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Page 1
PAGE # 1
PART : MATHEMATICS
SECTION – 1
Straight Objective Type (lh/ks oLrqfu"B izdkj)
This section contains 22 multiple choice questions. Each question has 4 choices (1), (2), (3) and (4) for its
answer, out of which Only One is correct.
bl [k.M esa 20 cgq&fodYih iz'u gSaA izR;sd iz'u ds 4 fodYi (1), (2), (3) rFkk (4) gSa] ftuesa ls flQZ ,d lgh gSA
1. Let
? ?
?
?
3
2
6 3
x sin 1 x sin
dx x cos
= f(x) ? ? c x sin 1
1
6
? ?
?
then find value of ?
?
?
?
?
? ?
?
3
f
ekuk
? ?
?
?
3
2
6 3
x sin 1 x sin
dx x cos
= f(x) ? ? c x sin 1
1
6
? ?
?
, rks ?
?
?
?
?
? ?
?
3
f dk eku gS&
(1) 4 (2) –2 (3)8 (4) –4
Ans. (2)
Sol. sin x = t
cos x dx = dx
I =
? ?
?
?
3
2
6 3
t 1 t
dt
=
?
?
?
?
?
?
?
?
3
2
6
7
t
1
1 t
dt
Put
3
6
r
t
1
1 ? ? j[kus ij ?? dr r
2
1
t
dt
2
7
?
?
?
? ?
2
1
–
r
dr r
2
1
2
2
r + c = c
x sin
1 x sin
2
1
3
1
6
6
?
?
?
?
?
?
?
?
?
?
? = ? ? c x sin 1
x sin 2
1
3
1
6
2
? ? ?
f(x) = x ec cos
2
1
2
? and vkSj 3 ? ?
?
?
?
?
?
? ?
?
3
f = – 2
2. If y(x) is a solution of differential equation 0 y 1
dx
dy
x 1
2 2
? ? ? ? , such that
2
3
2
1
y ? ?
?
?
?
?
?
, then
;fn y(x), vody lehdj.k 0 y 1
dx
dy
x 1
2 2
? ? ? ? dk gy gS rFkk
2
3
2
1
y ? ?
?
?
?
?
?
, rc
(1)
2
1
2
1
y ? ?
?
?
?
?
?
?
?
?
(2)
2
3
2
1
y ?
?
?
?
?
?
?
?
?
(3)
2
1
2
1
y ?
?
?
?
?
?
?
?
?
(4)
2
1
2
1
y ? ?
?
?
?
?
?
Ans. (3)
Page 2
PAGE # 1
PART : MATHEMATICS
SECTION – 1
Straight Objective Type (lh/ks oLrqfu"B izdkj)
This section contains 22 multiple choice questions. Each question has 4 choices (1), (2), (3) and (4) for its
answer, out of which Only One is correct.
bl [k.M esa 20 cgq&fodYih iz'u gSaA izR;sd iz'u ds 4 fodYi (1), (2), (3) rFkk (4) gSa] ftuesa ls flQZ ,d lgh gSA
1. Let
? ?
?
?
3
2
6 3
x sin 1 x sin
dx x cos
= f(x) ? ? c x sin 1
1
6
? ?
?
then find value of ?
?
?
?
?
? ?
?
3
f
ekuk
? ?
?
?
3
2
6 3
x sin 1 x sin
dx x cos
= f(x) ? ? c x sin 1
1
6
? ?
?
, rks ?
?
?
?
?
? ?
?
3
f dk eku gS&
(1) 4 (2) –2 (3)8 (4) –4
Ans. (2)
Sol. sin x = t
cos x dx = dx
I =
? ?
?
?
3
2
6 3
t 1 t
dt
=
?
?
?
?
?
?
?
?
3
2
6
7
t
1
1 t
dt
Put
3
6
r
t
1
1 ? ? j[kus ij ?? dr r
2
1
t
dt
2
7
?
?
?
? ?
2
1
–
r
dr r
2
1
2
2
r + c = c
x sin
1 x sin
2
1
3
1
6
6
?
?
?
?
?
?
?
?
?
?
? = ? ? c x sin 1
x sin 2
1
3
1
6
2
? ? ?
f(x) = x ec cos
2
1
2
? and vkSj 3 ? ?
?
?
?
?
?
? ?
?
3
f = – 2
2. If y(x) is a solution of differential equation 0 y 1
dx
dy
x 1
2 2
? ? ? ? , such that
2
3
2
1
y ? ?
?
?
?
?
?
, then
;fn y(x), vody lehdj.k 0 y 1
dx
dy
x 1
2 2
? ? ? ? dk gy gS rFkk
2
3
2
1
y ? ?
?
?
?
?
?
, rc
(1)
2
1
2
1
y ? ?
?
?
?
?
?
?
?
?
(2)
2
3
2
1
y ?
?
?
?
?
?
?
?
?
(3)
2
1
2
1
y ?
?
?
?
?
?
?
?
?
(4)
2
1
2
1
y ? ?
?
?
?
?
?
Ans. (3)
PAGE # 2
Sol. 0
x 1
dx
y 1
dy
2 2
?
?
?
?
? sin
–1
y + sin
–1
x = c
At x =
2
1
, y =
2
3
? c =
2
?
? sin
–1
y = cos
–1
x
Hence vr% y
?
?
?
?
?
?
?
?
2
1
= sin
2
1
2
1
cos
1
?
?
?
?
?
?
?
?
?
?
3.
2
x
1
2
2
0 x
2 x 7
2 x 3
lim
?
?
?
?
?
?
?
?
?
?
?
is equal to
2
x
1
2
2
0 x
2 x 7
2 x 3
lim
?
?
?
?
?
?
?
?
?
?
?
dk eku gS&
(1) e
–2
(2) e
2
(3) e
2/7
(4) e
3/7
Ans. (1)
Sol. Let L =
2
x
1
2
2
0 x
2 x 7
2 x 3
lim
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
1
2 x 7
2 x 3
x
1
lim
2
2
2
0 x
e =
?
?
?
?
?
?
?
?
?
?
?
?
?
2 x 7
x 4
x
1
lim
2
2
2
0 x
e =
2
2
4
e e
?
?
?
4. In a bag there are 5 red balls, 3 white balls and 4 black balls. Four balls are drawn from the bag. Find
the number of ways of in which at most 3 red balls are selected
,d Fksys esa 5 yky xsansa] 3 lQsn xsansa] vkSj 4 dkyh xsansa gS Fksys ls 4 xsansa fudkyh tkrh gS vf/kd ls vf/kd 3 yky xsansa
pqus tkus ds Øep;ksa dh la[;k gSA
(1) 450 (2) 360 (3) 490 (4) 510
Ans. (3)
Sol. 0 Red, 1Red, 2 Red, 3 Red
Number of ways Øep;ksa dh la[;k =
4
7
C +
3
7
1
5
C . C +
2
7
2
5
C . C +
1
7
3
5
C . C = 35 + 175 + 210 + 70 = 490
5. Let f(x) = {(sin (tan
–1
x) + sin (cot
–1
x)}
2
–1 where |x| > 1 and ? ? ) x ( f sin
dx
d
2
1
dx
dy
1 ?
? .
If ? ?
6
3 y
?
? then ? ? 3 y ? =
Ekkuk f(x) = {(sin (tan
–1
x) + sin (cot
–1
x)}
2
–1 tgk¡ |x| > 1 rFkk ? ? ) x ( f sin
dx
d
2
1
dx
dy
1 ?
? .
;fn ? ?
6
3 y
?
? rc ? ? 3 y ? =
(1)
6
5 ?
(2)
6
? ?
(3)
3
?
(4)
3
2 ?
Ans. (2)
Page 3
PAGE # 1
PART : MATHEMATICS
SECTION – 1
Straight Objective Type (lh/ks oLrqfu"B izdkj)
This section contains 22 multiple choice questions. Each question has 4 choices (1), (2), (3) and (4) for its
answer, out of which Only One is correct.
bl [k.M esa 20 cgq&fodYih iz'u gSaA izR;sd iz'u ds 4 fodYi (1), (2), (3) rFkk (4) gSa] ftuesa ls flQZ ,d lgh gSA
1. Let
? ?
?
?
3
2
6 3
x sin 1 x sin
dx x cos
= f(x) ? ? c x sin 1
1
6
? ?
?
then find value of ?
?
?
?
?
? ?
?
3
f
ekuk
? ?
?
?
3
2
6 3
x sin 1 x sin
dx x cos
= f(x) ? ? c x sin 1
1
6
? ?
?
, rks ?
?
?
?
?
? ?
?
3
f dk eku gS&
(1) 4 (2) –2 (3)8 (4) –4
Ans. (2)
Sol. sin x = t
cos x dx = dx
I =
? ?
?
?
3
2
6 3
t 1 t
dt
=
?
?
?
?
?
?
?
?
3
2
6
7
t
1
1 t
dt
Put
3
6
r
t
1
1 ? ? j[kus ij ?? dr r
2
1
t
dt
2
7
?
?
?
? ?
2
1
–
r
dr r
2
1
2
2
r + c = c
x sin
1 x sin
2
1
3
1
6
6
?
?
?
?
?
?
?
?
?
?
? = ? ? c x sin 1
x sin 2
1
3
1
6
2
? ? ?
f(x) = x ec cos
2
1
2
? and vkSj 3 ? ?
?
?
?
?
?
? ?
?
3
f = – 2
2. If y(x) is a solution of differential equation 0 y 1
dx
dy
x 1
2 2
? ? ? ? , such that
2
3
2
1
y ? ?
?
?
?
?
?
, then
;fn y(x), vody lehdj.k 0 y 1
dx
dy
x 1
2 2
? ? ? ? dk gy gS rFkk
2
3
2
1
y ? ?
?
?
?
?
?
, rc
(1)
2
1
2
1
y ? ?
?
?
?
?
?
?
?
?
(2)
2
3
2
1
y ?
?
?
?
?
?
?
?
?
(3)
2
1
2
1
y ?
?
?
?
?
?
?
?
?
(4)
2
1
2
1
y ? ?
?
?
?
?
?
Ans. (3)
PAGE # 2
Sol. 0
x 1
dx
y 1
dy
2 2
?
?
?
?
? sin
–1
y + sin
–1
x = c
At x =
2
1
, y =
2
3
? c =
2
?
? sin
–1
y = cos
–1
x
Hence vr% y
?
?
?
?
?
?
?
?
2
1
= sin
2
1
2
1
cos
1
?
?
?
?
?
?
?
?
?
?
3.
2
x
1
2
2
0 x
2 x 7
2 x 3
lim
?
?
?
?
?
?
?
?
?
?
?
is equal to
2
x
1
2
2
0 x
2 x 7
2 x 3
lim
?
?
?
?
?
?
?
?
?
?
?
dk eku gS&
(1) e
–2
(2) e
2
(3) e
2/7
(4) e
3/7
Ans. (1)
Sol. Let L =
2
x
1
2
2
0 x
2 x 7
2 x 3
lim
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
1
2 x 7
2 x 3
x
1
lim
2
2
2
0 x
e =
?
?
?
?
?
?
?
?
?
?
?
?
?
2 x 7
x 4
x
1
lim
2
2
2
0 x
e =
2
2
4
e e
?
?
?
4. In a bag there are 5 red balls, 3 white balls and 4 black balls. Four balls are drawn from the bag. Find
the number of ways of in which at most 3 red balls are selected
,d Fksys esa 5 yky xsansa] 3 lQsn xsansa] vkSj 4 dkyh xsansa gS Fksys ls 4 xsansa fudkyh tkrh gS vf/kd ls vf/kd 3 yky xsansa
pqus tkus ds Øep;ksa dh la[;k gSA
(1) 450 (2) 360 (3) 490 (4) 510
Ans. (3)
Sol. 0 Red, 1Red, 2 Red, 3 Red
Number of ways Øep;ksa dh la[;k =
4
7
C +
3
7
1
5
C . C +
2
7
2
5
C . C +
1
7
3
5
C . C = 35 + 175 + 210 + 70 = 490
5. Let f(x) = {(sin (tan
–1
x) + sin (cot
–1
x)}
2
–1 where |x| > 1 and ? ? ) x ( f sin
dx
d
2
1
dx
dy
1 ?
? .
If ? ?
6
3 y
?
? then ? ? 3 y ? =
Ekkuk f(x) = {(sin (tan
–1
x) + sin (cot
–1
x)}
2
–1 tgk¡ |x| > 1 rFkk ? ? ) x ( f sin
dx
d
2
1
dx
dy
1 ?
? .
;fn ? ?
6
3 y
?
? rc ? ? 3 y ? =
(1)
6
5 ?
(2)
6
? ?
(3)
3
?
(4)
3
2 ?
Ans. (2)
PAGE # 3
Sol. 2y = sin
–1
f(x) + C = sin
–1
(sin(2tan
–1
x)) + C ? C
3
2
sin sin
6
2
1
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
? ?
?
?
?
?
? ?
?
?
?
?
?
3 3
C ? C = 0
for 3 – x ? , 2y = sin
–1
?
?
?
?
?
?
?
?
?
?
?
?
?
? ? ?
6
2
sin + 0 ?? ? 2y =
3
? ?
?
?
?
?
?
? ? ?
?
6
y
6. If 2
1 –x
+ 2
1+x
, f(x), 3
x
+ 3
–x
are in A.P. then minimum value of f(x) is
;fn 2
1 –x
+ 2
1+x
, f(x), 3
x
+ 3
–x
lekUrj Js.kh esa gS rc f(x) dk U;wure eku gS&
(1) 1 (2) 2 (3) 3 (4) 4
Ans. (3)
Sol. f(x) =
?
?
?
?
?
?
?
?
? ? ?
? ? ?
2
3 3 2 2
x x x 1 x 1
Using AM ? GM
f(x) ? 3
7. Which of the following is tautology
fuEu esa ls dkSulh iqu%#fDr gS \
(1) (p ? (p ? q)) ? q (2) q ? p ? (p ? q)
(3) p v (p ? q) (4) (p ? (p ? q)
Ans. (1)
Sol.
p q p ? q p ? (p ? q) (p ?(p ?q)) ? q q ? p ?(p ?q) p ? q p ? (p ?q ) p ? q p ? (p ? q)
T T T T T T T T T T
T F F F T T F T T T
F T T F T F F F T F
F F T F T T F F F F
8. A is a 3 × 3 matrix whose elements are from the set { –1, 0, 1}. Find the number of matrices A such that
tr(AA
T
) = 3. Where tr(A) is sum of diagonal elements of matrix A.
A ,d 3 × 3 Øe dk vkO;wg gS] ftlds vo;o leqPp; { –1, 0, 1} ls fy, x;s gSA rc vkO;wg A dh la[;k Kkr djks
tks bl izdkj gS fd tr(AA
T
) = 3 tgk¡ tr(A) vkO;wg A ds fod.kZ ds vo;oksa dk ;ksx gSA
(1) 572 (2) 612 (3) 672 (4) 682
Ans. (3)
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