1. The period of the function, f(x) = [sin 3x] + |cos 6x| is : ( [.] denotes the greatest integer less than or equal to x)(a) (b) 32(c) 2 (d) 22. The function f(x) = [ ]sec1xx x , where [x] denotes the greatest integer less than or equal to x is defined for allx 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) = ) x ( log 2x is defined for x belonging to :(a) (- , 0) (b) (1,) (c) (0, ) (d) none of these4. If f(x) = , 145g and3x cos . x cos3x sin x sin 2 2 ,_

,_

+ + ,_

+ + then (gof)(x) =(a) 1 (b) -1 (c) x (d) none of these5. Let [x] denote the greatest integer x . The domain of definition of function 2 ] x [x 4) x ( f2+ is(a) ] 2 , 1 [ ) 2 , ( (b) [0, 2] (c) [-1, 2] (d) (0, 2)6. The domain of definition of the function

,_

4x x 5log ) x ( f210 is(a) [1, 4] (b) (1, 4) (c) (0, 5) (d) [0, 5]7. The range of the function x 3 cos 21) x ( fis(a) 1]1

0 ,31(b) R (c) 1]1

1 ,31(d) none of these8. The domain of definition of the function x | x |1) x ( f is(a) R (b) (0, ) (c) (- ,0) (d) none of theseLEVEL - 1 (Objective)FUNCTION9. The function ) 1 x x log( ) x ( f 2+ + is(a) an even function (b) an odd function (c) periodic function (d) none of these10. The function ( ) ) 1 x x log( cos ) x ( f 2+ + is(a) even (b) odd (c) constant (d) none of these11. Thr period of the function f(x) = sin4x+ cos4x is(a) (b) /2 (c) 2 (d) none of these12. Which of the following functions is an odd function(a) f(x) = constant (b) f(x) = sinx + cosx (c) ( ) ) 1 x x log( sin ) x ( f 2+ + (d) f(x) = 1 + x + 2x313. 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 these14. The function x tan x xx cos x sin) x ( f24 4++ is(a) even (b) odd (c) periodic with period (d) periodic with period 215. 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, ) (b) 1]1

2,23(c) ,_

1 ,32(d) 1]1

1 ,3217. The range of x | x | is(a) (0,) (b) [0,) (c) (-,0) (d) (-,0]18. If

,_

+1 101 10x ) x ( fx 2x 22 then f is(a) an even function (b) an odd function (c) neither even nor odd (d) cannot be determined19. The domain of the function f(x) = ) x 1 ( log110 + ) 2 x ( + is(a) [-3, -2] excluding (-2.5) (b) [0, 1] excluding 0.5(c) [-2, 1], excluding 0 (d) None of these20. The period of x cos ] x [ x x cos 2 4e + + is ([.] denotes the greatest integer function) (a) 2 (b) 1 (c) 0 (d) -121. The domain of the function f(x) =

,_

++ + x 2x 1sin ) x cos(sin ) x (log sin2121(a) {x : 1 < x < 2} (b) {1} (c) Not defined for any value x (d) {-1, 1}22. If [.] denotes the greatest integer function then the domain of the real valued function is | 2 x x | log 2] 2 / 1 x [ +(a)

,_

,23(b) ) , 2 ( 2 ,23

,_

(c) ) , 2 ( 2 ,21

,_

(d) None of these23. The domain of the function f(x) = 6) 1 x ( 2) 2 x (32x2 52 8 4 + is(a) (0, 1) (b) [3, ) (c) (1, 0) (d) none24. A function whose graph is symmetrical about the y-axis is given by(a) ) 1 x x ( log ) x ( f 2e + + (b) f(x + y) = f(x) + f(y) for all x, y R(c) f(x) = cos x + sin x (d) none of these25. If

,_

+ ,21x , 1 x x ) x ( f 2 then value of x satisfying f(x) = f -1(x) is(a) 1 (b) 2 (c) 21(d) none of these26. If ,_

5 x1 xlog ) x ( f 4 . 0 and g(x) = x2 - 36, then Df/g is(a) } 6 { ~ ) 0 , ( (b) } 6 , 1 { ~ ) , 0 ( (c) } 6 { ~ ) , 1 ( (d) } 6 { ~ ) , 1 [ 27. The domain of the real-valued function f(x) = loge |loge x| is(a) ) , 1 ( ) 1 , 0 ( (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 even29. The inverse of the function ) 1 a , 0 a ( ) 1 x x ( log y 2a > + + is(a) ) a a (21 x x (b) not defined for all x(c) defined for only positive x (d) none of these30. 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 , then(a) ] 5 , 4 [ a (b) ] 5 , 4 [ a (c) ) 5 , 4 [ a (d) none of these31. If f(x) = , e | x n |cos ........ | x 2 |cos | x |cos ] x [ x + + + + then period of f(x) is(a) 1 (b) n1(c) 1 n1 n22+(d) n ...... 3 . 2 . 1132. Let

,_

+ ,_

+ ,_

4| x | 2tan4| x | 2cos4| x | 2sin ) 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(sin x) + cos(cos x) is (c) The range of the function f(x) = cos (3 sin x) is [- 1, 1](d) None of these34. Let 9x cos 4 ) x ( f22 . Then(a) ] 1 , 1 [ R , , ,3D f f ,_

(b) ] 2 , 2 [ R , , ,3D f f ,_

(c) ] 4 , 4 [ R and ,3 3, D f f ,_

1]1

(d) ] 4 , 0 ( R ,3, D f f ,_

35. The range of the function ) , 1 ( x ), x x xe sin( ) x ( f 2 ] x [ + where [x] denotes the greatest integer function is:(a) (b) [0, 1] (c) [-1, 1] (d) R36. The domain of f(x) = log2 log3 / 4log (tan-1 x)-1 =(a) R (b) ) , / 4 ( (c) (0, 1) (d) None of these37. If f(x) is a periodic function of the period then f(x + a), where a is a constant, is a periodic function of theperiod(a) 1 (b) (c) a(d) none of these38. If ] 1 x [1| x |11 ) x x ( cos ) x ( f22 1+

,_

+ then domain of f(x) is (where [.] is the greatest integer)(a) 1]1

+25 1, 2 (b) 1]1

+25 1, 2 (c)

,_

22 1, 2 (d) none of these39. 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) ) 0 , ( R ), 1 , 1 ( R 2 1 (b) ] 1 , 1 [ R ], 0 , ( R 2 1 (c) ] 0 , ( R ), 1 , 1 ( R 2 1 (d) ] 0 , ( R ], 1 , 1 [ R 2 1 40. If x 1x 1log ) x ( f+ , then(a) f(x) is even (b) f(x1).f(x2) = f(x1 + x2) (c) ) x x ( f) x ( f) x ( f2 121 (d) f(x) is odd1. Find the value of x for which, 0) 4 x () 2 x ( ) 1 x )( 1 x 2 () x ( f43 2> 2. Find the values of x for which 0) 1 x () 4 x ( ) 3 x ( ) x 1 ( ) 2 x () x ( f2 2+ .3. Solve 12 xx | 3 x |>++ +.4. Find the domain of the function;

,_

+ | x sin | log3) 23 x 8 x ( log log ) x ( f22| x sin |5. Find domain for

,_

+5 x1 xlog ) x ( f 4 . 0.6. Find domain for x log3| x | 2 1cos y | 1 x |1+

,_

.7. Find domain for

,_

] 1 x [cos ] x [sin ) x ( f where [ ] denotes greatest integer function.8. Find the range for x ] x [ 1] x [ xy+ 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 x1 xf ) x ( f .11. Find whether the given function is even or odd function, where 21 x) x tan x (sin x) x ( f1]1

++ .where [ ] denotesgreatest integer function.LEVEL - 2 (Subjective)312. If ) , 2 [ ) , 1 [ : f is given by x1x ) x ( f + then find ) x ( f 1 . (assume bijective).13. Let ) , 4 / 3 [ ) , 2 / 1 [ : f , 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 ( log1) x ( f22 / 1 + 16. Find the domain of single valued function y = f(x) given by the equation 10x + 10y = 10.17. Let ,_

2, 0 x, then find the solution of the function x tan log1) x ( fx sin .18. Find the range of log3(log1/2(x2 + 4x + 4)).19. Find the domain & range of : 22x9sin 3 ) x ( f .20. Find the inverse of following functions:(i) ] 3 , 3 [ x ), 3 / x ( sin ) x ( f 1 [assuming bijective](ii) ] 3 , 1 [ x ), 1 x 3 x ( ln ) x ( f 2 + + . [assuming bijective]21.' 1 x 0 , x0 x 1 , 1 x) x ( f2 and g(x) = sinx. Find h(x) = f(|g(x)|) + |f(g(x))|.22. If , x ] cos[ x ] cos[ ) x ( f 2 2 + where [x] stands for the greatest integer function, then evaluate) 4 / ( f and ) ( f ), ( f ), 2 / ( f .23. A cubic expression f(x) satisfies the condition

,_

,_

+x1f ) x ( fx1f ) 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, R y , x 2 ) xy ( f ) y ( f ) x ( f ) y ( f ) x ( f + + . If f(2) = 5 thenprove that f(5) = 26.25. If for non-zero x, 5x1x1bf ) x ( af

,_

+ where b a then find f(x).1. The domain of the function 1 x ) x 1 log( ) x ( f 2 + is(a) [-1, 1] (b) ) , 1 ( (c) (0, 1) (d) ] 1 , ( 2. The range of the function 22xx 1) x ( f + is equal to(a) [0, 1] (b) (0, 1) (c) ) , 1 ( (d) ) , 1 [ 3. The curves 2 x 3 x y and 2 | x | 3 | x | y 2 3 2 3+ + + + have the same graph for(a) x > 0 (b) 0 x (c) all x except 0 (d) all x4. Domain of the function 73 x12x1) x ( f x sin21++ + is(a) (b) R - {0} (c) R (d) None of these5. The domain of definition of the function ) 1 x ( log e 3 y 1 x2 is(a) ) , 1 ( (b) ) , 1 [ (c) R ~ {1} (d) ) , 1 ( ) 1 , ( 6. The range of the function f(x) = cos [x], where 2x2< 0, then domain of the function y = log (ax3 + (a + b)x2 + (b + c)x + c) is(a) ;'a 2b~ R2(b)

,_

;' } 1 x | x {a 2b~ R(c)

,_

;' ] 1 , (a 2b~ R (d) none of these8. Which of the following functions is an even function?(a) x xx xa aa a) x ( f+ (b) 1 a1 a) x ( fxx+ (c) 1 a1 ax ) x