QUESTION BANK ON
QUESTION BANK ON
FUNCTIONS
FUNCTIONS
AND
AND
INVERSE TRIGONOM
INVERSE TRIGONOMETRY
ETRY FUNCTIO
FUNCTIONS
NS
Time
Time
Limit
Limit
:
:
5 Sitting
5 Sitting
Each of
Each of
60 Mi
60 Mi
nutes
nutes
duration approx.
duration approx.
TARGET IIT JEE 2007
TARGET IIT JEE 2007
M A T H E M A T I C S
M A T H E M A T I C S
BANSAL CLASSES
BANSAL CLASSES
There are 95 questions in this question bank. There are 95 questions in this question bank. Only one alternative is correct.
Only one alternative is correct. Q.1
Q.1 Let Let f f be be a a real real valuvalued ed funcfunction tion such such thatthat
f (x) + f (x) +
x x 2002 2002 f f 2 2 = 3x= 3xfor all x > 0. Find f (2). for all x > 0. Find f (2).
((AA) ) 1100000 0 ((BB) ) 2200000 0 ((CC) ) 3300000 0 ((DD) ) 44000000 Q.2
Q.2 SolSolutiution on set set of of the the eqequatuation, ion, cocoss11 x – sin x – sin11 x = cos x = cos11
((
xx 33))
((AA) ) iis s a a uunniit t sseet t ((BB) ) ccoonnssiisstts s oof f ttwwo o eelleemmeennttss ((CC) ) ccoonnssiisstts s oof f tthhrreee e eelleemmeenntts s ((DD) ) iis s a a vvooiid d sseett
Q
Q..3 3 IIff ff( ( ) xx)
2 2 3 taan t n 3 5 xx
5 1 1
ccooss66xx ; g(x) is a fu; g(x) is a function having nction having the same time period the same time period as that of f(x), as that of f(x), then whichthen which of the following can be g(x).of the following can be g(x). (A) (sec
(A) (sec223x + cosec3x + cosec223x)tan3x)tan2233x x ((BB) ) 2 2 ssiinn33x x + + 33ccooss33xx (C) 2
(C) 2 1 1
ccoos s x22 33x + + ccoosseecc33x x ((DD) ) 3 3 ccoosseecc33x x + + 2 2 ttaann33xx Q.4Q.4 Which Which one one of of the the followinfollowing g depicts depicts the the graph graph of of an an odd odd functionfunction??
(A) (B)
(A) (B)
(C) (D)
(C) (D)
Q.5
Q.5 The The sum sum of of the the infinfiniinite te terterms ms of of the the seriserieses cot cot11 11 33 4 4 2 2
+ cot + cot11 22 33 4 4 2 2
+ cot + cot11 33 33 4 4 2 2
+ ... is equal to : + ... is equal to : (A) tan(A) tan –1 –1 ((11) ) ((BB) ) ttaann –1 –1((22) ) ((CC) ) ttaann –1 –1 ((33) ) ((DD) ) ttaann –1 –1 (4) (4) Q.6
Q.6 DomDomain ain of of defdefinitinition ion of of the the funfunctioction n f f (x) (x) = = loglog 1010··33xx22
99xx11
11 ++ coscos11((11
xx)) is is((AA) ) [[00, , 11] ] ((BB) ) [[11, , 22] ] ((CC) ) ((00, , 22) ) ((DD) ) ((00, , 11))
Q.
Q.7 7 ThThe e vavalulue e of of tatann11 11
2 2 tatan n A22A
+ + tatann11(cotA)+tan(cotA)+tan11(cot(cot33A) A) for for 0 0 < < A A < < ((
/4) is/4) is (A) 4 tanQ Q..8 8 LLeett
f f xx tt tt xx g gxx tt tt xx aannd d h h x x f f x x g g xx ( ( ) ) mmaaxx. . ssiin n :: ( ( ) ) mmiinn. . ssiin n :: ( ( ) ) ( ( ) ) ( ( ))
0 0 0 0where [ ] denotes greatest integer function, then the range of h(x) is where [ ] denotes greatest integer function, then the range of h(x) is ((AA) ) {{00, , 11} } ((BB) ) {{11, , 22}}
((CC) ) {{00, , 11, , 22} } ((DD) ) {{
3,3,
2,2,
1, 0, 1, 2, 3}1, 0, 1, 2, 3} Q.9Q.9 The The perperiod iod of of the the funfunctiction on f(xf(x) ) = = sin sin 22
x + sinx + sin
xx 3 3
+ + sinsin
xx 5 5
isis ((AA) ) 2 2 ((BB) ) 6 6 ((CC) ) 115 5 ((DD) ) 3300 Q.1Q.10 0 ThThe e vavalulue e ofof sseec c ssiin n
ssiin n ccoos s cco oss
1 1 5050 11 9 9 31 31 9 9
is is equal equal toto (A) sec (A) sec1010 9 9
(B) sec (B) sec
9 9 ((CC) ) 1 1 ((DD) ) ––11 Q.1Q.11 1 The The domaidomain n of of definidefinition tion of of the the functifunctionon , , ff (x) = arc cos(x) = arc cos 3 3 7 7 88 1 1 2 2 2 2 x x xx x x
where where [[ *] d*] denoteenotes ths thee greatest integer function, is :greatest integer function, is :
((AA) ) ((11, , 66) ) ((BB) ) [[00, , 66) ) ((CC) ) [[00, , 11] ] ((DD) ) ((
2, 5]2, 5] Q.12Q.12
= sin = sin11
cocos s si
sinn11 xx
and and
= cos = cos11
sin sin co
coss11 xx
, then , then ::(A) tan
(A) tan
= cot = cot
(B) (B) tantan
= =
cot cot
(C) tan(C) tan
= tan = tan
(D) (D) tantan
= =
tan tan
Q.1 Q.13 3 GiGiveven n f f (x(x) ) == x x 1 1 8 8 x x 1 1 8 8
and and g g (x) (x) == f f (cos(cosxx)) 4 4 )) x x (sin (sin f f 4 4
then then g(x) g(x) isis (A) periodic with period(A) periodic with period
//2 2 ((BB) ) ppeerriiooddiic c wwiitth h ppeerriioodd
(C) periodic with period 2(C) periodic with period 2
(D) (D) aperiodicaperiodic Q.Q.14 14 IIf x f x = = tatann1111
cos cos11
11
2 2 + + sinsin 11 1 1 2 2 ; y = cos ; y = cos 1 1 2 2 1 1 8 8 1 1 cos cos
then then :: (A) x =(A) x =
y y ((BB) ) y y ==
x x ((CC) ) ttaann x x ==
(4(4/3/3)) y y (D(D) ta) tann x x = = (4(4/3/3)) yy Q.15Q.15 In the In the square square ABCD with ABCD with side side AB = AB = 2 , 2 , two pointwo points M & ts M & N are oN are on the n the adjacent sides adjacent sides of the of the square suchsquare such that MN is parallel to the diagonal BD. If x is the distance of MN from the vertex A and that MN is parallel to the diagonal BD. If x is the distance of MN from the vertex A and f (x) = Area (
f (x) = Area (
AMN) , AMN) , then rangthen range of e of f (x) f (x) is :is : (A)(A)
0 0 ,, 22
((BB) ) ((0 0 , , 2 ] 2 ] ((CC))
0 0 2 ,,2 22
(D)(D)
0 0 2 ,,2 33
Q.16Q.16 ccoos s ccoos s cco
os s
ttaan n ttaan
n
1 1 11 7 7 8 8 7 7
has the value equal to has the value equal to
((AA) ) 1 1 ((BB) ) ––1 1 ((CC)) coscos
7Q.17
Q.17 The The domain domain of of the the definition definition of of the the function f(x) = function f(x) = sinsin11
x x
55
2 2 ++
lloog g ( 1010( ))
1 1 6 6
xx is :is : ((AA) ) ((77, , 77) ) ((BB) ) ((
7,7,
3)3)
( (
3, 7)3, 7) (C) [ (C) [
7,7,
3]3]
[3, 5) [3, 5)
((55, , 66) ) ((DD) ) ((
3, 3)3, 3)
(5, 6) (5, 6) Q.1Q.18 8 ThThe e vavalulue e ofof ttaan n
ssiin n ttaan n
ssiinn 4 4 1 1 2 2 44 1 1 2 2 1 1 11 1 1
aa b b aa b b , , where where (0 (0 < < a a < < b), b), isis (A) (A) b b aa 2 2 (B)(B) aa b b 2 2 (C)(C) b b aa b b 2 2 22 2 2
(D) (D) b b aa aa 2 2 22 2 2
Q.Q.19 19 LLeett f f be a function satisfying be a function satisfying f f (xy) = (xy) = f f ((yyxx)) for all positive real numbers x and y. If for all positive real numbers x and y. If f f (30) = 20, then the (30) = 20, then the value of
value of f f (40) is (40) is
((AA) ) 115 5 ((BB) ) 220 0 ((CC) ) 440 0 ((DD) ) 6600
Q.20
Q.20 NumbNumber er of of real real value value of of x sax satisfytisfying ing the the equatequation, ion, arcarctantan x x x
x
11
+ + arcarcsinsin x x x
x
1 1
11 = =
2 2 isis
((AA) ) 0 0 ((BB) ) 1 1 ((CC) ) 2 2 ((DD) ) mmoorre e tthhaan n 22 Q.2
Q.21 1 LeLet f t f (x(x) ) = = sisinn22x + cosx + cos44x + 2 x + 2 and and g (x) = cos (cos xg (x) = cos (cos x) + cos (sin x) also let period ) + cos (sin x) also let period of f (x) and g of f (x) and g (x) be(x) be
T
T11 and T and T22 respectively then respectively then (A) T
(A) T11 = 2T = 2T22 (B) (B) 2T2T11 = T = T22 (C) T(C) T11 = = TT22 (D) (D) TT11 = 4T = 4T22 Q.22
Q.22 NumbNumber er of of solutisolutions ons of of the the equatequation ion 2 2 cotcot –1 –12 + cos2 + cos –1 –1(3/5) = cosec(3/5) = cosec –1 –1x x isis
((AA) ) 0 0 ((BB) ) 1 1 ((CC) ) 2 2 ((DD) ) mmoorre e tthhaan n 22
Q.23
Q.23 The The domadomain in of of defindefinition ition of of the the funcfunction tion : : f f (x) (x) == l l n (n ( x x 22
5 5 xx 2424 – – x x – – 2) 2) isis(A) (–
(A) (–
, , ––33] ] ((BB) ) ((– –
, –3 ], –3 ]
[8,[8,
) ) ((CC))
,, 2828
99 (D) (D) nonenone
Q.24
Q.24 The The periperiod od of of the the funcfunction tion f(x) f(x) = s= sinin coscosxx
2 2
+ c+ cos(sinx) os(sinx) equalequal (A)(A)
22 (B) (B) 22
(C)(C)
(D) (D) 44
Q.
Q.25 25 IIf x f x = = cocoss –1 –1 (cos 4) ; ; y (cos 4) y = sin= sin –1 –1 (sin 3) then which of (sin 3) then which of the following holds ?the following holds ? ((AA) ) x x – – y y = = 1 1 ((BB) ) x x + + y y + + 1 1 = = 00
((CC) ) x x + + 22y y = = 2 2 ((DD) ) ttaan n ((x x + + yy) ) = = – – ttaann77
Q.2
Q.26 6 LeLet f t f (x(x) ) == ee{{ee||xx||sgnsgnxx}} and g and g (x) (x) == ee[[ee||xx||sgnsgnxx]] , x, x
R where { x } R where { x } and [ ] denotes the fractional part andand [ ] denotes the fractional part and integral part functions respectively. Also h (x) =integral part functions respectively. Also h (x) = l l nn
f f ((xx))
++l l nn
gg((xx))
then then for for all all real real x, h x, h (x) (x) isis ((AA) ) aan n ooddd d ffuunnccttiioon n ((BB) ) aan n eevveen n ffuunnccttiioonn(C)
(C) neineithether r an an odd odd nonor r an an eveven en fufunctnction ion (D) (D) botboth h odd odd as as welwell l as as eveeven n fufunctnctionion
Q.27
Q.27 The The numbnumber er of of solutsolutions ions of of the the equaequation tion tantan –1 –1
3 3 x x + tan + tan –1 –1
2 2 x x = tan = tan –1 –1 x x isis ((AA) ) 3 3 ((BB) ) 2 2 ((CC) ) 1 1 ((DD) ) 00Q.28
Q.28 Which Which of tof the he followfollowing ing is the is the solutiosolution sen set of t of the the equatioequation n 2 c2 cosos –1 –1x = cotx = cot –1 –1
2 2 2 2 x x 1 1 x x 2 2 1 1 x x 2 2 ? ? ((AA) ) ((00, , 11) ) ((BB) ) ((––11, , 11) ) – – {{00} } ((CC) ) ((––11, , 00) ) ((DD) ) [[––11, , 11]] Q.29Q.29 SupSuppospose e thathatt f f is a periodic function with period is a periodic function with period 2 2 1 1
and that
and that f f (2) = 5 and f (2) = 5 and f
99 44 = = 2 2 thenthen f (–3) – ff (–3) – f
11 44 has has the the value value equal equal toto((AA) ) 2 2 ((BB) ) 3 3 ((CC) ) 5 5 ((DD) ) 77
Q.30
Q.30 ThThe e vavalulue e ofof ttaan n
ttaann
1 1 11 11 2 2 5 5 2 2 66 11 66 is equal : is equal :
(A) (A) 6 6
(B) (B) 4 4
(C) (C) 3 3
(D) none (D) none Q.31Q.31 GivGiven f en f (x) (x) ==(x+1)(x+1)CC(2x– 8)(2x– 8) ; g (x) =; g (x) =(2x – 8)(2x – 8)CC(x + 1)(x + 1) and and h (x) h (x) = f = f (x) . (x) . g g (x) , (x) , then which then which of the of the followingfollowing holds ?
holds ? (A)
(A) The The domain of domain of 'h' 'h' isis
(B) Th(B) The range e range of of 'h' is 'h' is {– 1}{– 1} (C) The domain of (C) The domain of 'h' is {x / x > 3 or x < – 3 ;'h' is {x / x > 3 or x < – 3 ; xx
II (D) The range (D) The range of of 'h' is {1}'h' is {1} Q. Q.32 32 ThThe e susumm n n
1 1 tan tan11 44 2 2 22 4 4 22 n n n n
nn
is is equal equal to to :: (A) tan (A) tan11 2 2 1 1 + tan + tan11 3 3 2 2 (B) (B) 4 4 tantan11 1 1 ((CC))
2 2 (D) sec(D) sec 11
22 Q.33Q.33 RangRange e of of the the functifunction on f f (x) (x) = = tantan –1 –1
[ [ ] x x ] [ [ ] x x ] | | ||xx x x
22
1122 is iswhere [*] is the greatest integer function. where [*] is the greatest integer function.
(A) (A) 11 4 4 ,,
LL
NNMM
I I
K K J J
(B)(B) 1 1 4 4 22RR
SS
TT
UU
VV
WW
,,
gg
(C)(C) 1 1 4 4 ,,22RR
SS
TT
UU
VV
WW
(D)(D) 1 1 4 4,,22LL
NNMM
OO
QQPP
Q.3Q.34 4 Let [x] deLet [x] denote thnote the greatese greatest integer t integer in x . in x . Then in Then in the intervathe interval [0, 3] l [0, 3] the numbthe number of soler of solutions outions of the eqf the equation,uation, x
x22
3x 3x + [x] + [x] = = 0 0 is :is :((AA) ) 6 6 ((BB) ) 4 4 ((CC) ) 2 2 ((DD) ) 00 Q.35
Q.35 The The rangrange oe of vf values alues of of p fp for or which which the the equatequation, ion, sin sin coscos –1 –1
ccooss((ttaan n 11 ))
x
x = = p p has has a a solution solution is:is:
(A) (A)
1 1 2 2 1 1 2 2 ,, ((BB) ) [[00, , 11) ) ((CC)) 11 2 2 ,,11
(D) (D) (– (– 1, 1, 1)1) Q.3 Q.36 6 LeLet t f f (x(x) ) ==
irrational irrational is is x x if if x x rational rational is is x x if if 0 0 and and g g (x) (x) ==
rational rational is is x x if if x x irrational irrational is is x x if if 0 0Then the function (f – g) x is Then the function (f – g) x is
((AA) ) ooddd d ((BB) ) eevveenn
Q.37
Q.37 NumbNumber er of of valuvalue e of of x sx satisfyatisfying ing the the equequation ation sinsin –1 –1
x x 5 5 + sin + sin –1 –1
x x 12 12 = = 2 2
is is((AA) ) 0 0 ((BB) ) 1 1 ((CC) ) 2 2 ((DD) ) mmoorre e tthhaan n 22
Q.38
Q.38 ConsidConsider a er a real real valuevalued fud function nction f(x) f(x) such such thatthat 11 1 1 ee ee x x f f xx ( ( )) (
( )) = x. The values of 'a' and = x. The values of 'a' and 'b' for which'b' for which
f (a) + f (b) = f f (a) + f (b) = f a a bb ab ab
11 is is satisfied satisfied areare (A) a
(A) a
(– (–
, 1); b, 1); b
R R ((BB) ) aa
(– (–
, 1); b, 1); b
(–1, (–1,
)) (C) a(C) a
(–1, 1) ; b(–1, 1) ; b
[[––11, , 11] ] ((DD) ) aa
(–1, 1); b (–1, 1); b
(–1, 1) (–1, 1) Q.39Q.39 ThThe e vavalulue e of of tantan
)) 3 3 (( cot cot 2 2 1 1 11 equals equals (A) (A)
33
1010
11 (B)(B)
1010
33
11 (C)(C) 33
1010
(D)(D) 1010
33
Q.40Q.40 The The periperiod od of of the the funfunctioction n coscos 22x x + + coscos2x is 2x is ::
(A)
(A)
(B)(B)
22 (C) (C) 22
(D) (D) none none of of thesetheseQ.41
Q.41 The The real real valvalues ues of of x satx satisfyisfying ing tantan –1 –1
x x 2 2 x x – – tantan –1 –1
x x 4 4 – tan – tan –1 –1
x x 2 2 x x = = 0 0 areare (A) ± (A) ± 2 2 1 1 (B) ± (B) ± 22 (C) (C) ±± 44 22 (D) (D) ± ± 22 Q.42Q.42 Which of Which of the following the following is true is true for a for a real valued real valued function function y y = f = f (x) , (x) , defined defined on [ on [ – a – a , a]?, a]? (A) f (x) can be expressed as a sum or a difference of two even functions
(A) f (x) can be expressed as a sum or a difference of two even functions (B)
(B) f (x) can be expressed as a sum f (x) can be expressed as a sum or a difference of two odd or a difference of two odd functionsfunctions (C)
(C) f (x) can be expressed as a sum or a f (x) can be expressed as a sum or a difference of an odd and an difference of an odd and an even functioneven function (D) f (x) can never 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 Q..443 3 ccooss 22 11 7 7 1 1 tan tan
equals equals(A) sin (4cot
(A) sin (4cot –1 –133) ) ((BB) ) ssiinn((33ccoott –1 –144) ) ((CC) ) ccooss((33ccoott –1 –144) ) ((DD) ) ccooss((44ccoott –1 –13)3) Q.44
Q.44 LeLet f(t f(x) x) = = sinsin [ [ ]]aa x (where x (where [ ] d[ ] denotes the enotes the greatest integer greatest integer function) . function) . If If f is pf is periodic with eriodic with fundamentalfundamental period
period
, then a belongs to :, then a belongs to :((AA) ) [[22, , 33) ) ((BB) ) {{44, , 55} } ((CC) ) [[44, , 55] ] ((DD) ) [[44, , 55)) Q.45
Q.45 The The rangrange oe of tf the he funcfunctiontion, f, f(x) (x) = c= cotot –1 –1loglog0 0 5..5
4 4 22 2 2 33 x x
xx
is:is: (A) (0, (A) (0,
) ) ((BB)) 00 33 4 4 ,,
(C)(C) 3 3 4 4
,,
(D)(D)
2 2 3 3 4 4 ,,
Q.46Q.46 Which Which of of the fthe following ollowing is the is the solution solution set of set of the ethe equation quation sinsin –1 –1x = cosx = cos –1 –1x + sinx + sin –1 –1(3x – 2)?(3x – 2)?
(A) (A) 11 2 2,,11
(B)(B) 1 1 2 2,,11
(C)(C) 1 1 3 3,,11
(D)(D) 1 1 3 3,,11
Q.47
Q.47 Which Which of tof the fhe followollowing ing funcfunctions tions are nare not hoot homogmogeneoeneous ?us ?
(A)
(A) x + x + y y coscos yy
x x (B)(B) xy xy x x y
y22 (C)(C) yysinsinxx yy x x cos cos y y x x
(D) (D) xx y y l l nn y y x x
+ + y y x xl l nn x x y y
Q.48Q.48 Which Which of thof the foe following llowing is the is the solution solution set oset of the f the equation equation 3cos3cos –1 –1x = cosx = cos –1 –1(4x(4x33 – 3x)? – 3x)?
((AA) ) [[––11, , 11] ] ((BB))
1 1 3 3 1 1 3 3 ,, (C)(C) 11 3 3,,11
(D)(D) 1 1 2 2,,11
Q.49Q.49 The The funfunctioction f n f : : RR
R, R, defined as defined as f(x) =f(x) = x x xx x x xx 2 2 2 2 6 6 1010 3 3 33
is :is : (A(A) ) ininjejectctivive e bubut t nonot t susurjrjecectitive ve (B(B) ) susurjrjecectitive ve bubut t nonot t ininjjecectitiveve (C
(C) ) ininjejectctivive e as as wewell ll as as susurjrjecectitive ve (D) (D) neneititheher r ininjejectctivive e nonor r susurjrjecectitiveve Q.50
Q.50 The The solutsolution ion of of the the equaequation tion 2co2coss –1 –1x = sinx = sin –1 –1((22x x 11
22x x )) ((AA) ) [[––11, , 00] ] ((BB) ) [[00, , 11] ] ((CC) ) [[––11, , 11] ] ((DD))
,,11
2 2 1 1 Q.51Q.51 The The periperiod od of of the the funcfunctiontion f f (x) = sin(x + (x) = sin(x + 3 – [x + 3 ] 3 – [x + 3 ] ), where [ ] denotes the greatest integer function is), where [ ] denotes the greatest integer function is (A) 2
(A) 2
+ + 3 3 ((BB) ) 22
((CC) ) 1 1 ((DD) ) 33 Q.Q.52 52 IIf f tatann –1 –1x + tanx + tan –1 –1 2x + tan 2x + tan –1 –13x =3x =
, then, then((AA) ) x x = = 0 0 ((BB) ) x x = = 1 1 ((CC) ) x x = = ––1 1 ((DD) ) xx
Q.53Q.53 If f(If f(x x + + ayay, x, x
ay) ay) = axy = axy then then f(x, y) f(x, y) is equal to :is equal to : (A) (A) x x yy 2 2 22 4 4
(B) (B) x x yy 2 2 22 4 4
((CC) ) 4 x4 xy y ((DD) ) nnoonnee Q.54Q.54 The The set oset of vaf values lues of of x fox for wr which hich the eqthe equation uation coscos –1 –1x + cosx + cos –1 –1 xx xx
2 2 1 1 2 2 3 3 33 2 2
==
33 holds good is holds good is
((AA) ) [[00, , 11] ] ((BB)) 00 11 2 2 ,,
(C)(C)
1 1 ,, 2 2 1 1 (D) {–1, 0, 1} (D) {–1, 0, 1} Q.55Q.55 The The rangrange e of of the the funcfunction tion y y == 22 x x 9 9 8 8
isis (A) (– (A) (–
,,
) ) – – {± {± 3} 3} (B(B))
,, 9 9 8 8 (C) (C)
9 9 8 8 ,, 0 0 (D) (D) (–(–
, 0), 0)
,, 9 9 8 8 Q.56Q.56 The The domain domain of of definitiodefinition n of of the the functiofunction n f f (x) (x) == loglog11 coscoscotcot loglog secsectantan 2 2 2 2 2 2 11 2 2 2 2 2 2 2 2 55 33 55 x x ec ec xx x x x x
isis (A) R – {n (A) R – {n
, n, n
II} } ((BB) ) R R – – {{((22n n + + 11))
2 2 , , nn
I} I} (C) R – {n (C) R – {n
, (2n + 1), (2n + 1)
2 2 , , nn
II} } ((DD) ) nnoonnee Q.57Q.57 The The solutiosolution n set set of of the the equatioequation n sinsin –1 –1 11
22x
x + + coscos –1 –1x = cotx = cot –1 –1 1 1
22
xx xx
– – sinsin –1 –1xx (A (A) ) [–[–1, 1, 1] 1] – – {0{0} } (B(B) ) (0(0, , 1]1] UU {{––11} } ((CC) ) [[––11, , 00))UU {1{1} } ((DD) ) [[––11, , 11]]Q.58
Q.58 Given Given the the graphs graphs of of the the two two functions, functions, y y = = f(x) f(x) & & y y = = g(x). g(x). In In thethe adjacent figure from point A
adjacent figure from point A on the graph of the function on the graph of the function y = f(x)y = f(x) corresponding to the given value of the independent variable (say corresponding to the given value of the independent variable (say xx00), ), aa straight line is drawn parallel to the X-axis to intersect the bisector of straight line is drawn parallel to the X-axis to intersect the bisector of the first and the
the first and the third quadrants at point B . From tthird quadrants at point B . From the point B a straighthe point B a straight line parallel to the Y-axis is drawn to intersect the graph of the function 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 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
the X-axis, to intersect the line NN
at D . If the st at D . If the straight line NNraight line NN
is is paralparallel to lel to YY-axis, -axis, then then the cothe co-ord-ordinates inates of thof the poe point D int D areare (A) f(x
(A) f(x00), g(f(x), g(f(x00)))) ((BB))xx00, g(x, g(x00)) (C) x
(C) x00, g(f(x, g(f(x00))) ) ((DD) ) ff((xx00), f(g(x), f(g(x00)))) Q.59
Q.59 ThThe e vavalue lue of siof sinn –1 –1(sin(2cot(sin(2cot –1 –1(( 22 – 1– 1))) is))) is
(A) – (A) –
4 4 (B)(B)
4 4 (C)(C) 3 3 4 4
(D) (D) 77 4 4
Q.60Q.60 The The funfunctioction f n f : : [2,[2,
))
Y Y defined by defined by f(x) = xf(x) = x22
4x + 5 is both one-one and onto if : 4x + 5 is both one-one and onto if :((AA) ) Y Y = = R R ((BB) ) Y Y = = [[11,,
) ) ((CC) ) Y Y = = [[44,,
) ) ((DD) ) [[55,,
)) Q.61Q.61 If If f(xf(x) ) = = cocosecsec –1 –1(cosecx) and cosec(cosec(cosecx) and cosec(cosec –1 –1x) are equal functions then maximum range of values of x isx) are equal functions then maximum range of values of x is
(A) (A)
2 2 ,, 1 1 1 1 ,, 2 2 (B)(B)
2 2 ,, 0 0 0 0 ,, 2 2 (C) (C)
,,
11
11,,
(D)(D)
11,,00
00,,11 Q.Q.62 62 IIf 2 f 2 f(f(xx22) + 3 f(1/x) + 3 f(1/x22) = x) = x22
1 (x 1 (x
0) then f(x 0) then f(x22) is :) is : (A) (A) 11 5 5 4 4 2 2
x x x x (B)(B) 1 1 5 5 2 2
x x x x (C)(C) 5 5 1 1 2 2 4 4 x x x x
(D)(D)
2 2 33 5 5 4 4 22 2 2 x x xx x x
Q.63Q.63 Sum Sum of of the the roots roots of of the the equatequation, ion, arc arc cot cot x – x – arc arc cot cot (x (x + + 2) 2) == 12 12
is is (A) (A) 33 ((BB) ) 2 2 ((CC) ) – – 2 2 ((DD) ) –– 33 Q.64Q.64 RangRange e of of the the funcfunction tion f f (x) (x) ==
} } x x { { 1 1 } } x x { {
where where {x} {x} denotes denotes the the fractional fractional part part function function isis((AA) ) [[0 0 , , 11) ) ((BB))
2 2 1 1 ,, 0 0 (C)(C)
2 2 1 1 ,, 0 0 (D)(D)
2 2 1 1 ,, 0 0 Q.65Q.65 RangRange e of of the the funfunction ction sgn sgn [[ l l n (xn (x22 – x + 1) ] is – x + 1) ] is
((AA) ) {{––11, , 00, , 11} } ((BB) ) {{––11, , 00} } ((CC) ) – – {{11} } ((DD) ) {{––11, , 11}} Q.66
Q.66 Number Number of of solution(solution(s) s) of of the the equation cosequation cos –1 –1(1 – x) – 2cos(1 – x) – 2cos –1 –1x =x =
2 2
is is ((AA) ) 3 3 ((BB) ) 2 2 ((CC) ) 1 1 ((DD) ) 00 Q.Q.67 67 LLeett f f (x) and (x) and g g (x) be functions which take integers as arguments. Let (x) be functions which take integers as arguments. Let f f (x + y) = (x + y) = f f (x) + (x) + g g (y (y) + ) + 8 8 for for all integer
all integer x x and and y y. Let. Let f f (x) = (x) = x x for all for all negative integnegative integersers x x, and let, and let g g (8) = 17. The value of (8) = 17. The value of f f (0) is (0) is ((AA) ) 117 7 ((BB) ) 9 9 ((CC) ) 225 5 ((DD) ) – – 1177
Q.68
Q.68 There There exists exists a pa positivositive ree real nal numbeumber x r x satisfysatisfying ing cos(tcos(tanan –1 –1x) = x) = x. The x. The value value of of coscos –1 –1
2 2 x x22 is is (A) (A) 10 10
(B) (B) 5 5
(C) (C) 5 5 2 2
(D) (D) 5 5 4 4
Q.69Q.69 The The domdomain ain of of the the funfunctionction, , f(x) f(x) ==
44xx22 44xx 33 3 3 x x 2 2 2 2 x x x x 5 5 .. 0 0 log log 5 5 .. 0 0 x x
is :is : (A) (A)
11
2 2 ,, (B) (B) [1, [1, 3]3] (C) (C) 11 2 2 11 3 3 2 2 , , ,,
(D)(D)
11
2 2 1 1 2 2 1 1 2 2 11 3 3 2 2 , , , , ,, Q. Q.70 70 cocoss –1 –1
5 5 2 2 sin sin 5 5 7 7 cos cos 2 2 1 1 is equal to is equal to (A) (A) 20 20 23 23
(B) (B) 20 20 13 13
(C) (C) 20 20 33 33
(D) (D) 20 20 17 17
Q.Q.71 71 LLeett f f (x) be a function with two properties (x) be a function with two properties (a
(a) ) for for any any two two reareal l nunumbember r x x and and yy, , f f (x (x + + y) y) = = x x ++ f f (y) and (y) and ((bb) ) f f ((00) ) = = 22..
The value of
The value of f f (100), is (100), is
((AA) ) 2 2 ((BB) ) 998 8 ((CC) ) 11002 2 ((DD) ) 110000 Q.
Q.72 72 LLeett f f be a funbe a function such ction such thatthat f f (3) = 1 and (3) = 1 and f f (3x) = x + (3x) = x + f f (3x – 3) for all (3x – 3) for all x. Then the value ofx. Then the value of f f (300) is (300) is ((AA) ) 5500550 0 ((BB) ) 4499550 0 ((CC) ) 5511551 1 ((DD) ) nnoonnee
Q.
Q.73 73 IIff f f (x) is an invertible function, and (x) is an invertible function, and g g (x) = 2 (x) = 2 f f (x) + 5, then the value of (x) + 5, then the value of g g –1 –1(x), is(x), is
(A) 2 (A) 2 f f –1 –1((xx) ) – – 5 5 ((BB)) 5 5 )) x x (( 2 2 1 1 1 1
f f (C)(C) 22 ((xx)) 55 1 1 11
f f (D)(D)
2 2 5 5 x x 1 1 f f Q.Q.74 74 IIff f f (2x + 1) = 4x (2x + 1) = 4x22 + 14x, then the sum of the roots of + 14x, then the sum of the roots of f f (x) = 0, is (x) = 0, is
((AA) ) 99//4 4 ((BB) ) 5 5 ((CC) ) – – 99//4 4 ((DD) ) – – 55 Q.7
Q.75 5 IIf f y y == f f (x) is a one-one function and (5, 1) is a point on its graph, which one of the following statements (x) is a one-one function and (5, 1) is a point on its graph, which one of the following statements is correct?
is correct?
(A) (1, 5) is a point on the graph of the inverse function y =
(A) (1, 5) is a point on the graph of the inverse function y = f f –1 –1(x)(x) (B)
(B) f f (5) = f (1) (5) = f (1)
(C) the graph of the inverse function y =
(C) the graph of the inverse function y = f f –1 –1(x) will be (x) will be symmetric about the y-axissymmetric about the y-axis
(D)
(D) f f
f f 11((55))
= 1= 1 Q.76Q.76 DomaiDomain n of of defindefinition ition of of the the functiofunction n f f (x) (x) ==
4 4 x x 3 3 x x 4 4 3 3 2 2 x x x x
is is (A) (– (A) (–
, , 00] ] ((BB) ) [[00,,
)) (C) (– (C) (–
, –1), –1)
[[00, , 44) ) ((DD) ) ((––
, 1), 1)
(1, 4) (1, 4) Q.77Q.77 SuSuppppososee f f and and g g are both are both linear functions, withlinear functions, with f f (x) = – 2x + 1 and (x) = – 2x + 1 and f f
g g ((xx))
= = x. x. The The sum sum of of the the slopeslope and the y-intercept ofand the y-intercept of g g , is, is
Q.78
Q.78 The The rangrange e of of the the functfunctionion f f (x) = (x) =
5 5 x x 3 3 4 4 x x
is is (A) (A)
3 3 1 1 ,, 0 0 (B)(B)
3 3 1 1 ,, 6 6 1 1 6 6 1 1 ,, 0 0 (C) (C) (–(–
, 0), 0)
(0, (0,
) ) (D(D) ) (0(0,,
)) Q.Q.79 79 IIff f f (x, y) = (x, y) =
max(max(xx,,yy))
min(min( xx,,yy)) andand g g (x, y) = max(x, y) – min(x, (x, y) = max(x, y) – min(x, y), theny), then
)) 75 75 .. 1 1 ,, 4 4 (( ,, 2 2 3 3 ,, 1 1 g g g g f f equals equals ((AA) ) – – 00..5 5 ((BB) ) 00..5 5 ((CC) ) 1 1 ((DD) ) 11..55 Q.80Q.80 The The domadomain ain and nd rangrange oe of tf the he functfunction ion f(x) f(x) = co= cosecsec –1 –1 loglog secsec sec sec 3 3 44 1 1 22 2 2 x x x x are respectively are respectively (A) R ; (A) R ;
2 2 2,, 2 (B) (B) R R + + ; ; 00 2 2 ,,
(C) (C)
2 2 ,, 0 0 }; }; n n 2 2 { { 2 2 n n 2 2 ,, 2 2 n n 2 2 (D)(D) {{00}} 2 2 ,, 2 2 }; }; n n 2 2 { { 2 2 n n 2 2 ,, 2 2 n n 2 2
More than one alternatives are correct. More than one alternatives are correct. Q.81
Q.81 The The valvalues ues of of x ix in [n [–2–2
, , 22
], for wh], for which the gich the graph of raph of the function the function y y == 11 1 1
sin sin sin sin x x xx – secx and – secx and y = – y = – 11 1 1
sin sin sin sin x x xx + + secx, coincide secx, coincide areare
(A) (A)
2 2 33 2 2 3 3 2 2 22
, ,
,,
(B)(B)
33
2 2 2 2 22 3 3 2 2
, , ,, (C) (C)
2 2 2,,2 (D) (D) [–2[–2
, , 22
] ] ––
2 2 3 3 2 2 ,, Q.Q.82 82 ssiinn-1-1(sin3) + sin(sin3) + sin-1-1 (sin4) + sin (sin4) + sin-1-1(sin5) when (sin5) when simplified simplified reduces toreduces to
((AA) ) aan n iirrrraattiioonnaal l nnuummbbeer r ((BB) ) a a rraattiioonnaal l nnuummbbeer r ((CC) ) aan n eevveen n pprriimme e ((DD) ) a a nneeggaattiivve e iinntteeggeer r Q.83
Q.83 The The grapgraphs ohs of wf which hich of of the the followfollowing ing pairs pairs diffediffer .r . (A)
(A) y y == sinsin
tan tan x x x x 1 1
22 ++ cos cos cot cot x x x x 1 1
22 ; ; y y = = sinsin 2x2x (B) y(B) y = = tantan x cotx cot x x ; ; y y = = sinsin x cosecx cosec xx (C) y =
(C) y =
cos xcos x
+ +
sin xsin x
; ; y y == sseec c ccooss sseec c ccoossx x eeccxx x x eeccxx
(D)(D) none none of of thesethese
Q.84 Q.84 If If f(f(x) x) = = cocoss 11 2 2 2 2
x + sinx + sin
1 1 2 2 2 2
xx , , [x] [x] denoting denoting the the greatest greatest integer integer function, function, thenthen((AA) ) f f ((00) ) = = 1 1 ((BB) ) ff
3 3
= = 11 3 3 1
1 (C) (C) ff
2 2
= = 0 0 ((DD) ) ff((
) = ) = 00Q.85
Q.85 ThThe e vavalue lue of coof coss 11
2 2 14 14 5 5 1 1 c coos s ccooss is : is : (A) cos (A) cos
77
5 5
(B) sin (B) sin
10 10
(C) (C) coscos 22 5 5
(D)(D)
cos cos 33 5 5
Q.86Q.86 The The functiofunctions ns which which are are aperioaperiodic dic are are ::
((AA) y ) y = = [[x x + + 11] ] ((BB) ) y y = = ssiin n xx22 (C) y (C) y = sin= sin22x x ((DD) ) y y = = ssiinn11xx where
where [x] denotes g[x] denotes greatest integer functionreatest integer function Q
Q..887 7 ttaann11 xx , , tatann11 y, tan y, tan11 z z are are in A.Pin A.P. . and and xx , y, y , z , z are are also also in A.Pin A.P. . (y(y
0 0 , , 11 ,,
1) then1) then ((AA) ) x , x , y , y , z z aarre e iin n GG..PP. . ((BB) ) x , x , y , y , z z aarre e iin n HH..PP..((CC) ) x x = = y y = = z z ((DD) ) ((xx
y) y)22 + (y+ (y
z) z)22 + (z+ (z
x) x)22 = = 00Q.88
Q.88 Which Which of thof the foe following llowing functifunction(s) on(s) is/are is/are periodperiodic wiic with pth perioderiod
. . (A) f(x) =(A) f(x) =
sinxsinx
(B) f(x) (B) f(x) = = [x [x ++
] ] ((C) C) ff((x) x) = = ccos(sos(sinx) inx) (D(D) ) f(f(x) x) = = cocoss22xx(where
(where [[ .. ] deno] denotes the tes the greatest greatest integer integer function)function) Q.89
Q.89 For For the the equatequation ion 2x 2x = = tan(2tan(2tantan –1 –1a) + 2tan(tana) + 2tan(tan –1 –1a + tana + tan –1 –1aa33), which of the following is invalid?), which of the following is invalid?
(A) a
(A) a22x x + + 22a a = = x x ((BB) ) aa22 + + 2a2ax x + + 1 1 = = 0 0 (C(C) ) aa
0 0 ((DD) ) aa
–1, 1 –1, 1 Q.90Q.90 Which Which of the of the functifunctions dons defined efined below below are are one-oone-one fune functionnction(s) ?(s) ? (A)
(A) f(xf(x) ) = = (x(x ++ 1) 1) , , ( ( xx
11) ) ((BB) ) gg((xx) ) = = x + (x + (11//xx) ) ( ( x x > > 00)) (C)(C) h(x) h(x) = = xx22 + 4x + 4x
55, , ((x x > > 00) ) ((DD) ) ff((xx) ) = = eexx, ( x, ( x
0) 0) Q.Q.91 91 IIf f cocoss –1 –1x + cosx + cos –1 –1y + cosy + cos –1 –1z =z =
, then, then (A) x(A) x22 + y + y22 + z + z22 + 2xyz = 1 + 2xyz = 1 (B) 2(sin
(B) 2(sin –1 –1x + sinx + sin –1 –1y + siny + sin –1 –1z) = cosz) = cos –1 –1x + cosx + cos –1 –1y + cosy + cos –1 –1zz
(C) xy + yz + zx = x + y + z – 1 (C) xy + yz + zx = x + y + z – 1 (D) (D) xx x x
11
+ + yy y y
1 1 + + zz zz
11
> > 66 Q.92Q.92 Which Which of the of the followfollowing ing homoghomogeneoeneous funus functionctions are os are of def degree gree zero zero ?? (A) (A) xx y y l l nn y y x x ++ y y x x l l nn x x y y (B) (B) x x x x yy y y x x yy ( ( )) ( ( ))
(C)(C) xy xy x x 2 2
yy22 (D) (D) x x sinsin y y xx
y cos y cos y y x xQ.93
Q.93 The The valvalue ue of of tantan –1 –1 xx x x sin sin cos cos
1 1
– – tantan –1 –1
xx
coscos
sin sin
is, is, forfor
00,,22
; x; x
R R ++ , is , is ((AA) ) iinnddeeppeennddeennt t oof f x x ((BB) ) iinnddeeppeennddeennt t ooff
(C) (C)
2
2 ––
(D) (D) none none of of thesetheseQ
Q..994 4 DD
[ [
1, 1] is the domain of the following functions, state which of them has the inverse.1, 1] is the domain of the following functions, state which of them has the inverse. (A)(A) f(x) = f(x) = xx22 (B) (B) g(x) = g(x) = xx33 (C(C) h) h(x(x) ) = = sisin n 22x x (D(D) k) k(x(x)= )= sisin n ((
x/2)x/2)Q.9
Q.95 5 Which Which of of the the following following function(function(s) s) have have no no domain?domain? (A) f(x) = log
(A) f(x) = logx – 1x – 1(2 – [x] – [x](2 – [x] – [x]22) ) where [x] denotes the where [x] denotes the greatest integer function.greatest integer function.
(B) g(x) = cos
(B) g(x) = cos –1 –1(2–{x}) where {x} denotes the fractional part (2–{x}) where {x} denotes the fractional part function.function. (C) h(x) = (C) h(x) = l l nn l l n(cosx)n(cosx) (D) f(x) = (D) f(x) =
1 1 sseecc-1-1 s g n es g n exxQ Q . . 1 1 B B Q Q . . 2 2 C C Q Q . . 3 3 A A Q Q . . 4 4 D D Q Q . . 5 5 B B Q Q . . 6 6 C C Q Q . . 7 7 A A Q Q . . 8 8 C C Q Q . . 9 9 D D Q Q . . 1 1 0 0 D D Q Q . . 1 1 1 1 A A Q Q . . 1 1 2 2 A A Q Q . . 1 1 3 3 A A Q Q . . 1 1 4 4 C C Q Q . . 1 1 5 5 B B Q Q . . 1 1 6 6 B B Q Q . . 1 1 7 7 C C Q Q . . 1 1 8 8 C C Q Q . . 1 1 9 9 A A Q Q . . 2 2 0 0 C C Q Q . . 2 2 1 1 C C Q Q . . 2 2 2 2 A A Q Q . . 2 2 3 3 A A Q Q . . 2 2 4 4 D D Q Q . . 2 2 5 5 D D Q Q . . 2 2 6 6 A A Q Q . . 2 2 7 7 A A Q Q . . 2 2 8 8 A A Q Q . . 2 2 9 9 B B Q Q . . 3 3 0 0 A A Q Q . . 3 3 1 1 D D Q Q . . 3 3 2 2 D D Q Q . . 3 3 3 3 C C Q Q . . 3 3 4 4 C C Q Q . . 3 3 5 5 B B Q Q . . 3 3 6 6 A A Q Q . . 3 3 7 7 B B Q Q . . 3 3 8 8 D D Q Q . . 3 3 9 9 A A Q Q . . 4 4 0 0 D D Q Q . . 4 4 1 1 B B Q Q . . 4 4 2 2 C C Q Q . . 4 4 3 3 A A Q Q . . 4 4 4 4 D D Q Q . . 4 4 5 5 C C Q Q . . 4 4 6 6 A A Q Q . . 4 4 7 7 B B , , C C Q Q . . 4 4 8 8 D D Q Q . . 4 4 9 9 D D Q Q . . 5 5 0 0 D D Q Q . . 5 5 1 1 C C Q Q . . 5 5 2 2 B B Q Q . . 5 5 3 3 A A Q Q . . 5 5 4 4 C C Q Q . . 5 5 5 5 D D Q Q . . 5 5 6 6 C C Q Q . . 5 5 7 7 C C Q Q . . 5 5 8 8 C C Q Q . . 5 5 9 9 B B Q Q . . 6 6 0 0 B B Q Q . . 6 6 1 1 A A Q Q . . 6 6 2 2 D D Q Q . . 6 6 3 3 C C Q Q . . 6 6 4 4 C C Q Q . . 6 6 5 5 A A Q Q . . 6 6 6 6 C C Q Q . . 6 6 7 7 A A Q Q . . 6 6 8 8 C C Q Q . . 6 6 9 9 D D Q Q . . 7 7 0 0 D D Q Q . . 7 7 1 1 C C Q Q . . 7 7 2 2 A A Q Q . . 7 7 3 3 D D Q Q . . 7 7 4 4 D D Q Q . . 7 7 5 5 A A Q Q . . 7 7 6 6 C C Q Q . . 7 7 7 7 C C Q Q . . 7 7 8 8 B B Q Q . . 7 7 9 9 D D Q Q . . 8 8 0 0 C C Q Q . . 8 8 1 1 A A , , C C Q Q . . 8 8 2 2 B B , , D D Q Q . . 8 8 3 3 A A , , B B , , C C Q Q . . 8 8 4 4 A A , , B B , , C C Q Q . . 8 8 5 5 B B , , C C , , D D Q Q . . 8 8 6 6 A A , , B B , , D D Q Q . . 8 8 7 7 A A , , B B , , C C , , D D Q Q . . 8 8 8 8 A A , , C C , , D D Q Q . . 8 8 9 9 B B , , C C Q Q . . 9 9 0 0 A A , , C C , , D D Q Q . . 9 9 1 1 A A , , B B Q Q . . 9 9 2 2 A A , , B B , , C C Q Q . . 9 9 3 3 A A , , C C Q Q . . 9 9 4 4 B B , , D D Q Q . . 9 9 5 5 A A , , B B , , C C , , D D
