functions objective question bank

7/28/2019 Functions Objective Question bank

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QUESTION BANKQUESTION BANKQUESTION BANKQUESTION BANK

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FUNCTIONSFUNCTIONSFUNCTIONSFUNCTIONS

JEE ADVANCE


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Q.B on Functions [2]

Q.1 Let f1(x) =

otherwise0

1xfor1

1x0forx

!

dd

and f2

(x) =f1

( x) for all x

f3 (x) = f2(x) for all xf

4(x) =f

3( x) for all x

Which of the following is necessarily true?

(A)f4

(x) =f1

(x) for all x (B)f1

(x) = f3

(x) for all x

(C)f2

(x) =f4

(x) for all x (D)f1

(x) +f3

(x) = 0 for all x

Q.2 Domain of definition of the function f (x) = log

19310 1x2x + )x1(cos 1 is

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

Q.3 The set of all real values of a so that the range of the function y = 1x

ax2

is R, is

(A) [1, f) (B) ( f, 1) (C) (1, f) (D) ( f, 1]

Q.4 The period of the function f (x) =|xcosxsin|

|xcos||xsin|

is

(A) S/2 (B) S/4 (C) S (D) 2S

Q.5 In the square ABCD with side AB = 2 , two points M and N are on the adjacent sides of the square

such that 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) (B) (0 , 2 ] (C) (D)

Q.6 f (x) =nx

x

land g (x) =

x

nxl. Then identify the CORRECT statement

(A))x(g

1andf(x) are identical functions (B)

)x(f

1and g(x) are identical functions

(C) f (x) . g (x) = 1 0x ! (D) 1)x(g.)x(f

1 0x !

Q.7 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.8 The domain and range of the function f(x) = cosec1

are respectively

(A) R ;

S S(B) R+ ;

S

(C)

S

S

S

SS

S 2,0};n2{2n2,2n2 (D) }0{2,2};n2{2n2,2n2

SS

S

S

SS

S


3/15


Q.9 A function f(x) = x21 + x is defined from D1 o D2 and is onto. If the set D1 is its completedomain then the set D

2is

(A)

f

2

1, (B) (f, 2) (C) ( f, 1) (D) ( f, 1]

Q.10 Which of the following function is surjective but not injective

(A) f : Ro R f (x) = x4 + 2x3 x2 + 1 (B) f : R o R f (x) = x3 + x + 1

(C) f : Ro R+ f (x) = 2x1 (D) f : Ro R f (x) = x3 + 2x2 x + 1

Q.11 Let f (x) =1x

2

; g (x) = cos x and h (x) = 3x then the range of the composite functionfogoh, is

(A) R+ (B) R {0} (C) [1, f) (D) R + {1}

Q.12 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 ggf

equals

(A) 0.5 (B) 0.5 (C) 1 (D) 1.5

Q.13 The range of the function f(x) =12x11x2

)10x7x(5xne2

2)2x(x2

l

is

(A) ),( ff (B) ),0[ f (C)

f,

2

3(D)

4,2

3

Q.14 If the solution set forf(x) < 3 is (0, f) and the solution set forf(x) > 2 is (f, 5), then the true solution

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

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

Q.15 Let f(x) = -

irrationalisxif0

rationalisxif1

A function g (x) which satisfies x f (x)d g (x) for all x is(A) g(x) = sin x (B) g (x) = x (C) g (x) = x2 (D) g (x) = | x |

Q.16 The graph of the function y = g (x) is shown.

The number of solutions of the equation211)x(g , is

(A) 4 (B) 5

(C) 6 (D) 8

Q.17 Consider the functions

f : X o Y and g : Y o Zthen which of the following is/are incorrect?

(A) If f and g both are injective then gof : X o Z is injective(B) If f and g both are surjective then gof : X o Z is surjective

(C) If gof : X o Z is bijective then f is injective and g is surjective.(D) none


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Q.18 Range of the function f (x) = tan1 [ ] [ ] | |x x xx

21

2is

where [*] is the greatest integer function.

(A)1

4,fNM (B)

1

42STV f, (C)

1

42,ST (D)

1

42,NM

Q.19 Which of the following statements are incorrect?

I Iff(x) andg(x) are one to one thenf(x) +g(x) is also one to one.

II Iff(x) andg(x) are one-one thenf(x) g(x) is also one-one.

III Iff(x) is odd then it is necessarily one to one.

(A) I and II only (B) II and III only (C) III and I only (D) I, II and III

Q.20 Let f : Ao B and g : B o C be two functions and gof : Ao C is defined. Then which of the followingstatement(s) is true?

(A) If gof is onto then f must be onto.

(B) If f is into and g is onto then gof must be onto function.

(C) If gof is one-one then g is not necessarily one-one.

(D) If f is injective and g is surjective then gof must be bijective mapping.

Q.21 Consider the function g (x) defined as

1x)x(g )12(

2008

= (x + 1)(x2 + 1)(x4 + 1)......

1x

20072 1.

the value of g (2) equals

(A) 1 (B) 22008 1 (C) 22008 (D) 2

Q.22 Let

-o

-

34R

34R:f be a function defined as f(x) =

4x3x4

. The inverse of f is the map

g : R

-

3

4o R

-

3

4is given by

(A) g(y) =y43

y3

(B) g(y) =

y34

y4

(C) g(y) = y43y4

(D) g(y) = y34y3

Q.23 Let F (x) =

t

d

1xif|x|x

1x1if]x1[]x1[

1xif|x|x

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

(A) even (B) odd

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

Q.24 Let f (k) =2009

kand g(k) = 44

4

))k(f())k(f1(

)k(f

then the sum

2009

0k

)k(g is equal :

(A) 2009 (B) 2008 (C) 1005 (D) 1004


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Q.25 The domain of definition of the function f (x) =

is

(A) R {nS, n I} (B) R {(2n + 1)S

, n I}

(C) R {nS, (2n + 1)S

, n I} (D) none

Q.26 If for all x different from both 1 and 0 we havef1(x) =

1x

x

, f

2(x) =

x1

1

, and for all integers n t 1,

we havefn + 2

(x) =

evenisnif)x(ff

oddisnif)x(ff

21n

11n

then f

4(x) equals

(A) x (B) x 1 (C)f1(x) (D)f

2(x)

Q.27 If f(x) = x2 + bx + c andf(2 + t) =f(2 t) for all real numbers t, then which of the following is true?

(A)f(1)


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Q.33 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.34 If f(x) is a function from Ro R, we say that f (x) has propertyI if f (f (x) ) = x for all real number x, and we say that f (x) has property

II if f (f(x)) = x for all real number x.

How many linear functions, have both property I and II?

(A) exactly one (B) exactly two (C) exactly three (D) infinite

Q.35 The function f (x) is defined by f (x) = cos4x + K cos22x + sin4x, where K is a constant. If the function

f (x) is a constant function, the value of k is

(A) 1 (B) 1/2 (C) 0 (D) 1/2 (E) 1

Q.36 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.37 Let f (x) = (3x + 2)2 1, f < x d3

2. If g(x) is the function whose graph is the reflection of the

graph of f(x) with respect to line y = x, then g(x) equals

(A) 1x,1x23

1t (B) 1x,1x2

3

1t

(C) 2x,2x13

1

t (D) 2x,2x131

t

Q.38 Let two functions f(x) and g(x) are defined on Ro R such that f (x) =-

rationalx,x2

irrationalx,x2

2

and g(x) =-

rationalx,x

irrationalx,x22

2

. Then the function f + g : Ro R is

(A) injective as well as surjective. (B) injective but not surjective.

(C) surjective but not injective. (D) neither surjective nor injective.

Q.39 If f (x) = 2x + 1 then the value of x satisfying the equation

116)x(ffff)x(fff)x(ff)x(f is equal to(A) 2 (B) 3 (C) 4 (D) 6


7/15


Q.40 Which of the following graphs best represent the function f (x) = x [x]?

(where [x] denotes the largest integer less than or equal to x.)

(A) (B)

(C) (D)

Q.41 For the function f (x) =1e

1ex

x

, if n(d) denotes the number of integers which are not in its domain and

n(r) denotes the number of integers which are not in its range, then n(d) + n(r) is equal to

(A) 2 (B) 3 (C) 4 (D) Infinite

Q.42 Which of the following equations have the same graphs?

I.

y = x 2II.

)2x(

)4x(y

2

III.

(x + 2)y = x

2

4(A) I and II only. (B) I and III only.

(C) II and III only. (D) All the equations have different graphs.

Q.43 If g(x) =7

1

74 xx4cos2

1x2cos2xcos4

, then the value of )100(gg is equal to

(A) 1 (B) 0 (C) 1 (D) 100

Let f (x) = x2

2x 1 x R. Let f : (f, a] o [b, f), where 'a' is the largest real number for whichf (x) is bijective.

Q.44 The value of (a + b) is equal to

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

Q.45 Let f : R o R, g (x) = f (x) + 3x 1, then the least value of function y = g(| x |) is(A) 9/4 (B) 5/4 (C) 2 (D) 1

Q.46 Let f : [a, f) o [b, f), then f1(x) is given by

(A) 1 + 2x (B) 1 3x (C) 1 2x (D) 1 + 3x

Q.47 Let f : R o R, then range of values of k for which equation f (| x |) = k has 4 distinct real roots is(A) ( 2, 1) (B) ( 2, 0) (C) ( 1, 0) (D) (0, 1)


8/15


Consider a quadratic function f(x) = ax2 + bx + c, (a, b, c R, a z 0) and satisfying the followingconditions.

(i) f (x 4) = f (2 x) x R and f (x) t x x R

(ii) f (x) d2

2

1x

x (0, 2)

(iii) The minimum value of f (x) is zero.

Q.48 The value of the leading coefficient of the quadratic polynomial is

(A) 1/4 (B) 1/3 (C) 1/2 (D) 1

Q.49 f ' (1) has the value equal to

(A) 1/4 (B) 1/3 (C) 1/2 (D) 1

Let S denotes the set consisting of four functions and S = { [x], sin1

x, |x|, {x} }where {x} denotes fractional part and [x] denotes greatest integer function. Let A, B, C are subsets of S.

Suppose

A: consists of odd function(s)

B : consists of discontinuous function(s)

and C : consists of non-decreasing function(s) or increasing function(s).

If f (x) A C ; g (x) B C ; h (x) B but not C and l(x) neither A nor B nor C.Then answer the following.

Q.50 The function l(x) is

(A) Periodic (B) Even (C) Odd (D) neither odd nor even

Q.51 The range of )x(fg is(A) {1, 0, 1} (B) {1, 0) (C) {0, 1} (D) {2, 1, 0, 1}

Q.52 The range of )x(hf is

(A)

S

2,0 (B)

S

2,0 (C)

S

2,0 (D)

S

2,0

An even periodic function f : Ro R with period 4 is such that

f(x) =

ddd2x1;x

1x0;)x|,x(|.max 2

Q.53 The value of {f(x)} at x = 5.12 (where { } represents fractional part), is

(A) {f (7.88) } (B) {f (3.26) } (C) { f (2.12) } (D) { f (5.88) }

Q.54 The equation of circle with centre lies on the curve f(x) at x = 9 and touches x-axis, is

(A) x2 + y2 14x 2y + 49 = 0 (B) x2 + y2 18x 4y + 84 = 0

(C) x2 + y2 18x 2y + 81 = 0 (D) x2 + y2 18x + 2y + 81 = 0

Q.55 If g(x) = |3sin x|, then the number of solutions of f(x) = g(x) for x (6, 6), are(A) 5 (B) 7 (C) 3 (D) 9


9/15


Q.56 Let f(x) = x2 + x + 1 x R and g(x) = (fof)(x), thenStatement-1: Range of g(x) is same as the range of f(x).

Statement-2: Domain of g(x) is same as the domain of f(x).

(A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.

(B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.

(C) Statement-1 is true, statement-2 is false.(D) Statement-1 is false, statement-2 is true.

Q.57 Statement-1: The function f(x) = x4 + 2x + 3 defined from R to R is not injective.

Statement-2: Every polynomial function of even degree defined from R to R is always not injective.



(C) Statement-1 is true, statement-2 is false.

(D) Statement-1 is false, statement-2 is true.

Q.58 Consider the function f (x) = ln ln

x

e

e4

x 3

Statement-1: The range of the function f (x) is R+

Statement-2: For two positive reals a and b, ab2

bat





Q.59 Statement-1: f is an even function, g and h are odd functions, all 3 being polynomials.

Given f (1) = 0, f (2) = 1, f (3) = 5, g (1) = 1, g (3) = 2, g (5) = 3, h (1) = 3,

h (3) = 5 and h (5) = 1. The value of )1(gfh)3(fhg)1(hgf is equal tozero.

Statement-2: If a polynomial function P(x) is odd then P(0) = 0.





Q.60 Let g : Ro R defined by g(x) = {ex}, where {x} denotes fractional part function.Statement-1 : g(x) is periodic function.

Statement-2 : {x} is periodic function.




(D) Statement-1 is false, statement-2 is true


10/15


Q.61 Statement-1: The function f (x) =

S

]x[2

3tan where [x] is the greatest integer function, is aperiodic.

Statement-2: g (x) = [x] is aperiodic





Q.62 Statement-1: If x1

< x2

< x3

........... < x2n 1

< x2n

and x R, n N then the least value of the function

f (x) = |x x1| + |x x

2| + |x x

3| + ............+ |x x

2n| is equal to

n

1i

iin )xx( .

Statement-2: Least value of |x x1| is zero.





Q.63 Let f : R o R defined by f(x) = cos S[x], where [x] denotes the greatest integer function less than orequal to x.

Statement-1 : f(x) is aperiodic function.

Statement-2 : [x] is aperiodic function.





Q.64 Statement 1: The equation x2 4x + 3 = [ x ] + [ x ] has two distinct real solution,

where [ x ] denotes largest integer less than or equal to x.

Statement 2: [ x ] + [ x ] =-

otherwise,1

Ix,0

where [ x ] denotes largest integer less than or equal to x.



(C) Statement-1 is true, statement-2 is false.(D) Statement-1 is false, statement-2 is true.

Q.65 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) = cos1(2{x}) where {x} denotes the fractional part function.

(C) h(x) = ln ln(cosx)

(D) f(x) =


11/15


Q.66 Which of the following option is/are correct?

(A) tan1(x2) = cot1

2x

1is true for all x R {0}

(B) 2 cos1

2

2

x1

x1= 2S no real solution

(C) Iff(x) = cos1 | x | + sec1 | x | thenf(x) is even as well as odd.

(D) Iff(x) = sin1(tan2x + cot2x) + cosec1(sin2x + cosec2x) then domain off(x) is R

- S

2

nnI.

Q.67 Which of the following function (s) is/are Transcendental?

(A) f (x) = 5 sin x (B) f (x) =2 3

2 12sin x

x x (C) f (x) = x x2 2 1 (D) f (x) = (x

2 + 3).2x

Q.68 The graphs of which of the following pairs differ.

(A) y =

+

; y = sin2x

(B) y = tanx cotx ; y = sinx cosecx

(C) y = ~cos x~ + ~sin x~ ; y =

(D) none of these

Q.69 Which of the following function(s) is/are periodic with period S?(A) f(x) = ~sinx~ (B) f(x) = [x + S] (C) f(x) = cos (sin x) (D) f(x) = cos2x(where [ . ] denotes the greatest integer function)

Q.70 The values of x in [2S, 2S], for which the graph of the function y =

secx and

y =

+ secx, coincide are

(A)

2

3

2

3

22S

S SS, ,

(B)

(C)

(D) [2S, 2S]

-

Q.71 Identify the statement(s) which is/are incorrect ?

(A) the function f(x) = cos (cos1 x) is neither odd nor even

(B) the fundamental period of f(x) = cos (sinx) + cos (cos x) is S(C) the range of the function f(x) = cos (3 sinx) is [ 1, 1]

(D) the function f(x, y) = + x is a homogeneous function of degree 1.


12/15


Q.72 Which of the following function(s) would represent a non singular mapping.

(A)f: Ro R f (x) = | x | Sgn x where Sgn denotes Signum function(B) g : Ro R g (x) = x3/5

(C) h : Ro R h (x) = x4 + 3x2 + 1

(D) k : Ro R k (x) =2xx

6x7x32

2

Q.73 If the functionf(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)

Q.74 Let f(x) = sgn(arc cot x) + tan

S]x[

2, where [x] is the greatest integer function less than or equal to x.

Then which of the following alternatives is/are true?

(A) f (x) is many one but not even function (B) f(x) is periodic function

(C) f(x) is bounded function (D) Graph of f(x) remains above x-axis

Q.75 The graph of the function y =f(x) is as follows.Which of the following graphs represents the function mentioned against them?

(A) y = | f (x) | (B) y = f ( | x | )

(C) y = f ( | x | ) (D) y =

2

1( | f (x) | f (x) )

Q.76 Let f : Ro R be a function defined as f (x) = x + [x]. (Where [x] denotes the greatest integer less thanor equal to x). Which of the following hold(s) good ?

(A) f (x) is aperiodic (B) f (x) is not surjective(C) f (x) is neither odd nor even (D) f (x) is injective

Q.77 Let f(x) = [x]2 + [x + 1] 3, where [x] denotes greatest integer less than or equal to x, then which of the

following statement(s) is/areCORRECT?

(A) f(x) is many one function.

(B) f(x) vanishes for atleast three values of x.

(C) f(x) is neither even nor odd function.

(D) f(x) is aperiodic.


13/15


Q.78 Let f : A o B and g : B o C be two functions and gof : A oC is defined. Then which of thefollowing statement(s) is/are incorrect?

(A) If gof is into then g must be into.

(B) If gof is onto then f must be onto.

(C) If gof is one-one then f must be one-one.

(D) If gof is bijective then both f and g must be bijective.

Q.79 Consider the function f (x) = x1x , then which of the following is/areCORRECT?

(A) Range of f (x) is 2,1 .

(B) f is many one.

(C) f is either even or odd.

(D) Range of f (x) is identical to range of g (x) =

S

4xcos2 .

Q.80 Let f : [ 1, 1] onto [3, 5] be a linear polynomial. Which of the following can be true?

(A)

2

1f =

2

7(B) f1

4

15=

4

1(C) f(0) z 4 (D)

2

1f

2

1f = 8

Q.81 Let f : R o R defined by f (x) = Min. ( | x |, 1| x |)Then which of the following hold(s) good?

(A) Range of f is (f, 1] (B) f is aperiodic.(C) f is neither even nor odd. (D) f is neither injective nor surjective.

Q.82 Which of the following statement(s) is(are) correct?

(A) If f is a one-one mapping from set A to A, then f is onto.

(B) If f is an onto mapping from set A to A, then f is one-one.

(C) Let f and g be two functions defined from Ro R such that gof is injective, then f must beinjective.

(D) If set A contains 3 elements while set B contains 2 elements, then total number of functions from

A to B is 8.

Q.83 Which of the following are identical functions?

(A) f (x) = sgn 1x (B) g (x) = sin2 (ln x) + cos2 (ln x)

(C) h (x) = xcosxsin2 11 S

(D) k (x) = xtanxsec 22

(where [ x ] denotes greatest integer less than or equal x, {x} denotes fractional part of x and sgn x

denotes signum function of x respectively.)


14/15


Q.84 Polynomial P(x) contains only terms of odd degree. When P(x) is divided by (x 3), the remainder is 6.

If P(x) is divided by (x2 9) then remainder is g (x). Find the value of g (2).

Q.85 Let f(x) = x100. If f(x) is divided by x2 + x, then the remainder is r(x). Find the value of r(10).

Q.86 If f : R o R be an injective mapping and p, q, r are non-zero distinct real quantities satisfying

rq

qpf

r

pf and

p

rf

r

qf .

If the graph of g(x) = px2 + qx + r passes through M (1, 6) then find the value of q.

Q.87 Let P(x) = x4 + ax3 + bx2 + cx + d, where a, b, c, d R.Suppose P(0) = 6, P(1) = 7, P(2) = 8 and P(3) = 9, then find the value of P(4).

Q.88 Let f (x) = ax9x2 . Find the number of integers in the range of a so that f (x) = 0

has 4 distinct real root.

Q.89 The polynomial R(x) is the remainder upon dividing x2007 by x2 5x + 6. If R(0) can be expressed as

ab(ac bc), find the value of (a + b + c).

Q.90 Number of integer in the range of the function,

f (x) =2

xsin

S+ 2x16 + x + log2 )2x(x .


15/15


Q.1 B Q.2 C Q.3 B Q.4 C

Q.5 B Q.6 A Q.7 C Q.8 C

Q.9 D Q.10 D Q.11 C Q.12 D

Q.13 A Q.14 D Q.15 D Q.16 D

Q.17 D Q.18 C Q.19 D Q.20 C

Q.21 D Q.22 B Q.23 A Q.24 C

Q.25 C Q.26 C Q.27 B Q.28 D

Q.29 C Q.30 C Q.31 D Q.32 C

Q.33 C Q.34 B Q.35 B Q.36 C

Q.37 A Q.38 D Q.39 B Q.40 B

Q.41 C Q.42 D Q.43 D Q.44 B

Q.45 C Q.46 A Q.47 A Q.48 A

Q.49 D Q.50 B Q.51 D Q.52 B

Q.53 A Q.54 C Q.55 B Q.56 D

Q.57 A Q.58 D Q.59 A Q.60 D

Q.61 D Q.62 B Q.63 D Q.64 A

Q.65 ABCD Q.66 ABC Q.67 ABD Q.68 ABC

Q.69 ACD Q.70 AC Q.71 ABC Q.72 AB

Q.73 ABC Q.74 ABCD Q.75 ABCD Q.76 ABCD

Q.77 ABCD Q.78 ABD Q.79 AB Q.80 ABD

Q.81 BD Q.82 CD Q.83 ACD Q.84 4

Q.85 10 Q.86 8 Q.87 34 Q.88 17

Q.89 2011 Q.90 1

functions objective question bank

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