MATHEMATICS

PART-A
Select the correct alternative : (Only one is correct)
Q.1 Let f be a real valued function suchthat
f (x) + 





x
2002
2 f = 3x
for all x > 0. Find f (2).
(A) 1000 (B) 2000 (C) 3000 (D) 4000
Q.2 PointAliesontheliney=2x andthesumofitsabscissaandordinateis12.PointBlies onthex-axisand
the lineAB is perpendicular to the line y= 2x. Let 'O' be the origin.The areaof thetriangleAOB is
(A) 20 (B) 40 (C) 60 (D) 80
Q.3 Minimumperiodofthefunction
f (x) = | sin32x | + | cos32x | is
(A)  (B)
2

(C)
4

(D)
4
3
Q.4 The number k is such that  
)
k
20
tan(
arc
)
2
tan(
arc
tan  =k. The sum ofall possible values of k is
(A) –
40
19
(B) –
40
21
(C) 0 (D)
5
1
Q.5 Let A = the number of ways of selecting a committee of 5 from a group of 9 persons.
B = the number of permutations of the word PRELEPCO taken all ata time.
C=thenumberofwaysinwhich8peoplecanbearrangedinalineiftwoparticularpeoplemust
stand next to each other.
The value of
C
AB
equals
(A) 63 (B) 126 (C) 360 (D) none
Q.6 Whichoneof thefollowingdepictsthegraph ofanodd function?
(A) (B)
(C) (D)

Q.7 If 349(x + iy) =
100
3
2
3









2
i
and x = ky then the value of k equals (x, y  R)
(A) 3 (B) – 3 (C)
3
1
(D) –
3
1
Q.8 If
2
1
z
z
is purelyimaginarythen
2
1
2
1
z
z
z
z


is equal to :
(A) 1 (B) 2 (C) 3 (D) 0
Q.9 If sin =
13
12
, cos = –
13
5
, 0 <  < 2. Consider the following statements.
I.  = cos–1 






13
5
II.  = sin–1 





13
12
III.  =  – sin–1 





13
12
IV.  = tan–1 






5
12
V.  =  – tan–1 





5
12
thenwhichofthefollowingstatementsaretrue?
(A) I, IIand IV only (B) IIIand V only
(C) I and III only (D) I, IIIand V only
Q.10 Domain ofdefinitionofthefunction f(x) =log 1
9
3
·
10 1
x
2
x

 
 + )
x
1
(
cos 1

 is
(A) [0, 1] (B) [1, 2] (C) (0, 2) (D) (0, 1)
Q.11 Two pointsA(x1, y1) and B(x2, y2) are chosen on the graph of f (x) = ln x with 0 < x1 < x2.The points
C andDtrisect linesegmentABwithAC <CB.ThroughC ahorizontal lineis drawn to cutthecurveat
E(x3, y3). If x1 = 1 and x2 = 1000 then the value of x3 equals
(A) 10 (B) 10 (C) (10)2/3 (D) (10)1/3
Q.12 The period of the function f(x) = sin 2x + sin
x
3





 + sin
x
5





 is
(A) 2 (B) 6 (C) 15 (D) 30
Q.13 TriangleABC has BC = 1 andAC = 2.The maximum possible value of theangleAis
(A)
6

(B)
4

(C)
3

(D)
2

Q.14 The sum
n 


1
tan1 






 1
n
n
3
2 is equal to
(A) 2
cot
4
3 1



(B) 3
cot
2
1



(C)  (D) 2
tan
2
1




Q.15 If f (x) = 2x3 + 7x – 5 then f–1(4) is
(A) equal to 1 (B) equal to 2 (C) equal to 1/3 (D)nonexistent
Q.16 Let z be a complex number, then the region represented by the inequality z + 2< z + 4is
given by
(A) Re (z) >  3 (B) Im(z) < 3
(C) Re (z) <  3 & Im (z) >  3 (D) Re (z) <  4 & Im (z) >  4
Q.17 Asquare OABC is formed byline pairs xy= 0 and xy + 1 = x + ywhere 'O' is the origin.Acircle with
centre C1 inside thesquare is drawn to touch the line pair xy= 0 and another circle with centre C2 and
radius twice that of C1, is drawn to touch the circle C1 and the other line pair.The radius of the circle
with centre C1 is
(A)
 
1
2
3
2

(B)
 
1
2
3
2
2

(C)
 
1
2
3
2

(D)
2
3
1
2 
Q.18 Given f (x) =
x
1
8
x
1
8



and g (x) = )
x
(cos
f
4
)
x
(sin
f
4
 then g(x) is
(A) periodic with period /2 (B) periodicwith period 
(C) periodic with period 2 (D) aperiodic
Q.19 LetABC be a triangle right angled at C. The value of a
log
·
a
log
a
log
a
log
b
c
c
b
b
c
c
b



 
(b + c  1, c– b  1) equals
(A) 1 (B) 2 (C) 3 (D) 1/2
Q.20 Thesumoftheinfinitetermsoftheseries
cot 1 1
3
4
2






 + cot 1 2
3
4
2






 + cot 1 3
3
4
2






 + ..... is equal to :
(A) tan–1 (1) (B) tan–1 (2) (C) tan–1 (3) (D) tan–1 (4)
Q.21 The period of thefunction f (x) =
|
x
cos
x
sin
|
|
x
cos
|
|
x
sin
|


is
(A) /2 (B) /4 (C)  (D) 2
Q.22 Number ofeven numbers between 1000and 9999 havingalldistinct digits, is
(A) 1848 (B) 2296 (C) 2298 (D) 2300
Q.23 Suppose, f (x, n) = 







n
1
k
x
x
k
log , then the value of x satisfying the equation f (x, 10) = f (x, 11), is
(A) 9 (B) 10 (C) 11 (D) none

Q.24 If  be a complex cube root of unity, then the value of
1
1 2
1
1
1
2





  
is :
(A) 0 (B) 1 (C)  1 (D) none
Q.25 The value of tan1 1
2
2
tan A





 +tan 1(cotA)+tan1(cot3A) for 0 <A< (/4) is
(A) 4 tan1 (1) (B) 2 tan1 (2) (C) 0 (D) none
Q.26 The maximum value of function f (x) = – (ex/2 + e–x/2),is
(A) 0 (B) – 1 (C) – 2 (D) – e
Q.27 Let f(x)=lnx& g(x)=
x x x x
x x
4 3 2
2
2 3 2 2
2 2 1
   
 
.Thedomainofthecompositefunctionfog(x) is: (A)
(, ) (B) [0, ) (C) (0, ) (D) [1, )
Q.28 In a ABC, a  b  c. If
C
sin
B
sin
A
sin
c
b
a
3
3
3
3
3
3




= 27, then the maximum value of 'a'
(A) 3/2 (B) 2 (C) 3 (D) cannot be determined
Q.29 The value of sec sin sin cos cos
 






  












1 1
50
9
31
9
 
is equal to
(A) sec
10
9

(B) sec

9
(C) 1 (D) –1
Q.30 Let f beafunctionsatisfying f (xy)=
y
)
x
(
f
forall positive real numbersx and y. If f (30)=20,then the
value of f (40) is
(A) 15 (B) 20 (C) 40 (D) 60
Q.31 If thex intercept of the liney= mx +2 is greaterthan 1/2 then the gradient ofthe linelies in the interval
(A) (–1, 0) (B) (–1/4, 0) (C) (– , – 4) (D) (– 4, 0)
Q.32 If  be a complex cube root of unity, then the number;
(1   2)3 + ( 1  2)3 + (2   1)3 is :
(A) divisible by3 but not by8 (B) divisible by8 but not by3
(C) divisible byboth 3 & 8 (D) none of these
Q.33 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 respectivelythen
(A) T1 = 2T2 (B) 2T1 = T2 (C) T1 = T2 (D) T1 = 4T2
Q.34 Given i = 1
 and (x + yi)2 = 45 + ai, where x and yare positive integers, then the smallest possible
valueforawhichsatisfiestheinequalities?
(A) 10 < a < 20 (B) 20 < a < 30 (C) 30 < a < 40 (D) 40 < a < 50

Q.35 If x = tan1 1  cos1 






1
2
+ sin1
1
2
; y = cos
1
2
1
8
1
cos 










 then :
(A) x = y (B) y = x (C) tan x =  (4/3) y (D) tanx = (4/3)y
Q.36 The periodof the function f(x)=sin cos
x
2





 +cos(sinx) equal
(A)

2
(B) 2 (C)  (D) 4
Q.37 Let the function f : D  R, f (x)=  
)
1
x
2
(
log
log
log 8
3
1
5  where Dis themaximumdomain of f(x).
If S represents the sum ofthe absolute values ofall integers from D.Then the value ofS, is
(A) 15 (B) 10 (C) 6 (D) 3
Q.38 A function f (x) = x
2
1 + x is defined from D1  D2 and is onto. If the set D1 is its complete
domain then the set D2 is
(A) 







2
1
, (B) (– , 2) (C) (– , 1) (D) (– , 1]
Q.39 A triangle hassides of length 13, 30 and 37. The radius of the inscribed circle is
(A) 7 + 2 (B)
2
9
(C) 7 – 2 (D)
2
7
Q.40 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.41 If Pistheaffix ofzintheArganddiagram&Pmovesso that
z i
z

 1
isalwayspurelyimaginary, then the
locus of z is :
(A) circle centre
1
2
1
2
,





 , radius
1
2
(B) circle centre  






1
2
1
2
, , radius
1
2
(C) circle centre (2,2) and radius 1/2 (D) none of these
Q.42 Let f (x) =
}
x
sgn
e
{ |
x
|
e and g (x) =
]
x
sgn
e
[ |
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 forall real x, h (x) is
(A) anodd function (B)anevenfunction
(C) neitheran odd nor aneven function (D) both odd as well as even function

Q.43 Number ofways in which4married couples can beseated round a table such that men andwomen are
alternate and not all women adjacent to her husband, is
(A) 144 (B) 132 (C) 120 (D) 36
Q.44 Let f(x) = (x + 1)² 1, (x 
 
1) . Then the set S = {x: f(x) = f1
(x)} is
(A) 0 1
3 3
2
3 3
2
, , ,

   










i i
(B) {0, 1, 1}
(C) {0, 1} (D)empty
Q.45 If | x – 2 | = p where x < 2 then (x – p) equals
(A) – 2 (B) 2 (C) 2 – 2p (C) 2p – 2
Q.46 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.47 Range of the function f (x) = tan–1
[ ] [ ] | |
x x x
x
    
2
1
2
is
where[*] is thegreatest integerfunction.
(A)
1
4
,
L
N
M I
K
J (B)
1
4
2
R
S
T
U
V
W
 
, g (C)
1
4
2
,
R
S
T
U
V
W (D)
1
4
2
,
L
N
M O
Q
P
Q.48 The vertices ofa triangleA, B and C have the coordinates as (b, c), (–a, 0) and (a, 0) respectively.The
coordinates of thepoint of concurrencyof its perpendicularbisectors are
(A) 






 

c
2
a
c
b
,
0
2
2
2
(B) 






 

a
2
b
a
c
,
0
2
2
2
(C) 






 

b
2
c
b
a
,
0
2
2
2
(D) none
Q.49 The sum i + 2i2 + 3i3 + ........ + 2002i2002, where i = 1
 is equal to
(A) – 999 + 1002i (B) – 1002 + 999i (C) – 1001 + 1000i (D) – 1002 + 1001i
Q.50 What percent of the domain ofthe function f (x) =
4
2
|
5
x
2
|
9
x
9



consists ofpositive numbers.
(A) 40% (B) 50% (C) 30% (D) 65%
Q.51 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.52 Write the correct order sequence in respect of followingfourstatements, where 'T' stands for true and
'F' stands for False.
I If a and b be positive real number different from 1, then logba + logab  2.
II If
89
log19
n = (89)2 then n = 361.
III Two different real numbers y and z are the roots of the quadratic equation ax2 + c = 0 with
a, c  0 then y3 + z3 = 0.
IV The quadratic equation x | x | + px + q = 0 where p, q  R, q  0 has at least one real root.
(A) F T T F (B) F T T T (C) T F T T (D) T T F F
Q.53 Let f (x) = 


irrational
is
x
if
x
rational
is
x
if
0
and g (x) = 


rational
is
x
if
x
irrational
is
x
if
0
Thenthefunction(f– g)x is
(A) odd (B)even
(C) neither odd nor even (D) odd as well as even
Q.54 Let f(x)=sin [ ]
a x (where [ ]denotesthegreatestintegerfunction) . If fis periodicwith fundamental
period , then a belongs to :
(A) [2, 3) (B) {4, 5} (C) [4, 5] (D) [4, 5)
Q.55 If x =
2
2
10 
and y=
2
2
10 
then the value of log2(x2 + xy + y2), is
(A) 0 (B) 2 (C) 3 (D) 4
Q.56 For x  R, the function f (x) satisfies 2 f (x) + f (1 – x) = x2 then the value of f (4) is equal to
(A)
3
13
(B)
3
43
(C)
3
23
(D) none
Q.57 Number of solution(s) of the equation cos–1(1 – x) – 2cos–1x =
2

is
(A) 3 (B) 2 (C) 1 (D) 0
Q.58 Let , 2 be complex cube roots of unity.Then the determinant
x
x
x



1
1
1
2
2
2
 
 
 
reduces to:
(A) x3 (B) 2x3 (C) 3 x3 (D) none
Q.59 The function f(x) = cot1
( )
x x
 3 + cos1
x x
2
3 1
  is defined on the set S, where S =
(A) {0, 3} (B) (0, 3) (C) {0,  3} (D) [ 3, 0]

Q.60 The domainof thefunction f(x) =
 
arc x
x x
cot
2 2

, where [x] denotesthe greatest integernotgreater than
x,is:
(A) R (B) R  {0}
(C) R   
  

n n I
: { }
0 (D) R  {n : n  I}
Q.61 The sum of the infinite series
3
2
log
+
9
4
log
+
27
8
log
+
81
16
log
+ ...... , is
(A) log 8 (B)
2
2
log
(C)
4
2
log
3
(D)
2
2
log
5
Q.62 There exists a positive real number x satisfying cos(tan–1x) = x. Thevalue of cos–1








2
x2
is
(A)
10

(B)
5

(C)
5
2
(D)
5
4
Q.63 The range of thefunction, f(x) = cot–1
 
log .
0 5
4 2
2 3
x x
  is:
(A) (0, ) (B) 0
3
4
,






 (C)
3
4


,





 (D)
 
2
3
4
,






Q.64 Number ofways in which 5 boys and 5 girls can be seated alternativelyon a round table if a particular
boys and a particular girl are neveradjacent to each other in anyarrangement, is
(A) 288 (B) 552 (C) 1584 (D) 1728
Q.65 If f (x + ay, x  ay) = axy then f (x, y) is equal to :
(A)
x y
2 2
4

(B)
x y
2 2
4

(C) 4 xy (D) none
Q.66 Let[x]denotethegreatestintegerinx.Thenintheinterval[0,3]thenumberofsolutionsoftheequation,
x2  3x + [x] = 0 is :
(A) 6 (B) 4 (C) 2 (D) 0
Q.67 Let z z
c
1 2
2
  then z z z z
1 2
2
1 2
2
   =
(A) c2 (B) c2/2 (C) 2 c2 (D) none
Q.68 Number of value of x satisfyingthe equation sin–1 





x
5
+ sin–1 





x
12
=
2

is
(A) 0 (B) 1 (C) 2 (D) more than 2

Q.69 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)
correspondingtothegivenvalueoftheindependent 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 Ba straight
lineparallel totheY-axis is drawnto intersect thegraphofthefunction
y = g(x)at C.Again astraight lineis drawnfrom thepoint C parallel to
the X-axis, to intersect the line NN at D . If the straight line NN  is
parallel toY-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.70 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.71 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.72 The domainof the function, f(x) =   3
x
4
2
x
4
3
x
2
2
x
x
5
.
0
log
5
.
0
x 




 is :
(A)  






1
2
, (B) [1, 3]
(C)
1
2
1
3
2
, ,





  





 (D) 





 





  






1
2
1
2
1
2
1
3
2
, , ,
Q.73 Let k bethe valueso thatthe functiondefined byf(x)=
k
x
5
x


will beits own inverse, thenthevalueof
tan k, to the nearest tenths, is
(A) – 1.6 (B) – 0.9 (C) 1.2 (D) 2.4
Q.74 The function f (x) is defined byf(x) = cos4x + K cos22x + sin4x, where K is a constant.If the function
f (x) is a constant function, thevalueof k is
(A) – 1 (B) – 1/2 (C) 0 (D) 1/2 (E) 1
Q.75 If w is an imaginarycube root of unity, then the value of  
sin w w
10 23
4
 








is :
(A) 
3
2
(B) 
1
2
(C)
1
2
(D)
3
2

Q.76 Theset ofvalues of x, satisfyingtheequation tan2(sin–1x) >1 is
(A) [–1, 1] (B) 






2
2
2
2
,
(C) (–1, 1) – 






2
2
2
2
, (D) [–1, 1] – 






2
2
2
2
,
Q.77 Definethefunctionf(n)wherenisanonnegativeintegersatisfyingf(0)=1andf(n)isdefinedforn > 0
as f (n) = n · 


1
n
0
i
)
(
f i . Let 2m be the highest power of 2 that divides f (20). The value of m is
(A) 18 (B) 19 (C) 20 (D) 21 (E) 22
Q.78 Let f (x) =
x
1
x

and let g(x)=
x
1
x
r

. Let S be the set of all real numbers r such that
f (g(x))= g(f (x))for infinitelymanyreal number x. The numberof elements in setS is
(A) 1 (B) 2 (C) 3 (D) 5
Q.79 If for all x different from both 1 and 0 we have f1(x) =
1
x
x

, f2(x) =
x
1
1

, and for all integers n  1,
we have fn + 2(x) =



 
  even
is
n
if
)
x
(
f
f
odd
is
n
if
)
x
(
f
f
2
1
n
1
1
n


then f4(x) equals
(A)x (B) x – 1 (C) f1(x) (D) f2(x)
Q.80 If | x | – x + y = 10 and x + | y | + y = 12 then x + y is equal to
(A)
5
26
(B) 0 (C) –
5
26
(D)
5
42
Q.81 ..
..........
5
3
5
3
Lim
n 

equals
(A) (15)2/3 (B) (15)1/3 (C) (45)1/2 (D) (45)1/3
Q.82 For n = 1, 2, 3, ......, let Tn = 1 + 2 + ....... + n. Which of the following statements is correct
(A) There is no value of n for whichTn is a positive power of 2. (A positive power of two is an integer
of the form 2k where k is apositive integer)
(B) There is exactlyone value of n for which Tn is a positive powerof 2.
(C) There are exactlytwovalues of n for which Tn is a positive power of 2.
(D) There are morethan two but finitelymanyvalues ofn for which Tn is a positive powerof 2.

Q.83 If x2 + bx + a = 0 has roots 'm' and 'n' (m > n) then the roots of the quadratic equation
(x – 2)2 + b(x – 2) + a = 0 are
(A) m – 2 and n – 2 (B) m + 2 and n + 2
(C) 2 – m and 2 – n (D) m and n
Q.84 If (a + ib)5 =  + i where a, b, ,   R then (b + ia)5 equals
(A) a + i) (B)  + i (C)  – i (D)  – i
Select the correct alternative : (More than one are correct)
Q.85 If the function f (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.86 Whichofthefollowinghomogeneous functions areofdegreezero ?
(A)
x
y
ln
y
x
+
y
x
ln
x
y
(B)
x x y
y x y
( )
( )


(C)
xy
x y
2 2

(D) x sin
y
x
 y cos
y
x
Q.87 The equation of the line making equal intercepts on the axes & touching z
2 1
2

where z = x + iy is :
(A) x + y = 1 (B) x  y = 1 (C) x  y = 1 (D) x  y = 1
Q.88 The value of sin
1
2
3
4
1
cot












 + cos
1
2
3
4
1
cot












 is equal to
(A) 1 (B)
3 2
10
(C) 2 sin
1
2
3
4
1
1 1
cot cot ( )
 






 





 (D) 2 sin   






 
tan ( ) tan
1 1
1
1
2
4
3
Q.89 Let f(x) =
1
x
& g(x) =
1
x
then :
(A) f(g(x))&g(f(x))havedifferent domains (B) f(g(x)) & g(f(x)) havethe same range
(C) f(g(x))is a bijectivemapping (D) g(f(x)) is neither odd nor even .
Q.90 If f(x)is a polynomial functionsatisfyingthecondition f(x) .f(1/x)= f(x) +f(1/x)and f(2) =9 then
(A) 2 f(4) = 3 f(6) (B) 14 f(1) = f(3) (C) 9 f(3) = 2 f(5) (D) f(10) = f(11)
Q.91 Suppose that thefunctions f (x)andg (x) satisfythesystem of equation
f (x) + 3g(x) = x2 + x + 6
2 f (x) + 4 g (x) = 2x2 + 4 for every x.
The value of 'x' for which f (x) = g (x) can be equal to
(A) – 2 (B) 2 (C) – 5 (D) 5

Q.92 tan1x , tan1y, tan1z 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.93 D  [1, 1] is the domain ofthefollowingfunctions, statewhich 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.94 Which ofthefollowingexpressionsarenot thetrigonometricforms ofanycomplexnumber?
(A)  3 cos sin
 
4 4







i (B) 2 cos sin
 
4 4







i
(C) 2 sin cos
2
3
2
3
 







i (D) cos

3
+ i sin
2
3

Q.95 Identifythepair(s)offunctions whichare identical .
(A) y = tan (cos 1 x); y =
1 2
 x
x
(B) y = tan (cot 1 x) ; y =
1
x
(C) y = sin (arc tan x); y =
x
x
1 2

(D) y = cos (arc tan x) ; y = sin (arc cot x)
Q.96 For the equation 2x = tan(2tan–1a) + 2tan(tan–1a + tan–1a3), which of the followingis invalid?
(A) a2x + 2a = x (B) a2 + 2ax + 1 = 0 (C) a  0 (D) a  –1, 1
Q.97 Suppose f (x) = sin x and g (x)= 1 – x . Which of the following composite functions have the same
range?
(A)fog (B)gof (C)fof (D) gog
Q.98 Thegraphs ofwhichofthefollowingpairs differ.
(A) y =
sin
tan
x
x
1 2

+
cos
cot
x
x
1 2

; y = sin 2x
(B) y = tanx cot x ; y = sin x cosecx
(C) y = cos x + sin x ; y =
sec cos
sec cos
x ecx
x ecx

(D) none of these
Q.99 Thefunctions whichareaperiodicare:
(A) y = [x + 1] (B) y= sin x2 (C) y = sin2 x (D) y = sin1 x
where [x] denotes greatest integerfunction

Q.100 The value of cos
1
2
14
5
1
cos cos





















is :
(A) cos 






7
5

(B) sin

10





 (C) cos
2
5






 (D)  cos
3
5







Q.101 Let x,ybereals satisfying sinx+cos y=1 and sin y+ cos x =–1. Then which ofthefollowingmust
be correct?
(A) sin(x + y) = 0 (B) cos(x – y) = 0 (C) cos(x + y) = 1 (D) cos 2x = cos 2y
Q.102 Whichofthefollowingfunction(s)is/areperiodicwith period.
(A) f(x) = sinx (B) f(x) = [x + ] (C) f(x) = cos(sinx) (D) f(x) = cos2x
(where [.] denotes the greatest integer function)
PART-B
SUBJECTIVE
Q.103 Solve thesystemofequations
log10(x3) + log10(y2) = 11
log10(x2) – log10(y3) = 3.
Q.104 If f (x) = x2 – x + 2; g (x) = ax + b and  
)
x
(
g
f = 9x2 – 3x + 2, determine all possible ordered pairs
(a,b)whichsatisfythisrelationship.
Q.105 A flight of stairs has 10 steps. A person can go up the steps one at a time, two at a time, or any
combination of1's and 2's. Find the total number ofways in which the person can go up the stairs.
Q.106 TriangleABC has incentre at point I.Also, one median is lineCD, and theinscribed circle is tangent to
sideABatpoint D.If thelengthofCD=3andlength ofAB=8.Findthedistanceofthevertex andfrom
theincentre.
Q.107 Find all triplets of natural numbers (a, b, c) such that a, b and c are in geometric progression and
a + b + c = 111.
Q.108 Letxbethesumofthefollowinginfiniteseries:
x = 3
1
1
+ 3
3
1
+ 3
5
1
+ 3
7
1
+ 
Evaluatethesum
3
1
1
+ 3
2
1
+ 3
3
1
+ 3
4
1
+ 
intermsofx.
Q.109 Findthegeneralsolutionoftheequation
 2
x
sin )
2
(
log = logsin x4 sin3x

Q.110 Find thesolution set ofinequality, 2
x
3
  3.
Q.111 Given that tan 







x
4
=
2
1
then find the value of the expression y=
x
2
cos
1
x
cos
x
2
sin 2


.
Q.112 Let p, q, r be the roots of the equation x3 – 14x2 + 26x – 4 = 0. Find the value of the expression

























r
1
1
q
1
1
p
1
1 .
Q.113 If sin x =
b
a
b
a


where 0 < a < b then find the value of x
cos
x
cot 2
2
 .
Q.114 The lengths of the sides of a triangle are log1012, log1075 and log10n,where n  N. Find the number of
possible values of n.
Q.115 Let P bea point on the line 3x + 4y= 12 which is closest to the origin. Find the sum ofthe abscissa and
ordinate ofthe point P.
Q.116 Find the equationof a line passingthrough(– 4, –2)having equal intercept on the coordinate axes.
Q.117 Let f (x) = x
x
x
x
a
a
a
a




, where a > 0 and a  1. If f (p) = 2, then find f (2p).
Q.118 Find the sum of the roots of the equation 1
2
2
2 2
x
222
1
x
111
2
x
333


 

 .
Q.119 Given tan  · tan  =
a
b
, show that the expression F =


 2
2
cos
b
sin
a
1
+


 2
2
cos
b
sin
a
1
is
independent of  and  and has the value equal to
ab
b
a 
.
Q.120 Let  and  be two different values of x satisfying the equation a cos x + b sin x = c with a2 + b2  0,
prove that 4 cos2
2

cos2
2

= 2
2
2
b
a
)
c
a
(


.

ANSWER KEY
Q.1 B Q.2 D Q.3 C Q.4 A Q.5 B Q.6 D Q.7 D
Q.8 A Q.9 D Q.10 C Q.11 A Q.12 D Q.13 A Q.14 A
Q.15 A Q.16 A Q.17 C Q.18 A Q.19 B Q.20 B Q.21 C
Q.22 B Q.23 C Q.24 A Q.25 A Q.26 C Q.27 A Q.28 C
Q.29 D Q.30 A Q.31 D Q.32 C Q.33 C Q.34 B Q.35 C
Q.36 D Q.37 C Q.38 D Q.39 B Q.40 B Q.41 A Q.42 A
Q.43 B Q.44 C Q.45 C Q.46 D Q.47 C Q.48 A Q.49 D
Q.50 A Q.51 A Q.52 B Q.53 A Q.54 D Q.55 C Q.56 C
Q.57 C Q.58 A Q.59 C Q.60 C Q.61 C Q.62 C Q.63 C
Q.64 D Q.65 A Q.66 C Q.67 A Q.68 B Q.69 C Q.70 D
Q.71 C Q.72 D Q.73 A Q.74 B Q.75 C Q.76 C Q.77 C
Q.78 B Q.79 C Q.80 A Q.81 D Q.82 A Q.83 B Q.84 B
Q.85 A,B,C Q.86 A,B,C Q.87 A,D Q.88 B,C,D
Q.89 B,C,D Q.90 B,C Q.91 A,D Q.92 A,B,C,D
Q.93 B,D Q.94 A,B,C,D Q.95 A,B,C,D Q.96 B,C
Q.97 B,D Q.98 A,B,C Q.99 A,B,D Q.100 B,C,D
Q.101 A,D Q.102 A,C,D
Q.103 x = 1000, y = 10 Q.104 (3, 0) and (– 3, 1) Q.105 89
Q.106 5/3 Q.107 (37, 37, 37); (1, 10, 100); (100, 10, 1)
Q.108 8x/7 Q.109 x = n + (–1)n
6

Q.110 x  (– , – 3]  






,
5
3
Q.111 – 5/6 Q.112 –
4
9
Q.113 2
2
a
b
ab
4

Q.114 893 Q.115
25
84
Q.116 2y = x and x + y + 6 = 0 Q.117 5/4 Q.118
111
2