− log0.

3 ( x − 1)
1. The domain of the function f(x)= is 10. The range of the functin
x 2 + 2x + 8
2
(A) (1, 4) (B) (–2, 4) (C) (2, 4) (D) [2, ∞) f(x) = log 2 ( 2– log2 (16 sin x + 1)) is

2. The domain of the function (A) (–∞, 1) (B) (–∞, 2) (C) (–∞, 1] (D) (–∞, 2]

  1   11. If [2 cos x] + [sin x] = –3, then the range of the

f(x) = log1/2  − log2  1 +  − 1 is
 4
x  
  function, f(x) = sin x + 3 cos x in [0, 2π] is
(A) 0 < x < 1 (B) 0 < x ≤ 1 (C) x ≥ 1 (D) null set
2 (where [ * ] dentoes greatest integer function)
3. If q – 4 p r = 0, p > 0, then the domain of the
3 2 (A) [–2, –1) (B) (–2, –1]
function, f(x) = log (px + (p + q) x + (q + r) x + r) is
(C) (–2, –1) (D) [–2, – 3)
 q   q 
(A) R – −  (B) R – ( −∞,−1] ∪ − 
12. The range of the function f(x) = 7 – xPx – 3 is
 2p    2p 
(A) {1, 2, 3} (B) {1, 2, 3, 4, 5, 6}
  q 
(C) R– ( −∞,−1] ∩ −  (D) none of these (C) {1, 2, 3, 4} (D) {1, 2, 3, 4, 5}
  2p 
x
cos 1 1
4. If domain of f(x) is (– ∝, 0] then domain of 2
2 x x
f(6{x} – 5{x} + 1) is 1 cos − cos
13. Range of the function f(x)= is
(where {*} represetns fractional part function) 2 2
x
− cos 1 −1
 1 1 2
(A) ∪ n + 3 ,n + 2 
  (B) (–∞, 0)
n∈Ι (A) [0, 2] (B) [0, 4] (C) [2, 4] (D) [1, 3]

 1  14. In the square ABCD with side AB = 2, two points

(C) ∪ n + 6 n + 1
n∈Ι
(D) None of these M & N are on the adjacent sides of the square such
that MN is parallel to the diagonal BD. If x is the
5. Find domain of the function distance of MN from the vertex A and
f(x) = Area (∆AMN), then range of f(x) is
 2x − 1 
f(x) = − log x + 4  log2  (A) (0, 2 ] (B) (0, 2] (C) (0,2 2 ] (D) (0, 2 3 ]
2 
3+x 

(A) (–4, –3) ∪ (4, ∞) (B) (– ∞, –3) ∪ (4, ∞) 15. Let f be a real valued function defined by
(C) (– ∞, – 4) ∪ (3, ∞) (D) None of these e x − e −|x|
f(x) = then the range of f(x) is
e x + e|x|
6. The domain of the function log1/ 3 log4 ([ x ]2 − 5) is
 1
(where [x] denotes greatest integer function) (A) R (B) [0, 1] (C) [0, 1) (D) 0, 
 2
(A) [–3, –2) ∪ [3, 4) (B) [–3, –2) ∪ (2, 3]
sin2 x + 4 sin x + 5
(C) R – [–2, 3) (D) R – [–3, 3] 16. If f(x) = , then range of f(x) is
2 sin2 x + 8 sin x + 8
7. Let f(x) = (x12 – x9 + x4 – x + 1)–1/2. the domain of
the function is 1  5  5  5 
(A) (1, + ∞) (B) (–∞, –1) (C) (–1, 1) (D) (–∞, ∞) (A)  , ∞  (B)  ,1 (C)  ,1 (D)  , ∞ 
2  9  9  9 
8. Range of f(x) = 4 + 2 + 1 is x x
17. The number of solution(s) of the equation
(A) (0, ∞) (B) (1, ∞) (C) (2, ∞) (D) (3, ∞)
[x] + 2{–x} = 3x, is/are
(where [ * ] represents the greatest integer function
9. Range of f(x) = log 5 { 2 (sin x –cos x) + 3} is
and { * } denotes the fractional part of x)
 3 (A) 1 (B) 2 (C) 3 (D) 0
(A) [0, 1] (B) [0, 2] (C) 0,  (D) None of these
2  
18. The number of solutions of the equation 25. The function f : [2, ∞) → Y defined by
2
[sin–1 x] = x – [x] is f(x) = x – 4x + 5 is both one–one & onto if
(where [ * ] denotes the greatest integer function) (A) Y = R (B) Y = [1, ∞) (C) Y = [4, ∞) (D) Y = [5, ∞)
(A) 0 (B) 1 (C) 2 (D) infinitely many
26. Let f : R → R be a function defined by
19. The sum 2x 2 − x + 5
f(x) = then f is
 1  1 1  1 2  1 3   1 1999  7 x 2 + 2x + 10
 2  +  2 + 2000  +  2 + 2000  +  2 + 2000  + ...... +  2 + 2000 
          (A) one – one but not onto
is equal to (B) onto but not one – one
(where [ * ] denotes the greatest integer function) (C) onto as well as one – one
(A) 1000 (B) 999 (C) 1001 (D) None of these (D) neither onto nor one – one

20. Which of the following represents the graph of 27. Let f : R → R be a function defined by
3 2
f(x) = sgn ([x + 1]) f(x) = x + x + 3x + sin x. Then f is
(A) one – one & onto (B) one – one & into
(C) many one & onto (D) many one & into

–1
1
–1
1 4a − 7 3 2
(A) (B) 28. If f(x) = x + (a – 3) x + x + 5 is a one–
3
–1 –1
one function, then
(A) 2 ≤ a ≤ 8 (B) 1 ≤ a ≤ 2
(C) 0 ≤ a ≤ 1 (D) None of these

–1
1 1 29. Let f: (e, ∞) → R be defined by f(x) = ln (ln(ln x)),
(C) (D) –1
1 then
–1 1
–1 (A) f is one one but not onto
(B) f is onto but not one – one
(C) f is one–one and onto
21. If f(x)=2 sin2θ+4 cos (x+θ) sin x. sin θ+cos (2x+2θ) (D) f is neither one–one nor onto
π 
then value of f2(x) + f2  − x  is 30. If f(x) = 2[x] + cos x, then f: R → R is
 4  (where [ * ] denotes greatest integer function)
(A) 0 (B) 1 (C) –1 (D) x2 (A) one–one and onto (B) one–one and into
(C) many–one and into (D) many–one and onto
22. If A, B, C are three decimal numbers and
p = [A + B + C] and q = [A] + [B] + [C] then maximum 31. If f : R → S, defined by f(x) = sin x – 3 cosx + 1,
value of p – q is (where [ * ] represents greatest
is onto, then the interval of S is
integer function).
(A) [0, 3] (B) [–1, 1] (C) [0, 1] (D) [–1, 3]
(A) 0 (B) 1 (C) 2 (D) 3
32. The function f : R → R defined by f(x) = 6x + 6|x| is
23. Let f(x) = ax2 + bx + c, where a, b, c are rational (A) one-one and onto (B) many-one and onto
and f : Z → Z, where Z is the set of integers. Then (C) one-one and into (D) many-one and into
a + b is
(A) a negative integer (B) an integer 33. If the real-valued function f(x) = px + sinx is a
(C) non-integral rational number (D) None of these bijective function, then the set of all possible values
of p ∈ R is
24. Which one of the following pair of functions are (A) R – {0} (B) R (C) (0, ∞) (D) None of these
identical ?
(ln x)/2 34. Let S be the set of all triangles and R+ be the set
(A) e and x
–1
of positive real numbers. Then the function, f : S→R+,
(B) tan (tan x) & cot–1 (cot x) f(∆) = area of the ∆, where ∆ ∈ S is
2 4
(C) cos x + sin x and sin2 x + cos4x (A) injective but not surjective
|x| (B) surjective but not injective
(D) and sgn (x) where sgn(x) stands for signum
x (C) injective as well as surjective
function. (D) neither injective nor surjective
35. Let ‘f’ be a function from R to R given by 42. If y = f (x) satisfies the condition
x −4
2
f(x) = . Then f(x) is  1 2 1
x2 + 1 fx +  =x + 2 (x ≠ 0) then f(x) equals
 x x
(A) one-one and into (B) one-one and onto
2 2 2 2
(C) many-one and into (D) many-one and onto (A) – x + 2 (B) – x – 2 (C) x + 2 (D) x – 2

36. Function f : (– ∞, 1) → (0, e5] defined by 43. If f(1) = 1 and f(n + 1) = 2f(n) + 1 if n ≥ 1, then
−( x −3 x + 2 )
2 f(n) is equal to
f(x) = e is n
(A) 2 + 1 (B) 2
n n
(C) 2 – 1 (D) 2
n–1
–1
(A) many one and onto (B) many one and into
(C) one one and onto (D) one one and into 44. A function f : R → R satisfies the condition,
2 4
x f(x) + f(1 – x) = 2x – x . Then f(x) is
2 2 2 4
 π (A) – x – 1 (B) –x + 1 (C) x – 1 (D) – x + 1
x : R →  0, 
–1 +
37. If f(x) = cot
 2
45. A real valued function f(x) satisfies the functional
2
and g(x) = 2x – x : R → R. Then the range of the equation f(x – y) = f(x) f(y) – f(a – x) f(a + y) where
function f(g(x)) wherever define is a is a given constant and f(0) = 1, f(2a – x) is equal to
(A) f(–x) (B) f(a) + f(a – x) (C) f(x) (D) –f(x)
 π  π π π π
(A)  0,  (B)  0,  (C)  ,  (D)  
46. If f : R → R satisfies f(x + y) = f(x) + f(y), for all
 2  4 4 2  4
n
+ x
38. f(x) = |x – 1|, f : R →R ; g(x) = e , g : [–1, ∞)→R x, y → R and f(1) = 7, then ∑ f (r ) is
r =1
If the function fog(x) is defined, then its domain and
range respectively are 7n 7(n + 1) 7n(n + 1)
(A) (0, ∞) & [0, ∞) (B) [–1, ∞) & [0, ∞) (A) (B) (C) 7n(n+1) (D) .
2 2 2
 1  1   1 + sin x 
(C) [–1, ∞) &  1 − ,∞ (D) [–1, ∞) &  − 1, ∞  47. The function f(x) = log   is
 e   e   1 − sin x 
(A) even (B) odd
− 1 if x<0
 (C) neither even nor odd (D) both even & odd
x=0
39. Let g(x) = 1 + x – [x] and f(x) = 
0 if
,
 1 if x>0 ax − 1
 48. If the graph of the function f(x) = is
then ∀ x, fog(x) equals x (a x + 1)
n

symmetric about y–axis, then n is equal to

(where [ * ] represents greatest integer function).
(A) 2 (B) 2/3 (C) 1/4 (D) –1/3
(A) x (B) 1 (C) f(x) (D) g(x)
49. The graph of the function y = f(x) is symmetrical
40. Let f: [0, 1] → [1, 2] defined as f(x) = 1 + x and
about the line x = 2, then
g : [1, 2] → [0, 1] defined as g(x) = 2 – x then the
(A) f(x + 2) = f(x – 2) (B) f(2 + x) = f(2 – x)
composite function gof is
(C) f(x) = f(–x) (D) f(x) = –f(–x)
(A) injective as well as surjective
(B) Surjective but not injective 50. If f(–x) = –f(x), then f(x) is
(C) Injective but non surjective (A) neither odd nor even (B) an odd function
(D) Neither injective nor surjective (C) an even function (D) periodic function

41. Let f & g be two functions both being defined  x 2 + 1

51. If g : [–2, 2] → R where g(x)=x3+tan x +  
x+ | x |  p 
from R → R as follows f(x) = and be an odd function , then the value of the parameter
2
x for x<0 P is
g(x) =  . Then (A) –5 < P < 5 (B) P < 5 (C) P>5 (D) None of these
x
2
for x≥0
52. It is given that f(x) is an even function and satisfy the
(A) fog is defined but gof is not
(B) gof is defined but fog is not xf ( x 2 )
relation f(x) = then the value of f(10) is
(C) both fog & gof are defined but they are unequal 2 + tan 2 x.f ( x 2 )
(D) both gof & fog are defined and they are equal
(A) 10 (B) 100 (C) 50 (D) None of these
53. Fundamental period of f(x) = sec (sin x) is
(A) π/2 (B) 2π (C) π (D) a periodic

54. If f(x) = sin [a] x has π as its fundamental period

then (where [ * ] denotes the gratest integer function)
(A) a = 1 (B) a = 9 (C) a ∈ [1, 2) (D) a ∈ [4, 5)

55. The fundamental period of the function,

f(x) = x + a – [x + b] + sin πx + cos 2πx + sin 3πx +
cos 4πx + ..... + sin (2n – 1) πx + cos 2 nπx
or every a, b ∈ R is
(where [ * ] denotes the greatest integer function)
(A) 2 (B) 4 (C) 1 (D) 0
π πx π
56. The period of sin [x] + cos + cos [x],
4 2 3
where [x] denotes the integral part of x is
(A) 8 (B) 12 (C) 24 (D) Non–periodic

57. The fundamental period of function

 1  2
f(x) = [x] +  x +  +  x +  – 3x + 15
 3  3
(A) 1/3 (B) 2/3 (C) 1 (D) Non–periodic

58. Which one of the following is true.

16 x − 1
(A) f(x) = is an odd function
4x
(B) f(x) = sin |x| is an odd function
(C) if sin x + cos a x is periodic then ‘a’ is irrational
(D) if f1 (x), f2 (x) are periodic then their sum function
will always be periodic

59. Let f(x) = x (2 – x), 0 ≤ x ≤ 2. If the definition of ‘f’

is extended over the set, R – [0, 2] by f(x + 2) = f(x),
then ‘f’ is a
(A) periodic function of period 1
(B) non-periodic function
(C) periodic function of period 2
(D) periodic function of period 1/2

60. Let f(2, 4) → (1, 3) be a function defined by

x –1
f(x) = x –   , then f (x) is equal to
2
(where [ * ] denotes the greatest integer function)
x
(A) 2x (B) x +   (C) x + 1 (D) x – 1
2

61. The mapping f : R → R given by

3 2
f(x) = x + ax + bx + c is a bijection if
2 2 2 2
(A) b ≤ 3a (B) a ≤ 3b (C) a ≥ 3b (D) b ≥ 3a
1. D 2. D 3. B 4. A 5. A 6. A 7. D 8. B

9. B 10. D 11. D 12. A 13. C 14. B 15. D 16. C

17. C 18. B 19. A 20. A 21. B 22. C 23. B 24. C

25. B 26. D 27. A 28. A 29. C 30. C 31. D 32. D

33. D 34. B 35. C 36. D 37. C 38. B 39. B 40. A

41. D 42. D 43. C 44. B 45. D 46. D 47. B 48. D

49. B 50. B 51. C 52. D 53. C 54. D 55. A 56. C

57. A 58. A 59. C 60. C 61. B