Maths Funtion & Inverse

8/12/2019 Maths Funtion & Inverse

http://slidepdf.com/reader/full/maths-funtion-inverse 1/10

Only one alternative is correct.

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

f (x) +

x

2002f 2 = 3x

for all x > 0. Find f (2).(A) 1000 (B) 2000 (C) 3000 (D) 4000

Q.2 Solution set of the equation , cos1 x – sin1 x = cos1(x 3 )(A) is a unit set (B) consists of two elements(C) consists of three elements (D) is a void set

Q.3 If f x x x( ) tan cos 2 3 5 1 6 ; g(x) is a function having the same time period as that of f(x), then which

of the following can be g(x).(A) (sec23x + cosec23x)tan23x (B) 2 sin3x + 3cos3x

(C) 2 1 32 cos x + cosec3x (D) 3 cosec3x + 2 tan3x

Q.4 Which one of the following depicts the graph of an odd function?

(A) (B)

(C) (D)

Q.5 The sum of the infinite terms of the series

cot 1 13

42

+ cot 1 2

3

42

+ cot 1 3

3

42

+ ..... is equal to :

(A) tan –1 (1) (B) tan –1 (2) (C) tan –1 (3) (D) tan –1 (4)

Q.6 Domain of definition of the function f (x) = log 193·10 1x2x + )x1(cos 1 is

(A) [0, 1] (B) [1, 2] (C) (0, 2) (D) (0, 1)

Q.7 The value of tan1 1

22tan A

+ tan 1(cot A) + tan 1(cot3A) for 0 < A < (/4) is

(A) 4 tan1 (1) (B) 2 tan1 (2) (C) 0 (D) none



Q.8 Let

f x t t x

g x t t x

and h x f x g x

( ) max. sin :

( ) min. sin :

( ) ( ) ( )

0

0

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}

Q.9 The period of the function f(x) = sin 2x + sin x3

+ sin x5 is

(A) 2 (B) 6 (C) 15 (D) 30

Q.10 The value of sec sin sin cos cos

1 150

9

31

9

is equal to

(A) sec10

9

(B) sec

9

(C) 1 (D) –1

Q.11 The domain of definition of the function

, f

(x) = arc cos 3 7 8

1

2

2

x x

x

where [

*] denotes the

greatest integer function, is :(A) (1, 6) (B) [0, 6) (C) [0, 1] (D) ( 2, 5]

Q.12 = sin 1 cos sin1 x and = cos 1 sin cos1 x , then :

(A) tan = cot (B) tan = cot (C) tan = tan (D) tan = tan

Q.13 Given f (x) =x1

8

x1

8

and g (x) = )x(cosf

4

)x(sinf

4 then g(x) is

(A) periodic with period /2 (B) periodic with period (C) periodic with period 2 (D) aperiodic

Q.14 If x = tan1 1 cos1

1

2 + sin1

1

2 ; y = cos

1

2

1

81cos

then :

(A) x = y (B) y = x (C) tan x = (4/3) y (D) tan x = (4/3) y

Q.15 In the square ABCD with side AB = 2 , two points M & N are on the adjacent sides of the square suchthat MN is parallel to the diagonal BD. If x is the distance of MN from the vertex A and f (x) = Area ( AMN) , then range of f (x) is :

(A) 0 2, (B) (0 , 2 ] (C) 0 2 2, (D) 0 2 3,

Q.16 cos cos cos tan tan

1 1

7

8

7

has the value equal to

(A) 1 (B) –1 (C) cos 7

(D) 0



Q.17 The domain of the definition of the function f(x) = sin1x

5

2 + log ( )10

16

x is :

(A) (7, 7) (B) ( 7, 3) ( 3, 7)(C) [ 7, 3] [3, 5) (5, 6) (D) ( 3, 3) (5, 6)

Q.18 The value of tan sin tan sin 4

1

2 4

1

21 1

1

a

b

a

b, where (0 < a < b), is

(A) b

a2(B)

a

b2(C)

b a

b

2 2

2

(D)

b a

a

2 2

2

Q.19 Let f be a function satisfying f (xy) =y

)x(f for all positive real numbers x and y. If f (30) = 20, then the

value of f (40) is(A) 15 (B) 20 (C) 40 (D) 60

Q.20 Number of real value of x satisfying the equation, arc tan x x 1 + arc sin x x 1 1 =2

is

(A) 0 (B) 1 (C) 2 (D) more than 2

Q.21 Let f (x) = sin2x + cos4x + 2 and g (x) = cos (cos x) + cos (sin x) also let period of f (x) and g (x) beT1 and T2 respectively then(A) T1 = 2T2 (B) 2T1 = T2 (C) T1 = T2 (D) T1 = 4T2

Q.22 Number of solutions of the equation 2 cot –12 + cos –1(3/5) = cosec –1 x is(A) 0 (B) 1 (C) 2 (D) more than 2

Q.23 The domain of definition of the function : f (x) = ln ( x x2 5 24 – x – 2) is

(A) (– , –3] (B) (– , –3 ][8, ) (C)

,

28

9 (D) none

Q.24 The period of the function f(x) = sin cosx

2

+ cos(sinx) equal

(A)2

(B) 2 (C) (D) 4

Q.25 If x = cos –1 (cos 4) ; y = sin –1 (sin 3) then which of the following holds ?(A) x – y = 1 (B) x + y + 1 = 0(C) x + 2y = 2 (D) tan (x + y) = – tan7

Q.26 Let f (x) = }xsgne{|x|

e and g (x) = ]xsgne[

|x|

e , x R where { x } and [ ] denotes the fractional part and integral part functions respectively. Also h (x) = ln )x(f + ln )x(g then for all real x, h (x) is

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

Q.27 The number of solutions of the equation tan –1

3

x + tan –1

2

x = tan –1 x is

(A) 3 (B) 2 (C) 1 (D) 0



Q.28 Which of the following is the solution set of the equation 2 cos –1 x = cot –1

2

2

x1x2

1x2 ?

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

Q.29 Suppose that f is a periodic function with period2

1 and that f (2) = 5 and f 49 = 2 then

f (–3) – f 41 has the value equal to

(A) 2 (B) 3 (C) 5 (D) 7

Q.30 The value of tan tan

1 11

2

5 2 6

1 6 is equal :

(A)6

(B)

4

(C)

3

(D) none

Q.31 Given f (x) = (x+1)C(2x– 8) ; g (x) = (2x – 8) C(x + 1) and h (x) = f (x) . g (x) , then which of the followingholds ?(A) The domain of 'h' is

(B) The range of 'h' is {– 1}(C) The domain of 'h' is {x / x > 3 or x < – 3 ; Ix(D) The range of 'h' is {1}

Q.32 The sumn

1

tan 1 4

2 24 2

n

n n is equal to :

(A) tan1

2

1+ tan1

3

2(B) 4 tan 1 1 (C)

2

(D) sec 1 2

Q.33 Range of the function f (x) = tan –1 [ ] [ ] | |x x xx

21

2 is

where [*] is the greatest integer function.

(A)1

4,NM (B)

1

42STVW , (C)

1

42,ST (D)

1

42,NM

Q.34 Let [x] denote the greatest integer in x . Then in the interval [0, 3] the number of solutions of the equation,x2 3x + [x] = 0 is :(A) 6 (B) 4 (C) 2 (D) 0

Q.35 The range of values of p for which the equation, sin cos –1 cos(tan )1 x = p has a solution is:

(A)

1

2

1

2, (B) [0, 1) (C)1

2 1,

(D) (– 1, 1)

Q.36 Let f (x) =

irrationalisxif x

rationalisxif 0and g (x) =

rationalisxif x

irrationalisxif 0

Then the function (f – g) x is(A) odd (B) even(C) neither odd nor even (D) odd as well as even



Q.37 Number of value of x satisfying the equation sin –1

x

5 + sin –1

x

12 =

2

is

(A) 0 (B) 1 (C) 2 (D) more than 2

Q.38 Consider a real valued function f(x) such that1

1

e

e

f x

f x

( )

( ) = x. The values of 'a' and 'b' for which

f (a) + f (b) = fa b

ab

1

is satisfied are

(A) a (– , 1); b R (B) a (– , 1); b (–1, )(C) a (–1, 1) ; b [–1, 1] (D) a (–1, 1); b (–1, 1)

Q.39 The value of tan

)3(cot

2

1 1 equals

(A) 1103

(B) 1310

(C) 103 (D) 310

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

(A) (B) 2 (C) 2 (D) none of these

Q.41 The real values of x satisfying tan –1

x

2x – tan –1

x

4 – tan –1

x

2x = 0 are

(A) ±2

1(B) ± 2 (C) ± 24 (D) ± 2

Q.42 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 functions(B) f (x) can be expressed as a sum or a difference of two odd functions

(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.43 cos 21

71tan

equals

(A) sin (4cot –13) (B) sin(3cot –14) (C) cos(3cot –14) (D) cos(4cot –13)

Q.44 Let f(x) = sin [ ]a x (where [ ] denotes the greatest integer function) . If f is periodic with fundamental

period , then a belongs to :(A) [2, 3) (B) {4, 5} (C) [4, 5] (D) [4, 5)

Q.45 The range of the function, f(x) = cot –1 log .0 5 4 22 3x x is:

(A) (0, ) (B) 03

4,

(C)3

4

,

(D)

2

3

4,

Q.46 Which of the following is the solution set of the equation sin –1x = cos –1x + sin –1(3x – 2)?

(A)1

21,

(B)1

21,

(C)1

31,

(D)1

31,



Q.47 Which of the following functions are not homogeneous ?

(A) x + y cosy

x(B)

xy

x y 2 (C)yxsiny

xcosyx

(D)x

y ln

y

x

+y

xln

x

y

Q.48 Which of the following is the solution set of the equation 3cos –1x = cos –1(4x3 – 3x)?

(A) [–1, 1] (B)

1

3

1

3, (C)

1

31,

(D)1

21,

Q.49 The function f : R R, defined as f(x) =x x

x x

2

2

6 10

3 3

is :

(A) injective but not surjective (B) surjective but not injective(C) injective as well as surjective (D) neither injective nor surjective

Q.50 The solution of the equation 2cos –1x = sin –1 (2x 1 2 x )

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

1,2

1

Q.51 The period of the function f (x) = sin(x + 3 – [x + 3 ] ), where [ ] denotes the greatest integer function is(A) 2 + 3 (B) 2 (C) 1 (D) 3

Q.52 If tan –1x + tan –1 2x + tan –13x = , then(A) x = 0 (B) x = 1 (C) x = –1 (D) x

Q.53 If f(x + ay, x ay) = axy then f(x, y) is equal to :

(A)x y2 2

4

(B)

x y2 2

4

(C) 4 xy (D) none

Q.54 The set of values of x for which the equation cos –1x + cos –1x

x2

1

23 3 2

=3

holds good is

(A) [0, 1] (B) 0 12

,

(C)

1,21 (D) {–1, 0, 1}

Q.55 The range of the function y = 2x9

8

is

(A) (– , ) – {± 3} (B)

,9

8(C)

9

8,0 (D) (– , 0)

,9

8

Q.56 The domain of definition of the function f (x) = logcot

coslog

tan

sec1

2

2

2 1

2

2

22 5 3 5

x

ec x

x

x

is

(A) R – {n, n I} (B) R – {(2n + 1) 2

, n I}

(C) R – {n, (2n + 1) 2

, n I} (D) none

Q.57 The solution set of the equation sin –11 2 x + cos –1x = cot –1

1 2

x

x – sin –1x

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



Q.58 Given the graphs of the two functions, y = f(x) & y = g(x). In theadjacent figure from point A on the graph of the function y = f(x)corresponding to the given value of the independent variable (say x0), astraight 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 straightline parallel to the Y-axis is drawn to intersect the graph of the functiony = 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))

Q.59 The value of sin –1(sin(2cot –1( 2 – 1))) is

(A) –4

(B)4

(C)3

4

(D)

7

4

Q.60 The function f : [2, ) Y defined by f(x) = x2 4x + 5 is both one-one and onto if :(A) Y = R (B) Y = [1, ) (C) Y = [4, ) (D) [5, )

Q.61 If f(x) = cosec

–1

(cosecx) and cosec(cosec

–1

x) are equal functions then maximum range of values of x is

(A)

2,11,

2(B)

2,00,

2

(C) ,11, (D) 1,00,1

Q.62 If 2 f(x2) + 3 f(1/x2) = x2 1 (x 0) then f(x2) is :

(A)1

5

4

2

x

x(B)

1

5

2 x

x(C)

5

1

2

4

x

x(D)

2 3

5

4 2

2

x x

x

Q.63 Sum of the roots of the equation, arc cot x – arc cot (x + 2) =

12

is

(A) 3 (B) 2 (C) – 2 (D) – 3

Q.64 Range of the function f (x) =}x{1

}x{

where {x} denotes the fractional part function is

(A) [0 , 1) (B)

2

1,0 (C)

2

1,0 (D)

2

1,0

Q.65 Range of the function sgn [ ln (x2 – x + 1) ] is(A) {–1, 0, 1} (B) {–1, 0} (C) – {1} (D) {–1, 1}

Q.66 Number of solution(s) of the equation cos –1(1 – x) – 2cos –1x =2 is

(A) 3 (B) 2 (C) 1 (D) 0

Q.67 Let f (x) and g (x) be functions which take integers as arguments. Let f (x + y) = f (x) + g (y) + 8 for all integer x and y. Let f (x) = x for all negative integers x, and let g (8) = 17. The value of f (0) is(A) 17 (B) 9 (C) 25 (D) – 17



Q.78 The range of the function f (x) =5x

34x

is

(A)

3

1,0 (B)

3

1,

6

1

6

1,0 (C) (– , 0) (0, ) (D) (0, )

Q.79 If f (x, y) = )y,xmin()y,xmax( and g (x, y) = max(x, y) – min(x, y), then

)75.1,4(,2

3,1 gg f equals

(A) – 0.5 (B) 0.5 (C) 1 (D) 1.5

Q.80 The domain and range of the function f(x) = cosec –1 log sec

sec

3 4

1 2

2

x

x

are respectively

(A) R ;

2 2

, (B) R + ; 02

,

(C)

2

,0};n2{2

n2,2

n2 (D) }0{2

,2

};n2{2

n2,2

n2

More than one alternatives are correct.

Q.81 The values of x in [–2, 2], for which the graph of the function y =1

1

sin

sin

x

x – secx and

y = –1

1

sin

sin

x

x + secx, coincide are

(A)

23

2

3

22

, , (B)

3

2 2 2

3

2

, ,

(C)

2 2,

(D) [–2, 2] –

2

3

2,

Q.82 sin-1(sin3) + sin-1 (sin4) + sin-1(sin5) when simplified reduces to(A) an irrational number (B) a rational number (C) an even prime (D) a negative integer

Q.83 The graphs of which of the following pairs differ .

(A) y =sin

tan

x

x1 2 +

cos

cot

x

x1 2 ; y = sin 2x

(B) y = tan x cot x ; y = sin x cosec x

(C) y = cos x + sin x ; y = sec cossec cos

x ecxx ecx

(D) none of these

Q.84 I f f (x) = cos1

22

x + sin

1

22 x , [x] denoting the greatest integer function, then

(A) f (0) = 1 (B) f3

=

1

3 1(C) f

2

= 0 (D) f() = 0



Q.85 The value of cos 1

2

14

51cos cos

is :

(A) cos

7

5

(B) sin

10

(C) cos

2

5

(D) cos 3

5

Q.86 The functions which are aperiodic are :(A) y = [x + 1] (B) y = sin x2 (C) y = sin2 x (D) y = sin1 xwhere [x] denotes greatest integer function

Q.87 tan1 x , tan1 y , tan1 z are in A.P. and x , y , z are also in A.P. (y 0 , 1 , 1) then(A) x , y , z are in G.P. (B) x , y , z are in H.P.(C) x = y = z (D) (x y)2 + (y z)2 + (z x)2 = 0

Q.88 Which of the following function(s) is/are periodic with period .(A) f(x) = sin x (B) f(x) = [x + ] (C) f(x) = cos (sin x) (D) f(x) = cos2x(where [ . ] denotes the greatest integer function)

Q.89 For the equation 2x = tan(2tan –1a) + 2tan(tan –1a + tan –1a3), which of the following is invalid?(A) a2x + 2a = x (B) a2 + 2ax + 1 = 0 (C) a 0 (D) a –1, 1

Q.90 Which of the functions defined below are one-one function(s) ?(A) f(x) = (x + 1) , ( x 1) (B) g(x) = x + (1/x) ( x > 0)(C) h(x) = x2 + 4x 5, (x > 0) (D) f(x) = e x, ( x 0)

Q.91 If cos –1x + cos –1y + cos –1z = , then(A) x2 + y2 + z2 + 2xyz = 1(B) 2(sin –1x + sin –1y + sin –1z) = cos –1x + cos –1y + cos –1z(C) xy + yz + zx = x + y + z – 1

(D) xx

1 + y

y

1+ z

z

1

> 6

Q.92 Which of the following homogeneous functions are of degree zero ?

(A)x

y ln

y

x+

y

x ln

x

y(B)

x x y

y x y

( )

( )

(C)xy

x y2 2(D) x sin y

x y cos y

x

Q.93 The value of tan –1 x

x

sin

cos

1

– tan –1

x

cos

sin

is, for 0

2,

; x R + , is

(A) independent of x (B) independent of

(C)2

– (D) none of these

Q.94 D [

1, 1] is the domain of the following functions, state which of them has the inverse.(A) f(x) = x2 (B) g(x) = x3 (C) h(x) = sin 2x (D) k(x)= sin (x/2)

Q.95 Which of the following function(s) have no domain?(A) f(x) = logx – 1(2 – [x] – [x]2) where [x] denotes the greatest integer function.(B) g(x) = cos –1(2–{x}) where {x} denotes the fractional part function.(C) h(x) = ln ln(cosx)

(D) f(x) =

1

sec -1 sgn e x

Maths Funtion & Inverse

Documents

Transcript of Maths Funtion & Inverse