Functions revision worksheet (t) part 2

FUNCTIONS WORKSHEET (T) PART 2
STHITPRAGYA SCIENCE CLASSES, GANDHIDHAM
ADVANCED MATHEMATICS FOR JEE | BITSAT| GUJCET| OLYMPIADS | IX, X, XI, XII, XII +
MISHAL CHAUHAN (M.Tech, IIT Delhi)
Address 1: Near Gayatri Mandir, Opp. PGVCL Office, Shaktinagar Address 2: Sec-5, G.H.B, Gandhidham
Contact: 9879639888 Email:sthitpragyaclasses@gmail.com
Q 51. Let f(x) =4, x < -1
-4x, -1 ≤ x ≤ 0.
If f(x) isan evenfunctioninRthenthe definitionof f(x) in(0,+) is
(a) f(x) = 4x, 0 < x ≤ 1 (b) f(x) = 4x,0 < x ≤ 1 (c) f(x) = 4, 0 < x ≤ 1 (d) none of these
4, x > 1 -4, x > 1 4x, x > 1
Q 52. If 2 x
f(x) x sin
2

 , |x| < 1
x |x|,|x|1 thenf(x) is
(a) an evenfunction (b) an oddfunction (c) a periodicfunction (d) none of these
Q 53. The periodof the functionf(x) =
x
sin
2
+ |cos x|is
(a) 2 (b)  (c) 4 (d) none of these
Q 54. If f(x) isa periodicfunctionof the periodkthenf(kx +a), where ais a constant,isa periodic
functionof the period
(a) k (b) 1 (c)
k
a
(d) none of these
Q 55. The periodof the functionf(x) =4cos(2x + 3) is
(a) 2 (b)
2

(c)  (d) none of these
Q 56. The periodof the functionf(x) =
x x
3sin 4cos
3 4
 
 is
(a) 6 (b) 24 (c) 8 (d) 2
Q 57. Let f(x) = cos px ,where p= [a] = the greatestintegerlessthanorequal toa. If the periodof
f(x) is  then
(a) a  [4, 5] (b) a = 4, 5 (c) a  [4,5) (d) none of these
Q 58. Let f(x) =cos 3x + sin 3x . Thenf(x) is
(a) a periodicfunctionof period2 (b) a periodicfunctionof period 3
(c) not a periodicfunction (d) none of these

Q 59. The function
x x
f(x) sin cos
n! (n 1)!
 
 

is
(a) not periodic (b) periodic,withperiod2(n!)
(c) periodic,withperiod(n+1) (d) none of these
Q 60. The functionf(x)=x – [x] + cos x,where [x] =the greatestintegerlessthanorequal tox,is a
(a) periodicfunctionof indeterminate period (b) periodicfunctionof period2
(c) nonperiodicfunction (d) periodicfunctionof period1
Q 61. Let f(x) =nx + n – [nx + n] +
x
tan
2

, where [x] isthe greatestinteger ≤x and n  N. It is
(a) a periodicfunction of period1 (b) a periodicfunctionof period4
(c) not periodic (d) a periodicfunctionof period2
Q 62. Let f(x) =x(2 – x),0 ≤ x ≤ 2. If the definitionof f isextendedoverthe setR – [0, 2] byf(x + 2)
= f(x) the f isa
(a) periodicfunction of period1 (b) nonperiodicfunction
(c) periodicfunctionof period2 (d) periodicfunctionof period
1
2
Q 63. If 2 2 5
f(x) sin x sin x cosx.cos x and g 1
3 3 4
 
     
     
     
     
then(gof)(x) is
(a) a polynomialof the firstdegree insinx,cosx
(b) a constant function
(c) a polynomialof the seconddegree insinx,cosx
(d) none of these
Q 64. If f(x) = xn
,n  N and (gof)(x)=ng(x) theng(x) canbe
(a) n |x| (b) 3 . 3
x (c) ex
(d) log|x|
Q 65. If g{f(x)|=|sinx| and f{g(x)} = 2
(sin x) then
(a) 2
f(x) sin x,g(x) x
  (b) f(x) sinx,g(x) | x |
 
(c) 2
f(x) x ,g(x) sin x
  (d) f andg cannotbe determined

Q 66. If
1
f(x)
1 x


, x  0, 1, thenthe graphof the functiony= f{f(f(x))},x >1, is
(a) a circle (b) an ellipse (c) a straightline (d)apair of straight
lines
Q 67. If f(x) isa polynomial functionof the seconddegree suchthatf(-3) =6, f(0) = 6 and f(2) = 11
thenthe graph of the functionf(x) cutsthe ordinate x = 1 at the point
(a) (1, 8) (b) (1, 4) (c) (1, -2) (d) none of these
Q 68. Let f(x) be afunctionwhose domainis[-5,7].Let g(x) = |2x + 5|. Thenthe domainof (fog)(x)
is
(a) [-5, 1] (b) [-4,0] (c) [-6, 1] (d) none of these
Q 69. Let f : (-,1]  (-,1] suchthat f(x) = x(2 – x).Thenf-1
(x) is
(a) 1 1 x
  (b) 1 1 x
  (c) 1 x
 (d) none of these
Q 70. If f(x) = 3x – 5 thenf-1
(x)
(a) is givenby
1
3x 5

(b) isgivenby
x 5
3

(c) doesnot existbecause f isnotone-one (d) doesnotexistbecause f isnot onto
Q 71. If the functionf:[1, +)  [1, +) is definedbyf(x)=2x(x-1)
thenf-1
(x) is
(a)
x(x 1)
1
2

 
 
 
(b) 2
1
(1 1 4log x)
2
  (c) 2
1
(1 1 1 4log x)
2
  (d) not defined
Q 72. If the functionf : R  R be suchthat f(x) = x – [x],where [y] denotesthe greatestintegerless
than or equal toy, thenf-1
(x) is
(a)
1
x [x]

(b) x – [x] (c) not defined (d) none of these
Q 73. The inverse functionof the function
x x
x x
e e
f(x)
e e





is
(a)
1 1 x
log
2 1 x


(b)
1 2 x
log
2 2 x


(c)
1 1 x
log
2 1 x


(d) none of these
Q 74. The graph of a real-valuedfunctionf(x) isthe following.
The functionis

y = 2x
y = 0
Y
O X
(a) f(x) = x - |x| (b) f(x) = x + |x| (c) f(x) = 2x (d) none of these
Q 75. If f(x + y,x – y) = xythenthe arithmeticmeanof f(x,y) andf(y,x) is
(a) x (b) y (c) 0 (d) none of these
Q 76. The graph of the functiony= f(x) issymmetrical aboutthe linex = 2.
Then
(a) f(x + 2) = f(x – 2) (b) f(2 + x) = f(2 – x) (c) f(x) = f(-x) (d) none of these
Choose the correct options.One or more optionsmay be correct.
Q 77. Let f(x) =x2
,0 < x < 2
2x – 3, 2 ≤ x < 3
x + 2, x  3. Then
(a)
3 3
f f f f
2 2
 
 
 
   

 
 
   
   
 
 
 
(b)
5 5
1 f f f f
2 2
 
 
 
   
 
 
 
   
   
 
 
 
(c) f{f(1)} = f(1) = 1 (d) none of these
Q 78. If f(x) = cos2
x + cos2
x
3

 

 
 
- cos x . cos x
3

 

 
 
then
(a) f(x) isan evenfunction (b) f f
8 4
 
   

   
   
(c) f(x) isa constant function (d) f(x) isnotperiodicfunction
Q 79. If one of the roots of x2
+ f(a) . x + a = 0 is equal tothe thirdpowerof the otherfor real a
then
(a) the domainof the real-valuedfunctionf isthe setof non-negative real numbers
(b) 1/ 4 1/ 2
f(x) x (1 x )
   (c) 1/ 4 3/ 4
f(x) x x
  (d) none of these

Q 80. If f is an evenfunctiondefinedonthe interval (-5,5) thena value of x satisfyingthe equation
x 1
f(x) f
x 2

 
  

 
is
(a)
1 5
2
 
(b)
2 5
2
 
(c)
1 5
2
 
(d)
3 5
2
 
Q 81. Let f(x) =[x] = the greatestintegerlessthanorequal tox and g(x) = x – [x].Thenforany two
real numbersx and y
(a) f(x + y) = f(x) +f(y) (b) g(x + y) = g(x)+g(y) (c) f(x + y) = f(x) +f{y+ g(x)} (d) none of
these
Q 82. Let x  N and letx be a perfectsquare.Letf(x) =the quotientwhenx isdividedby5and g(x)
= the remainderwhenx isdividedby5.Then x = f(x) + g(x) holdsforx equal to
(a) 0 (b) 16 (c) 25 (d) none of these
Q 83. If f(x) = 27x3
+ 3
1
x
and , are the roots of 3x +
1
x
= 2 then
(a) f() = f() (b) f() = 10 (c) f() = -10 (d) none of these
Q 84. If f(x) = sin-1
(sinx) then
(a) f(x) =  - x,0 ≤ x ≤
2

(b) f(x) =  - x,
2

≤ x ≤ 
(c) f(x) = x,0 ≤ x ≤  (d) f(x) = -x,-
2

≤x ≤ 0
Q 85. If ex
+ ef(x)
= e thenfor f(x)
(a) domain= (-,1) (b) range = (-,1) (c) domain= (-,0] (d) range = (-,1]
Q 86. If f(x) isan oddfunctionthen
(a)
f( x) f(x)
2
 
is an evenfunction
(b) [|f(x)|+1] iseven,where [x] =the greatestinteger ≤x
(c)
f(x) f( x)
2
 
is neithereve norodd (d) none of these
Q 87. Let f(x) =sec-1
[1+ cos2
x] where [.] denotesthe greatestintegerfunction.Then
(a) the domainof f is R (b) the domainof f is [1,2]
(c) the range of f is [1, 2] (d) the range of f is{sec-1
1,sec-1
2}

Q 88. If f(x) andg(x) are twofunctionsof x such that f(x) + g(x) = ex
and g(x) – g(x) = e-x
then
(a) f(x) isan oddfunction (b) g(x) isan oddfunction
(c) f(x) isan evenfunction (d) g(x) isan evenfunction
Q 89. Let f(x) =
2
2
4cos x
9

 . Then
(a) the domainof f is ,
3

 


 
(b) the range of f is[-1, 1]
(c) the domainof f is , ,
3 3
 
   
   
 
 
   
(d) the range of f is[-4, 4]
Q 90. Let f(x + y) = f(x) + f(y) forall x,y  R. Then
(a) f(x) isan evenfunction (b) f(x) isanodd function
(c) f(0) = 0 (d) f(n) = nf(1),n  N
Q 91. Let f(x) =[x]2
+ [x + 1] – 3, where [x] =the greatestinteger ≤x.Then
(a) f(x) isa many-one andintofunction (b) f(x) = 0 for infinite numberof valuesof x
(c) f(x) = 0 for onlytworeal values (d) none of these
Q 92. Let f and g be functionsfromthe interval [0, ) tothe interval [0, ) f beinganincreasing
functionandg beinga decreasingfunction.If f{g(0)} =0 then
(a) f{g(x)} f{g(0)} (b) g{f(x)} ≤g{f(0)} (c) f{g(2)} = 0 (d) none of these
51a 52b 53a 54b 55c 56b 57c 58c 59d 60c
61d 62c 63b 64d 65a 66c 67a 68c 69b 70b
71b 72c 73a 74b 75c 76b 77abc 78abc 79ab 80abcd
81c 82bc 83ac 84b 85ab 86ab 87ad 88bc 89cd 90bcd
91ab 92bc

Functions revision worksheet (t) part 2

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Functions revision worksheet (t) part 2