2. C/Page 2 SPACE FOR ROUGH WORK / ÚUȤ ·¤æØü ·ð¤ çÜ° Á»ã
PART A — MATHEMATICS Öæ» A — »ç‡æÌ
1. A complex number z is said to be
unimodular if ?z?51. Suppose z1 and z2
are complex numbers such that
1 2
1 2
2
2
z z
z z
2
2
is unimodular and z2 is not unimodular.
Then the point z1 lies on a :
(1) circle of radius 2 .
(2) straight line parallel to x-axis.
(3) straight line parallel to y-axis.
(4) circle of radius 2.
2. The normal to the curve, x212xy23y250,
at (1, 1) :
(1) meets the curve again in the fourth
quadrant.
(2) does not meet the curve again.
(3) meets the curve again in the second
quadrant.
(4) meets the curve again in the third
quadrant.
3. The sum of first 9 terms of the series
3 3 3 3 33
1 2 1 2 31
....
1 1 3 1 3 5
1 1 1
1 1 1
1 1 1
is :
(1) 192
(2) 71
(3) 96
(4) 142
1. °·¤ âçןæ â´Øæ z °·¤×æÂæ´·¤è ·¤ãÜæÌè ãñ ØçÎ
?z?51 ãñÐ ×æÙæ z1 ÌÍæ z2 °ðâè âçןæ â´Øæ°¡ ãñ´
ç·¤ 1 2
1 2
2
2
z z
z z
2
2
°·¤×æÂæ´·¤è ãñ ÌÍæ z2 °·¤×æÂæ´·¤è
Ùãè´ ãñ, Ìæð çÕ´Îé z1 çSÍÌ ãñ Ñ
(1) 2 ç˜æ’Øæ ßæÜð ßëžæ ÂÚUÐ
(2) x-¥ÿæ ·ð¤ â×æ´ÌÚU °·¤ ÚðU¹æ ÂÚUÐ
(3) y-¥ÿæ ·ð¤ â×æ´ÌÚU °·¤ ÚðU¹æ ÂÚUÐ
(4) 2 ç˜æ’Øæ ßæÜð ßëžæ ÂÚUÐ
2. ß·ý¤ x212xy23y250 ·ð¤ çÕ´Îé (1, 1) ÂÚU
¥çÖÜÕ Ñ
(1) ß·ý¤ ·¤æð ÎæðÕæÚUæ ¿ÌéÍü ¿ÌéÍæZàæ ×ð´ ç×ÜÌæ ãñÐ
(2) ß·ý¤ ·¤æð ÎæððÕæÚUæ Ùãè´ ç×ÜÌæÐ
(3) ß·ý¤ ·¤æð ÎæðÕæÚUæ çmÌèØ ¿ÌéÍæZàæ ×ð´ ç×ÜÌæ ãñÐ
(4) ß·ý¤ ·¤æð ÎæðÕæÚUæ ÌëÌèØ ¿ÌéÍæZàæ ×ð´ ç×ÜÌæ ãñÐ
3. Ÿæð‡æè
3 3 3 3 33
1 2 1 2 31
....
1 1 3 1 3 5
1 1 1
1 1 1
1 1 1
·ð¤
Âý‰æ× 9 ÂÎæð´ ·¤æ Øæð» ãñ Ñ
(1) 192
(2) 71
(3) 96
(4) 142
3. SPACE FOR ROUGH WORK / ÚUȤ ·¤æØü ·ð¤ çÜ° Á»ãC/Page 3
4. Let f (x) be a polynomial of degree four
having extreme values at x51 and x52.
If
20
( )
1 3
x
f x
lim
x→
1 5 , then f (2) is equal
to :
(1) 4
(2) 28
(3) 24
(4) 0
5. The negation of ~ s Ú (~ r Ù s ) is equivalent
to :
(1) s Ù r
(2) s Ù ~ r
(3) s Ù (r Ù ~ s)
(4) s Ú (r Ú ~ s)
6. If
1 2 2
A 2 1 2
a 2 b
5 2 is a matrix satisfying
the equation AAT59I, where I is 333
identity matrix, then the ordered pair
(a, b) is equal to :
(1) (22, 21)
(2) (2, 21)
(3) (22, 1)
(4) (2, 1)
4. ×æÙæ f (x) ƒææÌ 4 ·¤æ °·¤ ÕãéÂÎ ãñ çÁâ·ð¤
x51 ÌÍæ x52 ÂÚU ¿ÚU× ×æÙ ãñ´Ð ØçÎ
20
( )
1 3
x
f x
lim
x→
1 5 ãñ, Ìæð f (2) ÕÚUæÕÚU ãñ Ñ
(1) 4
(2) 28
(3) 24
(4) 0
5. ~ s Ú (~ r Ù s ) ·¤æ çÙáðÏ â×ÌéËØ ãñ Ñ
(1) s Ù r
(2) s Ù ~ r
(3) s Ù (r Ù ~ s)
(4) s Ú (r Ú ~ s)
6. ØçÎ
1 2 2
A 2 1 2
a 2 b
5 2 °·¤ °ðâæ ¥æÃØêã ãñ Áæð
¥æÃØêã â×è·¤ÚU‡æ AAT59I, ·¤æð â´ÌécÅU ·¤ÚUÌæ ãñ,
Áãæ¡ I, 333 ·¤æ ̈â×·¤ ¥æÃØêã ãñ, Ìæð ·ý¤ç×Ì Øé‚×
(a, b) ·¤æ ×æÙ ãñ Ñ
(1) (22, 21)
(2) (2, 21)
(3) (22, 1)
(4) (2, 1)
4. C/Page 4 SPACE FOR ROUGH WORK / ÚUȤ ·¤æØü ·ð¤ çÜ° Á»ã
7. The integral 3
42 4
d
( 1)
x
x x
∫ 1
equals :
(1)
1
44
4
1
c
x
x
1
2 1
(2)
1
44
4
1
c
x
x
1
1
(3)
1
44
( 1) cx 1 1
(4)
1
44
( 1) cx2 1 1
8. If m is the A.M. of two distinct real
numbers l and n (l, n > 1) and G1, G2 and
G3 are three geometric means between l
and n, then 4 4 4
1 2 3G 2G G1 1 equals.
(1) 4 l2m2n2
(2) 4 l2mn
(3) 4 lm2n
(4) 4 lmn2
9. Let y(x) be the solution of the differential
equation
d
( log ) 2 log , ( 1).
d
y
x x y x x x
x
1 5 /
Then y(e) is equal to :
(1) 2e
(2) e
(3) 0
(4) 2
7. â×æ·¤Ü 3
42 4
d
( 1)
x
x x
∫ 1
ÕÚUæÕÚU ãñ Ñ
(1)
1
44
4
1
c
x
x
1
2 1
(2)
1
44
4
1
c
x
x
1
1
(3)
1
44
( 1) cx 1 1
(4)
1
44
( 1) cx2 1 1
8. ØçÎ Îæð çßçÖ‹Ù ßæSÌçß·¤ â´Øæ¥æð´ l ÌÍæ n
(l, n > 1) ·¤æ â×æ´ÌÚU ×æŠØ (A.M.) m ãñ ¥æñÚU l ÌÍæ
n ·ð¤ Õè¿ ÌèÙ »é‡ææðžæÚU ×æŠØ (G.M.) G1, G2 ÌÍæ
G3 ãñ´, Ìæð 4 4 4
1 2 3G 2G G1 1 ÕÚUæÕÚU ãñ Ñ
(1) 4 l2m2n2
(2) 4 l2mn
(3) 4 lm2n
(4) 4 lmn2
9. ×æÙæ ¥ß·¤Ü â×è·¤ÚU‡æ
d
( log ) 2 log , ( 1)
d
y
x x y x x x
x
1 5 /
·¤æ ãÜ y(x) ãñ, Ìæð y(e) ÕÚUæÕÚU ãñ Ñ
(1) 2e
(2) e
(3) 0
(4) 2
6. C/Page 6 SPACE FOR ROUGH WORK / ÚUȤ ·¤æØü ·ð¤ çÜ° Á»ã
13. Let
1 1 1
2
2
tan tan tan ,
1
x
y x
x
2 2 2
5 1
2
where
1
<
3
x? ? . Then a value of y is :
(1)
3
2
3
1 3
x x
x
1
1
(2)
3
2
3
1 3
x x
x
2
2
(3)
3
2
3
1 3
x x
x
1
2
(4)
3
2
3
1 3
x x
x
2
1
14. The distance of the point (1, 0, 2) from the
point of intersection of the line
12 2
3 4 12
yx z12 2
5 5 and the plane
x2y1z516, is :
(1) 13
(2) 2 14
(3) 8
(4) 3 21
13. ×æÙæ
1 1 1
2
2
tan tan tan ,
1
x
y x
x
2 2 2
5 1
2
Áãæ¡
1
<
3
x? ? ãñ, Ìæð y ·¤æ °·¤ ×æÙ ãñ Ñ
(1)
3
2
3
1 3
x x
x
1
1
(2)
3
2
3
1 3
x x
x
2
2
(3)
3
2
3
1 3
x x
x
1
2
(4)
3
2
3
1 3
x x
x
2
1
14. ÚðU¹æ
12 2
3 4 12
yx z12 2
5 5 ÌÍæ â×ÌÜ
x2y1z516 ·ð¤ ÂýçÌ‘ÀðUÎ çÕ´Îé ·¤è, çÕ´Îé (1, 0, 2)
âð ÎêÚUè ãñ Ñ
(1) 13
(2) 2 14
(3) 8
(4) 3 21
9. SPACE FOR ROUGH WORK / ÚUȤ ·¤æØü ·ð¤ çÜ° Á»ãC/Page 9
21. The set of all values of l for which the
system of linear equations :
2x122x21x35lx1
2x123x212x35lx2
2x112x2 5lx3
has a non-trivial solution,
(1) contains more than two elements.
(2) is an empty set.
(3) is a singleton.
(4) contains two elements.
22. If 12 identical balls are to be placed in 3
identical boxes, then the probability that
one of the boxes contains exactly 3 balls
is :
(1)
11
1
22
3
(2)
11
55 2
3 3
(3)
10
2
55
3
(4)
12
1
220
3
23. The sum of coefficients of integral powers
of x in the binomial expansion of
( )
50
1 2 x2 is :
(1) ( )501
2 1
2
1
(2) ( )501
3 1
2
1
(3) ( )501
3
2
(4) ( )501
3 1
2
2
21. l ·ð¤ âÖè ×æÙæð´ ·¤æ â×é‘¿Ø, çÁÙ·ð¤ çÜ° ÚñUç¹·¤
â×è·¤ÚU‡æ çÙ·¤æØ
2x122x21x35lx1
2x123x212x35lx2
2x112x2 5lx3
·¤æ °·¤ ¥Ìé‘ÀU ãÜ ãñ,
(1) ×ð´ Îæð âð ¥çÏ·¤ ¥ßØß ãñ´Ð
(2) °·¤ çÚU€Ì â×é‘¿Ø ãñÐ
(3) °·¤ °·¤Ü â×é‘¿Ø ãñÐ
(4) ×ð´ Îæð ¥ßØß ãñ´Ð
22. ØçÎ 12 °·¤ Áñâè »ð´Îð´, 3 °·¤ Áñâð Õ€âæð´ ×ð´ ÚU¹è ÁæÌè
ãñ´, Ìæð §Ù×ð´ âð °·¤ Õ€âð ×ð´ ÆUè·¤ 3 »ð´Îð´ ãæðÙð ·¤è
ÂýæçØ·¤Ìæ ãñ Ñ
(1)
11
1
22
3
(2)
11
55 2
3 3
(3)
10
2
55
3
(4)
12
1
220
3
23. ( )
50
1 2 x2 ·ð¤ çmÂÎ ÂýâæÚU ×ð´ x ·¤è Âê‡ææZ·¤èØ
ƒææÌæ𴠷𤠻é‡ææ´·¤æð´ ·¤æ Øæð» ãñ Ñ
(1) ( )501
2 1
2
1
(2) ( )501
3 1
2
1
(3) ( )501
3
2
(4) ( )501
3 1
2
2
10. C/Page 10 SPACE FOR ROUGH WORK / ÚUȤ ·¤æØü ·ð¤ çÜ° Á»ã
24. The integral
4 2
2 2
2
log
d
log log (36 12 )
x
x
x x x
∫ 1 2 1
is equal to :
(1) 6
(2) 2
(3) 4
(4) 1
25. If the function.
1 , 0 3
g( )
m 2 , 3 < 5
k x x
x
x x
1 [ [
5
1 [
is differentiable, then the value of k1m is :
(1) 4
(2) 2
(3)
16
5
(4)
10
3
26. Locus of the image of the point (2, 3) in
the line (2x23y14)1k (x22y13)50,
k e R, is a :
(1) circle of radius 3 .
(2) straight line parallel to x-axis.
(3) straight line parallel to y-axis.
(4) circle of radius 2 .
27. ( )( )
0
1 cos 2 3 cos
tan 4x
x xlim
x x→
2 1
is equal to :
(1)
1
2
(2) 4
(3) 3
(4) 2
24. â×æ·¤Ü
4 2
2 2
2
log
d
log log (36 12 )
x
x
x x x
∫ 1 2 1
ÕÚUæÕÚU ãñ Ñ
(1) 6
(2) 2
(3) 4
(4) 1
25. ØçΠȤÜÙ
1 , 0 3
g( )
m 2 , 3 < 5
k x x
x
x x
1 [ [
5
1 [
¥ß·¤ÜÙèØ ãñ, Ìæð k1m ·¤æ ×æÙ ãñ Ñ
(1) 4
(2) 2
(3)
16
5
(4)
10
3
26. çÕ´Îé (2, 3) ·ð¤ ÚðU¹æ
(2x23y14)1k (x22y13)50, k e R ×ð´
ÂýçÌçÕ´Õ ·¤æ çÕ´ÎéÂÍ °·¤ Ñ
(1) 3 ç˜æ’Øæ ·¤æ ßëžæ ãñÐ
(2) x-¥ÿæ ·ð¤ â×æ´ÌÚU ÚðU¹æ ãñÐ
(3) y-¥ÿæ ·ð¤ â×æ´ÌÚU ÚðU¹æ ãñÐ
(4) 2 ç˜æ’Øæ ·¤æ ßëžæ ãñÐ
27. ( )( )
0
1 cos 2 3 cos
tan 4x
x xlim
x x→
2 1
ÕÚUæÕÚU ãñ Ñ
(1)
1
2
(2) 4
(3) 3
(4) 2
15. SPACE FOR ROUGH WORK / ÚUȤ ·¤æØü ·ð¤ çÜ° Á»ãC/Page 15
37.
Two long current carrying thin wires, both
with current I, are held by insulating
threads of length L and are in equilibrium
as shown in the figure, with threads
making an angle ‘u’ with the vertical. If
wires have mass l per unit length then the
value of I is :
(g5gravitational acceleration)
(1)
0
gL
tan
pl
u
m
(2)
0
gL
sin
cos
pl
u
m u
(3)
0
gL
2sin
cos
pl
u
m u
(4)
0
gL
2 tan
p
u
m
38. A particle of mass m moving in the
x direction with speed 2v is hit by another
particle of mass 2m moving in the
y direction with speed v. If the collision is
perfectly inelastic, the percentage loss in
the energy during the collision is close to :
(1) 62%
(2) 44%
(3) 50%
(4) 56%
37.
Îæð ÂÌÜð ÜÕð ÌæÚUæð´ ×ð´ ÂýˆØð·¤ âð I ÏæÚUæ ÂýßæçãÌ ãæð ÚUãè
ãñÐ §‹ãð´ L ÜÕæ§ü ·ð¤ çßléÌÚUæðÏè Ïæ»æð´ âð ÜÅU·¤æØæ
»Øæ ãñÐ §Ù Ïæ»æð´ ×ð´ ÂýˆØð·¤ ·ð¤ mæÚUæ ª¤ŠßæüÏÚU çÎàææ âð
‘u’ ·¤æð‡æ ÕÙæÙð ·¤è çSÍçÌ ×ð´, Øð ÎæðÙæð´ ÌæÚU âæØæßSÍæ
×ð´ ÚUãÌð ãñ´Ð ØçÎ §Ù ÌæÚUæð´ ·¤è ÂýçÌ §·¤æ§ü ÜÕæ§ü
ÎýÃØ×æÙ l ãñ ÌÍæ g »éL¤ˆßèØ ˆßÚU‡æ ãñ Ìæð, I ·¤æ ×æÙ
ãæð»æ Ñ
(1)
0
gL
tan
pl
u
m
(2)
0
gL
sin
cos
pl
u
m u
(3)
0
gL
2sin
cos
pl
u
m u
(4)
0
gL
2 tan
p
u
m
38. x-çÎàææ ×ð´ 2v ¿æÜ âð ¿ÜÌð ãé° m ÎýÃØ×æÙ ·ð¤ °·¤
·¤‡æ âð, y-çÎàææ ×ð´ v ßð» âð ¿ÜÌæ ãé¥æ 2m ÎýÃØ×æÙ
·¤æ °·¤ ·¤‡æ, ÅU·¤ÚUæÌæ ãñÐ ØçÎ Øã â´ƒæÅ÷UÅU (ÅU€·¤ÚU)
Âê‡æüÌÑ ¥ÂýˆØæSÍ ãñ Ìæð, ÅU€·¤ÚU ·ð¤ ÎæñÚUæÙ ª¤Áæü ·¤æ ÿæØ
(ãæçÙ) ãæð»è Ñ
(1) 62%
(2) 44%
(3) 50%
(4) 56%
16. C/Page 16 SPACE FOR ROUGH WORK / ÚUȤ ·¤æØü ·ð¤ çÜ° Á»ã
39.
Given in the figure are two blocks A and B
of weight 20 N and 100 N, respectively.
These are being pressed against a wall by
a force F as shown. If the coefficient of
friction between the blocks is 0.1 and
between block B and the wall is 0.15, the
frictional force applied by the wall on block
B is :
(1) 150 N
(2) 100 N
(3) 80 N
(4) 120 N
40. Consider an ideal gas confined in an
isolated closed chamber. As the gas
undergoes an adiabatic expansion, the
average time of collision between
molecules increases as V
q
, where V is the
volume of the gas. The value of q is :
p
v
C
C
g 5
(1)
1
2
g 2
(2)
3 5
6
g 1
(3)
3 5
6
g 2
(4) 1
2
g 1
39.
Øãæ¡ ¥æÚðU¹ ×ð´ Îæð ŽÜæò·¤ (»éÅU·ð¤) A ¥æñÚU B ÎàææüØð »Øð
ãñ´ çÁÙ·ð¤ ÖæÚU ·ý¤×àæÑ 20 N ÌÍæ 100 N ãñ´Ð §‹ãð´,
°·¤ ÕÜ F mæÚUæ ç·¤âè ÎèßæÚU ÂÚU ÎÕæØæ Áæ ÚUãæ ãñÐ
ØçÎ ƒæáü‡æ »é‡ææ´·¤ ·¤æ ×æÙ, A ÌÍæ B ·ð¤ Õè¿ 0.1
ÌÍæ B ¥æñÚU ÎèßæÚU ·ð¤ Õè¿ 0.15 ãñ Ìæð, ÎèßæÚU mæÚUæ
ŽÜæò·¤ B ÂÚU Ü»æ ÕÜ ãæð»æ Ñ
(1) 150 N
(2) 100 N
(3) 80 N
(4) 120 N
40. °·¤ ¥æÎàæü »ñâ ç·¤âè Õ‹Î (â´ßëÌ), çßØé€Ì
(çßÜç»Ì) ·¤ÿæ ×ð´ âèç×Ì (ÚU¹è) ãñÐ §â »ñâ ×´ð´
L¤Î÷Ïæðc× ÂýâæÚU ãæðÙð ÂÚU, §â·ð¤ ¥‡æé¥æð´ ·ð¤ Õè¿ ÅU€·¤ÚU
·¤æ ¥æñâÌ ·¤æÜ (â×Ø) V
q
·ð¤ ¥ÙéâæÚU Õɸ ÁæÌæ ãñ,
Áãæ¡ V »ñâ ·¤æ ¥æØÌÙ ãñÐ Ìæð q ·¤æ ×æÙ ãæð»æ :
p
v
C
C
g 5
(1)
1
2
g 2
(2)
3 5
6
g 1
(3)
3 5
6
g 2
(4) 1
2
g 1
17. SPACE FOR ROUGH WORK / ÚUȤ ·¤æØü ·ð¤ çÜ° Á»ãC/Page 17
41. A rectangular loop of sides 10 cm and
5 cm carrying a current I of 12 A is placed
in different orientations as shown in the
figures below :
(a)
(b)
(c)
(d)
If there is a uniform magnetic field of
0.3 T in the positive z direction, in which
orientations the loop would be in (i) stable
equilibrium and (ii) unstable equilibrium ?
(1) (b) and (c), respectively
(2) (a) and (b), respectively
(3) (a) and (c), respectively
(4) (b) and (d), respectively
41. 10 cm ÌÍæ 5 cm ÖéÁæ¥æ𴠷𤠰·¤ ¥æØÌæ·¤æÚU ÜêÂ
(Âæàæ) âð °·¤ çßléÌ ÏæÚUæ, I 5 12 A, ÂýßæçãÌ ãæðU
ÚUãè ãñÐ §â Âæàæ ·¤æð ¥æÚðU¹ ×ð´ ÎàææüØð »Øð ¥ÙéâæÚU
çßçÖóæ ¥çÖçß‹Øæâæð´ (çSÍçÌØæð´) ×ð´ ÚU¹æ »Øæ ãñÐ
(a)
(b)
(c)
(d)
ØçÎ ßãæ¡ 0.3 T ÌèßýÌæ ·¤æ ·¤æð§ü °·¤â×æÙ ¿éÕ·¤èØ
ÿæð˜æ, ÏÙæˆ×·¤ z çÎàææ ×ð´ çßl×æÙ ãñ Ìæð, ÎàææüØð »Øð
緤⠥çÖçß‹Øæâ ×ð´, Øã Âæàæ (ÜêÂ) (i) SÍæØè
â´ÌéÜÙ ÌÍæ (ii) ¥SÍæØè â´ÌéÜÙ ×ð´, ãæð»æ?
(1) ·ý¤×àæÑ (b) ÌÍæ (c) ×ð´
(2) ·ý¤×àæÑ (a) ÌÍæ (b) ×ð´
(3) ·ý¤×àæÑ (a) ÌÍæ (c) ×ð´
(4) ·ý¤×àæÑ (b) ÌÍæ (d) ×ð´
19. SPACE FOR ROUGH WORK / ÚUȤ ·¤æØü ·ð¤ çÜ° Á»ãC/Page 19
44. On a hot summer night, the refractive
index of air is smallest near the ground and
increases with height from the ground.
When a light beam is directed horizontally,
the Huygens’ principle leads us to conclude
that as it travels, the light beam :
(1) bends upwards
(2) becomes narrower
(3) goes horizontally without any
deflection
(4) bends downwards
45. From a solid sphere of mass M and radius
R, a spherical portion of radius
R
2
is
removed, as shown in the figure. Taking
gravitational potential V50 at r5:, the
potential at the centre of the cavity thus
formed is :
(G5 gravitational constant)
(1)
2GM
R
2
(2)
GM
2R
2
(3)
GM
R
2
(4)
2GM
3R
2
44. »ýèc× «¤Ìé ·¤è »×ü ÚUæç˜æ ×ð´, Öê-ÌÜ ·ð¤ çÙ·¤ÅU, ßæØé ·¤æ
¥ÂßÌüÙæ´·¤ ‹ØêÙÌ× ãæðÌæ ãñ ¥æñÚU Öê-ÌÜ â𠪡¤¿æ§ü ·ð¤
âæÍ ÕɸÌæ ÁæÌæ ãñÐ ØçÎ, ·¤æð§ü Âý·¤æàæ-ç·¤ÚU‡æ-´éÁ
ÿæñçÌÁ çÎàææ ×ð´ Áæ ÚUãæ ãæð Ìæð, ã槻ð‹â ·ð¤ çâhæ‹Ì âð
Øã ÂçÚU‡ææ× ÂýæŒÌ ãæðÌæ ãñ ç·¤, ¿ÜÌð ãé°
Âý·¤æàæ-ç·¤ÚU‡æ ´éÁ Ñ
(1) ª¤ÂÚU ·¤è ¥æðÚU Ûæé·¤ ÁæØð»æÐ
(2) â´·é¤ç¿Ì (â´·¤è‡æü) ãæð ÁæØð»æÐ
(3) çÕÙæ çßÿæðçÂÌ ãé°, ÿæñçÌÁ çÎàææ ×ð´ ¿ÜÌæ
ÚUãð»æÐ
(4) Ùè¿ð ·¤è ¥æðÚU Ûæé·¤ ÁæØð»æÐ
45. °·¤ ÆUæðâ »æðÜð ·¤æ ÎýÃØ×æÙ M ÌÍæ ç˜æ’Øæ R ãñÐ
§ââð
R
2
ç˜æ’Øæ ·¤æ °·¤ »æðÜèØ Öæ», ¥æÚðU¹ ×ð´ ÎàææüØð
»Øð ¥ÙéâæÚU ·¤æÅU çÜØæ ÁæÌæ ãñÐ r5:(¥Ù‹Ì) ÂÚU
»éL¤ˆßèØ çßÖß ·ð¤ ×æÙ V ·¤æð àæê‹Ø (V50) ×æÙÌð
ãé°, §â Âý·¤æÚU ÕÙð ·¤æðÅUÚU (·ñ¤çßÅUè) ·ð¤ ·ð¤‹Îý ÂÚU,
»éL¤ˆßèØ çßÖß ·¤æ ×æÙ ãæð»æ Ñ
(G5 »éL¤ˆßèØ çSÍÚUæ¡·¤ ãñ )
(1)
2GM
R
2
(2)
GM
2R
2
(3)
GM
R
2
(4)
2GM
3R
2
20. C/Page 20 SPACE FOR ROUGH WORK / ÚUȤ ·¤æØü ·ð¤ çÜ° Á»ã
46. Monochromatic light is incident on a glass
prism of angle A. If the refractive index of
the material of the prism is m, a ray,
incident at an angle u, on the face AB
would get transmitted through the face AC
of the prism provided :
(1) 1 1 1
< cos sin A sin
2 2
u m 1
m
(2) 1 1 1
> sin sin A sin
2 2
u m 2
m
(3) 1 1 1
< sin sin A sin
2 2
u m 2
m
(4) 1 1 1
> cos sin A sin
2 2
u m 1
m
46. ·¤æ¡¿ ·ð¤ ç·¤âè çÂý’× ·¤æ ·¤æð‡æ ‘A’ ãñÐ §â ÂÚU
°·¤ß‡æèü Âý·¤æàæ ¥æÂçÌÌ ãæðÌæ ãñÐ ØçÎ, çÂý’× ·ð¤
ÂÎæÍü ·¤æ ¥ÂßÌüÙæ´·¤ m ãñ Ìæð, çÂý’× ·ð¤ AB Ȥܷ¤
ÂÚU, u ·¤æð‡æ ¥æÂçÌÌ Âý·¤æàæ ·¤è ç·¤ÚU‡æ, çÂý’× ·ð¤
Ȥܷ¤ AC âð ÂæÚU»Ì ãæð»è ØçÎ Ñ
(1) 1 1 1
< cos sin A sin
2 2
u m 1
m
(2) 1 1 1
> sin sin A sin
2 2
u m 2
m
(3) 1 1 1
< sin sin A sin
2 2
u m 2
m
(4) 1 1 1
> cos sin A sin
2 2
u m 1
m
23. SPACE FOR ROUGH WORK / ÚUȤ ·¤æØü ·ð¤ çÜ° Á»ãC/Page 23
50. An LCR circuit is equivalent to a damped
pendulum. In an LCR circuit the capacitor
is charged to Q0 and then connected to
the L and R as shown below :
If a student plots graphs of the square of
maximum charge ( 2
MaxQ ) on the capacitor
with time(t) for two different values L1 and
L2 (L1>L2) of L then which of the following
represents this graph correctly ? (plots are
schematic and not drawn to scale)
(1)
(2)
(3)
(4)
50. LCR (°Ü.âè.¥æÚU) ÂçÚUÂÍ ç·¤âè ¥ß×´çÎÌ ÜæðÜ·¤
·ð¤ ÌéËØ ãæðÌæ ãñÐ ç·¤âè LCR ÂçÚUÂÍ ×ð´ â´ÏæçÚU˜æ ·¤æð
Q0 Ì·¤ ¥æßðçàæÌ ç·¤Øæ »Øæ ãñ, ¥æñÚU çȤÚU §âð ¥æÚðU¹
×ð´ ÎàææüØð »Øð ¥ÙéâæÚU L ß R âð ÁæðÇ¸æ »Øæ ãñÐ
ØçÎ °·¤ çßlæÍèü L ·ð¤, Îæð çßçÖóæ ×æÙæð´, L1 ÌÍæ L2
(L1>L2) ·ð¤ çÜØð, â×Ø t ÌÍæ â´ÏæçÚU˜æ ÂÚU
¥çÏ·¤Ì× ¥æßðàæ ·ð¤ ß»ü 2
MaxQ ·ð¤ Õè¿ Îæð »ýæȤ
ÕÙæÌæ ãñ Ìæð çÙÙæ´ç·¤Ì ×ð´ âð ·¤æñÙ âæ »ýæȤ âãè ãñ?
(ŒÜæòÅU ·ð¤ßÜ ÃØßSÍæ ŒÜæòÅU ãñ´ ÌÍæ S·ð¤Ü ·ð¤ ¥ÙéâæÚU
Ùãè´ ãñ´)
(1)
(2)
(3)
(4)
24. C/Page 24 SPACE FOR ROUGH WORK / ÚUȤ ·¤æØü ·ð¤ çÜ° Á»ã
51. A solid body of constant heat capacity
1 J/8C is being heated by keeping it in
contact with reservoirs in two ways :
(i) Sequentially keeping in contact with
2 reservoirs such that each reservoir
supplies same amount of heat.
(ii) Sequentially keeping in contact with
8 reservoirs such that each reservoir
supplies same amount of heat.
In both the cases body is brought from
initial temperature 1008C to final
temperature 2008C. Entropy change of the
body in the two cases respectively is :
(1) 2ln2, 8ln2
(2) ln2, 4ln2
(3) ln2, ln2
(4) ln2, 2ln2
52. An inductor (L50.03H) and a resistor
(R50.15 kV) are connected in series to a
battery of 15V EMF in a circuit shown
below. The key K1 has been kept closed
for a long time. Then at t50, K1 is opened
and key K2 is closed simultaneously.
At t51ms, the current in the circuit will
be : (e5@150)
(1) 0.67 mA
(2) 100 mA
(3) 67 mA
(4) 6.7 mA
51. °·¤ ÆUæðâ ç´ÇU (ßSÌé) ·¤è çSÍÚU ª¤c×æ ÏæçÚUÌæ
1 J/8C ãñÐ §â·¤æ𠪤c×·¤æð´ (ª¤c×æ Ö´ÇUæÚUæð´) ·ð¤ â·ü¤
×ð´ ÚU¹·¤ÚU çÙÙ Îæð Âý·¤æÚU âð »×ü ç·¤Øæ ÁæÌæ ãñ,
(i) ¥Ùé·ý¤ç×·¤ M¤Â âð 2 ª¤c×·¤æð´ ·ð¤ â·ü¤ ×ð´
§â Âý·¤æÚU ÚU¹·¤ÚU ç·¤ ÂýˆØð·¤ ª¤c×·¤ â×æÙ
×æ˜ææ ×ð´ ª¤c×æ ÎðÌæ ãñ,
(ii) ¥Ùé·ý¤ç×·¤ M¤Â âð 8 ª¤c×·¤æð´ ·ð¤ â·ü¤ ×ð´
§â Âý·¤æÚU ÚU¹·¤ÚU ç·¤ ÂýˆØð·¤ ª¤c×·¤ â×æÙ
×æ˜ææ ×ð´ ª¤c×æ ÎðÌæ ãñ,
ÎæðÙæð´ çSÍçÌØæð´ ×ð´ ç´ÇU ·¤æ ÂýæÚ´UçÖ·¤ Ìæ 1008C ÌÍæ
¥ç‹Ì× Ìæ 2008C ãñÐ Ìæð, §Ù Îæð çSÍçÌØæð´ ×ð´ ç´ÇU
·¤è °‹ÅþUæòÂè ×ð´ ÂçÚUßÌüÙ ãæð»æ, ·ý¤×àæÑ
(1) 2ln2, 8ln2
(2) ln2, 4ln2
(3) ln2, ln2
(4) ln2, 2ln2
52. ÎàææüØð »Øð ÂçÚUÂÍ ×ð´, °·¤ ÂýðÚU·¤ (L50.03H) ÌÍæ
°·¤ ÂýçÌÚUæðÏ·¤ (R50.15 kV) ç·¤âè 15V çßléÌ
ßæã·¤ ÕÜ (§ü.°×.°È¤) ·¤è ÕñÅUÚUè âð ÁéǸð ãñ´Ð ·é´¤Áè
K1 ·¤æð ÕãéÌ â×Ø Ì·¤ Õ‹Î ÚU¹æ »Øæ ãñÐ §â·ð¤
Âà¿æÌ÷ â×Ø t50 ÂÚU, K1 ·¤æð ¹æðÜ ·¤ÚU âæÍ ãè
âæÍ, K2 ·¤æð Õ‹Î ç·¤Øæ ÁæÌæ ãñÐ â×Ø t51ms
ÂÚU, ÂçÚUÂÍ ×ð´ çßléÌ ÏæÚUæ ãæð»è Ñ (e5@150)
(1) 0.67 mA
(2) 100 mA
(3) 67 mA
(4) 6.7 mA
25. SPACE FOR ROUGH WORK / ÚUȤ ·¤æØü ·ð¤ çÜ° Á»ãC/Page 25
53. A uniformly charged solid sphere of radius
R has potential V0 (measured with respect
to :) on its surface. For this sphere the
equipotential surfaces with potentials
0 0 03V 5V 3V
, ,
2 4 4
and 0V
4
have radius R1,
R2, R3 and R4 respectively. Then
(1) 2R < R4
(2) R150 and R2 > (R42R3)
(3) R1 ¹ 0 and (R22R1) > (R42R3)
(4) R150 and R2 < (R42R3)
54. A long cylindrical shell carries positive
surface charge s in the upper half and
negative surface charge 2s in the lower
half. The electric field lines around the
cylinder will look like figure given in :
(figures are schematic and not drawn to scale)
(1)
(2)
(3)
(4)
53. R ç˜æ’Øæ ·ð¤ ç·¤âè °·¤â×æÙ ¥æßðçàæÌ ÆUæðâ »æðÜð ·ð¤
ÂëcÆU ·¤æ çßÖß V0 ãñ (: ·ð¤ âæÂðÿæ ×æÂæ »Øæ)Ð §â
»æðÜð ·ð¤ çÜØð, 0 0 03V 5V 3V
, ,
2 4 4
ÌÍæ 0V
4
çßÖßæð´
ßæÜð â×çßÖßè ÂëcÆUæð´ ·¤è ç˜æ’ØæØð´, ·ý¤×àæÑ
R1, R2, R3 ÌÍæ R4 ãñ´Ð Ìæð,
(1) 2R < R4
(2) R150 ÌÍæ R2 > (R42R3)
(3) R1 ¹ 0 ÌÍæ (R22R1) > (R42R3)
(4) R150 ÌÍæ R2 < (R42R3)
54. ç·¤âè ÜÕð ÕðÜÙæ·¤æÚU ·¤æðàæ ·ð¤ ª¤ÂÚUè Öæ» ×ð´ ÏÙæˆ×·¤
ÂëcÆU ¥æßðàæ s ÌÍæ çÙ¿Üð Öæ» ×ð´ «¤‡ææˆ×·¤ ÂëcÆU
¥æßðàæ 2s ãñ´Ð §â ÕðÜÙ (çâçÜ‹ÇUÚU) ·ð¤ ¿æÚUæð´
¥æðÚU çßléÌ ÿæð˜æ-ÚðU¹æØð´, Øãæ¡ ÎàææüØð »Øð ¥æÚð¹æð´ ×ð´ âð
緤⠥æÚðU¹ ·ð¤ â×æÙ ãæð´»è?
(Øã ¥æÚðU¹ ·ð¤ßÜ ÃØßSÍæ ¥æÚðU¹ ãñ ¥æñÚU S·ð¤Ü ·ð¤
¥ÙéâæÚU Ùãè´ ãñ)
(1)
(2)
(3)
(4)
27. SPACE FOR ROUGH WORK / ÚUȤ ·¤æØü ·ð¤ çÜ° Á»ãC/Page 27
57. Îæð â×æÿæè ÂçÚUÙæçÜ·¤æ¥æð´ ×ð´, ÂýˆØð·¤ âð I ÏæÚUæ °·¤ ãè
çÎàææ ×ð´ ÂýßæçãÌ ãæð ÚUãè ãñÐ ØçÎ, ÕæãÚUè ÂçÚUÙæçÜ·¤æ
·ð¤ ·¤æÚU‡æ, ÖèÌÚUè ÂçÚUÙæçÜ·¤æ ÂÚU ¿éÕ·¤èØ ÕÜ
1F
→
ÌÍæ ÖèÌÚUè ÂçÚUÙæçÜ·¤æ ·ð¤ ·¤æÚU‡æ, ÕæãÚUè ÂçÚUÙæçÜ·¤æ
ÂÚU ¿éÕ·¤èØ ÕÜ 2F
→
ãæð Ìæð Ñ
(1) 1F
→
ÕæãÚU ·¤è ¥æðÚU ß ¥ÚUèØ ãñ ÌÍæ 2F
→
50
ãñÐ
(2) 1 2F F 0
→ →
5 5
(3) 1F
→
ÖèÌÚU ·¤è ¥æðÚU ß ¥ÚUèØ (ç˜æ’Ø) ãñ ¥æñÚU
2F
→
ÕæãÚU ·¤è ¥æðÚU ß ¥ÚUèØ ãñÐ
(4) 1F
→
ÖèÌÚU ·¤è ¥æðÚU ß ¥ÚUèØ ãñ ÌÍæ 2F
→
50
ãñÐ
58. ç·¤âè °·¤â×æÙ ÌæÚU ·¤è ¥ÙéÂýSÍ·¤æÅU ·¤æ ÿæð˜æȤÜ
‘A’ ãñÐ §ââð ÕÙæØð »Øð °·¤ ÜæðÜ·¤ ·¤æ ¥æßÌü·¤æÜ
T ãñÐ §â ÜæðÜ·¤ ·ð¤ »æðÜ·¤ âð °·¤ ¥çÌçÚU€Ì M
ÎýÃØ×æÙ ÁæðǸ ÎðÙð âð ÜæðÜ·¤ ·¤æ ¥æßÌü·¤æÜ ÂçÚUßçÌüÌ
ãæð·¤ÚU TM ãæð ÁæÌæ ãñÐ ØçÎ §â ÌæÚU ·ð¤ ÂÎæÍü ·¤æ Ø´»
»é‡ææ´·¤ ‘Y’ ãæð Ìæð
1
Y
·¤æ ×æÙ ãæð»æ Ñ
(g5»éL¤ˆßèØ ˆßÚU‡æ)
(1)
2
M
T A
1
T Mg
2
(2)
2
MT A
1
T Mg
2
(3)
2
M MgT
1
T A
2
(4)
2
MT A
1
T Mg
2
57. Two coaxial solenoids of different radii
carry current I in the same direction. Let
1F
→
be the magnetic force on the inner
solenoid due to the outer one and 2F
→
be
the magnetic force on the outer solenoid
due to the inner one. Then :
(1) 1F
→
is radially outwards and 2F
→
50
(2) 1 2F F 0
→ →
5 5
(3) 1F
→
is radially inwards and 2F
→
is
radially outwards
(4) 1F
→
is radially inwards and 2F
→
50
58. A pendulum made of a uniform wire of
cross sectional area A has time period T.
When an additional mass M is added to
its bob, the time period changes to TM. If
the Young’s modulus of the material of the
wire is Y then
1
Y
is equal to :
(g5gravitational acceleration)
(1)
2
M
T A
1
T Mg
2
(2)
2
MT A
1
T Mg
2
(3)
2
M MgT
1
T A
2
(4)
2
MT A
1
T Mg
2
28. C/Page 28 SPACE FOR ROUGH WORK / ÚUȤ ·¤æØü ·ð¤ çÜ° Á»ã
59. From a solid sphere of mass M and radius
R a cube of maximum possible volume is
cut. Moment of inertia of cube about an
axis passing through its center and
perpendicular to one of its faces is :
(1)
2
4MR
3 3p
(2)
2
MR
32 2p
(3)
2
MR
16 2p
(4)
2
4MR
9 3p
60. When 5V potential difference is applied
across a wire of length 0.1 m, the drift
speed of electrons is 2.531024 ms21. If
the electron density in the wire is
831028 m23, the resistivity of the material
is close to :
(1) 1.631025 Vm
(2) 1.631028 Vm
(3) 1.631027 Vm
(4) 1.631026 Vm
59. ç·¤âè ÆUæðâ »æðÜð ·¤æ ÎýÃØ×æÙ M ÌÍæ §â·¤è ç˜æ’Øæ
R ãñÐ §â×ð´ âð ¥çÏ·¤Ì× â´Öß ¥æØÌÙ ·¤æ °·¤
€ØêÕ (ƒæÙ) ·¤æÅU çÜØæ ÁæÌæ ãñÐ §â €ØêÕ ·¤æ
ÁǸˆß ¥æƒæê‡æü ç·¤ÌÙæ ãæð»æ, ØçÎ, §â·¤è ƒæê‡æüÙ-¥ÿæ,
§â·ð¤ ·ð¤‹Îý âð ãæð·¤ÚU »é$ÁÚUÌè ãñ ÌÍæ §â·ð¤ ç·¤âè °·¤
Ȥܷ¤ ·ð¤ ÜÕßÌ÷U ãñ?
(1)
2
4MR
3 3p
(2)
2
MR
32 2p
(3)
2
MR
16 2p
(4)
2
4MR
9 3p
60. 0.1 m Ü´Õð ç·¤âè ÌæÚU ·ð¤ çâÚUæð´ ·ð¤ Õè¿ 5V çßÖßæ´ÌÚUU
¥æÚUæðçÂÌ ·¤ÚUÙð âð §Üð€ÅþUæòÙæð´ ·¤è ¥Âßæã ¿æÜ
2.531024 ms21 ãæðÌè ãñÐ ØçÎ §â ÌæÚU ×ð´ §Üð€ÅþUæòÙ
ƒæÙˆß 831028 m23 ãæð Ìæð, §â ·ð¤ ÂÎæÍü ·¤è
ÂýçÌÚUæðÏ·¤Ìæ ãæð»è, ֻܻ Ñ
(1) 1.631025 Vm
(2) 1.631028 Vm
(3) 1.631027 Vm
(4) 1.631026 Vm
36. C/Page 36 SPACE FOR ROUGH WORK / ÚUȤ ·¤æØü ·ð¤ çÜ° Á»ã
83. Sodium metal crystallizes in a body centred
cubic lattice with a unit cell edge of 4.29Å.
The radius of sodium atom is
approximately :
(1) 0.93Å
(2) 1.86Å
(3) 3.22Å
(4) 5.72Å
84. The standard Gibbs energy change at
300 K for the reaction 2A B C1ì is
2494.2 J. At a given time, the composition
of the reaction mixture is
1
[A]
2
5 , [B]52
and
1
[C]
2
5 . The reaction proceeds in
the : [R58.314 J/K/mol, e52.718]
(1) reverse direction because Q < Kc
(2) forward direction because Q > Kc
(3) reverse direction because Q > Kc
(4) forward direction because Q < Kc
83. âæðçÇUØ× ÏæÌé °·¤ ¥´ÌÑ·ð¤ç‹ÎýÌ ƒæÙèØ ÁæÜ·¤ ×ð´
ç·ý¤SÅUçÜÌ ãæðÌæ ãñ çÁâ·ð¤ ·¤æðÚU ·¤è Ü´Õæ§ü 4.29Å ãñÐ
âæðçÇUØ× ÂÚU×æ‡æé ·¤è ç˜æ’Øæ ֻܻ ãñ Ñ
(1) 0.93Å
(2) 1.86Å
(3) 3.22Å
(4) 5.72Å
84. 300 K ÂÚU ¥çÖç·ý¤Øæ 2A B C1ì ·¤è ×æÙ·¤
绎$Á ª¤Áæü 2494.2 J ãñÐ çΰ »° â×Ø ×ð´
¥çÖç·ý¤Øæ çןæ‡æ ·¤æ â´ƒæÅUÙ
1
[A]
2
5 ,
[B]52 ¥æñÚU
1
[C]
2
5 ãñÐ ¥çÖç·ý¤Øæ ¥»ýçâÌ ãæðÌè
ãñ Ñ [R58.314 J/K/mol, e52.718]
(1) çßÂÚUèÌ çÎàææ ×ð´ €Øæð´ç·¤ Q < Kc
(2) ¥»ý çÎàææ ×ð´ €Øæð´ç·¤ Q > Kc
(3) çßÂÚUèÌ çÎàææ ×ð´ €Øæð´ç·¤ Q > Kc
(4) ¥»ý çÎàææ ×ð´ €Øæð´ç·¤ Q < Kc
37. SPACE FOR ROUGH WORK / ÚUȤ ·¤æØü ·ð¤ çÜ° Á»ãC/Page 37
85. ¥æð$ÁæðÙæðçÜçââ ·¤ÚUÙð ÂÚU ·¤æñÙ âæ Øæñç»·¤
5 - ·¤èÅUæð - 2 - ×ðçÍÜ ãð€âæÙñÜ ÎðÌæ ãñ?
(1)
(2)
(3)
(4)
86. çÙÙçÜç¹Ì ×ð´ âð ·¤æñÙ âæ Øæñç»·¤ ÂýçÌ¥Ü Ùãè´ ãñ?
(1) ÚñUçÙçÅUÇUèÙ
(2) °ðÜéç×çÙØ× ãæ§ÇþUæ€âæ§ÇU
(3) çâ×ðçÅUÇUèÙ
(4) çȤÙçËÁÙ
85. Which compound would give
5 - keto - 2 - methyl hexanal upon
ozonolysis ?
(1)
(2)
(3)
(4)
86. Which of the following compounds is not
an antacid ?
(1) Ranitidine
(2) Aluminium hydroxide
(3) Cimetidine
(4) Phenelzine