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HOTS QUESTIONS FOR BRIGHT STUDENTS
CLASS - XII

SUB : MATHEMATICS

CHAPTER 1
RELATIONS AND FUNCTIONS
show that fof-1 is an identity function
...
If f(x) =
2
...

Ans
...
If f(x) = ax + b and g(x) = cx + d, then show that f[g(x)] – g[f(x)] is
equivalent to f(d) – g(b)
...
If A= {0,2,3} , find the number of binary operations on A
...
39
5
...
0
6
...

Ans- +2
7
...

Ans:
8
...


9
...
Let

Ans:x

...
Binary operation * on the set of Integers is defined as a * b = a + b + 1 for a, b Z, then
what is the inverse of – 2?
Ans
...
Find the Total number of equivalence relations defined in the set
S = {a, b, c}
Ans
...
Let A = {1, 2, 3}
...

Ans
...
Let f and g be two functions from R to R given by f(x) = 2x - 3,
g (x) = x3 + 5
...

5
...

6
...

7
...
Also find
Ans
...
R is a relation on non empty set A, prove that R is symmetric iff R -1=R
...
If the function f:R
given by
is onto function , find the set A
...
Let f: N  R be a function defined as f ( x ) = 4x2 + 12x + 15
...
Find the inverse of f
...
Let * be a binary operation on the set Q of rational numbers as follows :
(i) a * b = a2 + b2 (ii) a * b = a + ab (iii) a * b =

ab
(iv) a * b = a – b
4

Find which of the binary operations are commutative and which are associative
...
Consider f : R   5,   given by f ( x)  9 x 2  6 x  5
...
find
the inverse of f, where R+ is set of non-negative real nos
...
A binary operation * on the set {0, 1, 2, 3, 4, 5} is defined as
a*b =
show that zero is the identity element for this operation and each non
zero element ‘a’ of the set is invertible with 6 – a being the inverse of a
14
...

15
...


16
...


for all a,

CHAPTER 2
INVERSE TRIGONOMETRIC FUNCTIONS
(OF 1 MARKS)
1
...

Ans
...
Find the value of sin(2sin−1 0
...

3
...

Ans
...
Find the value of sin (sin 10)
Ans
...
If two angles of triangle are
and
find third angle
...
Find the value of
+
-1
-1
7
...
If cos x = tan x, show that sin (cos x) = x

)

(of 4 marks)

1
...
Prove that tan x  cos
 2 1  x2 
2


 1  x 2  1 x 2  
1
   cos 1 x 2
4
...
Solve for x if tan-1 (x+1)+ tan-1 x + tan-1 (x-1) = tan-1 3

6
...
Let A is an invertible square matrix of order 2 such that det(A)=10, find the Value
of det (

)
Ans
...
If A  
then find A100
...
299A

1 1
3
...
625
(of 4 marks)
1
...

2
...


 2 4 0 2 1 6 
The matrix X if 
X


1 3 1 3 3 2

where

1  53 18 
An X   
4  25 10 

(of 6 marks)
1
...
If a, b, c are all positive and are pth , qth and rth terms of G
...
, then find the value of

log a

p 1

log b

q 1

log c

r

(Ans-o)

1

3
...
P
...
Show that: 



1
1
tan  / 2
  tan  / 2

5
...

cos  
(Ans-o)

6
...
Using elementary operation , find inverse of matrix=
8
...
Three types of foods F1 ,F2, and F3 are available
...
One
unit of food F2 contains at one unit of vitamin A , 2 units of vitamin B and 1 unit
of vitamin C
...
Represents the above situation algebraically and find the
diet contains each types of food by using matrix method
...
A school wants to award its students for the values honesty , regularity and hard
work with a total cash award of ` 6000
...
The award money given
for honesty and hard work together is double the one given for regularity
...
Apart from these values, namely honesty, regularity

and hard work, suggest one more value which the school must include for award
...
If a  b  c  0 and c
bx
a  0 then prove that either x  0 or
b
a
cx

x

3 2
(a  b 2  c 2 )
2

11
...
Write the points where the function f ( x)  x  x  1 is not differentiable
...
Find the interval where the function log(1+x) is continuous
Ans: (-1,
| x  2|
3
...
Discuss the continuity of the function f(x) = 1/x
...

Ans:
dx

1
...
If y =

then show that (

3
...
If y =

, then show that ( 1-

5
...


6
...


Find

, if

8
...


9
...
Show that the function f(x) defined as follows is continuous but not differentiable
at x= 2 :
where f(x)=
11
...

Ans
...
Discuss the continuity of the function f(x) =

at x=0
...
Continuous at x= 0
12
...
If
dx
Ans: 0

1  cos 4 x
ifx  0

2
14
...

15
...

16
...
Find derivative of sinx + x with respect to x
...
If x= a(cos t+t sin t) and y= a(sin t- tcos t ), find d2y/dx2
...
If y = tan-1
20
...


sinx

prove that

=

then find

Ans:sec 3t/at

+
in terms of y and it’s first derivative
...
How will you interpret geometrically of a function y=f(x) at a point ( x0 , y0 )
...
slope of tangent
2
...
Find the values of a and b
...
a=1/2, b=-1
3
...
72 units/sec
3

(of 4 marks)
1
...
tan-1
2
...


3
...


, is strictly increasing

Ans: strictly decreasing in (-∞,1/e) and strictly

increasing in (1/e,∞)
4
...

5
...
For what values of x is the rate of increase of x  5x 2  5x  8 is twice the rate
of increase of x?
7
...
If the rate of change of a side
is R times the rate of change of the opposite angle
...

3

8
...

Ans: a=1, b=2
9
...
Find the equation of
the curve where their points of contact lie
...
Find the intervals in which function f given by
f(x) = sin x + cos x , 0
Is strictly increasing or strictly decreasing
...
A given quantity of metal to be cast into half cylinder with a rectangular base and
semicircular ends
...

2
...

3
...


4
...
x+2y=

, 2x+y=

5
...
2ab sq
...
A sheet of paper for a poster is 15000 sq
...
in area
...
wide and at the sides 4 cm
...
Find the dimensions of
the sheet to maximize the printed area
...
One side 150 cm and other side 100 cm
7
...
Show that the maximum capacity of
the box is

a3
6 3


...
An enemy helicopter is moving along the curve y=x2+7
...
Find the nearest distance
...

Ans:
9
...

10
...
One of the pieces is
to be made into a square and other into a circle
...

28
 112

Ans: 
cm
...

 4 
  4



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