MATHS

ASSIGNMENTS
Exercise - 01
CBSE FLASH BACK
1. Let ‘ f ‘ be the exponential function ex and ‘ g ‘ be logarithmic function  n x. Find (f + g) (1) .
[ 1994 ]

2. If f (x) = sin x , g (x) = cos x and h (x) = 2 x . ‘ f ‘ , ‘ g ‘ and ‘ h ‘ being real valued function .
Show that , ho(f g) = foh . [ 1995 ]

3. Find gof and fog , If f (x) = 8 x3 and g (x) = x 1/3 . [ 1995 ]

1
4. If f (x) = x 2 + 3 and g (x) = , find :
x 1
(i) fog (ii) gof (iii) fof (iv) gog
[ 1997 ]

5. Let ‘ f ‘ be the greatest integer function and ‘ g ’ be the absolute value function , find the value of :
5  5
(gof)   – (fog)   . [ 1998 ]
3  3

6. If f (x) = x , (x > 0) and g (x) = x2 – 1 , find if fog = gof . [ 2002 ]

x 1
7. If f (x) = , (x  1 , – 1) , show that fof–1 is an identity function. [ 2002 ]
x 1

5x  3  5
8. If f (x) = ,  x   , show that f {f (x)} is an identity function. [ 2002 ]
4x  5  4

9. If f (x) = ex and g (x) =  n x , show that fog = gof , given x > 0 . [ 2003 ]

7 3 3x  4 3 2

10. If f : R –    R –   be defined as f (x) = and g : R –    R –   be defined
5 5 5x  7 5 5

7x  4 3 2
as g (x) = . Show that fog = I A and gof = I B , where A = R –   , B = R –   .
5x  3 5
  5
I A (x) = x ,  x  A and I B (x) = x  x  B are identity functions.
[ 2005 ]

11. Consider the function , f : {1 , 2 , 3}  {a , b , c} given by f (1) = a , f (2) = b and f (3) = c .

Find , ( f –1)–1 . Show that ( f –1)–1 = f . [ 2006 ]
 Exercise - 02
OBJECTIVE
1. Complete solution set of the inequality x (ex – 1) (x + 2)(x – 3)2  0 is :
(A) [–2, 3] (B) (– 2, 0]
(C) (– , – 2]{0, 3} (D) (– , – 2)[0, 3]
2. If log1/2 (x 2 – 5 x + 7) > 0 , then exhaustive range of values of x is :
(A) (–  , 2)  (3 , ) (B) (2 , 3)
(C) (–  , 1)  (1 , 2)  (2 , ) (D) None of these

 x 
3. Solution set of log( x 2 )   x   0 is :
 x 
 
(A) (–  , 0)  (1 , 2) (B) (–  , 1)  (2 , )
(C) (–  , – 1)  (0 , 1) (D) (–  , – 2]  (0 , 1)
4. If sin x + cos x = sin x + cos x, then x belongs to the quadrant,
(A) I or III (B) II or IV (C) I or II (D) III or IV

1
5. The domain of the function y =
x x
(A) (–  , 0) (B) (–  , 0] (C) (–  , – 1] (D) (–  , + )

1 x
6. The domain of the function f (x) = is :
x 2

(A) (–  , – 1)  (1 ,  ) (B) (–  , – 2)  (2 ,  )
(C) (– 2 , – 1]  [1 , 2) (D) none of these

 1  x2 
7. The number of solutions of the equation , sin–1   =  sec (x – 1) is/are :

 2x  2
(A) 1 (B) 2 (C) 3 (D) infinite

x 2  7 x  10
8. Let f(x) = , then the range of f (x) is :
x2  5 x  6
(A) R (B) R – {1} (C) R – {3) (D) R – {1, 3}

x2
9. If the function f : R  A given by f (x) = is a surjection, then A is :
x2  1
(A) R (B) [0, 1] (C) (0, 1] (D) [0, 1)
10. If f : R  R be a function such that f(x) = x + x + 3x + sin x. Then :
3 2

(A) f is oneone into (B) f is oneone and onto

(C) f is many one and into (D) f is many one and onto

  3 
11. The number of roots of the equation , cot x = + x in    , is :
2  2 
(A) 3 (B) 2 (C) 1 (D) infinite
12. Total number of solution of 2x + 3x + 4x – 5x = 0 is/are :
(A) 1 (B) 2 (C) 3 (D) none of these
 3 x 5 x
13. Period of the function , sin + cos is
2 2
(A) 2  (B) 10  (C) 8  (D) 5 
14. f:[2, )  (– , 4], where f(x) = x(4 – x) then f –1 (x) is

(A) 2 – 4x (B) 2 + 4x (C) – 2 + 4x (D) – 2 – 4x

15. If f : R  R where f (x) = ax + cos x is an invertible function then

(A) a  (– 2, 1][1, 2) (B) a  [– 2, 2]
(C) a  (– , – 1][1, ) (D) a  [– 1, 1]

x
16. If f(x) = , then (fof of) (x) is
1  x2

3x x 3x
(A) (B) (C) (D) none of these
2
1 x 2 1 3x 1  x2

 1
17. If 3 f (x) – f   = log x4, then f(e–x) is
x

1
(A) 1+ x (B) (C) x (D) – x
x

18. If f (x) + 2 f (1 – x) = x2 + 1  x  R then f(x) is

1 (x2 + 4 x – 3)
(A) 3 2 (x2 + 4x – 3)
(B) 3

1 (x2 – 4 x + 3)
(C) 3 2 (x2 – 4x + 3)
(D) 3

n
19. Let ‘ f ’ be a function satisfying f (x + y) = f (x) . f (y) for all x , y  R . If f (1) = 3 , then  f (r) is
r 1
equal to

(A) 3
2 (3 – 1)
n
(B) 3
2 n (n + 1) (C) 3n + 1  3 (D) None of these

20. If f (x + 2y, x – 2y) = xy, then f (x, y) equals :

x2  y2 x2  y2 x2  y2 x2  y2
(A) (B) (C) (D)
8 4 4 2
 Exercise - 03
SUBJECTIVE
(1) Find the domain of definitions of the following functions.
1. f (x) = log7 log5 log3 log2 (2 x3 + 5 x2 – 14 x)
2. f(x) = log10(1  log10(x2  5x + 16)) .
3. f(x) = logx sin x .

 
x  3 ( x  2)
4. f (x) = logx
( x  1) x

5. The function f (x) is defined on the interval [0 , 1] . Find the domain of definition of the functions :
(a) f (sin x) (b) f (2 x + 3)

x 3  2  x 
6. f (x) =
x ( x  2)

(2) Find the range of the following functions.

x 1  2 
1. f (x) = . 2. f (x) = sin–1  x  x  1 
x2  2 x  3  

1
3. f (x) = x2 + 2
x 1

(3) Find the domain and range of the following functions ,

1. f (x) =
x
1 x
. 2. y = log 5
 2 (sin x  cos x)  3 
3. f (x) = cos (sin x ) . 4. y = sec –1 (log3 tan x + logtan x 3)

(4) Identical function.

Find for what values of x, the following functions be identical.

 x  1
1. f (x) =  n (x – 1) –  n (x – 2) and g (x) =  n  
 x  2

1
2. f (x) = logx e , g (x) =
log e x

3. f (x) = sec2 x – tan2 x; g(x) = cosec2 x  cot 2 x .

4. f (x) = sin(cos1 x); g(x) = cos(sin1 x) .

x2
5. f (x) = , g (x) = x.
x
 MATHS
(5) Types of functions.
Classify the following functions f (x) defined in R  R as injective , surjective , both or none.

x 2  4 x  30
1. f (x) = 2. f (x) = x3 – 6 x2 + 11 x – 6
x 2  8 x  18

3. f : A  B where f (x) = x2 – 2 x + 2 , A  (– 1 , 2) and B  [1 , 5]

x2
4 Let A = R – {3}, B = R – {1} and let f : A  B defined by f (x) = is ‘ f ‘ bijective ?
x3

5. Let A = { x : – 1  x  1 } = B . For each of the following functions from A to B, find whether

it is surjective , injective or bijective.
x
(a) f (x) = (b) g (x) = sin x (c) i (x) = x2 (d) j (x) = sin  x
2

(6) Inverse function.

Compute the inverse of the following function

 x ,   x  1
10 x  10  x  2
1. f : R  (–1,1) y = 2. f : R  R , f (x) =  x , 1  x  4 .
10 x  10  x  2x , 4  x , 


3. Let g : R  R be given by g (x) = 3 + 4x. If gn(x) = gog .... og ( x ) , show that gn(x) = (4n  1) + 4n x .
n times
If gn(x) denotes the inverse of gn(x) , prove that the above formula hold for all negative integers.

(7) Even or Odd function.

Find whether the following functions are even or odd or neither even nor odd.

1.
 2
f (x) = log  x  1  x  2. f (x) =

x ax  1 
x
  a 1

3. f (x) = (x  1) 
2 1/ 3

+ ( x  1)2 
1/ 3
4. f(x) = x 1 x 2  1 x  x2

(8) Periodic function.

Find the period for each of the following functions.
1. f (x) = sin x+ cos x


2. Let f (x) = 2  cos x +  2 sin x  + 3 . If is the fundamental period of f (x), find  .
2
 MATHS
(9) Composite Function.
1. Let f (x) = max. {1 + sin x , 1 – cos x , 1}  x  [0 , 2 ] and
g (x) = max. { 1 , x –1}  x  R.
Determine f {g (x)} and g {f (x)} in term of ‘ x‘ .

 1 , 2  x  0
2. Let f (x) be defined on [– 2 , 2] and is given by , f (x) =  and
x 1 , 0  x  2
 
g (x) = f x  f ( x ) , then find g (x) .

3. If f (x) = – 1 + x – 2, 0  x  4 and g (x) = 2 – x, – 1  x  3 .

Then find fog (x) and gof (x) . Draw rough sketch of the graphs of fog (x) and gof (x) .

(10) Functional equation.

1
1. The function f (x) has the property that for each real number ‘x’ in its domain , is also in its domain
x
 1
and f (x) + f   = x . Find the largest set of real numbers that can be in the domain of f (x) ?
x

2. If f (2 x + 1) = 4 x2 + 14 x , then find the sum of the roots of the equation , f (x) = 0 .

 1 1
3. If for nonzero x , a f (x) +b f   = – 5 , where a  b , then find f (x).
x x

4. If function , f (x) is satisfying 2 f (sin x) + f (cos x) = x for all x  R , then express f (sin x) as a
polynomial in x .

2n  1
ax  r 
5. If f (x) = x
a  a
(a > 0) . Evaluate  2 f   .
 2n 
r 1

(11) Graphs
Draw the graph of each of the following function.

1. y = |1 – | x2 – 2 || 2. y = ex + e–x 3. y = | n | x ||
4. y = min {| x |, | x – 2 |, 2 – | x – 1 | } 5. y = sgn (x – | x | )
 Exercise - 04
OBJECTIVE
1. The domain of the function , sec1 [x2  x + 1] , is given by [ where [.] is greatest integer function ]
(A) (–  , ) (B) (–  , 0]  [1 , ) (C) [2 , )(D) None of these

1
2. The domain of the function , f (x) = where { . } denotes fractional part, is :
sin x   sin (   x ) 
n
(A) [0 , ] (B) R – ,nI (C) (0 , ) (D) None of these
2

3. The domain of f (x) =  x   1  x2 ; where [ . ] denotes the greatest integer function is :

(A) (–  . – 2)  [1 , )  
(B)   ,  2  [1 , )

(C) (–  . – 3)  [1 , ) (D)   ,  3   [1 , )
4. The domain of f (x) = cos (sin x ) + log x  x  ; { . } denote the fractional part, is :

 
(A) [1 , ) (B) (0 , 2 ) – [1 , ) (C)  0 ,  – {1} (D) (0 , 1)
 2 

1
5. Let f (x) = , [ . ] denotes the greatest integer function , then domain of f (x) is :
x  1  [x]
(A) (–1, 1) (B) (– , 1) (C) (–  , – 1) (D) None of these

6. Let f (x) = cot (5  3 x ) cot (5 )  cot (3 x )   cot 3 x  1 , then domain is :

 n  
(A) R –   ,nI (B) (2 n + 1) , nI
 3  6

 n n  5   n  5 
(C) R –  ,  ,nI (D) R –   , nI
 3 3   3 

sin x cos x
7. Let f (x) = – , then range of f (x) is :
2
1  tan x 1  cot 2 x
(A) [1 , 0] (B) [0 , 1] (C) [1 , 1] (D) none of these

8. Total number of positive real values of ‘ x ‘ satisfying 2 [ x ] = x + { x } , where [] and {} denotes the
greatest integer function and fractional part respectively, is :
(A) 2 (B) 1 (C) 0 (D) 3

9. The complete set of values of ' a ' for which 4  x 2 – ex + a = 0 has one and only one positive
solution is:
 1 2  2 1   1 2
(A)  2 , e  (B)   e , 2  (C)  2 , e  (D) None of these
e   e  e 
 MATHS

10. Total number of solutions of the equation , sin  x =  n x are :

(A) 8 (B) 10 (C) 9 (D) 6
11. f : R  (6 , ) , f (x) = x2 – (a – 3) x + a + 6 , then the values of ' a ' for which the function is onto is
(A) (1, 9) (B) [1, 9] (C) {1, 9} (D) none of these
x5 , xQ
12. Let f : R  R , be a function such that f(x) =  , then
 x  5 , x  Qc
(A) f is one–one and onto (B) f is one–one and into
(C) f is many one and into (D) f is many one and onto

x , if ' x' is rational

13. On [ 0, 1 ] , f (x) is defined as f (x) =  . Then for all x  [0 , 1] , f (f (x)) is :
 1  x , if ' x' is irrational
(A) constant (B) 1 + x (C) x (D) none of these

 
14. Let f : R  0 ,  (where R is the set of real numbers) be a function defined by ,
 2
f (x) = tan1 (x 2 + x + a) . If ‘ f ‘ is onto then ‘ a ’ equals :
(A) 0 (B) 1 (C) 1/2 (D) 1/4

15. Fundamental period of the function, f (x) = cos (tan x + cot x) . cos (tan x  cot x), is
 
(A) (B) (C)  (D) 2 
4 2

16. f(x) is an odd function and g(x) is neither odd nor even , then
(A) f (x) + g (x) is neither even nor odd (B) f (x) + g (x) is even
(C) f (x) + g (x) is odd (D) none of these
17. If f : I  I be defined by f(x) = [x + 1], where [.] denotes the greatest integer function, then f 1(x) is
equal to
1 1
(A) x  1 (B) [x + 1] (C) (D)
 x  1 x 1

 2 n x  2 
18. If f (x) = log e2 x   and g (x) = {x} then range of g (x) for the existence of f (g (x)) is :
 x 

 1  1   2  1   3  1 
(A)  0 ,  ~  2  (B)  0 ,  ~  2  (C)  0 ,  ~  2  (D) None of these
 e e   e e   e e 

19. If ‘ f ’ is a real valued function not identically zero, satisfying f (x + y) + f (x – y) = 2 f(x) . f (y)
 x , y  R , then f (x) is :
(A) odd (B) even
(C) neither even nor odd (D) None of these

20. If f : R  R , g : R  R , be two given functions then h (x) = 2 min (f (x)  g (x), 0) equals :
(A) f (x) + g (x)  g (x)  f (x) (B) f (x) + g (x) + g (x)  f (x)
(C) f (x)  g (x) + g (x)  f (x) (D) f (x)  g (x)  g (x)  f (x)
 MATHS

Exercise - 05
SUBJECTIVE
(1) Find the domain of definitions of the following functions.
  
 1  +
f (x) = log2   log1/ 2 1  log10 log10 x   log10 4  log10 x   log10 3
1.




 
xº
sin 100  


1
2. if f (x) = x2  5 x  4 + 7 x  5  2 x  +   n  72  x  
2

3. f (x) = log x cos 2  x 

4. f (x) = x12  x 9  x 4  x  1

 1  x2 
5. f (x) = sin 1 log2 x  + cos (sin x ) + sin–1  
 2x 
 

3 1 2 (2 x  1) !
6. f (x) = – 5 cos x +
x 2  x 1
, where [ . ] denotes greatest integer function.

(2) Find the range of the following functions.

1.  
f (x) = tan–1 log 4 / 5 5 x 2  8 x  4 
x
2. f (x) = cos–1 log[ x ] , where [ . ] denotes greatest integer function .
x

3. f (x) = sin–1 x + cos–1 x + tan–1 x .

(3) Find the domain and range of the function ,

x2  3 x  2
1. f (x) = .
x2  x  6

ex
2. y= , find range only for x  0 , where [ . ] denotes greatest integer function .
1 [x]

3. y= (1  cos x ) (1  cos x ) ....  .

  1 
4. y =   n  sin x 2  x  1   , where [ . ] denotes greatest integer function .
  
 MATHS
(4) Types of functions.
Classify the following functions f (x) defined in R  R as injective , surjective, both or none.
1. Find the set of values of ‘ a ‘ for which the function f : R  R given by :
f (x) = x3 + (a + 2) x2 + 3 a x + 5 is one  one.

 1 
2. f :  ,     R , f (x) = (x2 + x + 5) (x2 + x – 3) .
 2 

 
3. Let f : X  Y be a function defined by f (x) = a sin  x   + b cos x + c . If ‘ f ‘ is both oneone
 4
and onto, find sets X and Y .

4. Check whether f : (–  , 2)  (3 ,  )  R , f (x) =  n (x2 – 5 x + 6) is a bijective function or not. If

not then choose a suitable longest domain and codomain for which the above function becomes
bijective.

x2  3 x  c
5. Let f : R  R defined by f (x) = . Show that ‘ f ‘ is a many one function for all c.
x2  x  1

(5) Inverse Function

Compute the inverse of the following function
x
x 1
1. Compute the inverse of the function , f : R – {1}  R+ – {2} , f (x) = 2 .
1
Also find domain and range of f .

2. Find the minimum value of ‘a’ and ‘b’ for which f (x) = xx ; [a , )  [b ,  ) be an invertible function.

3. Let f (x) = x2 + 3x  3 , x  0 . n points x 1 , x2 , ....... , xn are so chosen on the xaxis that :

1
n 1 n  n n
 
(a)
n i 1

f –1 ( xi ) = fn  i 
x (b) 
f –1
( x i
) =  xi
 i 1  i 1 i 1
1
where f denotes the inverse of ‘ f ‘ . Find the A.M. of xi’s .

(6) Even or Odd functions.

Find whether the following functions are even or odd or none.
1. f (x) = x sin2 x – x 3 2. f (x) = sin x – cos x

x x
3. f (x) = x + +1 4. f (x) = sin x + cos x
e 1 2

 x 4 tan  x , x 1
5. f (x) =  2 . Prove that f (x) is an odd function .
 x x , x 1
 MATHS
(7) Periodic functions.
Find the period for each of the following function.
1. Prove that if the graph of the function, y = f(x), defined throughout the number scale, is
symmetrical about two lines x = a and x = b,(a < b), then the function is a periodic one.
OR
If f (a  x) = f (a + x) and f (b  x) = f (b + x) for all real x , where a , b (a < b) are constants , then prove
that f(x) is a periodic function.
2. Find out the integral values of ‘ n ‘ , if 3  is a period of the function :
5
f (x) = cos nx . sin   x .
n

3. Let ‘f’ be a real valued function defined for all real numbers ‘ x ‘ such that for some positive constant
1
‘a’ the equation , f (x + a) = + f ( x )   f ( x )2 holds for all x. Prove that the function f is periodic.
2

4. (a) f (x) = sin (x + sin x) (b) f (x) = sin (cos x ) + cos (sin x)
x x x x x
(c) f (x) = sin x + tan + sin 2 + tan 3 + ... + sin n  1 + tan n
2 2 2 2 2

(8) Composite Function.

1. Find fog (x) if f (x) = [ x ] + { x } 2 and g (x) = [ x ] + {x} and also find the range of fog (x) .

 1
2. Function ‘ f ‘ and ‘ g ‘ are defined by f (x) = sin x , x  R ; g (x) = tan x , x  R –  K    ,
 2
where K  I . Find : (a) periods of fog and gof (b) range of the function fog and gof .

3. If f (x) = – 1 +  x – 2 , 0  x  4
g (x) = 2 –  x , –1x3
Then find fog (x) and gof (x) .

 1  x 2 , x  0  tan x , 0  x  2
4. If f (x) =  and g (x) =   then find the fog (x) .
 3  x 2 , x  0  cos ec x ,  2  x  0

(9) Functional equations.

1. Let f (x) be a function with two properties :
(i) for any two real numbers ‘x’ an ‘y’ , f (x + y) = x + f (y) and
(ii) f (0) = 2
Find the value of f (100) .
2. Let ‘ f ‘ be a function such that f (3) = 1 and f (3 x) = x + f (3 x – 3) for all ‘ x ‘ . Then find the value of
f (300) .
3. A function f : R  R satisfies the condition , x2 f (x) + f (1 – x) = 2 x – x4 . Find f (x) .
4. If for all real values of u and v , 2 f (u) cos v = f (u + v) + f (u  v), prove that, for all real values
of x
(a) f(x) + f(x) = 2a cos x . (b) f ( – x ) + f (– x) = 0
(c) f ( – x) + f (x) = – 2 b sin x
Deduce that f (x) = a cos x  b sin x , where a , b are arbitrary constants .
 MATHS

 2 2 
5. Let f (x) + f (y) = f  x 1  y  y 1  x  . Prove that f (4 x3  3 x) + 3 f (x) = 0 ,
 

 1 1   1   1 
 x   ,  also prove that f (x) = 0 ,  x   1 ,     , 1
 2 2  2  2 

(10) Graphs.
Draw the graph of each of the following function.

1. y=1+ sin x 2. y = log x

 3 
3. y = max {1 – x, 1 + x, 2} 4. y = min  e x , , 1  e x  , 0  x  1
 2 

11. A is a point on the circumference of a circle. AB and AC divide the area of the circle into three
equal part. If the angle BAC is the root of the equation, f(x) = 0 then find one such f(x).
12. Find the integral solutions to the equation [x] [y] = x + y. Show that all the non-integral
solutions lie on exactly two lines. Determine these lines.
 MATHS

Exercise - 06
IIT NEW PATTERN QUESTIONS
Section I
Fill in the blanks :
1. The number of solutions of 2x – x2 + 1 = 0 is ________ .
2. Solution of the equation x + [y] + {z} = 3.1
{x} + y + [z] = 4.3
[x] + {y} + z = 5.4
( where [ . ] denotes the greatest integer function { . } denotes fractional part ) is ________ .
3. The domain and the range of f (x) = sin–1x + cos–1x + tan–1 x + sec–1 x + cot–1 x + cosec–1 x is
________ .

4. The number of solution of , 2 x + 3  (cos x – 1) = 0 is ________ .

1
5. The domain of f (x) = is ________ .
1
3x  x 1
2

f (x)  f (y)
6. If f (x + y) =  x , y  R , ‘ f ‘ is even/odd/even as well as odd/neither .
3

Section II
More than one correct :
1. Which of the following pairs of functions are not identical ?

(A) f (x) = x 2 and g (x) = x  2

(B) f (x) = sec (sec x) and g (x) = cosec (cosec1 x)
1

1  cos 2 x
(C) f (x) = and g (x) = cos x
2

(D) f (x) = tan1 x + cot1 x & g (x) =
2
2. Which of following pairs of functions are identical ?
1
 n sec x
(A) f (x) = e and g (x) = sec1 x
(B) f (x) = tan (tan1 x) and g (x) = cot (cot 1 x)
(C) f (x) = sgn (x) and g (x) = sgn (sgn (x))
(D) f (x) = cot 2 x . cos2 x and g (x) = cot 2 x  cos2 x

 1  1
3. If f(x) is a polynomial function satisfying the condition f (x) . f   = f (x) + f   and f (2) = 9 then
x x
(A) 2 f (4) = 3 f (6) (B) 14 f (1) = f (3) (C) 9 f (3) = 2 f (5) (D) f (10) = f (11)
 MATHS
 x2  2 x  3 
4. The domain of definition of the function ,  2 tan  x log2 tan  x   4 x2  4 x  3  ,

 1 1
where [ . ] denotes the greatest integer function is , n  , n  ; n  I then
 4 2
(A) n = 0 (B) n  4 (C) n 4 (D) none of these
5. The period of the function,
f (x) = x + a  [x + b] + sin  x + cos 2  x + sin 3  x + cos 4  x + ...... + sin (2 n  1)  x + cos 2 n  x
for every a , b  R is :
(A) 2 (B) 4 (C) 1 (D) 0

Section III
Condition/Result
Each question has a conditional statement followed by a result statements.
If condition  result, then condition is sufficient and
If result  condition, then condition is necessary
If condition is necessary as well as sufficient for the result, mark (A)
If condition is necessary but not sufficient for the result, mark (B)
If condition is sufficient but not necessary for the result, mark (C)
If neither necessary nor sufficient for the result, mark (D)
Consider the following example :
Condition : a > 0, b > 0
Result : a+b>0
Here, if a > 0 and b > 0, then it always implies that a + b is positive but if a + b is positive, then
a and b both need not to be positive. So condition implies result but result does not always implies
condition hence condition is sufficient but not necessary for the result to be hold. So answer is ‘C’.
1. Condition : f (x) is periodic
Result : f (x) is manyone
2. Condition : f  (x) is positive in the domain of f(x)
Result : f (x) is oneone
3. Condition : f : R  R , g : R  R both are one to one function .
Result : f (x) + g (x) is one to one function.
4. Condition : f : R  R , g : R  R , f (x) and g (x) both are odd functions
Result : f (x) – g(x) is an odd function
5. Condition : f (x) and g (x) both are periodic functions
Result : f (x) . g (x) is a periodic function.

Section IV
Assertion/Reason
Each question contains STATEMENT - 1 (Assertion) and STATEMENT - 2 (Reason). Each question
has 4 choices (A), (B), (C) and (D) out of which ONLY ONE is correct.
(A) Statement - 1 is True, Statement - 2 is True, Statement - 2 is a correct explanation for
Statement - 1
(B) Statement - 1 is True, Statement - 2 is True ; Statement - 2 is NOT a correct explanation
for Statement - 1
(C) Statement - 1 is True, Statement - 2 is False
(D) Statement - 1 is False, Statement - 2 is True
1. Statement  1 (A) : Let f : R  R be a function defined by ,
f (x) = x3 + 3 x + sin x , then f –1 (3 + 3 ) = 
Statement  2 (R) : Graph of f (x) and its inverse are mirror image of each other in mirror y = x
 MATHS
2. Statement  1 (A) : f (x) = sin x + (ex – sin x) is neither even nor odd function.
Statement  2 (R) : Sum of odd and neither even nor odd is always neither even nor odd.

3. Statement  1 (A) : f (x) = {x}sin x is periodic function {.} denotes fractional part of x
with period 2.
Statement  2 (R) : f (x) and g (x) both are periodic then f(x).g(x) is periodic with period
equal to LCM of their periods.

4. Statement  1 (A) : The range of the function , f (x) = sin2 x + p sin x + q where p > 2 , will

p2
be real numbers between q – and q + p + 1 .
4
Statement  2 (R) : The function , g (t) = t 2 + p t + 1 where t  [– 1 , 1] and p > 2 , will attain
minimum and maximum values at – 1 and 1 .

5. Statement  1 (A) : The inverse of a strictly increasing exponential function is a logarithmic

function that is strictly decreasing.
Statement  2 (R) :  n x is inverse function of ex .

Section V
Comprehensions
Write Up I
Mr. X is a teacher of mathematics. His students want to know the ages of his son's S1 and S2. He
told that their ages are 'a' and 'b' respectively such that f (x + y) – a x y = f (x) + by2  x , y  R
after some time students said that information is insufficient, please give more information .
Teacher says that f (1) = 8 and f (2) = 32 .
1. The age of S1 will be
(A) 4 (B) 8 (C) 16 (D) 32
2. The age of S2 will be
(A) 4 (B) 8 (C) 16 (D) 32
3. The function f(x) is
(A) even (B) odd
(C) neither even nor odd (D) periodic as well as odd

Write Up II
A function is called oneone if distinct elements has distinct images in codomain, otherwise
manyone. Function is called onto if codomain = Range otherwise into function. A function which is
both oneone and onto is called Bijective function
1. If f (x) : A  B defined by ex + ef (x) = e is onto , then the set A and B respectively are :
(A) (–  , 0) , (–  , 1) (B) (–  , 1) , (–  , 0)
(C) (–  , 0) , (–  , 0) (D) (–  , 1) , (–  , 1)

3
2. Let f (x) = x – sin x and g (x) = x , then :
(A) gof is oneone but fog is manyone (B) fog is oneone but gof is manyone
(C) both gof and fog are oneone (D) both gof and fog are manyone

 
3. Let f : R   0 ,  is defined by f (x) = tan–1 (x2 + x + a) , then the set of values of ‘ a ‘ for which
 2
f is onto is :
 1
(A) [0 ,  ) (B) [2 , 1) (C)   (D) ( –  , )
 4
 MATHS
Write Up III
Let f (x) be a function defined on a domain D of real numbers and T > 0 is such that x , x + T  D
and f (x + T) = f (x) , then T is called a period of f (x) . Answer the following questions.

1. The function , f (x) = k cos x + k 2 sin x has period if K equals to :
2
(A) 1 (B) 2 (C) 3 (D) 1/2
 1 , when ' x' is rational
2. The period of the function , f (x) =  is :
 0 , when ' x' is irrational
(A) 1 (B) 2 (C) 2 (D) does not exist
3. If f (x + a) + f (x + b) = f (x + c) and a < c < b are in A.P. , then the period of ‘ f ‘ is :
(A) b – a (B) 2 (b – a) (C) 3 (b – a) (D) 6 (b – a)

Write Up IV
 3 1 
Let y = f (x) is a parabola whose vertex is at  ,  and axis is parallel to yaxis, that the
2 4 

parabola is passing through (0 , 2) . Let g (x) = f x  , h (x) = g ( x ) ,  (x) = g ( x ) – [ x ] ,

where [ . ] denotes greatest integer function.
1. The number of real roots of equation , g (x) = 0 is :
(A) 2 (B) 3 (C) 4 (D) 6
2. If  (x) + a = 0 has exactly two roots , then ‘ a ‘ belongs to :
 3
(A) (0 , 2) (B) (– 1 , 0)  0 , 
4
(C) (2 , ) (D) (– 1 , 2)


3. The number of solutions of h  (x) = 0 is :

(A) 1 (B) 2 (C) 3 (D) 4

Section VI
Subjectives
1. The number of values of ‘ x ‘ which satisfy the domain of 16 – x
C2x – 1 + 20 – 3xP4x – 5 , .

2. Find the period of the real values function satisfying , f (x) + f (x + 4) = f (x + 2) + f (x + 6) .

x  x  1
3. If [ x ] =   +   , where [ . ] denotes the greatest integer function , then find the value.
2  2 

 101  1   101  2   101  4   101  8 

 2  +  4  +  8  +  16  + ... = ?.
       

4. Find the number of solutions of the equation ,  y   y   = 2 cos x , where

1
y=
3
 sin x   sin x   sin x    (where [ . ] denotes the greatest integer function).
Section VII
Match the Column
1. Match the following function with the domain
Function Domain

(A) f (x) = sin–1 x + 4  x2 + x2  1 (p) {–1, 1}

(B) f (x) = sin  cos x  (q) R – {–2, –3}

sin 2 x  4 sin x  3
(C) f (x) = (r) R
sin 2 x  5 sin x  6
(D) f (x) = log3(x + 4) + log4(1 – x2) (s) (–1, 1)

2. Match the following function with the range

Function Range
(A) f (x) = log3 (x2 + 1) (p) (– , 0]
(B) f (x) = log 1 (x2 + 1) (q) (1, 3]
7

1
(C) f (x) = (r) (0, )
x

x2  3
(D) f (x) = (s) [0, )
x2  1

3. Match the column

Column I Column II
(A) f : R  [–1, 1], f(x) = sin(x + sin x), is (p) one  one
(B) f:R R, f(x) = n (x + 1 x 2 ), is (q) onto
(C) f : [–1, 1]  [–1, 1], f(x) = x cos  x, is (r) odd
(D) f : R  R, f(x) = esinx + e–sin x, is (s) periodic

4. Column I Column II
(A) If f 2 (x) is even then f (x) (p) is even
1
(B) If f (x) = x [x2] +
1  x2
(where [.] denotes the greatest integer function) (q) is odd
x3  x 2 2 x2  x3
x 1 x2
(C) If e + e (r) is neither even nor odd
(D) If 2 x f (x) = f (2 x2 – 1) then f (x) (s) may not be even or odd
(where f (x) is not identically zero)
 Exercise - 07
AIEEE FLASH BACK
2x
1. The range of the function , f (x) = , x  2 is :
2x
(A) R (B) R – { – 1 } (C) R – { 1 } (D) R – { 2 }
[ 2002 ]

 
2. The function , f (x) = log  x  ( x 2  1)  is :
 
(A) an odd function (B) a periodic function
(C) neither an even nor an odd function (D) an even function [ 2003 ]
3. A function ‘ f ‘ from the set of natural numbers to integers defined by ,

 n  1 , when ' n' is odd


f (x) =  2n is :
  2 , when ' n' is even

(A) onto but not one-one (B) one-one and onto both
(C) one-one but not onto (D) neither one-one nor onto [ 2003 ]

n
4. If f : R  R satisfies f (x + y) = f (x) + f (y) , for all x , y  R and f (1) = 7 , then  f (r) is :
r 1

7 (n  1) 7 n (n  1) 7n
(A) (B) 7 n (n + 1) (C) (D) [ 2003 ]
2 2 2

5. The range of the function , f (x) = 7 – xPx – 3 is :

(A) {1 , 2 , 3 , 4} (B) {1 , 2 , 3 , 4 , 5 , 6} (C) {1 , 2 , 3 , 4 , 5} (D) {1 , 2 , 3}
[ 2004 ]

6. If f : R  S , defined by f (x) = sin x – 3 cos x + 1 , is onto , then the interval of ‘ S ’ is :

(A) [ 0 , 1] (B) [ – 1 , 1] (C) [ 0 , 3] (D) [ – 1 , 3]
[ 2004 ]
7. The graph of the function , y = f (x) is symmetrical about the line x = 2 , then :
(A) f (x) = f (– x) (B) f (2 + x) = f (2 – x)
(C) f (x + 2) = f (x – 2) (D) f (x) = – f (– x) [ 2004 ]

sin –1 ( x  3)
8. The domain of the function , f (x) = is :
9  x2
(A) [1 , 2] (B) [2 , 3) (C) [2 , 3] (D) [1 , 2) [ 2004 ]

2x
9. Let f : (– 1 , 1)  B be a function defined by f (x) = tan–1 , then ‘ f ‘ is both one-one and onto
1  x2
when B is the interval :

         
(A)  0 ,  (B)  0 ,  (C)  – ,  (D)  – , 
 2   2   2 2  2 2
[ 2005 ]
10. A real valued function , f (x) satisfies the functional equation ,
f (x – y) = f (x) . f (y) – f (a – x) f (a + y) for some given constant ‘a’ and f (0) = 1 .
Then f (2 a – x) is equal to :
(A) f (x) (B) – f (x) (C) f (– x) (D) f (a) + f (a – x)
[ 2005 ]

  
11. The largest interval lying in   ,  for which the function ,
 2 2

 – x2 x  
 f (x)  4  cos –1   1  log (cos x )  is defined as :
 2  

       
(A)   ,  (B)   ,  (C)  0  (D) [0, ]
 2 2  2 2  2
[ 2007 ]
12. Let f : N  Y be a function defined as f(x) = 4x + 3 where Y = |y  N : y = 4x + 3 for some x  N|. Show
that f is invertible and its inverse is [ 2008 ]

y3 y3 y3 3y  4

(A) g(y) = 4 + (B) g(y) = (C) g(y) = (D) g(y) =
4 4 4 3

13. For real x, let f(x) = x3 + 5x + 1, then - [ 2009 ]

(A) f is one – one but not onto R
(B) f is onto R but not one – one
(C) f is one – one and onto R
(D) f is neither one – one nor onto R

14. Let f(x) = (x + 1)2 –1, x > –1

Statement – 1 : [ 2009 ]
The set {x : f(x) = f (x)} = {0, –1}.
–1

Statement – 2 :
f is a bijection.
(A) Statement -1 is true, Statement -2 is true; Statement -2 is a correct explanation for Statement -1
(B) Statement -1 is true, Statement -2 is true; Statement -2 is not a correct explanation for Statement -1.
(C) Statement -1 is true, Statement -2 is false.
(D) Statement -1 is false, Statement -2 is true.

1
15. The domain of the function f(x) =
| x |  x is [ 2011 ]

(A) (–) (B) (0, ) (C) (–, 0) (D) (–) – {0}

16. Given A = sin2  + cos4  for all value of  , then : [ 2011 ]
3 13 3 13
(A) 1  A  2 (B) A1 (C) A2 (D) A
4 6 4 6
 Exercise - 08
IIT FLASH BACK (OBJECTIVE)
(A) Fill in the blanks :

1. y = 10x is the reflection of y = log10 x in the line whose equation is ________ . [IIT - 82]

 2 
 
2. The values of f (x) = 3 sin   x 2  lie in the interval ________ . [IIT  83]
 16 
 

 x 2 
3. The domain of the function f (x) = sin–1  log 2 is given by ________ . [IIT  84]
 2 


4. Let A be a set of n distinct elements . Then the total number of distinct functions from A to A is
________ and out of these ________ are onto functions . [IIT  85]


 ( 4  x 2 ) 
5. If f (x) = sin  n   , then the domain of f (x) is ________and its range is ________ .
 (1  x ) 
 
[IIT  85]

6. There are exactly two distinct linear functions , ________ and ________ , which map [ 1 , 1] onto
[ 0 , 2] . [IIT  89]

7. If ‘ f ‘ is an even function defined on the interval (5, 5), then 4 real values of ‘ x ‘ satisfying the
 x  1
equation , f (x) = f   are ________ , ________ , ________ and ________ . [IIT  96]
 x  2

    5
8. If f (x) = sin2 x + sin2  x   + cos x cos  x   and g   = 1 , then gof (x) = 
 3   3  4
[IIT  96]

(B) True or False :

1. If f (x) = (a – xn)1/n where a > 0 and n is a positive integer , then f [ f (x) ] = x. [IIT  83]

x 2  4 x  30
2. The function f (x) = is not one to one . [IIT  83]
x 2  8 x  18

3. If f 1(x) and f 2(x) are defined on domains D 1 and D 2 respectively, then f 1(x) + f 2(x) is defined
on D1  D2 . [IIT  88]
(C) Multiple choice questions with one or more than one correct answer :
x2
1. If y = f (x) = x  1 , then :
(A) x = f (y) (B) f (1) = 3
(C) y increases with ‘ x ‘ for x < 1 (D) f is a rational function of x [IIT  84]
2. Let g(x) be a function on [1, 1]. If the area of the equilateral triangle with two of its
3
vertices at (0, 0) and [x, g (x)] is , then the function g(x) is
4

(A) g (x) = ± 1  x 
2
(B) g (x) = 1  x 
2
(C) g (x) = – 1  x 
2
(D) g (x) = 1  x 
2

[IIT  89]
3. If f (x) = cos [ 2 ] x + cos [ –2 ] x , where [ x ] stands for the greatest integer function, then :
(A) f (/2) = – 1 (B) f () = 1 (C) f (–) = 0 (D) f (/4) = 2
[IIT  91]
4. In this question each entry in column 1 is related to exactly one entry in column 2. Write the
correct letter from column 2 against the entry number in column 1. Let the function defined in
  
column 1 have domain   ,  and codomain (–  ,  ) .
 2 2
Column 1 Column 2
(i) 1 + 2x (A) Onto but not oneone
(ii) tan x (B) Oneone but not onto

(iii) x5 (C) Oneone and onto

(iv) cos ec x (D) Neither oneone nor onto

[IIT  92]

5. If g ( f (x)) = sin x and f ( g (x)) = sin x  2 , then :

(A) f (x) = sin2 x , g (x) = x (B) f (x) = sin x , g (x) = x
(C) f (x) = x , g (x) = sin x
2
(D) f and g cannot be determined [IIT  98]
6. If f (x) = 3 x  5 , then f 1
(x)
1 x5
(A) is given by (B) is given by
3x – 5 3
(C) does not exist because f is not oneone (D) does not exist because f is not onto
[IIT  98]

(D) Multiple choice questions with one correct answer :

1. Let R be the set of real numbers . If f : R  R , is a function defined by f(x) = x 2 then ‘ f ‘ is :
(A) Injective but not surjective (B) Surjective but not injective
(C) Bijective (D) None of these [IIT  79]
2. The entire graphs of the equation , y = x2 + kx – x + 9 is strictly above the xaxis if and only if
(A) k < 7 (B) – 5 < k < 7 (C) k > – 5 (D) None of these
[IIT  79]
1
3. The domain of definition of the function , y = + x  2 is :
log10 (1  x )
(A) (– 3 , – 2) excluding – 2.5 (B) [0 , 1] excluding 0.5
(C) [– 2 , 1) excluding 0 (D) None of these [IIT  83]
4. Which of the following functions is periodic ?
(A) f (x) = x – [x] where [x] denotes the greatest integer less than or equal to the real number x
(B) f (x) = sin 1 for x  0 , f (0) = 0 (C) f (x) = x cos x (D) none of these
[IIT  83]
5. Let f (x) =  x – 1, then
(A) f (x2) = (f(x))2 (B) f (x + y) = f (x) + f (y)
(C) f  x  =  f (x) (D) None of these [IIT  83]

6. If ‘ x ‘ satisfies  x – 1 + x – 2 + x – 3  6 , then :
(A) 0  x  4 (B) x  – 2 or x  4
(C) x  0 or x  4 (D) None of these [IIT  83]

1  x 
7. If f (x) = cos (  n x) , then f (x) . f (y) –  f    f ( x y )  has the value :
2   y  

1
(A) – 1 (B) (C) – 2 (D) none of these
2
[IIT  83]

8. If the function , f : [1 ,  )  [1 ,  ) is defined by f (x) = 2x (x  1) , then f 1(x) is :

x ( x  1)
 1
(A)  
2
(B)
1
2

1  1  4 log 2 x 
(C) 2 
1 1  1  4 log x
2  (D) Not defined [IIT  99]

9. The domain of definition of the function, f(x) given by the equation, 2x + 2y = 2 is

(A) 0 < x  1 (B) 0  x  1 (C) –  < x  0 (D) –  < x < 1
[IIT  2000]

 1 , x  0

10. Let g (x) = 1 + x  [ x ] and f (x) =  0 , x  0 . Then for all x , f (g (x)) is equal to :
1 , x 0

(A) x (B) 1 (C) f(x) (D) g(x) [IIT  2001]

1
11. If f : [1 ,  ) [2 ,  ) is given by , f (x) = x + , then f 1(x) equals :
x

x x2  4 x x x2  4
(A) (B) (C) (D) 1 – x2  4
2 1  x2 2
[IIT  2001]

log2 ( x  3)
12. The domain of definition of f (x) = is :
x2  3 x  2
(A) R { 1 ,  2} (B)  2 ,   (C) R {1, 2, 3} (D) (– 3 ,  ) – {– 1 , – 2}
[IIT  2001]
13. Let E = {1, 2, 3, 4} and F = {1, 2} . Then the number of onto functions from E to F is
(A) 14 (B) 16 (C) 12 (D) 8 [IIT  2001]

x
14. Let f (x) =
x 1
, x  – 1 . Then for what value of  is f( f(x)) = x

(A) 2 (B) – 2 (C) 1 (D)  1 [IIT  2001]

15. Suppose f (x) = (x + 1)2 for x  – 1 . If g (x) is the function whose graph is the reflection of the graph
of f (x) with respect to the line y = x , then g (x) equals : [IIT  2002]
1
(A) – x – 1, x  0 (B) , x > –1 (C) x  1 , x  – 1 (D) x – 1, x  0
( x  1)2
16. Let function f : R  R be defined by f (x) = 2x + sin x for x  R , then ‘ f ‘ is :
(A) onetoone and onto (B) onetoone but NOT onto
(C) onto but NOT onetoone (D) neither onetoone nor onto [IIT  2002]
17. Let f (x) = (x + 1) – 1 , 2
(x  – 1) . Then the set S = { x : f (x) = f (x)} is , if ‘ f ‘ is onto :
1

  3  i 3  3  i 3 
(A)  0 ,  1 , ,  (B) {0 , 1 , 1}
 2 2 
(C) { 0 ,  1} (D) empty [IIT  2003]

x2  x  2
18. Range of the function f (x) = ; x  R is :
x2  x  1
(A) (1 ,  ) (B) (1 , 11/7] (C) (1 , 7/3] (D) [1 , 7/5] [IIT  2003]

 1 
19. Domain of definition of the function , f (x) =  sin (2 x )   , is :
 6 

 1 1  1 1  1 1  1 1
(A)   ,  (B)   ,  (C)   ,  (D)  , 
 4 2  2 2  2 9  4 4
[IIT  2003]
x
20. If f : [ 0 ,  )  [ 0 ,  ) and f (x) = then ‘ f ‘ is :
1 x
(A) one  one and into (B) onto but not one  one
(C) one  one and onto (D) neither one  one nor onto [IIT  2003]
21. If f (x) = x2 + 2 b x + 2 c2 and g (x) = – x2 – 2 c x + b2 such that min. f (x) > max. g (x) , then the
relation between ‘ b ’ and ‘ c ’, is :
(A) no real value of ‘ b ’ and ‘ c ’ (B) 0 < c < b 2
(C) c < b 2 (D) c > b 2 [IIT  2003]
22. Let f(x) = sin x + cos x and g(x) = x – 1, then domain for which gof is invertible, is :
2

    2       
(A)  0 ,  (B)  ,  (C)   ,  (D)   , 
 2  2 3   2 3  4 4
[IIT 2004]
 x , if x is rational  0 , if x is rational
23. f (x) =  , and g(x) =  , then (f – g) : R  R is :
 0 , if x is irrational  x , if x is irrational
(A) one–one and into (B) neither one–one nor onto
(C) many one and on to (D) one–one and onto [IIT 2005]
24. Let f : X  Y be a function such that ,
f (C) = { f (x) ; x  C } , C  X and f –1 (D) = { f –1 (x) ; x  D } , D  Y then :
(A) f –1 (f (A)) = A only if A  D (B) f (f –1 (B)) = B only if B  D
(C) f (f (A)) = A
–1
(D) f (f –1 (B)) = B [IIT 2005]

25. f (x) = – f (x) where , f (x) is a double differentiable function and g (x) = f (x)
2 2
  x    x 
If F(x) =  f    +  g    and F(5) = 5 , then F(10) is equal to :
  2    2 
(A) 0 (B) 5 (C) 10 (D) 25 [IIT 2006]
26. Let f(x) = x2 and g(x) = sin x all x  R. Then the set of all x satisfying (fogogof) (x) = (gogof) (x), where (fog)
(x) = f(g(x)), is [IIT-2011]

(A) ± n , n  {0, 1, 2,...} (B) ± n , n  {1, 2,...}


(C) + 2n, n  {..., –2, –1, 0, 1, 2,...} (D) 2n, n  {..., –2, –1, 0, 1, 2,...}
2
 Exercise - 09
IIT FLASH BACK (SUBJECTIVE)
  
1. Given A =  x :  x   and f (x) = cos x – x (1 + x) ; find f (A) . [IIT  80]
 6 3 

2. Let ‘ f ‘ be a one–one function with domain { x , y , z } and range { 1 , 2 , 3 } . It is given that exactly one
of the following statements is true and the remaining two are false f (x) = 1 , f (y)  1 , f (z)  2.
Determine f –1(1) . [IIT  82]

n
3. Find the natural number ‘a’ for which  f (a + k) = 16(2n  1) where the function ‘ f ‘ satisfies the
k 1
relation f (x + y) = f (x). f (y) for all natural numbers x, y & further f (1) = 2. [IIT  92]

4. Let { x } and [ x ] denotes the fractional and integral part of a real number x respectively.
Solve , 4 { x } = x + [ x ]. [IIT  94]

x2
5. Find the domain and range of the real function , 2
. [REE 95]
x  8x  4

 x2  6 x  8
6. A function f : R  R , where R is the set of real numbers, is defined by , f (x) = . Find
  6 x  8 x2
the interval of values of  for which f is onto. Is the function onetoone for  = 3 ?
Justify your answer. [IIT  96]

7. Let f : { x , y , z }  { a , b , c } be a oneone function . It is known that only one of the following

statements is true :
(i) f (x)  b (ii) f (y) = b; (iii) f (z)  a
Find the function ‘ f ‘ .
[IIT  96]

8. If the functions f , g , h are defined from the set of real numbers R to R such that ;
0 , x  0
f (x) = x2 – 1 , x 2  1 , h (x) = 
g (x) = ; then find the composite function hofog and
x , x  0
determine whether the function fog is invertible & the function h is the identity function.
[REE  97]

9. If the functions f & g are defined from th e set of real numbers R to R such t hat :
f(x) = ex ; g(x) = 3x  2, then find functions fog & gof. Also find the domains of functions ,
(fog)1 and (gof)1. [REE  98]

10. Given X = { 1 , 2 , 3 , 4 } , find all oneone , onto mappings , f : X  X , such that :

f (1) = 1 ; f (2)  1 and f (4)  4 . [REE  2000]
 ANSWER SHEET
Exercise - 01 CBSE FLASH BACK
1. e 3. (gof) (x) = 2 x ; (fog) (x) = 8 x
2
 1  1 x 1
4. (i)   +3 (ii) 2 (iii) x4 + 6 x2 + 12 (iv) ]
 x  1 x 2 2x
5. zero 6.
11. f = { (a , 1) , (b , 2) and (c , 3)} and
–1
( f–1)–1 = { (1 , a) , (2 , b) and (3 , c)}

Exercise - 02 OBJECTIVE
1. C 2. B 3. D 4. A 5. A 6. C 7. A
8. D 9. D 10. B 11. A 12. A 13. A 14. B
15. C 16. B 17. D 18. C 19. A 20. A

Exercise - 03 SUBJECTIVE
(1) Domain of definitions
 1
1.   4 ,    (2 , ) 2. x  (2 , 3)
 2 
3. 2 K  < x < (2 k + 1)  but x  1 , where K is nonnegative integer
4. (1 , 2)  (3 , )
 3 
5. (a) 2 K   x  2 K  +  where K  I (b)   2 , 1 
 
6. [ – 3 , – 2]  ( 0 , 2)  ( 2 , 3]
(2) Range
 2 2  
1.  ,  2. 3,2 3. [1 ,  )
 4 4   

(3) Domain and range

1. D : R ; R : (– 1 , 1) 2. Df : x  R , Rf : [0 , 2]

3. Df : R , Rf :  cos1 , 1 
     2 
4. Df : x   n  , n    – n    nI Rf :  ,  –  
 2   4 3 3  2

(4) Identical function

n
1. (2 ,  ) 2. (0 , 1)  (1 ,  ) 3. R – , where ‘n’ is an integer
2
4. [1, 1] 5. R  {0}
(5) Types of functions
1. neither surjective nor injective 2. surjective but not injective
3. many one into 4. yes
5. (a) injective but not surjective (b) neither injective nor surjective
(c) neither injective nor surjective (d) surjective but not injective

(6) Inverse function

 x , x  1

1 1 x  x , 1  x  16
1. log x  (– 1 , 1) 2.
2 1 x  log x , 16  x
 2

(7) Even or Odd function

1. Odd 2. Even 3. Even 4. Even

(8) Periodic function


1. 2. 2
2

(9) Composite Function

1 3
 , 1  2  x  1 
2
 3 3
1  cos (1  x ) , 1
2
 x 1
4
 3
1  sin (1  x ) , 1
4
x0  1  f ( x) , f (x)  0
 
1. f {g (x)} = 1  sin 1 , 0x2 ; g {f (x)} =  1 , 0  f (x)  2
 3  f (x)  1 , f (x)  2
1  sin ( x – 1) , 2  x 1
4 
 3 3
1  cos ( x  1) , 1
4
 x 1
2
 3
 1 , 1  x  2  1
2

 x , 2 x  0

2. g (x) =  0 , 0  x 1
 2 ( x  1) , 1  x  2


x 1 , 0  x 1

  (1  x ) ,  1  x  0 3  x , 1 x  2
3. fog (x) =  gof (x) = 
 x 1 , 0x2  x 1 , 2x3
 5  x , 3x4
(10) Functional equation

1 a  5
1. {– 1 , 1} 2. –5 3. 2 2   bx  –
a b x  ab


4. x– 5. 2n  1
6

(11) Graphs

1. 2.

3. 4.

5.

Exercise - 04 OBJECTIVE
1. B 2. B 3. D 4. D 5. B 6. B 7. C

8. B 9. A 10. D 11. D 12. C 13. C 14. D

15. B 16. D 17. A 18. A 19. B 20. D

 Exercise - 05 SUBJECTIVE

(1) Domain of definitions

1. {x1000  x < 10000} 2. {1}

 1 3 
3.  0 ,    , 1   z  2  4. x  (–  ,  ) 5. {1} 6. { 1/2 }
 4 4 

(2) Range
       
1. Df : x  R , Rf :   ,  2.   3. 4, 4 
 2 4 2  

(3) Domain and range

1. D: {x x  R ; x  – 3 , x  2} ; R : {f (x) f (x) Î R , f (x)  1/5 , f (x)  1}

2. Df : R – [– 1 , 0) , Range (for x  0) : [1 , )
3. Df : x  R , Rf : [0 , 2] 4. Df : [– 1 , 0] , Rf : {0}

(4) Types of functions.

1. a  [1, 4] 2. Injective but not surjective

    a  b 2 
3. X =     ,    , Y = [c –  , c + ] , where  = tan–1   , = a2  b2  2 ab
 2 2   a 
 

4. Domain = (–  , 2) , Co-domain = (–  ,  )

(5) Inverse Function

log 2 x 1
1. f –1 (x) = , Domain : R+ – {0} , Range : R – {1} 2. a= , b = e–1/e
log2 x  1 e

n
1
3.
n
 xi , = 1
i1

(6) Even or Odd functions.

1. Odd 2. Neither odd nor even
3. Even 4. Neither even nor odd

(7) Periodic functions

2. ± 3 , ± 15 4. (a) 2  (b) 2  (c) 2n 

(8) Composite Function

1. fog (x) = x ; Range is (–  ,  )
2. (a) Period of fog is  , period of gof is 2 p
(b) Range of fog is [– 1 , 1] , range of gof is [– tan 1 , tan 1]

x 1 , 0  x 1

  (1  x ) ,  1  x  0 3  x , 1 x  2
3. fog (x) =  ; gof (x) = 
 x 1 , 0  x  2  x 1 , 2x3
 5  x , 3x4

 x , 1  x  0
 x , 0 x 1 
fof (x) =  ; gog (x) =  x , 0x2
 4  x , 3x4 4  x , 2  x  3


 2  cot 2 x ,    x  0
 2
4. fog (x) = 
 sec 2
x , 0 x 
2

(9) Functional equations

1. 102 2. 5050 3. f (x) = 1 – x2

(10) Graphs

1. 2.

3. 4.


11. f (x) = sin x + x – 12. x + y = 6, x + y = 0
3
 Exercise - 06 IIT NEW PATTERN
Section I
 3 
1. 3 2. x = 2 , y = 1.3 , z = 3.1 3. Df : {– 1 , 1} Rf :  
 2 

 31 
4. 5 5.  1 , 1  6. even as well as odd
 8 

Section II
1. A, C 2. B, C, D 3. B, C
4. A, B, C 5. A, B

Section III
1. C 2. D 3. D 4. C 5. D

Section IV
1. A 2. C 3. C 4. D 5. D

Section V
Write Up I
1. C 2. B 3. A

Write Up II
1. D 2. C 3. C

Write Up III
1. A 2. D 3. C

Write Up IV
1. C 2. B 3. B

Section VI
1. {2} 2. 8 3. 101 4. No Solution

Section VII
1. A p B r C r D s
2. A s B p C r D q
3. A  q, r, s B  p, q, r C  q, r Ds
4. A s B r C r D q

Exercise - 07 AIEEE FLASH BACK

1. B 2. A 3. B 4. C 5. D 6. D

7. B 8. B 9. C 10. B 11. C 12. C

13. C 14. B 15. C 16. B

 Exercise - 08 IIT-JEE FLASH BACK (OBJECTIVE)
(A) Fill in the blanks :
 3 
1. y=x 2. 0 ,  3. [ – 2 , – 1]  [ 1 , 2] 4. nn , n !
 2 

5. (2, 1), [1, 1] 6. f (x) = 1 + x ; f (x) = 1  x

1  5 1  5 3  5 3  5
7. , , , 8. 1
2 2 2 2

(B) True or False :

1. True 2. True 3. False

(C) Multiple choice questions with one or more than one correct answer :
1. AD 2. BC 3. AC
4. (i) B (ii) C (iii) D (iv) D 5. A 6. B

(D) Multiple choice questions with one correct answer :

1. D 2. B 3. C 4. A 5. D
6. C 7. D 8. B 9. D 10. B
11. A 12. D 13. A 14. D 15. D
16. A 17. C 18. C 19. A 20. A
21. D 22. D 23. D 24. B 25. A
26. A

Exercise - 09 IIT-JEE FLASH BACK (SUBJECTIVE)

3    1   
1.  1    f (A)  – 1   2. f 1(1) = y
2 6  6 2 3  3

3. a=3 4. x = 0 or 5/3

 1 
4  2 5  , 4  2 5   1
5. Df : R – Rf :    ,      ,
 4  20 

9
6.  =  , as domain is R  < – can not be 3,
8
Note : If f : { x : 8 x2 – 6 x –  = 0 and x  R }  R , then  [2 , 14] for onto function,
for  = 3 , f is not one to one.
7. { (x , b) , (y , a) , (z , c) }
8. ( hofog ) x = h (x2) = x2 for x  R , Hence ‘ h ’ is not an identity function , fog is not invertible
9. (fog)(x)= e3x2; (gof ) (x) = 3ex2 ; Df : x  R+ and Dg : x  (– 2 , )

10. { (1.1), (2, 3), (3, 4), (4, 2) } ; { (1, 1), (2, 4), (3, 2), (4, 3) } and
{ (1, 1), (2, 4), (3, 3), (4, 2) } ; { (1, 1), (2, 2), (3, 4), (4, 2) }