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Kendriya Vidyalaya (O.F),Dumdum
Mathematics Worksheet
Prepared by:- Mrs. S. Basu
Class:-XII-B & C
Relation & Function
1.
Let R be a relation defined on the set of Natural numbers N as.
R  {( x, y ) : x, y  N ,2 x  y  4}
Find the domain & range of R. Also verify whether R is (i) Reflexive,
(ii)Symmitric (iii) transitive.
2.
Prove
that
the
relation
a, bRc, d   a  d  b  c for
R
all
on
the
set
NXN
a, b, c, d  NXN is
an
defined
by
equivalence
relation.
Also find the equivalence classes
3.
2,3& 1,3
Let N denote the set of all natural numbers & R be the relation on
NXN defined by (a, b) R(c, d ) ad (b  c)  bc(a  d ) , Check whether R is an
equivalance relation on NXN.
4.
Let n be a fixed positive integer. Define a relation R on Z as follows :
(a, b)  R  a  b is divisible by n.
Show that R is an equivalence relation on Z
5.
Show
that
the
function
a, b  R, a  0 is a bijection.
f : R  R given
by
f ( x)  ax  b .
Where
6.
Show that the function f : R  R given by f ( x)  Cosxx  R is neither
on-one nor on to.
7.
Let
A  R  {2} & B  R  {1} . If
f x  
f : A B
is a mapping defined by
x 1
, show that f is bijective.
x2
8.
Show that the function f : R  R given by f ( x)  x 3  x is a bijection.
9.
If the function f : R  R be given by f ( x)  x 2  2 & g : R  R be given by
g ( x) 
10.
x
find gof & fog.
x 1
If f , g : R  R are defined Respectively by f ( x)  x 2  3x  1, g ( x)  2 x  3, ,
find (i) fog (ii) gof (iii) fof, (iv) gog
11.
 x, if x  Q
Let A  x  R : 0  x  1. If f : A  A is defined by f x   
then
1  x if x  Q
prove fof ( x)  x that for all x  A .
12.
Let A  R  {3} & B  R  {1} . Consider the function f : A  B defined by
f ( x) 
13.
x2
. Show that f is one. One & onto & hence find f
x3
.
Consider the function f : R   [9, ] given by f ( x)  5 x 2  6 x  9 . Prove
that f is invertible with f 1 ( y ) 
14.
1
54  5 y  3
.
5
Let f : N  N be a function defined as f ( x)  9 x 2  6 x  5 . Show that
f : N  S where s is the range of f, is invertible. Find the inverse of f &
hence find f 1 (43) & f 1 (163) .
15.
Discuss the commutativity & associativity of binary operation
* defined on Q by the, rule a * b  a  b  ab for all a, b  Q .
16.
If the binary operation * on the set Z is defined by a * b  a  b  5 , then
find the identity element with respect to *.
17.
Let * be a binary operation Qo (Set of non zero rational numbers)
defined by a * b 
3ab
a, b  Q . Show that * is commutative as well as
5
associative. Also find its identity elemenet if it exists.
18.
Let * be a binary operation, on the set of all non zero real numbers,
given by a * b 
ab
for all a, b  R  {0} . Write the value of k given by
5
2 * ( x * 5)  10 .
19.
Let A = RoXR, where RO denote the set of all non zero real numbers. A
binary operation 'O' is defined on A as follows : (a, b)O(c, d )  (ac, bc  d )
for all (a, b), (c, d )  RoXR .
(i) Show that 'O' is commutative & associative on A
(ii) Find the indentity element in A.
(iii) Find the invertible element in A.
Kendriya Vidyalaya (O.F),Dumdum
Mathematics Worksheet
Prepared by:- Mrs. S. Basu
Class:-XII-B & C
Inverse Trigonometric Function
1.
Evaluate each of the following : 7 

1
0
1
0
(a) Cos 1  Cos
 (b) Sin Sin (600 ) (c) Cos Cos(680 )
6






2.

 Cosx   
 x .
Express the following in simplest form tan  1

2
 1  Sinx  2
3.
 1  Sinx  1  Sinx  x

Prove that, Cot 1 
 0 x
2
 1  Sinx  1  Sinx  2
4.

 1 x2  1 x2
Prove that, tan 
2
2

 1 x  1 x
5.
3

4
3

x .
Simplify : Cos 1  Cosx  Sinx  where
4
4
5
5

6.
1
4

Evaluate : Cos Sin 1  Sec 1 
4
3

7.
Prove that : tan 2 ( Sec 1 2)  Cot 2 (Co sec 1 3)  11.
8.
If Sin Cot 1 ( x  1)  cos(tan 1 x) , then find x.
9.
3

Solve the following equation for x : Costan 1 x   Sin Cot 1 
4

10.
3
3
6

Prove that : Cos Sin 1  Cot 1  
5
2  5 13

1


  1
1 2
   Cos x 1  x  1

 2 2

11.
Prove that : tan 1
1
1
1 
 tan 1  tan 1 
2
5
8 4
12.
Prove that : tan 1
3
3
8 
 tan 1  tan 1 
4
5
19 4
13.
 8 
Solve for x : tan 1 x  2  tan 1 x  2  tan 1   x  0
 79 
14.
Solve for x : tan 1
15.
 x2
 x2 
Solve for x : tan 1 
  tan 

 x4
 x4 4
x
x 
 tan 1  ,0  x  6
2
3 4
Kendriya Vidyalaya (O.F),Dumdum
Mathematics Worksheet
Prepared by:- Mrs. S. Basu
Class:-XII B & C
: Matrics :
1.
2  1
 0 4
If A = 
&B= 
find 3 A2  2 B  I .


3 2 
 1 7 
2.
 Cos 2
If A = 
 Sin 2
3.
1 0 2 1
If 1 1 x 0 2 1 1  0 , find x .

 
2 1 0 1
4.
 3 1
IF A = 
, show that A2  5 A  7 I 2  0

 1 2
5.
  1

If A =  2  & B =  2  1  4 verify  AB   BA
 3 
6.
4 2  1
7  as the sum of a symmetric & skew
Express the matrix A = 3 5
1  2 1 
Sin 2 
, find A2 .

Cos 2 
symmetric matrix.
7.
1 3  y 0 5 6
Find the values of x & y if 2



0 x   1 2 1 8
8.
Cos
If A = 
 Sin 
 Sin  
is identify matrix, then write the value of  .
Cos 
9.
Construct a 2 x 2 matrix A = [Cuj] whose element aij are given by
  3i  j

if i  j
aij  
2
 (i  j ) 2 if i  j

2  2
1

0  by using elementary
10. Find the inverse of the matrix. A =  1 3
 0  2 1 
row transformations.