Question bank function ITF 387 kho tài liệu bách khoa

A I, II and IV only B III and V only C I and III only D I, III and V only Q.6 Which one of the following depicts the graph of an odd function?... Also hx ln fx ln gx then for all real x,

Q.1 Let f be a real valued function such that

f (x) +

x

2002

for all x > 0 The value of f (2), is

Q.2 The number k is such that tan arctan(2) arctan(20k) = k The sum of all possible values of k is

(A) –

40

19

(B) – 40

21

5 1

otherwise

1 x for

1 x 0 for

0 1

x x

x all for x

f

x all for x

f

Which of the following is necessarily true?

(A) f 4 x f 1 x forallx (B) f 1 x f 3 x forallx

(C) f 2 x f 4 x forallx (D) f 1 x f 3 x 0 forallx

Q.4 Domain of definition of the function f x log 10 3x 2 9x 1 1 cos 1 1 x is

13

5 cosθ , 13

12

I

13

5 cos

13

12 sin

13

12 sin

IV

5

12 tan

5

12 tan

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 Q.6 Which one of the following depicts the graph of an odd function?

Q.7 The sum

1 n

2 1

1 n n

3

4

(B) cot 3 2

1

2

1

Q.8 The value of tan2A tan cotA tan cot A for 0 A /4

2

1

Q.9 Given

x 1

8 x 1

8 x

x cos f

4 x

sin f

4 x

(A) periodic with period /2 (B) periodic with period

(C) periodic with period 2 (D) aperiodic

Q.10 sin 1 cossin 1x and cos 1 sincos 1x , then :

Q.11 The period of the function

x cos x sin

x cos x sin x f

Q.12 The value of

1 1

1

b

a sin 2

1 4

tan b

a sin 2

1 4

(A)

a

2

b

(B) b

a

(C)

b

a

b2 2

(D)

a 2

a

b2 2

Q.13 The sides of a triangle are 9, 12, 15 The area of the triangle formed by its income, centroid and orthocentre, is

2

1 1 x g and x cos x sin x

20 ,

10 satisfying the equation f x sgn g x , is

Q.15 Number of solutions of the equation 2cot 12 cos 1 3/5 cosec 1x is

x

nx x

g and nx

x x

(A) g x

1

and f(x) are identical functions (B)

x f

1 and g(x) are identical functions

x g x f 1

Q.17 The number of solutions of the equation tan x is

2

x tanh 3

x

Q.18 Let f (x) = sin2x + cos4x + 2 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

Q.19 Which of the following is the solution set of the equation 2 cos–1(x) = ?

x 1 x

1 x cot

2

2 1

(A) (0, 1) (B) (–1, 1) – {0} (C) (–1, 0) (D) [–1, 1]

Q.20 Let f x eexsgn x andg x eexsgn x,x where { x }and [ ] denotes the fractional part andR

integral part functions respectively Also h(x) ln f(x) ln g(x) then for all real x, h (x) is

(C) neither an odd nor an even function (D) both odd as well as even function

Q.21 Find the range of the function f x cot 1x sec 1 coses 1x

(A)

2

3 ,

2

3 , 4

5 4

3 , 2

2

3 , ,

2

3 , ,

2 Q.22 Which of the following function is surjective but not injective

(A) f:R"R f x x4 x3 x2 1 (B) f:R"R f x x3 x 1

x 1 x f R R :

Q.23

5

2 sin 5

7 cos 2

1

(A)

20

23

(B) 20

23

(C) 20

3

(D) 20 17

1 x

2 x

Q.25 There exists a positive real number x satisfying costan 1x x The value of

2

x cos

2 1 is

(A)

2

(D) 5 4

Q.26 If (x,y) max(x,y) min( x , y )andg(x,y) max(x,y) min(x,y),then

f

) 75 1 , 4 ( g , 2

3 , 1 g

x

2 tan 1 x

1 tan 1 x

1

Q.28 If the solution set for f (x) < 3 is (0, ) and the solution set for f (x) > – 2 is (– , 5), then the true solution

set for f(x) 2 # f (x) + 6, is

(A) (– , + ) (B) (– , 0] (C) [0, 5] (D) (– , 0] ! [5, )

Q.29 The range of the value of p for which the equation sincos 1 cos(tan 1x) p has a solution is :

(A)

2

1 , 2

1

2

1

(D) 1,1 Q.30 Let f x x 12 1, x 1.

# Then the set S x: (x) f 1(x) is

3 i 3 , 2

3 i 3 , 1 ,

3 y x 2

y 1

2

1

(C) 2

3

(D) 4 1

Q.32 2cotcot 1(3) cot 1(7) cot 1(13) cot 1(21) has the value equal to

Q.33 The graph of the function y = g (x) is shown

The number of solutions of the equation

2

1 1 ) x (

x 1

x r x g let and x 1

x x

f Let S be the set off all real numbers r such that

(x) g

f for infinitely many real number x The number of elements in set S is

Q.35 Number of natural solution(s) of the equation sin 1 sinx cos 1 cosx in 0,5 is

x

1 x 2 ] x [ ] x [ tan ) x (

where [*] is the greatest integer function

(A) ,

4

1

(B) ! ,2 4

1

(C) ,2 4

1

(D) ,2 4 1

Q.37 The set of values of ox, satisfying the equation tan2 sin 1x 1 is

2

2 , 2 2

2 , 2

2 1

,

2

2 , 2

2 ]

1 , 1 [

rational is

x if

irrational is

x if x

0 ) x ( g and rational is

x if

irrational is

x if x

0 ) x ( f

Let

Then the function (f - g) x is

(A) odd (B) even (C) neither odd nor even (D) odd as well as even

x

x 1 cot x cos x 1

2 1

1 2

1

(A) [ 1,1] { } (B) 0,1! 1 (C) 1,0 ! 1 (D) [-1, 1]

Q.40 The period of the function cos 2x cos xis:

40 tan 2 65 tan

(A) 0

40

Q.42 Range of the function

} { 1

} { ) x (

f where { } denotes the fractional part function is

2

1 ,

2

1 ,

2

1 , 0 Q.43 Consider the function g (x) defined as

1 1 x )

1 x ( 1 x ( 1 x ( 1 x

)

x

(

the value of g (2) equals

Q.44 The range of the function, (x) (1 sec-1x)(1 cos 1x) is

f

) 1 ( ,

1 (D) [0,(1 )2] Q.45 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

Q.46 Which of the following represents an odd function?

2 x e

) e 1 ( ) x (

(C) h(x) cos(cos 1x) (D) k(x) cot 1(cotx)

Q.47 Let f be a real valued function defined by

5

3 x cos 3

x 1 sin x

given by :

Q.48 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))

2

1

x the the radian measue of cot 1 `x cot 1y is

(A)

3

Q.50 Given f (x) is a polynomial function of x, satisfying f(x) f(y) = f(x) + f(y) + f(xy) - 2 and that f (2) = 5

then f (3) is equal to

Q.51 Let cos 1 x cos 1 x cos 1 x If x satisfies the cubic ax3 bx2` cs 1 0,then (a + b + c)

has the value equal to

Q.52

# 1 x if

x x

1 x 1 if ] x 1 [ ] x 1 [

1 x if x

x ) x ( f Let

where [x] denotes the greatest integer function then F(x) is

(C) neither odd nor even (D) even as well as odd

Q.53 The domain of the function

] x [ x

x cot arc x

f

2

2 , where [x] denotes the greatest integer not greater than

x, is :

(C) R & n:n 1 !{ } (D) R n:n }

2

1 tan 1 2

1 2 tan 1 1 then which one of the following can not be equal to

(A)

2

1 tan

5

1 sin 3

y=g(x)

y-f(x)

Q.55 The range of the function, (x) cot log0.5 x 2x 3 is:

4

3 ,

4

3

(D)

4

3 , 2

4

1 cos 2

6 4

1 cos

(A)

2

1

(B) 2

3

(C) 2

1

(D) 0 Q.57 If f x ay,x-ay axythen f x,y is equal to :

(A)

4

y

x2 2

(B)

4

y

x2 2

Q.58 If f(x) px q and f f f(x) 8x 21,where p and q are real number, then p + q equals

Q.59 The value of x satisfying the equation sin(tan 1x) coscot 1(x 1) is

(A)

2

1

(B) 2

1

(C) 2 1 (D) no finite value Q.60 If (x) 2tan x 5 1 cos x;g(x) is a function having the same period as that of f(x), then which

of the following can be g(x)

(A) (sec23x cosec23x)tan23x (B) 2sin x 3cos x

(C) 2 1 cos23x cosec3x (D) 3cosec3x 2tan3x

x t 0 : sin min )

x ( g

) x ( g ) x ( )

x ( h and

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 = 1 when x = 0 Q.62 The domain of the explicit form of the function is

Q.63 Range of the function is

Q.64 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

1 x 1 x

Paragraph for question nos 65 to 68

" b, a , : f Let R x 1 x 2 x ) x (

f

f (x) is bijective

Q.65 The value of (a + b) is equal to

Q.66 Let f:R"R,g(x) f(x) x 1, then the least value of function y g x is

Q.67 Letf:a, " b, ,thenf 1(x) is given by

Q.68 Letf:R" , then range of values of k for which equation R f x k has 4 distinct real roots is

(A) (- 2, - 1) (B) (- 2, 0) (C) (- 1, 0) (D) (0, 1)

[MULTIPLE OBJECTIVE TYPE]

Q.69 Which of the following function (s) is/are Transcendental ?

1 x x

x sin 2 ) x (

2 3 x ) x ( f Q.70 sin 1 sin3 sin 1 sin4 sin 1 sin5 when simplified reduces to

(A) an irrational number (B) a rational number

Q.71 The functions which are aperiodic are :

(A) y = [x + 1] (B) y = sin x2 (C) y = sin2 x (D) y = sin-1 x

where [x] denotes greatest integer function

Q.72 Which of the following pairs of functions are identical ?

(A) (x) en sec 1andg(x) sec 1x

(B) f x tan tan 1x andg(x) cot cot1x (C) (x) sgn(x)andg(x) sgn(sgn(x)) (D) f(x) cot2x.cos2xandg(x) cot2x cos2x Q.73 Which of the functions defined below are one-one function(s) ?

(A) (x) x 1, x# 1 (B) g(x) x 1/x (x 0)

(C) h(x) x2 4x 5,(x 0) (D) (x) e x, x#0

Q.74 The value of

5

14 cos cos 2

1

(A)

5

7

10

5

2

5

3 cos Q.75 If cos 1x cos 1y cos 1z , then

(A) x2 y2 z2 2xyz 1

(B) 2sin 1x sin 1y sin 1z cos 1x cos 1y cos 1z

(C) xy + yz + zx = x + y + z - 1

z

1 z y

1 y x

1

Q.76 Which of the following functions are homogeneous ?

(A) xsiny ysinx (B) y / x x / y

e y e

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] (B) Range of f(x) is ( , )

Q.78 Suppose f (x) = ax + b and g (x) = bx + a, where a and b are positive integers If f g(50) g (50) 28

then the product (ab) can have the value equal to

Q.79 Which of the following function(s) have the same domain and range ?

(A) (x) 1 x2 (B)

x

1 ) x (

Q.80 2tan tan 1(x) tan 1(x3) wherex R { 1, }isequalto

(A) 2

x 1

x 2

(B) tan2tan 1x (C) tancot 1( x) cot 1(x) (D) tan 2cot 1x Q.81 Which pair(s) of function(s) is/are equal ?

2 1

x 1

x 1 ) x ( g

; ) x tan 2 cos(

) x (

x 1

x 2 ) x

2

(C) n (sgn cot x ) n [ 1 x }]

e ) x ( g

; e

) x (

1 x

where {x} and [x] denotes the fractional part and integral part functions

Q.82 Which of the following function(s) would represent a non singular mapping

(A) f :R"R f(x) xSgnx where Sgn denotes Signum function

x (x) R

R

g

(C) h:R"R h(x) x4 3x2 1

(D)

2 x x

6 x 7 x 3 (x) R

R

2

k

Q.83 Let x1,x2,x3,x4 be four non zero numbers satisfying the equation

2 x

d tan x

c tan x

b tan x

a

(A) 4

1 i

x

(B) 4

1

0 x 1

4

1 i i

' (D) x1 x2 x3 x2 x3 x4 x3 x4 x1 x4 x1 x2 abcd Q.84 If the function f(x) = ax + b has its own inverse then the ordered pair (a, b) can be

l

f

Q.85 Let y (sinx sin2x sin x) (cosx cos2x cos x) then which of the following is correct ?

2 x when dx

dy

(B) value of y when

2

5 3 is 5 x

(C) value of y when

2

3 2 1 is 12

x (D) ysimplifiesto(1 2cosx)in[0, ] Q.86 Which of the following trigonometric equation(s) has/have no solution x R?

2 2 2

x

1 x x sin 2

x cos

5 5 e

where [ ] denotes greatest integer function

[MATCH THE COLUMN]

(A) The period of the function cos(sinx)equalsk

2

x cos sin ) x (

then k is equal to (B) The integral value(s) in the domain of definition of the function, (Q) 3

2 2 x 1

8 x 7 x cos arc ) x ( where [*] denotes the greatest integer functin, is (C) Let (x) sin [a] x If f is periodic with fundamental period , then (R) 4 the possible integral vlaue(s) of ‘a’ is/are

(where [ ] denotes the greatest integer function) (D) If the values of x satisfying the equation x 2 5[x] 6 0, then integral (S) 5 value of x, is/are

(where [ ] denotes the greatest integer function)

(A) "

n

1 n n 2 n

C

and the roots of g(x) = 0 are -1, 3, 5, 7 and 8.

Number of solutions of the euqation

x g

x

f

= 0 is

2 x 0 where x

sin

x cos x cos

x sin y Let

3 3

(R) 3/2 then the minimum value of y is

(D) A circle passes through vertex D of the square ABCD, and is tangent (S) 2

to the sides AB and BC If AB = 1, the radius of the circle can be expressed as p q 2, then p + q has the value equal to

f

Q.89 Column-I Column-II

4

3 4

, 4 4

3 ,

2

3

!

4 , 0

6

5 , 6

4

x sin ) x ( g , b ax 3 x (x) , , R :

(A) The possible integral values of ‘a’ for which f(x) is many one in (P) - 2

interval [-3, 5] is/are

(B) Let a = - 1 and gof(x) is defined for x [ 1,1] then possible (Q) - 1

integral values of b can be

(C) Leta 2, 8 the value(s) of which f(x) is surjective is/are (R) 0

(D) If a = 1, b = 2, then integers in the range of fog(x) is/are (S) 1

143

127

(C)

9

1 cos 2

1

4 3

(D)

8

1 cos arc 2

1

3 2

2 x

1 x g(x) and x

1 x (x)

Let f

Match the composite function given in Column-I with their respective domains given in Column-II

(A) f(x) sin sin 12x cosec(cosec12x) tan(tan 12x) (P) odd function

(B) g(x) sin 1{x}, where {x} denotes fractional part function (Q) injective mapping

integer only

Q.94 Find the value of x satisfying the equation,

1 5 log 3 x cot 2 log 2

1 1 x cot 5

10 1

10

Q.95 Let the straight line L:tan(cot 12)x y 4 be rotated through an angle cot-13 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