1. The period of the function, f(x) = [sin 3x] + |cos 6x| is : ( [.] denotes the greatest integer less than or equal to x) (a) π (b) 3 2π (c) 2π (d) 2 π 2. The function f(x) =
[ ]
sec− − 1 xx x , where [x] denotes the greatest integer less than or equal to x is defined for all x belonging to :
(a) R (b) R - {(-1, 1) ∪ {n : n ∈ I)} (c) R′ - (0, 1) (d) R′ - {n : n ∈N} 3. The function f(x) = logx2(x) is defined for x belonging to :
(a) (-∞, 0) (b) (1,∞) (c) (0, ∞) (d) none of these
4. If f(x) = 1, 4 5 g and 3 x cos . x cos 3 x sin x sin2 2 = +π + +π + then (gof)(x) =
(a) 1 (b) -1 (c) x (d) none of these
5. Let [x] denote the greatest integer ≤x. The domain of definition of function
2 ] x [ x 4 ) x ( f 2 + − = is (a) (−∞, −2)∪[−1, 2] (b) [0, 2] (c) [-1, 2] (d) (0, 2)
6. The domain of definition of the function
− = 4 x x 5 log ) x ( f 2 10 is (a) [1, 4] (b) (1, 4) (c) (0, 5) (d) [0, 5]
7. The range of the function
x 3 cos 2 1 ) x ( f − = is (a) − ,0 3 1 (b) R (c) 1 , 3 1 (d) none of these
8. The domain of definition of the function
x | x | 1 ) x ( f − = is
(a) R (b) (0,∞) (c) (-∞,0) (d) none of these
L E V E L - 1 (Objective)
9. The function f(x)=log(x+ x2 +1) is
(a) an even function (b) an odd function (c) periodic function (d) none of these 10. The function f(x)=cos
(
log(x+ x2 +1))
is(a) even (b) odd (c) constant (d) none of these
11. Thr period of the function f(x) = sin4x+ cos4x is
(a) π (b) π/2 (c) 2π (d) none of these
12. Which of the following functions is an odd function
(a) f(x) = constant (b) f(x) = sinx + cosx (c) f(x)=sin
(
log(x+ x2+1))
(d) f(x) = 1 + x + 2x3 13. The domain of definition of the functionf(x) = 7-xPx-3 is
(a) [3, 7] (b) {3,4,5,6,7} (c) {3,4,5} (d) none of these
14. The function x tan x x x cos x sin ) x ( f 2 4 4 + + = is
(a) even (b) odd (c) periodic with periodπ (d) periodic with period 2π 15. Range of f(x) = |x| + |x+1| is
(a) [0, ∞) (b) (0, ∞) (c) (1, ∞) (d) [1, ∞)
16. Range of tan-1x - cot-1x is
(a) (0,π) (b) − π π 2 , 2 3 (c) 1 , 3 2 (d) ,1 3 2 17. The range of |x|−x is (a) (0,∞) (b) [0,∞) (c) (-∞,0) (d) (-∞,0] 18. If + − = 1 10 1 10 x ) x ( f 2x x 2 2 then ‘f’ is
(a) an even function (b) an odd function (c) neither even nor odd (d) cannot be determined 19. The domain of the function f(x) =
) x 1 ( log 1 10 − + (x+2) is (a) [-3, -2] excluding (-2.5) (b) [0, 1] excluding 0.5
(c) [-2, 1], excluding 0 (d) None of these
20. The period of ecos4πx+x−[x]+cos2πxis ([.] denotes the greatest integer function)
(a) 2 (b) 1 (c) 0 (d) -1
21. The domain of the function f(x) = − + + − + x 2 x 1 sin ) x cos(sin ) x (log sin 2 1 2 1
22. If [.] denotes the greatest integer function then the domain of the real valued function log[x+1/2]|x2−x−2|is (a) ∞, 2 3 (b) ,2 (2, ) 2 3 ∞ ∪ (c) ,2 (2, ) 2 1 ∞ ∪ (d) None of these
23. The domain of the function f(x) = 6 3(x 2) 2(x 1) 2 x 2 52 8 4 + − − − − is (a) (0, 1) (b) [3, ∞) (c) (1, 0) (d) none
24. A function whose graph is symmetrical about the y-axis is given by
(a) f(x)= loge(x+ x2+1) (b) f(x + y) = f(x) + f(y) for all x, y ∈ R
(c) f(x) = cos x + sin x (d) none of these
25. If ∞ ∈ + − = , 2 1 x , 1 x x ) x (
f 2 then value of ‘x’ satisfying f(x) = f -1(x) is
(a) 1 (b) 2 (c) 2 1 (d) none of these 26. If − − = 5 x 1 x log ) x ( f 0.4 and g(x) = x2 - 36, then D f/g is (a) (−∞, 0) ~{−6} (b) (0, ∞) ~{1, 6} (c) (1,∞)~{6} (d) [1,∞)~{6} 27. The domain of the real-valued function f(x) = loge |loge x| is
(a) (0,1)∪ (1, ∞) (b) (0, ∞) (c) (e,∞) (d) (1, ∞)
28. If f(x) and g(x) are two functions of ‘x’ such that f(x) + g(x) = ex and f(x) - g(x) = e-x, then (a) f(x) is odd, g(x) is odd (b) f(x) is even, g(x) is even
(c) f(x) is even, g(x) is odd (d) f(x) is odd, g(x) is even 29. The inverse of the function y=loga(x+ x2+1) (a>0, a≠1) is
(a) (a a ) 2
1 x − −x
(b) not defined for all x (c) defined for only positive x (d) none of these
30. Let f(x) = cos p x, where p = [a] = the greatest integer less than or equal to ‘a’. If the period of f(x) is π, then (a) a∈[4,5] (b) a=[4,5] (c) a∈[4,5) (d) none of these
31. If f(x) = ex−[x]+|cosπx|+|cos2πx|+...+|cosnπx|, then period of f(x) is
(a) 1 (b) n 1 (c) 1 n 1 n 2 2 + − (d) n ... 3 . 2 . 1 1
32. Let − + − + − = − − − 4 | x | 2 tan 4 | x | 2 cos 4 | x | 2 sin ) x ( f 1 1 1 , then Df is (a) [-6, 6] (b) [6,∞) (c) [-6, 3] (d) [-3, 6]
33. Identify the statement(s) which is/are incorrect ?
(a) The function f(x) = cos(cos-1 x) is neither odd nor even (b) The fundamental period of f(x)=cos(sinx)+ cos(cos x) is π (c) The range of the function f(x) = cos (3 sin x) is [-1, 1] (d) None of these 34. Let 9 x cos 4 ) x ( f 2 2−π = . Then (a) , , ,R [ 1,1] 3 Df f = − π ∞ = (b) , , ,R [ 2, 2] 3 Df f = − π ∞ = (c) , and R [ 4,4] 3 3 , Df f = − π ∞ ∪ −∞ −π = (d) ,R (0,4] 3 , Df f = −∞ −π =
35. The range of the function f(x)=sin(xe[x]+x2−x), x∈(−1,∞) where [x] denotes the greatest integer function is:
(a) φ (b) [0, 1] (c) [-1, 1] (d) R
36. The domain of f(x) = log2 log3 log4/π (tan-1 x)-1 =
(a) R (b) (4/π, ∞) (c) (0, 1) (d) None of these
37. If f(x) is a periodic function of the period ‘λ’ then f(λx + a), where a is a constant, is a periodic function of the period (a) 1 (b) λ (c) a λ (d) none of these 38. If ] 1 x [ 1 | x | 1 1 ) x x ( cos ) x ( f 1 2 2 − + − + − = −
then domain of f(x) is (where [.] is the greatest integer)
(a) + 2 5 1 , 2 (b) + 2 5 1 , 2 (c) − − 2 2 1 , 2 (d) none of these
39. Let f(x) = sin x, g(x) = ln |x|. If the ranges of the composite functions fog and gof are R1 and R2 respectively, then (a) R1=(−1,1),R2 =(−∞,0) (b) R1=(−∞,0],R2 =[−1,1] (c) R1=(−1,1),R2 =(−∞,0] (d) R1=[−1,1],R2 =(−∞,0] 40. If x 1 x 1 log ) x ( f − + = , then (a) f(x) is even (b) f(x1).f(x2) = f(x1 + x2) (c) f(x x ) ) x ( f ) x ( f 2 1 2 1 = − (d) f(x) is odd
1. Find the value of x for which, 0 ) 4 x ( ) 2 x ( ) 1 x )( 1 x 2 ( ) x ( f 4 3 2 > − − − − =
2. Find the values of x for which 0
) 1 x ( ) 4 x ( ) 3 x ( ) x 1 ( ) 2 x ( ) x ( f 2 2 ≤ + − − − − = . 3. Solve 1 2 x x | 3 x | > + + + .
4. Find the domain of the function;
− + − = | x sin | log 3 ) 23 x 8 x ( log log ) x ( f 2 2 | x sin |
5. Find domain for
+ − = 5 x 1 x log ) x ( f 0.4 .
6. Find domain for log x
3 | x | 2 1 cos y −1 + |x−1| − = .
7. Find domain for
− π = ] 1 x [ cos ] x [sin ) x (
f where [ ] denotes greatest integer function.
8. Find the range for
x ] x [ 1 ] x [ x y + − −
= where [ ] denotes greatest integer function.
9. Find domain and range of the function y = loge(3x2 - 4x + 5).
10. If f is an even function, find the real values of x satisfying the equation + + = 2 x 1 x f ) x ( f .
11. Find whether the given function is even or odd function, where
2 1 x ) x tan x (sin x ) x ( f − π π + + = .where [ ] denotes
greatest integer function.
L E V E L - 2 (Subjective)
12. If f:[1, ∞)→[2,∞) is given by x 1 x ) x (
f = + then find f−1(x). (assume bijective).
13. Let f:[1/2, ∞) →[3/4,∞), where f(x) = x2
- x + 1. Find the inverse of f(x).
14. , x ] x [ 1 ) x ( f −
= where [ ] denotes greatest integral function less than or equals to x. Then find domain of f(x).
15. Find the domain of
) 13 x 7 x ( log 1 ) x ( f 2 2 / 1 − + =
16. Find the domain of single valued function y = f(x) given by the equation 10x + 10y = 10.
17. Let π ∈ 2 , 0
x , then find the solution of the function
x tan log 1 ) x ( f x sin − = .
18. Find the range of log3(log1/2(x2 + 4x + 4)).
19. Find the domain & range of : 2 2 x 9 sin 3 ) x ( f = π − .
20. Find the inverse of following functions:
(i) f(x)=sin−1(x/3), x∈[−3,3] [assuming bijective] (ii) f(x)=ln(x2+3x+1), x∈[1,3]. [assuming bijective]
21. ≤ ≤ ≤ ≤ − − = 1 x 0 , x 0 x 1 , 1 x ) x (
f 2 and g(x) = sinx. Find h(x) = f(|g(x)|) + |f(g(x))|.
22. If f(x)=cos[π2]x+cos[−π2]x, where [x] stands for the greatest integer function, then evaluate ) 4 / ( f and ) ( f ), ( f ), 2 / ( f π π −π π .
23. A cubic expression f(x) satisfies the condition = + x 1 f ) x ( f x 1 f ) x (
f , then prove that f(x) = 1 + x3or1 - x3.
If f(3) = 28. Then prove that f(2) = 9.
24. Let f(x) be a polynomial function satisfying, f(x)f(y)=f(x)+f(y)+f(xy)−2∀ x, y∈R. If f(2) = 5 then prove that f(5) = 26. 25. If for non-zero x, 5 x 1 x 1 bf ) x ( af = −
1. The domain of the function f(x)=log(1−x)+ x2−1 is
(a) [-1, 1] (b) (1, ∞) (c) (0, 1) (d) (−∞, −1]
2. The range of the function 2 2 x x 1 ) x ( f = + is equal to (a) [0, 1] (b) (0, 1) (c) (1,∞) (d) [1,∞)
3. The curves y=|x|3+3|x|2+2and y=x3+3x2+2 have the same graph for
(a) x > 0 (b) x≥0 (c) all x except 0 (d) all x
4. Domain of the function 7
3 x 1 2 x 1 ) x ( f 2 sin1x + − + + = − is
(a) φ (b) R - {0} (c) R (d) None of these
5. The domain of definition of the function y=3e x2−1 log (x−1) is
(a) (1, ∞) (b) [1,∞) (c) R ~ {1} (d) (−∞, −1)∪ (1, ∞)
6. The range of the function f(x) = cos [x], where
2 x 2 π < < π − , is
(a) {-1, 1, 0} (b) {cos 1, 1, cos 2} (c) {cos1, −cos1,1} (d) none of these 7. If b2 - 4ac = 0 and a > 0, then domain of the function y = log (ax3 + (a + b)x2 + (b + c)x + c) is
(a) − a 2 b ~ R 2 (b) ∪ ≥− − {x|x 1} a 2 b ~ R (c) ∪ −∞ − − ( , 1] a 2 b ~ R (d) none of these
8. Which of the following functions is an even function?
(a) x x x x a a a a ) x ( f − − − + = (b) 1 a 1 a ) x ( f x x − + = (c) 1 a 1 a x ) x ( f x x + − = (d) f(x) log
(
x x2 1)
2 + + = 9. If (log3 x) (logx 2x) (log2x y) = logx x2, then y is equal to(a) 9 (b) 18 (c) 27 (d) 81
L E V E L - 3
10. The range of the function 1 x x 1 x x ) x ( f 2 2 + + + − = is (a) R (b) [3, ∞) (c) ,3 3 1 (d) none of these
11. The domain of the function 2
x x 2 2 ) x ( f = − − is
(a) −3≤x≤ 3 (b) −1− 3≤x≤−1+ 3 (c) −2≤x≤2 (d) none of these 12. If the domain of the function f(x)=x2−6x+7is (−∞,∞), then range of the function is
(a) (−∞,∞) (b) [−2, ∞) (c) (-2, 3) (d) (−∞, −2)
13. The domain of the function
= − 2 x log sin ) x ( f 2 2 1 is
(a) −1≤x≤1 (b) 0≤x≤1 (c) 1≤x≤2 (d) none of these
14. If f(x) = cos (log x) then
+ − f(xy) y x f 2 1 ) y ( f ) x (
f has the value
(a) 1 (b)
2 1
(c) -2 (d) 0
15. The inverse of the function 2
e e e e ) x ( f x x x x + + − = −− is given by (a) 2 / 1 e 1 x 2 x log − − (b) 3 / 1 e x 3 1 x log − − (c) 2 / 1 e x 2 x log − (d) 2 e 1 x 1 x log − + −
16. The range of the function for real x of
x 3 sin 2 1 y − = is (a) y 1 3 1≤ ≤ (b) y 1 3 1≤ < − (c) y 1 3 1> > − (d) y 1 3 1> >
17. The domain of the function 2x x2 ]
x [
1 + −
, where [ . ] denotes greatest integer function.
(a) [1, 2] (b) [0, 2] (c) [0, 1) (d) [1, 2)
18. Domain of sin-1(log
3(x/3)) is
(a) [1, 9] (b) [-1, 9] (c) [-9, 1] (d) [-9, -1]
19. The range of f(x) = 7-xP x - 3 is
20. The value of n∈Z for which the function = n x sin nx sin ) x (
f has 4 as its period isπ
(a) 2 (b) 3 (c) 5 (d) 4
21. The domain of the function f(x) = log2(log3(log4 x)) is
(a) x < 4 (b) x > 4 (c) 0 < x < 2 (d) 2 < x < 4 22. If f(x) is an odd periodic with period 2, then f(4) is
(a) 0 (b) 2 (c) 4 (d) -4
23. If f(x) = 1 + xα , (α≠0) is the inverse of itself then the value of α is
(a) -2 (b) -1 (c) 0 (d) 2
24. If g(x) be a function defined on [-1, 1] and if the area of the equilateral triangle with two of its vertices at (0, 0) and (x, g(x))) is
4 3
, then the function is
(a) 2 x 1 ) x ( g =± − (b) g(x)=− 1−x2 (c) g(x)= 1−x2 (d) g(x) = 2 x 1+ 25. Which of the following functions is periodic
(a) f(x) = x - [x] (b) = ≠ = 0 x , 0 0 x , x 1 sin x ) x (
f (c) f(x) = x cos x (d) none of these
26. 2 x ) 3 x )( 1 x ( ) x ( f − − +
= is real valued in the domain
(a) (−∞, −1]∪[3, ∞) (b) (−∞, −1]∪(2, 3] (c) [−1, 2)∪[3, ∞) (d) none of these 27. The domain of definition of the function y(x) given by the equation 2x + 2y = 2 is
(a) 0<x≤1 (b) 0≤x≤1 (c) −∞<x≤0 (d) −∞<x<1 28. If the function f:[1, ∞) →[1, ∞) is defined by f(x) = 2x(x - 1) then f-1(x) is
(a) ) 1 x ( x 2 1 − (b)
(
1 1 4log x)
2 1 2 + + (c)(
1 1 4log x)
2 1 2 + − (d) not defined29. If log0.3(x - 1) < log0.09(x - 1), then x lies in the interval
(a) (2, ∞) (b) (1, 2) (c) (-2, -1) (d) none of these
30. If g(f(x)) = |sin x| and f(g(x)) = 2 ) x
(sin , then
(a) f(x)=sin2x, g(x)= x (b) f(x)=sinx, g(x)=|x| (c) f(x)=x2, g(x)=sin x (d) f and g cannot be determined
31. Let > = < − = − + = 0 x , 1 0 x , 0 0 x , 1 ) x ( f and ] x [ x 1 ) x (
g . Then for all x, f(g(x)) is equal to
(a) x (b) 1 (c) f(x) (d) g(x).
32. The domain of definition of
2 x 3 x ) 3 x ( log ) x ( f 2 2 + + + = is (a) R ~ {-1, -2} (b) (−2, ∞) (c) R ~ {-1, -2, -3} (d) (−3,∞) ~{−1, −2} 33. If , x 1 1 x x ) x ( f ≠− + α
= , then for what value of α is f(f(x)) = x ?
(a) 2 (b) − 2 (c) 1 (d) -1
34. The set of all real numbers x for which x2 - |x + 2| + x > 0, is
(a) (−∞, −2)∪(2, ∞) (b) (−∞, − 2) ∪( 2, ∞) (c) (−∞, −1) ∪(1, ∞) (d)
(
2,∞)
35. Range of the function , x R
1 x x 2 x x ) x ( f 2 2 ∈ + + + + = is (a) (1, ∞) (b) 7 11 , 1 (c) 3 7 , 1 (d) 5 7 , 1
ANSWER KEY
1. b 2. b 3. b 4. a 5. a 6. a 7. c 8. c 9. b 10. a 11. b 12. c 13. c 14. b 15. d 16. b 17. b 18. b 19. d 20. b 21. b 22. b 23. b 24. d 25. a 26. c 27. a 28. c 29. a 30. c 31. a 32. a 33. a 34. c 35. c 36. c 37. a 38. a 39. d 40. d LEVEL - 1 (Objective)ANSWER KEY
1. x∈(−∞,1/2)∪(2,∞)~{4} 2. x∈(−1,1]∪[3, ∞) 3. x∈(−5, −2)∪(−1, ∞) 4. π ∪ π π ∪ π ∈ , 5 2 3 2 3 , ) , 3 ( ) x ( f 5. f(x)∈(1, ∞) 6. x∈(0,1)∪(1, 2] 7. R ~ [1, 2) 8. Range = 2 1 , 0 9. Range is ∞ , 3 11 log 10. − + − − − + − − = 2 5 3 , 2 5 3 , 2 5 1 , 2 5 1 x11. f(x) is an odd function (if x≠nπ) and f(x) is an even function if (x=nπ). 12. 2 4 x x+ 2− 13. 4 3 x 2 1 − + 14. φ 15. (3, 4) 16. x∈(−∞,1) 17. π π 2 , 4 18. Range∈R 19. ∈−π π 3 , 3 Domain , ∈ 2 3 3 , 0 Range
20. (i) 3 sin x (ii)
2 e 5 3± + x − 21. ≤ ≤ < < − + − 1 x 0 , x sin 2 0 x 1 , 1 x sin x sin 2 2 22. 2 1 4 f , 0 ) ( f , 0 ) ( f , 1 2 f = π = π − = π − = π 23. 24. 25. ) b a ( 5 bx x a b a 1 ) x ( f 2 2 + − − − = LEVEL -2 (Subjective)
1. d 2. c 3. b 4. a 5. a 6. b 7. c 8. c 9. a 10. c 11. b 12. b 13. d 14. d 15. b 16. a 17. a 18. a 19. a 20. a 21. b 22. a 23. b 24. b 25. a 26. c 27. d 28. b 29. a 30. a 31. b 32. d 33. d 34. b 35. c
LEVEL - 3 (Questions asked from previous Engineering Exams)