1. Q.3 Thesetofrealvaluesof'x'satisfyingtheequality 





x
3
+ 





x
4
=5(where[ ]denotesthegreatestinteger
function)belongstotheinterval 





c
b
,
a where a, b,cNand
c
b
isinits lowest form. Findthevalueof
a + b + c + abc.

2. PART-C
SUBJECTIVE: [3 × 8 = 24]
Q.1 Let y = sin–1(sin 8) – tan–1(tan 10) + cos–1(cos 12) – sec–1(sec 9) + cot–1(cot 6) – cosec–1(cosec 7).
If y simplifies to a + b then find (a – b).

3. ROUGHWORK
PART-B
MATCH THE COLUMN [4 × 4 = 16]
INSTRUCTIONS:
Column-Iand column-IIcontainsfour entries each. Entries ofcolumn-Iareto bematchedwith some
entries of column-II. One or more than one entries of column-Imayhave the matching with the same
entries ofcolumn-IIandoneentryofcolumn-Imayhaveoneormorethan one matchingwith entries of
column-II.
Q.1 ColumnI ColumnII
(A) Constant function f (x) = c, c  R (P) Bound
(B) The function g (x) = 
x
1
t
dt
(x > 0), is (Q) periodic
(C) The function h(x) = arc tan x is (R) Monotonic
(D) The function k (x) = arccot x is (S) neither odd nor even
Q.2 ColumnI ColumnII
(A) cot–1
 
)
37
tan( 
 (P) 143°
(B) cos–1
 
)
233
cos( 
 (Q) 127°
(C) sin 














9
1
cos
2
1 1
(R)
4
3
(D) cos 













8
1
cos
arc
2
1
(S)
3
2

4. ROUGHWORK
Q.26 Acircleofradius320units istangenttotheinsideofacircleofradius 1000.Thesmallercircleis tangent
toadiameterofthelargercircleatthepointP.LeastdistanceofthepointPfrom thecircumferenceofthe
largercircleis
(A) 300 (B) 360 (C) 400 (D) 420
Select the correct alternative. (More than one are correct) [8 × 4 = 32]
Q.27 Inwhichofthefollowingcaseslimit exists attheindicatedpoints.
(A) f (x) =
x
]
|
x
|
x
[ 
at x = 0 where [x] denotes the greatest integer functions.
(B) f (x) =
x
1
x
1
e
1
e
x

at x = 0
(C) f (x) = (x – 3)1/5 Sgn(x – 3) at x = 3, where Sgn stands for Signum function.
(D) f (x) =
x
|
x
|
tan 1

at x = 0.
Q.28 LetAand B are two independent events. If P(A) = 0.3 and P(B) = 0.6, then
(A) P(A and B) = 0.18 (B) P(A) is equal to P(A/B)
(C) P(A or B) = 0 (D) P(A or B) = 0.72
Q.29 Let T be the triangle with vertices (0, 0), (0, c2) and (c, c2) and let R be the region between y= cx and
y = x2 where c > 0 then
(A)Area (R) =
6
c3
(B)Area of R =
3
c3
(C)
)
R
(
Area
)
T
(
Area
Lim
0
c 

= 3 (D)
)
R
(
Area
)
T
(
Area
Lim
0
c 

=
2
3

5. ROUGHWORK
Q.14 Number of real solution of equation 16 sin–1x tan–1x cosec–1x = 3 is/are
(A) 0 (B) 1 (C) 2 (D)infinite
Q.15 Length of the perpendicular from the centre of the ellipse 27x2 + 9y2 = 243 on a tangent drawn to it
whichmakes equal intercepts on thecoordinates axes is
(A)
2
3
(B)
2
3
(C) 2
3 (D) 6
Q.16 Let f (x) = cos–1










2
2
x
1
x
1
+ tan–1 





 2
x
1
x
2
where x  (–1, 0) then f simplifies to
(A) 0 (B)
4

(C)
2

(D) 
Q.17 Apersonthrowsfourstandardsix sideddistinguishable dice. Number ofways in which hecan throwif
the product of the four number shownon the upper faces is 144, is
(A) 24 (B) 36 (C) 42 (D) 48
Q.18 LetA=








z
y
x
r
q
p
c
b
a
and suppose that det.(A)= 2 then the det.(B) equals, where B=











r
c
2
z
4
q
b
2
y
4
p
a
2
x
4
(A) det(B) = – 2 (B) det(B) = – 8 (C) det(B) = – 16 (D) det(B) = 8
Q.19 The digit at the unit place of the number (2003)2003 is
(A) 1 (B) 3 (C) 7 (D) 9
Q.20 LetABCDEFGHIJKL be a regular dodecagon, then the value of
AF
AB
+
AB
AF
is
(A) 4 (B) 3
2 (C) 2
2 (D) 2

6. ROUGHWORK
PART-A
Select the correct alternative. (Only one is correct) [26 × 3 = 78]
Q.1 Number of zeros of the cubic f (x) = x3 + 2x + k  k  R, is
(A) 0 (B) 1 (C) 2 (D) 3
Q.2 The value of  



x
3
3
x
dr
)
1
r
)(
1
r
(
r
dx
d
Lim , is
(A) 0 (B) 1 (C)
2
1
(D)non existent
Q.3 There are two numbers x makingthe value of the determinant
x
2
4
0
1
x
2
5
2
1


equal to 86. The sum of
these twonumbers, is
(A) – 4 (B) 5 (C) – 3 (D) 9
Q.4 A function f(x)takesadomainDontoarangeR ifforeach yR,thereissomexDforwhichf (x) = y.
Numberoffunctionthat canbedefinedfrom thedomainD={1,2,3} onto therangeR = {4, 5} is
(A) 5 (B) 6 (C) 7 (D) 8
Q.5 Suppose f , f ' and f '' are continuous on [0, e] and that f ' (e) = f (e) = f (1) = 1 and 
e
1
2
dx
x
)
x
(
f
=
2
1
,
then the value of dx
x
n
)
x
(
'
'
e
1
l
f
 equals
(A)
e
1
2
5
 (B)
e
1
2
3
 (C)
e
1
2
1
 (D)
e
1
1
Q.6 Acircle with centre C (1, 1) passes through the origin and intersect thex-axis atA and y-axis at B.The
area of thepart of the circlethat lies in thefirst quadrant is
(A)  + 2 (B) 2 – 1 (C) 2 – 2 (D)  + 1