. (^) [
𝑎 0 0
0 𝑎 0
0 0 𝑎
]
(vi) Identity matrix : A square matrix in which elements in the diagonal are all 1 and rest are all zero is called
an identity matrix.
e.g. [
1 0 0
0 1 0
0 0 1
]
(vii) Zero matrix: A matrix is said to be zero matrix or null matrix if all its elements are zero. For example, [0],
[
0 0 0
0 0 0
0 0 0
]
Addition and subtraction of matrices: Two matrices A and B can be added or subtracted if they are of the
same order i.e. if A and B are two matrices of order 𝑚 × 𝑛 then A ± B is also a matrix of order m×n.
Multiplication of matrices : The product of two matrices A and B can be defined if the number of rows of B is
equal to the number of columns of A i.e. if A be an 𝑚 × 𝑛 matrix and B be an 𝑛 × 𝑝 matrix then the product
of matrices A and B is another matrix of order 𝑚 × 𝑝.
Minors and cofactors:
Minor of an element 𝑎𝑖𝑗 of a determinant is the determinant obtained by
deleting its i th row and j th column in which element 𝑎𝑖𝑗 lies. Minor of an element 𝑎𝑖𝑗is
denoted by 𝑀𝑖𝑗.
Cofactors: cofactors of an element 𝑎 𝑖𝑗 𝑑𝑒𝑛𝑜𝑡𝑒𝑑 by 𝐴 𝑖𝑗 and is defined by 𝐴 𝑖𝑗 = (− 1 )
𝑖+𝑗 𝑀 𝑖𝑗 where 𝑀 𝑖𝑗 is the
minor of 𝑎 𝑖𝑗 .
Adjoint of a Matrix : Let A = [
𝛼 𝛽
𝛾 𝛿
] be a Matrix of order 2 × 2
Then adj(A) = [
𝜹 −𝜷
−𝜸 𝜶
]
Again letA = [
𝑥 𝑦 𝑧
𝑝 𝑞 𝑟
𝑎 𝑏 𝑐
] be a Matrix of order 3 × 3
Then adj (A) =
[
|
𝑞 𝑟
𝑏 𝑐
| − |
𝑝 𝑟
𝑎 𝑐
| |
𝑝 𝑞
𝑎 𝑏
|
− |
𝑦 𝑧
𝑏 𝑐
| |
𝑥 𝑧
𝑎 𝑐
| − |
𝑥 𝑦
𝑎 𝑏
|
|
𝑦 𝑧
𝑞 𝑟
| − |
𝑥 𝑧
𝑝 𝑟
| |
𝑥 𝑦
𝑝 𝑞
| ]
𝑇
=
[
𝑞𝑐 − 𝑏𝑟 −(𝑝𝑐 − 𝑎𝑟) 𝑝𝑏 − 𝑎𝑞
−(𝑦𝑐 − 𝑏𝑧) 𝑥𝑐 − 𝑎𝑧 −(𝑏𝑥 − 𝑎𝑦)
𝑦𝑟 − 𝑞𝑧 −(𝑥𝑟 − 𝑝𝑧) 𝑥𝑞 − 𝑝𝑦
]
𝑇
= [
𝑞𝑐 − 𝑏𝑟 𝑏𝑧 − 𝑎𝑦 𝑦𝑟 − 𝑞𝑧
𝑎𝑟 − 𝑝𝑐 𝑥𝑐 − 𝑎𝑧 𝑝𝑧 − 𝑥𝑟
𝑝𝑏 − 𝑎𝑞 𝑎𝑦 − 𝑏𝑥 𝑥𝑞 − 𝑝𝑦
]
Inverse of a Matrix: Inverse of a Square Matrix A is defined as 𝑨
−𝟏
𝒂𝒅𝒋(𝑨)
|𝑨|
Note: If A be a given Square Matrix of order n then
(i) A(adj(A) = adj(A)A= |𝑨|𝑰where I is the Identity Matrix of order n.
(ii) A square Matrix A is said to be singular and non-singular according as |𝑨| = 𝟎 𝒂𝒏𝒅 |𝑨| ≠ 𝟎
(iii) |𝒂𝒅𝒋(𝑨)| = |𝑨|
𝒏−𝟏 (𝑭𝒐𝒓 𝒂 𝒔𝒒𝒖𝒂𝒓𝒆 𝑴𝒂𝒕𝒓𝒊𝒙 𝒐𝒇 𝒐𝒓𝒅𝒆𝒓 𝟑 × 𝟑 |𝒂𝒅𝒋(𝑨)|^ = |𝑨|
𝟐 )
IMPORTANT SOLVED PROBLEMS
Q1. If A=[
− 2
4
5
] , 𝐵 = [ 1 3 − 6 ] , Verify that (AB)′ = B′A′
Solution: - We have
If A=[
− 2
4
5
] , 𝐵 = [ 1 3 − 6 ]
Then AB =[
− 2
4
5
] [ 1 3 − 6 ]^ =^ [
− 2 − 6 12
4 12 − 24
5 15 − 30
]
Now A′ = [− 2 4 5 ] , B′= [
1
3
− 6
]
B′A′ = [
1
3
− 6
] [− 2 4 5 ] = [
− 2 4 5
− 6 12 15
12 − 24 − 30
] = (AB)′
Clearly (AB)′ = B′A′
Q2. If 𝑥 [
2
3
] + 𝑦 [
− 1
1
] = [
10
5
] then find the value of x and y.
Sol. Given 𝑥 [
2
3
] + 𝑦 [
− 1
1
] = [
10
5
]
[
2 𝑥
3 𝑥
] + [
−𝑦
𝑦
] = [
10
5
] or [
2 𝑥 − 𝑦
3 𝑥 + 𝑦
] = [
10
5
]
So 2x – y = 10 and 3x + y = 5 ; On solving we get x = 3 and y = - 4
Q3. If F(x) = [
𝑐𝑜𝑠𝑥 −𝑠𝑖𝑛𝑥 0
𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑥 0
0 0 1
] prove that F(x) F(y) = F(x+y)
Sol. Given F(x) = [
𝑐𝑜𝑠𝑥 −𝑠𝑖𝑛𝑥 0
𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑥 0
0 0 1
] so F(y) = [
𝑐𝑜𝑠𝑦 −𝑠𝑖𝑛𝑦 0
𝑠𝑖𝑛𝑦 𝑐𝑜𝑠𝑦 0
0 0 1
]
Hence F(x) .F(y) = [
𝑐𝑜𝑠𝑥 −𝑠𝑖𝑛𝑥 0
𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑥 0
0 0 1
] [
𝑐𝑜𝑠𝑦 −𝑠𝑖𝑛𝑦 0
𝑠𝑖𝑛𝑦 𝑐𝑜𝑠𝑦 0
0 0 1
] =
[
𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦 − 𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑦 −𝑐𝑜𝑠𝑥𝑠𝑖𝑛𝑦 − 𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑦 0
𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑦 + 𝑐𝑜𝑠𝑥𝑠𝑖𝑛𝑦 −𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑦 + 𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦 0
0 0 1
]
= [
cos(𝑥 + 𝑦) −sin(𝑥 + 𝑦) 0
sin(𝑥 + 𝑦) cos(𝑥 + 𝑦) 0
0 0 1
]
Hence F(x) F(y) = F(x+y)
Q4. Express the given Matrix as the sum of a symmetric and skew symmetric matrix
A= [
6 − 2 2
− 2 3 − 1
2 − 1 3
]
Sol. Here 𝐴
𝑇 = [
6 − 2 2
− 2 3 − 1
2 − 1 3
]
P =
1
2
(𝐴 + 𝐴
𝑇 ) =
1
2
[
12 − 4 4
− 4 6 − 2
4 − 2 6
] = [
6 − 2 2
− 2 3 − 1
2 − 1 3
]
Now 𝑃
𝑇 = 𝑃 so P =
1
2
(𝐴 + 𝐴
𝑇 ) (^) is a symmetric Matrix.
Also let Q = =
1
2
(𝐴 − 𝐴
𝑇 ) (^) =
1
2
[
0 0 0
0 0 0
0 0 0
] = [
0 0 0
0 0 0
0 0 0
]
𝑄
𝑇 = −𝑄 Hence Q is an Skew Symmetric Matrix.
Now P + Q = [
6 − 2 2
− 2 3 − 1
2 − 1 3
] + [
0 0 0
0 0 0
0 0 0
] = [
6 − 2 2
− 2 3 − 1
2 − 1 3
] = 𝐴
ASSIGNMENTS
Multiple choice questions
Choose the correct option
1.The matrix P = (
𝟎 𝟎 𝟒
𝟎 𝟒 𝟎
𝟒 𝟎 𝟎
) is a
(a) square matrix (b) diagonal matrix (c) Unit Matrix (c) none of these
2.Total number of possible matrix of order 3x3 with each entry 2 or 0 is
(a) 9 (b) 27 (c) 81 (d) 512
3.If A and B are two matrices of order 3xm and 3xn respectively, and m=n, then the order
of the matric (5A-2B) is
(a)mx3 (b)3x3 (c)mxn (d)3xn
4. If A and B are matrices are of the same order then (AB’-BA’) is a
(a) skew symmetric matrix (b) null matrix (c) symmetric matrix (d) unit matrix
5. Let A be a square matrix of order 3x3, then |kA| equal to
(a)k|A| (b)k
2 |A| (c) k
3 |A| (d)3k|A|
6. Which of the following is correct
(a) Determinant is a square matrix
(b) Determinant is a number associated to a matrix
©Determinant is a number associated to a square matrix.
(d)None of these
Answer:
1.a 2.d 3.d 4.a 5.c
VERY SHORT ANSWER TYPE QUESTIONS(VSA-2 MARKS )
1. If a matrix has 6 elements, what are the possible orders it can have?
- Construct a 3 × 2 matrix whose elements are given by aij = |i – 3j |
- If A = , B = , then find A – 2 B.
- If A = and B = , write the order of AB and BA.
- Find the matrices X and Y if 𝑋 + 𝑌 = [
5 2
0 9
] 𝑎𝑛𝑑 𝑋 − 𝑌 = [
3 6
0 − 1
]
- Solve: [𝑥 1 ]^ [
1 0
− 2 − 3
] [
𝑥
3
] = 0.
- Find the co-factor of 𝑎 12 in A =
- Find the adjoint of the matrix A =
- If A = , write A
- If A is square matrix satisfying A 2 = I, then what is the inverse of A?
- For what value of k , the matrix A = is not invertible?
- Evaluate 0 0
0 0
Sin Cos
Cos Sin
- What is the value of , where I is identity matrix of order 3?
- If A is non singular matrix of order 3 and = 3, then find
- For what valve of a, is a singular matrix?
Answer:
**1. 1 6, 6 1 , 2 3 , 3 2 2. 3. 4. 2 2, 3 3
- 46 8**
9 A
- 1 = - A 10. A - 1 = A 11 k = 17, 12.0 13.27 14.24 15.-4/
SHORT ANSWER TYPE QUESTIONS (SA-3 MARKS)
1. For the following matrices A and B, verify (AB)
T = B
T A
T , where A = , B =.
- Give example of matrices A & B such that AB = O, but BA ≠ O, where O is a zero matrix and
A, B are both non zero matrices.
- If B is skew symmetric matrix, write whether the matrix (ABA
T ) is symmetric or skew symmetric.
- If A = and I = , find a and b so that A
2
5.Verify A(adjA) = (adjA) A = I if
