Question bank function ITF 387 kho tài liệu bách khoa

Q.1 Q.2 Q.3 Let f be a real valued function such that 2002 = 3x f (x) + f x for all x > The value of f (2), is (A) 1000 (B) 2000 (C) 3000 (D) 4000 The number k is such that tan arc tan(2) arc tan(20k ) = k The sum of all possible values of k is 19 21 (A) – (B) – (C) (D) 40 40 x for x Let f x for x and f x f x for all x otherwise f3 x f x for all x f4 x f x for all x Which of the following is necessarily true? (A) f x f x for all x (C) f x f x for all x (B) f x (D) f x f3 f3 x log 10 3x x (C) (0, 2) Q.4 Domain of definition of the function f x (A) [0, 1] (B) [1, 2] Q.5 If Sinθ Q.6 Which one of the following depicts the graph of an odd function? x for all x for all x cos 1 x (D) (0, 1) is 12 , θ π Consider the following statements , cosθ 13 13 12 1 12 sin I θ cos II θ sin III θ 13 13 13 12 12 tan IV θ tan V θ 5 then which of the folloing statements are true ? (A) I, II and IV only (B) III and V only (C) I and III only (D) I, III and V only (A) (B) (C) (D) ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.7 tan The sum n n (A) Q.8 cot The value of tan is equal to n (B) cot tan 2A (A) tan 1 (B) tan 8 Given f x Q.10 sin cos sin x and (B) tan (A) tan cot Q.11 cot A (D) tan cot A for A (C) 2 tan / is (D) None 4 then g(x) is f sin x f cos x (B) periodic with period (D) aperiodic cos sin cos x , then : (C) tan cot tan (D) tan sin x cos x sin x cos x (C) /4 tan The period of the function f x (A) Q.12 and g x Q.9 x x (A) periodic with period / (C) periodic with period tan (C) (B) /2 The value of tan (A) b 2a sin (B) a b tan a 2b sin (C) (D) 1 a b , where (0 < a < b), is b2 a 2b (D) b2 a 2a Q.13 The sides of a triangle are 9, 12, 15 The area of the triangle formed by its income, centroid and orthocentre, is (A) (B) (C) 3/2 (D) Q.14 f x 10 , 20 (A) Q.15 Q.16 tan x , then the number of values of x in interval satisfying the equation f x sgn g x , is (B) 10 (C) 15 (D) 20 sin 23 x cos 22 x and g x Number of solutions of the equation cot cos / (A) (B) (C) cos ec 1x is (D) more than x l nx and g x Then identify the CORRECT statement l nx x 1 (B) and g(x) are identical functions (A) g x and f(x) are identical functions f x f x (C) f x g x x (D) f x g x x ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.17 The number of solutions of the equation tan (A) Q.18 Q.19 x (C) 1 (B) x tan x is (D) Let f (x) = sin2x + cos4x + and g (x) = cos (cos x) + cos (sin x) Also let period of f (x) and g (x) be T1 and T2 respectively then (A) T1 = 2T2 (B) 2T1 = T2 (C) T1 = T2 (D) T1 = 4T2 2x 1 ? Which of the following is the solution set of the equation cos–1(x) = cot 2x x (A) (0, 1) (B) (–1, 1) – {0} (C) (–1, 0) (D) [–1, 1] x x where { x }and [ ] denotes the fractional part and l n f(x) l n g(x) then for all real x, h (x) is (B) an even function (D) both odd as well as even function Q.20 Let f x e e sgn x and g x e e sgn x , x R integral part functions respectively Also h (x) (A) an odd function (C) neither an odd nor an even function Q.21 Find the range of the function f x , (A) 2 , ! , (C) 2 Q.22 Which of the following function is surjective but not injective (B) f : R " R (A) f : R " R f x x x x f x x3 (D) f : R " R f x x 2x (C) f : R " R Q.23 cos (A) Q.24 23 20 x (A) R Q.25 ;g x 23 20 cos x and h x (C) 10 If f ( x , y) 1, (A) - 0.5 max ( x , y) 20 (D) (C) 1, (C) min( x , y ) and g ( x , y) (B) 17 20 (D) R {1} x The value of cos (D) x2 is max( x , y) min( x , y), then , g ( 4, 1.75) equals (B) 0.5 The number of solutions of the equation tan (A) x x then tha range of the composite function fogoh, is (B) R {0} (B) x is equal to There exists a positive real number x satisfying cos tan x f g Q.27 1 (A) Q.26 sin (B) Let f x cos es x , ! , (B) 2 , ! , (D) 2 x2 f x cos cot x sec (C) 1 2x (C) (D) 1.5 tan 1 4x tan is x2 (D) ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.28 If the solution set for f (x) < is (0, ) and the solution set for f (x) > – is (– , 5), then the true solution set for f ( x ) # f (x) + 6, is (A) (– , + ) (B) (– , 0] Q.29 The range of the value of p for which the equation sin cos 1 , 2 (A) Q.30 Let f x x (A) 0, 1, 1, x # Then the set S i 3 i , 2 If x y 2 $x y (B) is a parabola, then $ (C) (B) (D) The graph of the function y = g (x) is shown , is (B) (D) Let f x f g(x) (A) rx Let S be the set off all real numbers r such that x x g f (x) for infinitely many real number x The number of elements in set S is (B) (C) (D) x and let g x Number of natural solution(s) of the equation sin (A) Q.36 (D) (C) (A) (C) Q.35 1, x : f ( x ) f ( x ) is The number of solutions of the equation g ( x ) Q.34 (D) cot cot (3) cot (7) cot (13) cot (21) has the value equal to (A) Q.33 p has a solution is : (D) empty (A) Q.32 cos(tan x ) (B) 0,1, (C) 0, Q.31 1 ,1 (C) (B) 0, (D) (– , 0] ! [5, ) (C) [0, 5] (B) Range of the function f ( x ) sin x cos cos x in 0,5 (C) tan [x ] [ x ] is (D) infinite x is x2 where [*] is the greatest integer function (A) Q.37 , (B) ! 2, (C) ,2 The set of values of ox, satisfying the equation tan sin x (A) [ 1,1] (C) 1,1 (B) 2 , 2 (D) ,2 is 2 , 2 (D) [ 1,1] 2 , 2 ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.38 x Let f ( x ) if x is irrational and g ( x ) if x is rational Then the function (f - g) x is (A) odd (B) even Q.39 Q.40 (B) The value of the angle tan 20 1 cos x cot x2 x 1, ! (C) sin x (D) [-1, 1] Range of the function f ( x ) (C) (D) none of these tan 650 tan 40 in degrees is equal to (B) 20 (C) 250 (D) 40 {x} where {x} denotes the fractional part function is {x} (B) 0, (A) 0,1 Q.43 x The period of the function cos x cos x is : (A) Q.42 (B) 0,1 ! (A) Q.41 if x is irrational if x is rational (C) neither odd nor even (D) odd as well as even The solution set of the equation sin (A) [ 1,1] {0} x (C) 0, (D) 0, Consider the function g (x) defined as g(x ) x ( 2008 1) ( x 1) ( x 1) ( x 1) x 2007 1 the value of g (2) equals (B) 2008 (A) Q.44 Q.46 (B) , , ! 4, (C) 1, (1 )2 (D) [0, (1 )2 ] Which of the following is true for a real valued function y = f (x), defined on [-a, a]? (A) f (x) can be expressed as a sum or a difference of two even function (B) f (x) can be expressed as a sum or a difference of two odd function (C) f (x) can be expressed as a sum or a difference of an odd and an even function (D) f (x) can never be expressed as a sum or a difference of an odd and an even function Which of the following represents an odd function? (A) f ( x ) Q.47 (D) The range of the function, f (x) (1 sec -1x) (1 cos 1x) is (A) Q.45 (C) 2008 (1 e x ) ex (B) g ( x ) sec (sec x ) (C) h ( x ) cos (cos x ) (D) k ( x ) cot (cot x ) Let f be a real valued function defined by f x sin 1 x cos x Then domain of f(x) is given by : (A) 4, (B) 0, (C) 3, (D) 5, ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.48 Q.49 Given the graphs of the two functions, y = f(x) & y = g(x) In the adjacent figure from point A on the graph of the function y = f(x) corresponding to the given value of the independent variable (say x0), a straight line is drawn parallel to the X-axis to intersect the bisector of the first and the third quadrants at point B From the point B a straight line parallel to the Y-axis is drawn to intersect the graph of the function y = g(x) at C Again a straight line is drawn from the point C parallel to the X-axis, to intersect the line NN% at D If the straight line NN% is parallel to Y-axis, then the co-ordinates of the point D are (A) f(x0), g(f(x0)) (B) x0, g(x0) (C) x0, g(f(x0)) (D) f(x0), f(g (x0)) (A) Q.50 and x y If x (B) (B) 24 Let cos x cos x has the value equal to (A) 24 Q.52 (C) 1` y-f(x) x cot y is (D) 4 Given f (x) is a polynomial function of x, satisfying f(x) f(y) = f(x) + f(y) + f(xy) - and that f (2) = then f (3) is equal to (A) 10 Q.51 the the radian measue of cot y=g(x) (C) 15 If x satisfies the cubic ax bx ` cs 0, then (a + b + c) cos 3x (B) 25 Let f ( x ) (D) none (C) 26 xx if [1 x ] [1 x ] xx if if x (D) 27 1 x x #1 where [x] denotes the greatest integer function then F(x) is (A) even (B) odd (C) neither odd nor even Q.53 (D) even as well as odd The domain of the function f x x, is : (A) R (C) R Q.54 x [x ] , where [x] denotes the greatest integer not greater than (B) R - {0} & n : n ! {0} 2 1 If tan (A) tan (C) arc cot x tan tan tan (D) R {n : n 1} , then which one of the following can not be equal to 1 (B) cot cot (D) tan sin 1 ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.55 Q.56 (A) 0, (B) 0, The value of cos cos 1 (A) Q.57 If f x ay, x- ay (A) Q.58 x2 y2 Q.61 (B) 3 , (C) cos (C) , (D) equals (D) x2 y2 (C) xy px q and f f f (x) (D) none 8x 21, where p and q are real number, then p + q equals (B) (C) (D) 11 The value of x satisfying the equation sin(tan x ) cos cot ( x 1) is (A) Q.60 (B) If f (x) axy then f x, y is equal to : (A) Q.59 cot log 0.5 x 2x is : The range of the function, f ( x ) (B) (C) (D) no finite value If f ( x ) tan 3x cos 6x ; g ( x ) is a function having the same period as that of f(x), then which of the following can be g(x) (A) (sec x cosec x) tan x (B) sin 3x cos 3x (C) cos x (D) cosec x tan x Let cosec x f (x) max sin t : t x g( x ) sin t : t x and h ( x ) f (x ) g ( x ) where [ ] denotes greatest integer function, then the range of h(x) is (A) {0, 1} (B) {1, 2} (C) {0, 1, 2} (D) {-3, -2, -1, 0, 1, 2, 3} [COMPREHENSION TYPE] Paragraph for question nos 62 to 64 Consider a function y = f (x) satisfying the equation tan-1y = tan-1x + C where y = when x = Q.62 The domain of the explicit form of the function is (A) Q.63 ,1 (C) ( 1,1) (D) 0, (B) (C) 1, (D) Range of the function is (A) R { 1} Q.64 (B) R { 1,1} 1, ,1 For the function y = f (x) which one of the following does not hold good ? (A) f (x) is injective (B) f (x) is neither odd nor even (C) f (x) is aperiodic (D) explicit form of f (x) is x x ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Paragraph for question nos 65 to 68 x R Let f : , a " b, , where ‘a’ is the largest real number for which Q.65 Q.66 Q.67 Let f ( x ) x x f (x) is bijective The value of (a + b) is equal to (A) - (B) - (D) Let f : R " R , g ( x ) f (x ) 3x 1, then the least value of function y g x is (A) - 9/4 (B) - 5/4 (C) - (D) - 1 Let f : a , " b, , then f ( x ) is given by (A) Q.68 (C) (B) x (C) x (D) x Let f : R " R , then range of values of k for which equation f x (A) (- 2, - 1) (B) (- 2, 0) (C) (- 1, 0) x k has distinct real roots is (D) (0, 1) [MULTIPLE OBJECTIVE TYPE] Q.69 Q.70 Q.71 Q.72 Which of the following function (s) is/are Transcendental ? (A) f ( x ) sin x (B) f ( x ) sin 3x x 2x (C) f ( x ) 2x (D) f ( x ) x 2x x2 sin sin sin sin sin sin when simplified reduces to (A) an irrational number (B) a rational number (C) an even prime (D) a negative integer The functions which are aperiodic are : (B) y = sin x (C) y = sin2 x (D) y = sin-1 x (A) y = [x + 1] where [x] denotes greatest integer function Which of the following pairs of functions are identical ? (A) f ( x ) Q.73 e x2 sec x and g ( x ) x 5, ( x The value of cos cos (A) cos Q.75 n sec (B) f x tan tan x and g ( x ) cot cot1 x (C) f ( x ) sgn ( x ) and g ( x ) sgn(sgn( x )) (D) f ( x ) cot x cos x and g ( x ) cot x cos x Which of the functions defined below are one-one function(s) ? (A) f ( x ) x , x # (B) g ( x ) x / x ( x 0) (C) h ( x ) Q.74 cos (B) sin 1 If cos x cos y cos z (A) x y z (D) f ( x ) e x , x # 0) 14 10 is : (C) cos (D) cos , then xyz (B) sin x sin y sin z (C) xy + yz + zx = x + y + z - cos x cos y cos z 1 y z #6 x y z Which of the following functions are homogeneous ? (A) x sin y y sin x (B) x e y / x y e x / y (C) x xy (D) x Q.76 (D) arc sin xy ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.77 Let tan f (x) cos x then which of the following do/does not hold good ? (A) Domain of f(x) is [-1, 1] (C) f (x) is bounded Q.78 Q.79 l x2 x (B) g ( x ) tan tan ( x ) tan ( x ) where x (A) Q.81 , ) Suppose f (x) = ax + b and g (x) = bx + a, where a and b are positive integers If f g (50) then the product (ab) can have the value equal to (A) 12 (B) 48 (C) 180 (D) 210 Which of the following function(s) have the same domain and range ? (A) f ( x ) Q.80 (B) Range of f(x) is ( (D) f(x) is odd (C) h ( x ) (D) tan cot x x2 x2 cos( tan x ) ; g ( x ) (B) f ( x ) 2x ; g( x ) sin cot x x2 (D) f ( x ) x a , a 0; g (x ) a x , a where {x} and [x] denotes the fractional part and integral part functions Q.82 f e n (sgn cot x) ; g( x) e n [1 { x }] Which of the following function(s) would represent a non singular mapping (A) f : R " R f (x) (B) g : R " R g (x) x 3/ (C) h : R " R h(x) x4 x2 k (x) x2 x x x2 (D) k : R " R Q.83 x (B) tan tan x (C) tan cot ( x ) cot ( x ) Which pair(s) of function(s) is/are equal ? (C) f ( x ) 28 R { 1,1}is equal to 2x x2 (A) f ( x ) (D) l ( x ) x g f (50) x Sgn x where Sgn denotes Signum function Let x1 , x , x , x be four non zero numbers satisfying the equation tan a x tan b x tan c x tan d x (A) xi a b c d xi i (B) i (C) 'x i abcd i (D) x1 x Q.84 x3 x2 x3 x4 x3 x4 x1 x x1 x abcd If the function f(x) = ax + b has its own inverse then the ordered pair (a, b) can be (A) (1, 0) (B) (-1, 0) (C) (-1, 1) (D) (1, 1) ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 10 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.85 (sin x sin x sin 3x ) (cos x cos x cos 3x ) then which of the following is correct ? Let y (A) dy when x dx is (C) value of y when x Q.86 12 (B) value of y when x is 2 x sin x (C) cos e x 5x Q.87 x2 x2 x R? (B) sin x sin x x (D) sin x [1 sin x ] [1 cos x ] where [ ] denotes greatest integer function [MATCH THE COLUMN] Column-I Column-II (A) The period of the function f ( x ) sin cos x cos(sin x ) equals k then k is equal to (B) The integral value(s) in the domain of definition of the function, f (D) y simplifies to (1 cos x ) in [0, ] Which of the following trigonometric equation(s) has/have no solution (A) cos is (P) (Q) (R) (S) 3x x f ( x ) arc cos x2 where [*] denotes the greatest integer functin, is (C) Let f ( x ) sin [a ] x If f is periodic with fundamental period , then the possible integral vlaue(s) of ‘a’ is/are (where [ ] denotes the greatest integer function) 5[ x ] 0, then integral (D) If the values of x satisfying the equation x value of x, is/are (where [ ] denotes the greatest integer function) Q.88 Column-I n Column-II n C2 n equals n (B) Let the roots of f(x) = are 2, 3, 5, and and the roots of g(x) = are -1, 3, 5, and (A) Lim n" Number of solutions of the euqation (Q) (R) 3/2 (S) f x = is gx sin x cos x where x , cos x sin x then the minimum value of y is (D) A circle passes through vertex D of the square ABCD, and is tangent to the sides AB and BC If AB = 1, the radius of the circle can be expressed as p q , then p + q has the value equal to (C) Let y (P) ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 11 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.89 Column-I (A) sin x cos x cos x sin x, x (B) sin x sin x 0, (C) tanx and x [ , ] , is (D) cos x sin x # and Q.90 Column-II Let : f : R " , f (x) x 3ax b, g ( x ) sin , , (P) , is x x π , is ! (Q) ,2 (R) 0, (S) 6 x ( , 4 ! , ! {0} , R ) Column-I Column-II (A) The possible integral values of ‘a’ for which f(x) is many one in (P) - interval [-3, 5] is/are (B) Let a = - and gof(x) is defined for x [ 1,1] then possible (Q) - integral values of b can be (C) Let a 2, (R) the value(s) of which f(x) is surjective is/are (D) If a = 1, b = 2, then integers in the range of fog(x) is/are (S) Q.91 Column-I (A) cot Column-II (P) 1430 tan( 37 ) (B) cos cos( 2330 ) Q.92 (C) sin cos (D) cos 1 arc cos (Q) 127 (R) (S) x and g(x) x x Match the composite function given in Column-I with their respective domains given in Column-II Column-I Column-II (A) fog (P) R - {-2, -5/3} (B) gof (Q) R - {-1, 0} (C) fof (R) R - {0} (D) gog (S) R - {-2, -1} Let f (x) x Q.93 Column-I (A) f (x) sin sin x 1 cosec (cosec x) tan(tan x) (B) g (x) sin {x}, where {x} denotes fractional part function (C) h (x) cosec (sgn x) Column-II (P) odd function (Q) injective mapping (R) range contian two integer only (D) k ( x ) cos ( sin x cos x ) (S) aperiodic ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 12 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) [SUJBECTIVE] Q.94 Q.95 Find the value of x satisfying the equation, log10 cot x log10 cot x log10 Let the straight line L : tan(cot 2) x y be rotated through an angle cot -1 about the point M(0, - 4) in anticlockwise sense After rotation the line become tangent to the circle which lies in 4th quadrant and also touches coordinate axes Find the sum of radii of all possible circles ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 13 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005)