MANAV RACHNA INTERNATIONAL SCHOOL
EROSGARDEN, CHARMWOODVILLAGE, DELHI-SURAJKUND ROAD, FARIDABAD
Session 2023 – 2024
GRADE XII
DATE: 07/07/2023
SUBJECT: MATHEMATICS
LEVEL - 1
TOPIC: Relations , Functions , Inverse Trigo , Matrices & Determinants
1.
2.
Check the following relations R and S for reflexivity, symmetry and transitivity.
(i)
aRb iff b is divisible by a, b, b  N
(ii)
1S 2 iff 1   2 , where 1 and  2 are straight lines in a plane.
Let a relation R1 on the set R of real numbers be defined as (a, b)  R 1  1  ab  0 for all a, b  R . Show
that R1 is reflexive and symmetric but not transitive.
3.
Determine whether each of the following relations are reflexive, symmetric and transition :
(i)
Relation R on the set A = {1, 2, 3, ......., 13, 14} defined as R = {(x, y) : 3x – y = 0}
(ii)
Relation R on the set N of all natural numbers defined as R = {(x, y) : y x + 5 and x < 4}
(iii)
Relation R on the set A = {1, 2, 3, 4, 5, 6} defined as R = {(x, y) : y is divisible by x}
(iv)
Relation R on the set Z of all integer defined as R = {(x, y) : x – y is an integer}
4.
Show that the relation R on R defined as R = {(a, b) : a  b}, is reflexive and transitive but not symmetric.
5.
Let S be the set of all points in a plane and R be a relation on S defined as R = {(P, Q) : Distance between P
and Q is less than 2 units. Show that R is reflexive and symmetric but not transitive.
6.
Let X = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Let R1 be a relation on X given by R1 = {(x, y) : x – y is divisible by 3} and R2 be
another relation on X given by R2 = {(x, y) : {x, y}  {1, 4, 7} or {x, y}  {2, 5, 8} or {x, y}  {3, 6, 9}.
Show that R1 = R2.
7.
Show that the relation R on the set R of all real numbers, defined as R  {(a, b) : a  b 2 } is neither
reflexive nor symmetric nor transitive.
8.
Let R be a relation on the set of all lines in a plane defined by (1 ,  2 )  R  line l1 is parallel to line l2.
Show that R is an equivalence relation.
9.
Show that the relation R defined on the set A of all triangle in a plane as R = {(T 1, T2) : T1 is similar to T2} is an
equivalence relation.
10.
Let n be a positive integer. Prove that the relation R on the set Z of all integers numbers defined by
(x, y)  R  x  y is divisible by n, is an equivalence relation on Z.
11.
Show that the relation R on the set A = {1, 2, 3, 4, 5}, given by R = {(a, b) : |a – b| is even}, 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}
12.
Show that the relation R on the set A  {x  Z : 0  x  12} , given by R  {(a, b) :| a  b | is a multiple of
4} is an equivalence relation. Find the set of all elements related to 1 i.e. equivalence class [1].
13.
Show that the function f : N  N, given by f(x) = 2x, is one-one but not onto
14.
Prove that f : R  R , given by f(x) = 2x, is one-one and onto.
15.
Show that the function f : R  R, defined as f(x) = x2, is neither one-one nor onto.
16.
Show that f : R  R, defined as f(x) = x3, is a bijection.
17.
Show that the function f : R 0  R 0 , defined as f (x) 
1
, is one-one onto, where R0 is the set of all nonx
zero real numbers. Is the result true, if the domain R0 is replaced by N with co-domain being same as R0?
18.
Prove that the greatest integer function f : R  R , given by f(x) = [x], is neither one-one nor onto, where
[x] denotes the greatest integer less than or equal to x.
19.
Show that the modulus function f : R  R, given by f(x) = |x| is neither one-one nor onto.
20.
Let C and R denote the set of all complex numbers and all real numbers respectively. Then show that
f : C  R given by f(z) = |z| for all z  C is neither one-one nor onto.
21.
Show that the function f : R  R given by f(x) = ax + b, where a, b  R , a  0 is a bijection.
22.
Show that the function f : R  R given by f(x) = cosx for all x  R , is neither one-one nor onto.
23.
Let A = R – [2] and B = R – [1]. If f : A  B is a mapping defined by f (x) 
24.
Let A and B be two sets. Show that f : A  B  B  A defined by f(a, b) = (b, a) is a bijection.
25.
Let A = [1, 2]. Find all one-to-one functions from A to A.
26.
Consider the identity function I N : N  N defined as, I N (x)  x for all x  N. Show that although IN is
x 1
, show that f is bijective.
x2
onto but I N  I N : N  N defined as (I N  I N )(x)  I N (x)  I N (x)  x  x  2x
27.
Consider the function f :[0,  / 2]  R given by f (x)  sin x and g :[0,  / 2]  R given by
g(x) = cosx. Show that f and g are one-one, but f + g is not one-one.
28.
Find the domain of the function f (x)  sin 1 (2x  3)
29.
Find the domain of 𝑓(𝑥) = 𝑠𝑖𝑛
(−𝑥 ).
30.
Find the domain of 𝑓(𝑥) = 𝑠𝑖𝑛
𝑥 + cos 𝑥
31.
If x, y, z  [ 1, 1] such that sin 1 x  sin 1 y  sin 1 z  
32.
Let x, y, z  [ 1, 1] be such that sin 1 x  sin 1 y  sin 1 z 
(i)
x 2018  y 2019  z 2020
(ii)
3
, find the value of x 2  y 2  z 2 .
2
3
. Find the values of
2
x 2016  y 2018  z 2020 
9
x
2016
y
2018
 z 2020



1 

2 
33.
Find the principal values of cos 1 sin  cos 1
34.
If x, y, z  [ 1, 1] such that cos 1 x  cos 1 y  cos 1 z  0, find x + y + z.
35.
If x, y, z  [ 1, 1] such that cos 1 x  cos 1 y  cos 1 z  3 , then find the values of

(1)
xy + yz + zx
(ii)

x(y + z) + y(z + x) + z (x + y)
  
 .
 2 
36.
Find the principal values of tan 1 sin  
37.
For the principal values, evaluate : cot sin 1 cos(tan 1 1 
38.
Which is greater, tan1 or tan 1 1?
39.
Find the minimum value of n for which tan 1
40.
Find the principal values of sec 1
41.
Find the domain of sec 1 (2x  1)
42.
Find the principal values of cosec 1 (2) and cosec 1  
43.
For the principal values, evaluate each of the following :




n 
 , nN
 4
2
and sec 1 ( 2)
3


(i)
tan 1 3  sec 1 ( 2)  cosec 1
2
3
2 

3
(ii)
2sec 1 (2)  2 cosec 1 ( 2)
3
x y z
, find the value of  
2
y z x
44.
If cosec 1 x  cosec 1 y  cosec 1 z  
45.
Find the set of values of cot 1 (1) and cot 1 (1)
46.
For the principal values, evaluate : cot 1 ( 3)  tan 1 (1)  sec 1 
47.
Evaluate each of the following :
 2 

 3
(i)
7 

cos 1  cos 
6 

(ii)
sin 1  sin( 600o ) 
48.
Construct a 3 × 2 matrix A = [aij] whose elements is given by
49.
For what values of x and y are the following matrices equal?
3y 
 x  3 y2  2
 2x  1
A
,
B



y2  5y 
6 
 0
 0
aij  eix sin jx .
50.
Find non-zero values of x satisfying the matrix equation:
 x 2  8 24 
 2x 2 
8 5x 
x

2

2



 4 4x 
6x 
 3 x


 10
 2 2 0 
 2 0 2 
and C  


 3 1 4
7 1 6 
51.
Find a matrix A such that 2A – 3B + 5C = O, where B  
52.
Two farmers Ram Kishan and Gurcharan Singh cultivate only three varities of rice namely Basmati, Permal
and Naura. The sale (in Rs.) of these varities 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 Rs.)
Basmati
Permal Naura
10, 000 20, 000 30, 000  Ram Kishan
A

50,000 30, 000 10, 000  Gurcharana Singh
October sales (in Rs.)
Basmati
Permal Naura
 5, 000 10,000 6, 000  Ram Kishan
B

 20, 000 10,000 10, 000  Gurcharana Singh
Find :
(i)
What were the combined sales in September and October for each farmer in each variety.
(ii)
What was the change in sales from September to October?
(iii)
If both farmers receive 2% profit on gross rupees sales, compute the profit for each farmer and for
each variety sold in October.
x
7
5  3 4   7 14 


y  3 1 2  15 14 
53.
Find x, y, z and t, if 2 
54.
If X and Y are 2 × 2 matrices, then solve the following matrix equations for X and Y.
 2 3
 2 2 
2X  3Y  
, 3X  2Y  


4 0
 1 5
55.
The monthly incomes of Aryan and Babban are in the ratio 3 : 4 and their monthly expenditures are in the
ratio 5 : 7. If each saves Rs. 15000 per month, find the monthly incomes using matrix method. This problem
reflects which value?
56.
If A  
57.
If A  
 1 1
a 1 
2
2
2
, B

 and (A  B)  A  B , find a and b.
2

1
b

1




 0 1
2
 , find x and y such that (xI + yA) = A.
 1 0 
58.
 1 3 2  1 

 
Find the value x such that [1 x 1] 2 5 1 2  0

 
15 3 2   x 
59.
 2 1
 1 8 10


If 1
0 A   1 2 5  , find A.




 3 4 
 9 22 15 
60.
There are two families A and B. There are 4 men, 6 women and 2 children in family A and 2 men, 2 women
and 4 children in family B. The recommended daily allowance for calories is :Man : 24000, Women : 1900,
child : 1800 and for proteins is : Man : 55 gm, Women : 45 gm and child 33 gm.
Represent the above information by matrices. Using matrix multiplication, calculate the total requirement
of calories and proteins for each of the two families.
61.
Use matrix multiplication to divide Rs. 30,000 in two parts such that the total annual interest at 9% on the
first part and 11% on the second part amounts Rs. 3060.
62.
Three schools A, B and C organized a mela for collecting funds for helping the rehabilitation of flood
victims. They sold hand made 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:
School Article
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 sailing the above articles. Also, find the total funds
collected for the purpose.
63.
 cos   sin  0 
If A  sin  cos 
0  , find adj A and verify that A(adj A) = (adj A)A = |A|I3.


 0
0
1 
64.
If A  
65.
1 3 3 
Find the inverse of A  1 4 3  and verify that A 1A  I3 .


1 3 4 
66.
Show that A  
67.
Find a 2 × 2 matrix B such that B 
68.
Find the matrix A satisfying the matrix equation 
2 3 
1
, show that A 1  A

19
5 2 
 2 3 
satisfies the equation x 2  6x  17  0. Hence, find A 1 .

3 4 
1 2   6 0 

.
1 4   0 6 
 2 1   3 2  1 0 
A


 3 2   5 3   0 1 
69.
4 5 / 2 
0 1 3 
1/ 2



1
If A  1 2 x and A  1/ 2 3
3 / 2  , find x, y.




 2 3 1 
1/ 2
y 1/ 2 
70.
If A  
71.
2 2 0


Find the matrix A such that | A |  2 and adj A  2 5 1


 0 1 1 
72.
 1 2 1 

Find the non-singular matrices A, if its is given that adj(A)  3
0 3


 1 4 1 
73.
Solve the following system of equations, using matrix method :
sin  
 cos 
T
1
 is such that A  A , find 

sin

cos



x + 2y + z = 7, x + 3z = 11, 2x – 3y = 1
74.
Show that the following system of equations is consistent.
2x – y + 3z = 5, 3x + 2y – z = 7, 4x + 5y – 5z = 9. Also, find the solution.
75.
 4 4 4   1 1 1 
Determine the product  7 1
3   1 2 2  and use it to solve the system of equations:



3 
 5 3 1  2 1
x – y + z = 4, x – 2y – 2z = 9, 2x + y + 3z = 1
76.
 1 2 3

Find A , where A  2 3
2  . Hence solve the system of equations


 3 3 4 
1
x  2y  3z  4, 2x  3y  2z  2, 3x  3y  4z  11