Function [Practice Question]

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P A C E

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Practice Question Question based on

Q.1

LEVEL –1 Q.6

Inequation

The inequality

2 x

Q.7

 2  (A)  ,    3 

x 3

< 2 is satisfied when x satisfies-

Question based on

Q.8

Q.4

Q.5

2x  3 3x  5

(B) x < 4

(C) 3 < x < 4

(D) x =

2

7 2

2

If x – 1  0 and x – x – 2  0, then x line in (B) (–1, 1) (D) {– 1}

Definition of function

Which of the following relation is a function ? (A) {(1,4), (2,6), (1,5), (3,9)} (B) {(3,3), (2,1), (1,2), (2,3)} (C) {(1,2), (2,2,), (3,2), (4,2)}

(C) (– , 3)  [10, ) (D) none of these these

Solution of

(A) x > 3

(A) (–1, 2) (C) (1, 2)

(A) (– , 3)  (10, ) (B) (3, 10)

Q.3

2

the interval/set

2  (B)     3 

 2  (C)  ,    (– , 0) (D) none of these  3  x4

2

If x + 6x – 27 > 0 and x – 3x – 4 < 0, then-

< 3 is true, when x belongs

to-

Q.2

FUNCTION

(D) {(3,1), (3,2), (3,3), (3,4)}

 3 is Q.9

If x, y  R, then which of the following rules is

 12  (A) 1,   7 

 5 12  (B)  ,   3 7 

not a function(A) y = 9 –x

(B) y = 2x

5   (C)   ,  3  

12  (D)  ,    7 

(C) y =

(D) y = x + 1

2

Question based on

2

x – |x|

2

2

Even and Odd function

Solution of (x – 1) (x + 4) < 0 is(A) (– , 1)

(B) (– , –4)

(C) (–1, 4)

(D) (1, 4)

Q.10

function(A) sin x (C) tan h x

Solution of (2x + 1) (x – 3) ( x + 7) < 0 is-

 1  (A) (– , –7)    , 3  2   1  (B) (– , – 7)   , 3   2   1  (C) (– , 7)    , 3  2 

Which one of the following is not an odd

Q.11

The function f(x) =

(B) tan x (D) None of these sin 4 x  cos 4 x x  tan x

is -

(A) odd (B) Even (C) neither even nor odd (D) odd and periodic

(D) (– , –7)  (3, ) 2624522 3 / 09 ; .www.iitianspace.com .www.iitians pace.com IIT - ian’s PACE ; ANDHERI / DADAR / CHEMBUR / THANE ; Tel : 26245223

1

Q.12

Q.20

f (x) = cos log (x + 1  x 2 ) is

f(x) = log cos 2x + tan 4x is-

(A) even function (B) odd function (C) neither even nor odd (D) constant Q.21 Q.13

A function whose graph is symmetrical about the y-axis is given by-

(C) f(x) = cos x + sin x (D) None of these

(C)

Q.15

Q.22

ax

Question based on

(B) tan x

2

ax

1 (D) x a 1

Q.23

In the following, odd function is (A) cos x

2

(D) 2/5

x

1

The period of the function f(x) = 2 cos

3

(x– )

(B) 4

(C) 2

(D) 

In the following which function is not periodic(C) cos x

x

 a x

(C) 2

(A) tan 4x

Which of the following is an even function?

1 (A) x x a 1

(B) 

(A) 6

(B) f(x + y) = f(x) + f(y) for all x, y  R

a

(A) /2

is -

(A) f(x) = loge (x + x 2  1 )

Q.14

The period of the function

2

(B) cos 2x 2

(D) cos x

Domain, Co-domain and Range of function

Domain of the function f(x) =

1 x2

(A) R

(B) (–2,  )

(C) [2, ]

(D) [0, ]

is-

x

(B) (e + 1)/(e – 1)

2

(C) x – |x|

(D) None of these

Q.24

2

The domain where function f(x) = 2x – 1 and g(x) = 1 – 3x are equal, is-

Q.16

2

The function f(x) = x – |x| is-

(A) {1/2}

(B) {2}

(A) an odd function

(C) {1/2, 2}

(D) {1/2, –2}

(B) a rational function (C) an even function

Q.25

The domain of the function log

(D) None of these Question based on

Q.17

Periodic function 4

Q.26

4

The period of sin x + cos x is (A) 

(B) /2

(C) 2

(D) None of these Q.27

Q.18

The period of function |cos 2x| is (A) 

Q.19

(B) /2

(C) 4

(D) 2

 x  + cos  x      2   2 

The period of function sin  is(A) 4

(B) 6

(C) 12

(D) 24

Q.28

(A) (3, )

(B) (– , 3)

(C) (0, 3)

(D) (–3, 3)

3 x 2

is-

–1

Domain of the function cos (4x –1) is(A) (0,1/2)

(B) [0,1/2]

(C) [1/2,2]

(D) None of these 2

Domain of the function log |x – 9| is(A) R

(B) R– [–3, 3]

(C) R – {–3, 3}

(D) None of these

The domain of the functionf (x) =

x  1 +

6  x is-

(A) (1, 6)

(B) [1, 6]

(C) [1, )

(D) (– , 6]

IIT - ian’s PACE ; ANDHERI / DADAR / CHEMBUR / THANE ; Tel : 26245223 / 09 ; .www.iitianspace.com

2

Q.29

The domain of the function

Q.37

If f : R

 R,

f(x) = (2  2x  x 2 ) is (A) – 3  x  (C) – 2  x Q.30

Q.31

3

2

image set of R under f is -

(B) –1– 3 x –1+ 3 (D) –2+ 3 x  –2– 3 –1

Domain of a function f(x) = sin 5x is-

 1 1  (A)   ,   5 5 

 1 1 (B)  ,   5 5

(C) R

(D)  0,

 

Q.38



5 

Q.39

The range of f(x) = sin

 2

[x] is -

(A) {–1, 1}

(B) {–1, 0, 1}

(C) {0, 1}

(D) [–1, 1]

Domain and range of f(x) =

(A) R, [–1, 1]

Q.41 | x 3| x 3

are

(B) R– {3}, {1, –1}

+

(C) R , R

Q.36

Q.42

(D) None of these

+

(B) R

(C) R 0

(B) [2, 16]

(C) [–1, 1]

(D) (2, 16]

For real values of x, range of function

(A)

1 3

The range of f(x) = cos 2x – sin 2x contains the (A) [2, 4]

(B) [–1, 1]

(C) [–2, 2]

(D) [–4, 4]

If the domain of the function f(x) =

(A) [–1, 1]

(B) {–1, 1}

(C) {1}

(D) {–1}

The domain of the function f(x) =

1 (C) – > y > – 1 3

x

1 x  [x]

(A) R

(B) R–Z

(C) Z

(D) None of these

be

is-

The range of the function (A) [5, 6]

(B) [5, 6)

(C) R

(D) None of these

Question based on

Q.43

Value of function

If f is a real function satisfying the relation f(x + y) = f (x) f(y) for all x, y  R and f(1) = 2, n

then a

 N, for which

 f (a  k ) = 16(2 – 1), n

k 1

is -

 y  1

|x|

(D) R

(A) (2, 16)

2  sin 3x

(D) None of these

–

Range of the function f(x) = 9 – 7 sin x is-

1



(B) {1}

f(x) = 2 + x – [x–3] is-

The domain of the function f(x) = sin 1/x is -

y=

2

[3, 7] then its range is-

respectively-

Q.35

If f : R R, f(x) = x , then {x|f(x) = –1} equals-

set -

Q.40

(D) None of these

(A) R

(D) None of these

  

(B)   ,   2 2

(C) R

Q.34

(C) {1, –1}

1 

–1

(A) 

Q.33

(B) (–1, –1)

(C)

x is-

Q.32

(A) {1, 1}

(A) {–1, 1}

The range of the function f : R R, f(x) = tan

   ,  2 2 

 1, when x  Q , then f(x) =   1, when x  Q

is given by 1 (B) –  y  1 3

(D)

1 3

(A) 2

(B) 4

(C) 3

(D) None of these

>y>1


3

Q.44

If f : R

 R,

 1, when x Q , then f(x) =   1, when x Q

which of the following statement is wrong ?

Question based on

Q.52

Mapping

If f : I  I,f (x) = x + 1, then f is 3

(A) f ( 2 )= –1

(B) f() = –1

(A) one-one but not onto

(C) f(e) = 1

(D) f( 4 )= 1

(B) onto but not one-one (C) One-one onto

Q.45

If f (x) =

x( x  1) 2

(D) None of these , then the value of f (x + 2) is-

(A) f(x) + f(x + 1) (C)

( x  1) 2

f(x +1)

(B) (D)

x2 x

f(x + 1)

( x  2) 2

Q.53

Function f : R  R, f(x) = x |x| is (A) one-one but not onto (B) onto but not one-one

f(x +1)

(C) one-one onto (D) neither one-one nor onto

Q.46

If f(x + ay, x – ay ) = axy, then f(x, y) equals(A)

x2  y2

(C) x

4 2

(B)

x 2  y2

Q.54

f : R  R, f(x) =

4 2

x2 1 x

2

, is -

(A) many-one function

(D) y

(B) odd function Q.47

If f(x) = cos (log x), then

(C) one-one function

f ( xy)  f ( x / y)

(D) None of these

f ( x )f ( y)

equals(A) 1 (C) 0 Q.48

(B) –1

Q.55

(D) 2

(C) 2x + 1

(D) 2x – 1

Q.50

Q.51

, then f is -

(C) neither one-one nor onto (D) both one-one and onto Q.56

Q.49

x

(B) onto but not one-one

equals(B) –1

1


If f(x) = |x| + |x – 1|, then for 0 < x < 1, f(x) (A) 1

If f : R 0  R 0, f(x) =

Function f : R  R, f(x) = x + |x| is

f(2x + 3y, 2x – 7y) = 20 x then f(x , y) equals to –

(A) one-one

(B) onto

(A) 7x – 3y

(B) 7x + 3y

(C) one-one onto

(D) None of these

(C) 3x – 7y

(D) x – 10y

(A) f(a) f(x)

(B) 1+ f(x)

  3  Function f :  ,  R, f(x) = tan x is  2 2 

(C) f(x)

(D) a f(x)

(A) one-one

(B) onto

(C) one-one onto

(D) None of these

If f(x) = log ax, then f(ax) equals-

Q.57

If f(x) = (ax – c)/(cx – a) = y, then f(y) equals(A) x

(B) 1/x

(C) 1

(D) 0

Q.58

  3  Function f :  ,  [–1,1], f(x) = sin x is  2 2  (A) one-one

(B) onto

(C) one-one onto

(D) None of these


4

Q.59

x

f : N  N where f(x) = x – (–1) then f ' is -

(A) one-one onto

(B) one-one into

(C) many-one onto

(D) many-one into

(A) one-one and into (B) many-one and into (C) one-one and onto (D) many-one and onto Q.60

x

Question based on

–x

If f : R R, f(x) = e + e , then f is -

Q.68


(B) (g + g) (x) = g(x)

(C) neither one-one nor onto

(C) (fog) (x) = (g + g) ( x)

(D) both one-one and onto

Q.62

(D) None of these

If f : R  [–1,1], f(x) = sin x, then f is(A) one-one onto

(B) one-one into

(C) many-one onto

(D) many-one into

Q.69

gof exists, when(A) domain of f = domain of g (B) co-domain of f = domain of g

If f : R R , f(x) = sin x + cos x , then f is -

(C) co-domain of g = domain of g


(D) co-domain of g = co-domain of f

2

2


Q.70

(C) neither one-one nor onto

 R, f(x) = x2 + 2x – 3 and g : R  R,

If f : R

g(x) = 3x – 4, then the value of fog (x) is-

(D) both one-one onto Q.63

If f(x) = 2x and g is identity function, then(A) (fog) (x) = g(x)


Q.61

Composite function

2

2

(A) 3x + 6x – 13

Which of the following functions from Z to

(B) 9x –18x + 5

2

(C) (3x– 4) + 2x – 3 (D) None of these

itself are bijections ? (A) f(x) = x

3

(B) f(x) = x + 2 2

(C) f(x) = 2x + 1 Q.64

(D) f(x) = x + x

Q.71

A = {x: –1 x 1} to itself are bijections ?

 x    2 

x

(B) g(x) = sin 

2

(C) h(x) = |x|

(D) k(x) = x

Q.72

(A) 0

(B) 

(C) – 

(D) Undefined +

If f : R

x

2

+

x

Q.66

Q.73

–x

(A) f(x) = |x| (C) f(x) = x Q.67

3

2

2

If f : R R, g : R  R and f(x) = 3x + 4 and

(C)

1 3

(B) 2x – 11

(2x – 11)

If f : R

(D) None of these

 R, g : R  R and g(x)

= x + 3 and

2

(B) f(x) = e

(fog) (x) = (x + 3) , then the value of f(–3) is -

(D) f(x) = sin x.

(A) –9

(B) 0

(C) 9

(D) None of these

If f :   , f(x) = x – x, then f is 2

2

ex

(D) e x . e x

(A) 2x – 1

Q.74 –x

1

2

(B) e x +

(gof) (x) = 2x – 1, then the value of g(x) is-

Which of the following function defined from R to R is onto ?

–2x

(C) e + e

R ; f(x) = e (C) f: [0,/2]  [–1,1]; f(x) = sin x (D) f : R  R: f(x) = cosh x (B) f : R

2

2

(A) e x + e x

Which of the following function is onto ? (A) f : R  R ; f(x) = 3

R +, f(x) = x2+ 1/x2 and g : R +  R +,

g(x) = e then (gof) (x) equals-

2x

Q.65

R  R,

g(x) = log x, then the value of (gof) (2) is -

Which of the following functions from

(A) f(x) =

 R, f(x) = x2 – 5x + 4 and g :

If f : R


5

Q.75

If f(x) = ax + b and g(x) = cx + d, then f(g(x)) = g(f(x)) is equivalent to(A) f(a) = g(c)

(B) f(b) = g(b)

(C) f(d) = g(b)

(D) f(c) = g(a)

Q.82

If f(y) =

If f:[0,1]  [0,1], f(x) =

1 x 1 x

, g(y) =

1 y2

y 1 y2

(A)

(C)

Q.77

1  4x  4 x

2

1  4x  4x

2

1  4x  4 x

2

(B)

. g : [0,1]  [0,1], (C) y

8 x (1  x ) (1  x )

2

wrong statement is –1

–1

Q.83

(D) None of these

If f, g, h are three functions in any set, then (A) (fog) = g of

Question based on

(B) gof  fog –1

Q.78

Q.79

If f(x) =

1 x 1 x

(A) sin 

(B) tan (/2)

(C) cot (/2)

(D) cosec 

(D)

(B) x

n

n

n

(D) (a – x)

 1  x  and g(x)   1  x 

If f (x) = log 

(A) –f(x) (C) [f(x)]

1 y2 1  y2

of gof is (A) {0}

(B) {–1, 1}

(C) {–1, 0, 1}

(D) [–1, 1]

Inverse function

2

If f : R  R, f(x) = x + 3, then pre-image of

1, when x Q If function f(x) =  , (fof) ( 4 ) 0, when x  Q the value will be(A) 0

(B) 2

(C) 1

(D) None of these

(B) {1}

(C) {–1}

(D) 

Which of the following functions has its inversex

+

(C) f : R 0 R , f(x) = |x| (D) f : [, 2] [–1,1], f(x) = cos x

Q.86

+

x

If function f : R R , f(x) = 2 , then f

–1

(x)

will be equal to(A) logx 2 (C) log2 x

 3x  x 3    1  3x 2  ,  

(D) None of these

(A) {1,–1}

(B) f : R R, f(x) = |x| + |x – 1|

=

(B) 3f(x) 3

1 y2

If f(x) = [x] and g(x) = cos (x), then the range

(A) f : R R, f(x) = a

n 1/n

then f[g(x)] is equal to-

Q.81

Q.85

If f(x) = (a – x ) , n  N, then f [f(x)] = (C) x

Q.80

–1

, then f [f (sin)] equals -

(A) 0

y

2 under f is –

(C) (fog)oh = fo(goh) –1

, then (fog)(y)

–1

Q.84

(D) (gof) = g of

1 y2

(B)

g(x) = 4x (1–x), then (fog) (x) equals1  4x  4 x 2

y

equals (A)

Q.76

y

Q.87

(B) log2 (1/x) (D) None of these

The inverse of the function f(x) =

 e x +2 x x e e ex

is given by 1/ 2

 x  1  (B) log    x  1 

1/ 2

 x  1  (D) log    3  x 

 x  2  (A) log    x  1   x  (C) log    2  x 

1/ 2

1/ 2


6

Q.88

If f : [1,

)  [2, ) is given by ƒ(x) = x +

1 x

–1

then f (x) equals (A)

(C)

Q.89

x x

2

4

2 x x

2

(B)

4

1 x2 x2  4

(D) 1 +

2

If f(x) = loge(x + 1  x 2 ), then f (A) log (x – 1  x 2 ) (B) (C)

Q.90

x

ex

 ex 2

(D)

ex

–1

(x) equals-

 e x 2

 e x e x  e x ex

3

If f(x) = x – 1 and domain of f = {0, 1, 2, 3}, –1

then domain of f is -

Q.91

(A) {0, 1, 2, 3}

(B) {1, 0, –7, –26}

(C) {–1, 0, 7, 26}

(D) {0, –1, –2, –3} 3 1/5

If f(x) = {4 – (x – 7) } , then its inverse is5 1/3

(B) 7 – (4 + x )

5 1/3

5 1/3

(D) None of these

(A) 7 – (4 – x )

(C) 7 + (4 – x ) Q.92

x

If f : R  R, f(x) = e & g : R R, g(x) = 3x – 2, –1

then the value of (fog) (x) is equal to (A) log (x – 2)

 x  3    2 

(C) log 

(B)

2  log x 3

(D) None of these


7

LEVEL- 2 Q.1

–1

The range of f (x) = sin (A) (0, /2] (C) [/3, /2]

Q.2

If f(x) =

1

 x  1 is (B) (0, /3] (D) [/6, /3] x

1

and g (x) =

x 1

2

x 1

Q.7

| sin x |  | cos x |

The period of f(x) =

| sin x – cos x |

(A) /2

(B) 

(C) 2

(D) None of these

, then Q.8

is -

sec1 x

The function f(x) =

x  [ x]

common domain of function is -

, where [x]

(A) {x | x < 1, x  R}

denotes the greatest integer less than or equal

(B) {x | x  0, x  1, x  R}

to x, is defined for all x belonging to -

(C) {1}

(A) R

(D) {–1}

(B) R – {(–1, 1)  {n : n  Z}} +

(C) R – (0, 1) +

(D) R – {n : n  N}

1/ 12

Q.3

 x   If f(x) =   1 | x | 

,x

 R then domain of

the function f(x) is -

Q.4

Q.9

(A) (–1,0]

(B) (– , –1)  [0, 1)

(C) (–1, ) – {1}

(D) None of these

–1

(A) [0, )

(B) [0, 3]

(C) [0, 1]

(D) [0, 2]

–1 under f is-

   n  I  (B) n  4  

(C) {n| n  I }

 n I  

x +cos

x=

 2

holds-

If f : R  R, f(x) = tan x, then pre-image of

  (A) n  4 

–1

The interval for which sin

Q.10

–1

The function f(x) = cos

 | x | 3     2 

–1

+ [loge (4 – x)] is defined for -

(D) None of these

(A) [–1, 0]  [1, 5]

 [1, 4] (C) [–5, –1]  ( [1, 4) – {3}) (B) [–5, –1]

Q.5

The domain of f(x) =

–1

–1

[cos(sin x )] + (1 – x) + sin

 x 2  1     2x   

equal to -

Q.6

Q.11

(A) R – {1}

(B) {–1}

(C) (1, )

(D) None of these

If f : R

 R,

(D) [1, 4] – {3}

 

g(x) =  1 

3

f(x) = x + 3, and g : R R, –1

+

2

+

Function f : R  R , f(x) = x + 2 & g : R

 R,

 then the value of gof (2) is  1  x  1

(A) 5/6

(B) 8/7

(C) 1/6

(D) 6/5

–1

g(x) = 2x + 1, then f og (23) equals(A) 2 (C) (14)

(B) 3 1/3

(D) (15)

Q.12 1/3

Period of function 2 + sin x + 3 + cos 2x is (where { } represent fractional part of x) (A) 2 (B) 1 (C) 3 (D) None of these {x}

{x/2}


8

Q.13

Let f : (4, 6)  (6, 8) be a function defined by

Q.21

f(x) = x + [x/2] where [ ] represent G.I.F. then

2000

x  r } is  {2000

–1

f (x) is equal to -

Q.14

(A) x – 2

(B) x – [x/2]

(C) – x – 2

(D) None of these

If f(x) = log

Q.15

Q.16

r 1

(A) x (C) {x}

1 x

, when – 1 < x1, x 2< 1, then 1 x f(x1) + f(x2) equals -

 x  x 2   (A) f  1  1  x1x 2   x  x 2   (C) f  1 1 x x  1 2  

 x  x 2   (B) f  1  1  x1x 2   x  x 2   (D) f  1 1 x x   1 2 

(C) (–2, 3)

(D) (– , –2)

Q.24

Period of f(x) = sin 3  {x} + tan  [x] where [ ] and {} represent of G.I.F and fractional part of x (B) 2

(C) 3

Q.20

is equal to (A) 3 (C) 6

cos (sin nx ) tan( x / n )

(n

 N) is 6 then n

(C) 2

(D) 4

The domain of function f(x) = log |log x| is(B) (1, ) (D) (– , 1) –1

–1

2

Domain of the function tan x + cos x is (A) R– [–1, 1] (B) R– (–1, 1) (C) (–1, 1) (D) [–1, 1]

Q.26

Which of the following functions are equal? (A) f(x) = x, g(x) = x 2 2 (B) f(x) = log x , g(x) = 2 log x 2 2 (C) f(x) = 1, g(x) = sin x + cos x (D) f(x) = x/x, g(x) = 1

+

If f : C R , f(z) = |z|, then f is (A) one-one but not onto (B) onto but not one-one (C) neither one-one nor onto (D) both one-one and onto

(B) –1

  5  =    3 

Q.25

If S be the set of all triangles and f : S  R ,

If period of

If f be the greatest integer function and g be the modulus function, then

(A) (0, ) (C) (0, 1)  (1, )

(D) 

f () = Area of  , then f is (A) One-one onto (B) one-one into (C) many-one onto (D) many-one into Q.19

(B) /6 (D) /2

(A) 1

2

(B) [–2, )

The period of f(x) = cos (sin x) + cos (cos x) is -

 5  (gof)    – (fog)  3 

If the domain of function f(x) = x – 6x + 7 is (A) (– , )

(B) [x] (D) x + 2001

(A) /3 (C) 

3(x – [x])

Period of the function f(x) = | sin x | + e (where [ ] represent G.I.F.) is (A) 1 (B) 2 (C) 1/3 (D) None of these

(A) 1 Q.18

Q.22

Q.23

(– , ), then the range of function is -

Q.17

If [x] and {x} represent the integral and fractional part of x respectively then value of

2

Q.27

f : N  N defined by f(x) = x + x + 1, x  N then f is (A) one-one onto (B) many-one onto (C) one-one but not onto (D) none of these

Q.28

Let f(x) = sin (x/2) + cos (x/2) and g(x) = 2 2 sec x – tan x. The two function are equal over the set -

2

(A)

2

 





(B) R – x : x  ( 2n  1) , n  Z 2   (B) 2 (D) 1

(C) R (D) None of these


9

Q.29

The domain of the function

 2 – | x |  + cos –1  2 – | x |  + tan –1      4   4 

–1

f(x) = sin

Q.36

(B) [–6, 6] (D) None of these

sin([x ]) x 2  2x  4

one is not true (A) f is periodic (C) f is many-one

 2 – | x |  is given by    4  (A) [– 3, 3] (C) [0, 6]

Let f(x) =

Q.37

, [.] = G.I.F., then which

(B) f is even (D) f is onto

The domain of function f(x) = log (3x –1) + 2 log (x +1) is -

Q.30

The domain of function

(A) [1/3, )

(B) [–1, 1/3]

1

(C) (–1, 1/3)

(D) None of these

f(x) =

log10 (3  x)

+

(A) [–2, 3) (C) [–3, 2]

Q.31

x  2 is -

(B) [–2, 3) – {2} (D) [–2, 3] – {2}

Domain of the function f(x) =

Q.38

( x  1) x

2

Q.33

   Domain and range of sin  log    

   is  

1 x

(B) (–2, 1), [–1, 1]

(C) (–2, 1), R

(D) None of these

Let f : R

 R

f(x) = x +

x 2 , then f is-

(B) surjective (D) None of these

Q.40

Q.41

3x

Q.42

3

(B)

(B) x (D) 3 log x

1  3x

2

(D) None of these

2

(B) 217 (D) None of these 3

If f (x) = x – x and g(x) = sin 2x, then(A) g [f(1)] = 1

(B) f (g (/12)) = – 3/8

(C) g {f(2)} = sin 2

(D) None of these

 R is defined by f(x) = cos 2x + sin4x x  R then the range of f (x) is -

f:R

(A) (3/4, 1)

(B) [3/4, 1)

(C) [3/4, 1]

(D) (3/4, 1)

The natural domain of the real valued function defined by f (x) =

If f : R  R f(x) = cos (5x + 2) then the value

x

If f (x) be a polynomial satisfying f (x). f(1/x)= f (x) + f (1/x) and f(4) = 65 then

for

If f (x) = e and g(x) = n x, x > 0, then (fog) (x) (A) 3x (C) log 3x

, then (fofof) (x) is equal to-

f(6) = ? (A) 176 (C) 289

be a function defined by

is equal to-

Q.35

Q.39

4  x 2  

(A) [–2, 1), (–1, 1)

(A) injective (C) bijective Q.34

1 x

(B) (– , –2)  (2, )

(C) (– ,–2)  (1, ) (D) (– ,) – {1, ±2}

2

2

3x

(C)

(A) (1, 2)

Q.32

1 x

4

is -

1 x

3x

(A)

x3

x

If f(x) =

x2 1 +

x 2  1 is-

(A) 1 < x < 

(B) –  < x < 

(C) –  < x <–1

(D) (– ,) – (–1, 1)

–1

of f (x) is (A)

cos 1 ( x)  2 5 cos  ( x) 1

(C)

5

2

–1

(B) cos (x) –2 (D) Does not exist

Q.43

If f(x) =

9  x2 sin 1 (3  x )

(A) [2, 3] (C) (2, 3]

, then domain of f is (B) [2, 3) (D) None of these


10

Q.44

 

Let f  x 

1 

1

 = x2 + 2 (x  0), then f(x) x  x

equals 2 (A) x – 2 (C) x Q.45

The range of the function f (x) = |x – 1| + |x – 2|, –1  x  3 is (A) [1, 3] (B) [1, 5] (C) [3, 5] (D) None of these

Q.53

The range of the function y = log3 (5 + 4x – x ) is (A) (0, 2] (B) (– , 2] (C) (0, 9] (D) None of these

2

(B) x –1

2

(D) None of these ( 2  x  x 2 ) and

Let f(x) = g(x) =

Q.52

 x +

1 x2

is given by (A) (–2 0] (C) [–1, 0]

. Then domain of f + g

Q.54

(B) [0, 1] (D) (0, 1)

2

Let f (x) =

9 9x

find value of Q.46

Q.47

–1

2

and f(x) + f (1–x) = 1 then

3  1  +  2  + f      1996   1996 

…… +

 1995  is   1996 

f 

(A) 998

(B) 997

(C) 997.5

(D) 998.5

If x = loga bc, y = log b ca, and z = logcab, then 1 x

Q.49

–1

The range of sin [x + 1/2] + cos [x – 1/2] where [ ] represent G.I.F. (A) {/2, } (B) {} (C) {/2} (D) None of these

1

Q.48

2

x

+

1 1 y

+

1 1 z

equals-

Q.55

The range of

f(x) = (1 – cos x ) (1 – cos x ) 1 – cos x .......  is -

(A) 1

(B) x + y + z

(C) abc

(D) ab + bc + ca

(A) [0, 1] (C) [0, 2]

(B) [0, 1/2] (D) None of these

The range of 5 cos x – 12 sin x + 7 is(A) [–6,20]

(B) [–3,18]

(C) [–6,15]

(D) None of these

The domain of the function log 2log 3log 4(x) is-

Q.50

(A) (1, )

(B) (2, )

(C) (3, )

(D) (4, )

Let f(x) =

x  [x] 1  [x ]  x

, then range of f(x) is

([.] = G.I.F.) -

Q.51

(A) [0, 1]

(B) [0, 1/2]

(C) [1/2, 1]

(D) [0, 1/2)

f(x) = log ( x – 3 + 5 – x ), x  R then domain of f(x) is (A) [3, 5] (B) [– , 3]  [5, ] (C) {3, 5} (D) None of these


11

LEVEL- 3 Q.1

The domain of definition of

Q.8

1  x  1  is –  2  x  5  x  36

2, then f (4) equals to-

f(x) = log 0.4 

(A) (x : x < 0, x  – 6} (B) (x : x > 0, x  1, x  6}

Q.9

(C) (x : x > 1, x  6}

Q.2

If f(x) is an odd periodic function with period (A) 0

(B) 2

(C) 4

(D) –4

Domain of the function 1  

x

f(x) = sin

The function f : R  R defined by

(A) [–5, –1]  [1, 5]

f (x) = (x – 1) (x – 2) (x – 3) is -

(B) [–5, 5]


(C) (–5, –1)  (1, 5)


log

 

5

5

(D) None of these

(C) both one and onto (D) neither one-one nor onto Q.10 Q.3

 x 2  1  p  is  

 R

(D) None of these

3

where f(x) = x + tan x +

a odd function then the value of Q.11

If f(x) = 3 sin

2 16

 x 2 , then values of f(x) lie

in

   ,  4 4 

(A) 

Let f : R  R be a function defined by f (x ) 

is -

(C) [–1, 1]  (– , –2)  (2, )

p where [ ] represent G.I.F. (A) – 5< p< 5 (B) p < 5 (C) p > 5 (D) None of these Q.6

2 | x |

(B) R – [–1, 1]

(B) (1, e) (D) (– e, 1)

If g : [–2, 2]

1 | x |

(A) R – [–2, 2]

x

f(e ) + f (n | x | ) is -

Q.4

Domain of f(x) =

The domain of f(x) is (0, 1) therefore domain of (A) (–1, e) (C) (–e, – 1)

  is 

2

(D) (x : x  1, x  6}



e | x|  e  x

(C) 0 ,

. Then x x e e



  2

3

(B) [–2, 2]

(D) None of these

(A) f is a bijection (B) f is an injection only

Q.12

The period of f (x) = sin

(C) f is a surjection only (D) f is neither an injection nor a surjection

x n!

+ cos

x ( n  1) !

is -

(A) non-periodic (B) periodic with period (2 ) n!

Q.7

The value of n f(x) =

(A) 2

sin nx

 x  sin    n 

I

for which the function

has 4 as its period is-

(B) 3

(C) 4

(C) periodic with period 2  (n + 1)! (D) periodic with period 2 (n + 1)



(D) 5


12

Q.13

The function f(x) = max. [1 – x, 1+x, 2] ; x  R is equivalent to -

1  x , x  1  (A) f ( x )  2 ,  1  x  1  1 x, x  1  1  x , x  1  (B) f ( x )  2 ,  1  x  1  1 x, x  1  1  x , x  1  (C) f ( x )  1,  1  x  1  1 x, x  1 

Q.20

–1



[– /2,

/2]

3

defined by f(x) = sin (3x – 4x ) is– (A) both one-one and onto (B) neither one-one nor onto (C) onto but not one-one (D) one-one but not onto Q.21

The function f satisfies the equation

 x  59   10x  30 for all real x  1.   x  1 

3f (x) + 2f 

The value of f (7) is -


The function f : [–1/2, 1/2]

9–x

The domain of the function f(x) = Px–5 is(A) [5, 7] (B) {5, 6, 7} (C) {3, 4, 5, 6, 7} (D) None of these

Q.22

(A) 8

(B) 4

(C) –8

(D) 11

The domain of the function 2

Q.15

f (x) = log3+x(x – 1) is -

9–x

The range of the function f(x) = Px–5 is (A) {1, 2, 3} (B) [1, 2] (C) {1, 2, 3, 4, 5} (D) None of these

(A) (–3, –1)  (1, ) (B) [–3, –1)  [1, ) (C) (–3, –2)  (–2, –1)  (1, )

Q.16

Domain of the function

  

 

f ( x )  log 2   log1/ 2 1 

Q.17

(D) [–3, –2)  (–2, –1)  [1, )

    1 is4 x   1

(A) (0, 1)

(B) (0, 1]

(C) [1, )

(D) (1, )

Statement type Questions Each of the questions given below consists of

The period of f(x) = [sin 5x] + |cos 6x| is (A)

 2

(B) 

(C) 2

(D)

2

Statement-I and Statement-II. Use the following key to choose the appropriate answer. (A)

5

Statement-I and Statement-II are true but Statement-II is the correct explanation of

Q.18

Period of f (x) = sin x + tan x 23

+ ... + sin

(A)  Q.19

x 2 n 1

(B) 2

+ tan

x 2

x 2n n

(C) 2 

+ sin

x 22

2 is (A) n (C)

1 n

x where n

(B)

is -

Statement-I and Statement-II are true but Statement-II is not the correct explanation of

(D)

Statement-I.

 2n

The period of f(x) = [x] + [2x] + … + [nx] – n ( n  1)

Statement-I

+ tan

 N and [ ] represent G.I.F.

(C)

Statement-I is true but Statement-II is false

(D)

Statement-I is false but Statement-II is true.

Q.23

Statement- I : The period of

f(x) = sin 2x cos [2x] – cos 2x si n [2x] is (B) 1 (D) None of these

1 2

Statement- II : The period of x – [x] is 1

Where [.] = G.I.F.


13

On the basis of above information, answer Q.24

Statement- I : If f(x) = |x – 1| + |x – 2| + |x – 3|

Where 2 < x < 3 is an identity function. Statement- II : f : A

 A

defined by f(x) = x

the following questionsQ.30

Which statement is correct(A) Period of f(x), g(x) and h(x) are same

is an identity function.

and value is Q.25

2

Statement- I : f : R  R defined by f(x) = sin x

3

(B) Period of f(x), g(x) and h(x) makes

is a bijection Statement- II : If f is both one and onto it is

the A.P. with common difference

bijection

 4

(C) Sum of periods of f(x), g(x) and h(x) is 3 Q. 26

Statement- I : f : R

by f(x) =

2x  1 3

–1

Then f (x) =

 R is a function defined

.


Which statement is correct regarding function j(x) and I(x)-

3x  1

(A) The domain of j(x) and I(x) are the same

2

(B) Range of j(x) and I(x) are the same

Statement- II : f(x) is not a bijection.

(C) The union of domain of j(x) and I(x) are all Q.27

Statement- I : If f is even function, g is odd

function then

f g

real numbers (D) None of these

, (g  0)is an odd function.

Statement- II : If f(–x) = –f(x) for every x of

Q.32

If the solution of equation I(x) – g(x) = 0 are x1, x2, x3, .... xn when x

and if f(–x) = f(x) for every x of its domain,

option is correct-

then f(x) is called an even function.

(A) x1, x2, x3 ... xn makes the A.P. with common difference

Q.28

Statement I : Function f(x) = sinx + {x} is

20 for x  [0, 10]

Statement II : sinx and {x} are both periodic

(C) Sum of all solutions of the given equation

with period 2  and 1 respectively.

Statement I : y = f(x) =

x 2  2x  4 x 2  2x  5

is 100 in the interval [0, 10 ] (D) (B) and (C) are correct , x

R Q.33

Range of f(x) is [3/4, 1) 2

Statement II : (x – 1) =

4y  3 1 y

(B) h(x) is one-one and onto function

.

(C) h(x) is onto function (D) h(x) is many one and into function Q.34

Passage :-

 [–1, 1] and g : R  [–1, 1].

Now f(x) = 2 cos2 x – 1, g(x) = cos 2x, h(x) = f(x) + g(x), I(x) = f(x) – g(x), j (x) =

f ( x ) g( x )

If h : R  [–2, 2], then (A) h(x) is one-one function

Passage Based Questions

Let here we define f : R



(B) Total no. of solutions of I(x) – g(x) = 0 is

periodic with period 2

Q.29

 [0, 10] then which

its domain, then f(x) is called an odd function

are 5 functions.

Domain and range of j(x) respectively (A) R and {1} (B) R and {0, 1} (C) R – {(2n + 1) /4}, n  I and {1} (D) R – {(2n + 1) /2}, n  I and {1}


14

Column Matching Questions Match the entry in Column 1 with the entry in Column 2. Q.35

Match the column

Column 1

Column 2 –1

(A) f(x) = {x}, the fractional(P) f (x) =

1 2

x

–x

(4 – 4 )

part of x (B) f(x) =

16

x

4

1 x

(Q) f is an even function

(C) f(x) = log4 ( x  x 2  1) (R) f is a periodic function (D) f(x) = x

3

x

3

x

1 1

(S) f is odd function


15

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