2023-24

HOLIDAY H.W
Class : XII Subject: Math

TOPICS :MATRICES AND DETERMINANTS

RELATIONS AND FUNCTIONS
INVERSE TRIGONOMETRIC FUNCTIONS

MCQ ( 1 mks each )

1. Given the relation R = {(1, 2), (2, 3), (3, 4)}, which of the following is true?
a) R is reflexive
b) R is symmetric
c) R is transitive
d) R is an equivalence relation

2.The set of all first coordinates of the ordered pairs in a relation is called the:
a) Domain
b) Range
c) Codomain
d) Image

3.Which of the following functions is injective (one-to-one)?

a) f(x) = x^2
b) f(x) = |x|
c) f(x) = 2x + 3
d) f(x) = sin(x)

4.Let f: A → B be a function. If every element in set B is associated with at least one element in set A, then
f is said to be:
a) Injective
b) Surjective
c) Bijective
d) None of the above

5.Let f: R → R be defined by f(x) = x^3. Which of the following is true about this function?
a) f(x) is injective
b) f(x) is surjective
c) f(x) is bijective
d) f(x) is neither injective nor surjective

6.If a function is both injective and surjective, it is called:

a) Monotonic
b) Bijective
c) Polynomial
d) Exponential

7.Let f: A → B and g: B → C be functions. The composite function (g ∘ f) is defined as:

a) g(f(x))
b) f(g(x))
c) f(x)g(x)
d) g(x)/f(x)
8.. Principal value of tan-1 (-1) is
(a) π/4
(b) −π/2
(c) 5π/4
(d) −π/4

9.The composite of a function f: A → B with its inverse f^(-1): B → A results in:

a) Identity function on A
b) Identity function on B
c) Composite function on A
d) Composite function on B

10.Let f: R → R be defined by f(x) = x2. Which of the following is true about the function f(x)?
a) f(x) is an onto function
b) f(x) is a one-to-one function
c) f(x) is a many-to-one function
d) f(x) is a constant function

11.The set of all first coordinates of the ordered pairs in a relation is called the:
a) Domain
b) Range
c) Codomain
d) Image

12.Which of the following functions is injective (one-to-one)?

a) f(x) = x 2
b) f(x) = |x|
c) f(x) = 2x + 3
d) f(x) = sin(x)

13.A matrix A = [aij]m×n is said to be symmetric if

(a) aij = 0
(b) aij = aji
(c) aij = aij
(d) aij = 1

14.A = [aij]m×n is a square matrix if

(a) m = n
(b) m < n
(c) m > n
(d) None of these

15. If A and B are square matrices then (AB)’ =

(a) B’A’
(b) A’B’
(c) AB’
(d) A’B’
 Short answer type : (3mks each )
1) For the set A = {1, 2, 3}, define a relation R in the set A as follows:
R = {(1, 1), (2, 2), (3, 3), (1, 3)}.

Write the ordered pairs to be added to R to make it the smallest equivalence relation.

2) Let A = {0, 1, 2, 3} and define a relation R on A as follows:

R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}.

Is R reflexive? Symmetric? Transitive?

3) Relation R on the set A = {1, 2, 3, ….., 14} is defined as R = {(x, y) : 3x – y = 0 }. Determine

whether the given relation is reflexive, symmetric and transitive.

4) Let S be the set of all points in a plane and R be a relation on S, defined by

R = {(P, Q) : distance between P and Q is less than 4.5 units}

Show that R is reflexive and symmetric but not transitive.

5) Prove that the relation R on Z defined by (a, b)  R  a − b is divisible by 5, is an equivalence

relation on Z.

6) Let Z be the set of all integers and R be the relation on Z, defined as

R = (a, b) : a, b  Z and a – b is divisible by 5}.

Prove that R is an equivalence relation.

7) Show that the relation S in the set R of a real number, defined as

S = (a, b ) : a, b  R and a  b 3 , is neither reflexive, nor symmetric, nor transitive.

8) Show that the relation S in the set A =

x  Z : 0  x  12, is given by S = (a, b ) : a, b  Z and a − b is divisible by 4 , is an equivalence
relation.
 9) Prove that the relation R in the set = {1,2,3,4,5} given by R = (a, b ) : a − b is evenis an
equivalence
relation.

10) Show that the relation R in the set of real number, defined as

S = (a, b ) : a  b 2 , is neither reflexive, nor symmetric, nor transitive.

11) Let T be the set of all triangles in a plane with R a relation in T given by R = {(T1,T2): T1  T2}.
Show that R is an equivalence relation.

12) Let A = {1, 2}. How many onto functions from A to A are possible? Also, write them.

13) Let A = {1, 2, 3, 4} and B = {5, 6, 7, 8, 9}. Let f : A → B be defined by

f = {(1, 5), (2, 6), (3, 7), (4, 8)}.
Show that f is one – one but not onto.

14) Show that the function f : N → N defined by f (n) = 2n + 3  n  N , is not bijective.

 4 4 4x + 3
15) Consider f : R - −  → R −   given by f ( x) = . Show that f is bijective.Find the
 3 3 3x + 4
Inverse of f and hence find f −1 (0) and x such that f −1 (x) = 2.
16) If f : R → R is defined by f(x) = x2 + 3, prove that f is not invertible.

 12  4  56 
17) cos−1   + cos−1   = tan −1  
 13  5  33 
1 1 1 1 
18) tan −1   + tan −1   + tan −1   + tan −1   =
 3 5 7 8 4

19) Let E = {1, 2, 3, 4} and F = {1, 2} Then, find the number of onto functions from E to F?

20)Let A = {1, 2, 3, 4,…. n} How many bijective function f : A → B can be defined?

21)What type of relation is ‘less than’ in the set of real numbers?

22) If f : R → R such that f(x) = 3x then Is function f bijective ?

23) Find the principal values of the following :

a) sec −1 2( )
b) sec −1 (−2)
1
c) cos−1  
2
d) cot (1)
−1


24) Solve: sin −1 x = + cos−1 x
6
2
25) Solve: tan −1 x + 2 cot−1 x =
3
3
26) Solve: tan −1 (2 x ) + tan −1 (3x ) = n + , where n  Z .
4
 27) Solve: tan −1 (x + 1) + tan −1 (x − 1) = tan −1
8
.
31
 1 2  x 
28) If 2 x 3     = 0, find x.
− 3 0  3
SECTION - C
Long answer type ( 4 mks each)

1) Express the following matrix as the sum of a symmetric and skew symmetric matrix and verify
your result.

 3 − 2 − 4
A =  3 − 2 − 5
− 1 1 2 
2 0 1
2) If A = 2 1 3 , find A2 – 5A + 4I and hence find a matrix X such that
1 − 1 0

A2 – 5A + 4I + X = 0
3)For the following matrices A and B, verify that

1
( AB ) = B A . A = − 4 , B = − 1 2 1
/ / /

 3 
4)Three school A, B and C organized a mela for collecting funds for helping the flood victims. They
sold handmade fans, mats and plates from recycled material at a cost of Rs 25, Rs 100 and Rs 50 each. The
number of articles sold are given below.

Article/School A B C

Hand fans 40 25 35

Mats 50 40 50

Plates 20 30 40

Find the funds collected by each school separately by selling the above articles. Also, find the total
funds collected for the purpose.

1 − 1 a 1 
 and B =
2 2 2
5)If A =  b − 1 and (A+ B) = A + B , then find the value of a and b.
2 − 1  

6)A trust fund, Rs 35000 is to be invested in two different types of bonds. The first bond pays 8% interest
per annum which will be given to orphanage and second bond pays 10 % interest per annum which will
be given to an NGO. Using matrix multiplication, determine how to divide Rs 35000 among two types
of bonds if the trust fund obtains an annual total interest of Rs 3200.
  
7)For a 2 x 2 matrix A = aij , whose elements are given by aij =
(i + 2 j )2 , write the value of a21.
4
1 − 2  1 2
8)Solve the matrix equation A = 
1 4  − 1 3

9) by using concept of inverse (ii) without using concept of inverse.

 4  − 4 8 4
 
10)Find A, if 1 A =  − 1 2 1
3  − 3 6 3
CASE STUDY – I
Two farmers Ramakishan and Gurucharan Singh cultivate only three varieties of rice namely Basmati, Permal and
Naura. The sale (in rupees) of these varieties of rice by both the farmers in the month of September and October are
given by the following matrices A and B September sales (in Rupees)

1. The total sales in September and October for each farmer in each variety can be represented as:
a. A+B b. A-B c. A> 𝐵 d. A< 𝐵
2. What is the value of 𝐴23?
a. 10000 b. 20000 c. 30000 d. 40000
3. The decrease in sales from September to October is given by _______ .
a. A+B b. A-B c. A> 𝐵 d. A< 𝐵
4. If Ramkishan receives 2% profit on gross sales, compute his profit for each variety sold in October. a.
Rs. 100, Rs. 200 and Rs. 120 b. Rs. 100, Rs. 200 and Rs. 130
c. Rs. 100, Rs. 220 and Rs. 120 d. Rs. 110, Rs. 200 and Rs. 120
5. If Gurucharan receives 2% profit on gross sales, compute his profit for each variety sold in September.
a. Rs. 100, Rs. 200, Rs. 120 b. Rs. 1000 , Rs. 600, Rs. 200
c. Rs. 400, Rs. 200, Rs. 120 d. Rs. 1200, Rs. 200, Rs. 120

CASE STUDY – II
Three schools SAJS LKO, SAJS GZB and SAJS KNP decided to organize a fair for collecting money for
helping the flood victims. They sold handmade fans, mats, and plates from recycled material at a cost of
Rs. 25, Rs.100 and Rs. 50 each respectively. The numbers of articles sold are given as:
 School /Article SAJS LKO SAJS GZB SAJS KNP
Handmade fans 40 25 35
Mats 50 40 50
Plates 20 30 40

Based on the information given above, answer the following questions:

1. What is the total money (in Rupees) collected by the school SAJS LKO?
a. 700 b. 7,000 c. 6,125 d. 7,875

2. What is the total amount of money (in Rs.) collected by schools SAJS GZB and SAJS KNP?
a. 14,000 b. 15,725 c. 21,000 d. 13,125

3. What is the total amount of money collected by all three schools SAJS LKO, SAJS GZB and SAJS
KNP?
a. Rs. 15,775 b. Rs. 14,000 c. Rs. 21,000 d. Rs. 17,125

4. If the number of handmade fans and plates are interchanged for all the schools, then what is the total
money collected by all schools?
a. Rs. 18,000 b. Rs. 6,750 c. Rs. 5,000 d. Rs. 21,250

5. How many articles (in total) are sold by three schools?

a. 230 b. 130 c. 430 d. 330