astronomy olympiad practice test solutions

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DATE : 06-11-2014 STAGE - I

OLYMPIAD PREPARATORY TEST-3

(OPT-3)

SUBJECT : ASTRONOMY

HINTS & SOLUTIONS

1. (A)

2. (B)

x = A sin(!t) + Asin " #

$%&

' ()!

3

2t + Bsin(!t + *)

B = A & * =34( to make the resultant 0.

3. (D)tan2+ + 3 = 3sec+, sec2+ – 3sec+ + 2 = 0

, (sec+ – 1) (sec+ –2) = 0

, sec+ = 1 or sec+ = 2

, + = 2n( or + = 2n( ±3

(, n. I

Hence two solutions for 0 / + / 90º

(d) is correct

4. (D)

5. (D)

cos + =|q||p|

q.p ; cos 0 =p

2

qp

)p(.

2

qp

)

" #

$%&

' )

, , tan 0 =q.pp

sinpq2 )

+

cos 1 = |p|2

qp

)p(.2

qp

2

" #

$%&

' 2

=

, tan 1 =q.pp

sinpq2 2

+

tan 0 + tan 1 = pq sin + "" #

$%%&

'

2)

))2

)q.pp()q.pp(

q.ppq.pp22

22

=+2

+222 cosqp

sinpq2(B) is correct

6. (D)

2 / 1

22

F

Er

.T

1 2

3

4

56

7

8

9(

9

2 / 1

2

2 / 1

22 L

1

TL

Q 22

34

567

8:3

4

567

8

9

9= L

7. (B)

8. (B)

9. (B)

º20sin2

º70cos7 +

º65tanº45tanº25tan2

º35eccosº55cos3

= º70cos2

º70cos7

+ 1.)º65tanº25tan(º35sin2

º55cos3= 2

7+ 2

3=5

(B) is correct

10. (C)

F = !r2

ii 210

(

;; F = 2

0

21

rc4

qq

(

22

22

2

2

2

0

0

qi

rF

Fr

q

i

F

<::

=

; =

iq

Fr =

iq

w

v = iR =qW ; R =

iqw

11. (C)

12. (A)

>U = U2 – U

1

= UD – U

C

= q3 3

4

567

8" #

$%&

' )2"

#

$%&

' )

50

Kq

40

Kq

10

Kq

40

Kq 2121

= Kq2q

3 ×

50

4

=0

3

4q(= q

0 × 210

1252

2< = 8q

2

mailto:[email protected]

http://www.resonance.ac.in/






13. (C)

14. (C)A = { 2,4,6,8 ....... 400}B = { 3,6,9,12 ....... 750}A ? B = { 6 ,12, 18 , ..... , 396}, n (A ? B) = 66and n (A@B) = n(A) +n(B) – n(A?B)= 200 + 250 – 66 = 384

(C) is correct15. (A)

16. (B)

3 sin + = v2 cos +

v2 = 3 tan 30º = 1

(B) is correct

17. (A)

Energy incident =20kq

2

1

Energy refracted after first reflection

= )r1(kq2

1 220 2

Energy refracted after second reflection

= )r1)(r1(kq2

1 2220 22

2tkA

2

1 =

2220 )r1(q

2

K2

q0 = 1

At2 = (1 – r 2)2

At = (1 – r 2)

18. (B)

n = 1360

2+

; (n + 1) =+

360

+ = =1n

2

)(

or n =+(2

; + =n

2(

19. (C)Let a is first term and d is common difference

a3 + a

7 = 6 , (a + 2d) + (a + 6d) = 6 .........(1)

a3a

7 = 8 , (a + 2d) (a + 6d) = 8 .........(2)by solving (1) and (2)

a + 2d = 2 and a + 6d = 4or

a + 2d = 4 and a + 6d = 2

Case I IfABC

:)

:)

4d6a

2d2a, d =

2

1 and a = 1

, S16

=2

1634

567

8)

2

1).15()1(2 = 8 3

4

567

82

19 = 76

Case II IfABC

:)

:)

2d6a

4d2a, d = –

2

1 , a = 5

, S16

=2

16 3

4

567

8"

#

$%&

' 2)

2

1)15()5(2 =8 3

4

567

82

2

1510 = 80 –60=20

(C) is correct.

20. (A)v

avt = 0.5

2

1t

2

)9.0(g2)4.0(g2:

""

#

$

%%

&

' )

t =)32(g2.0

1

)

=)5(8.92.0

1

<

=7

1

21. (A)2x2 + 2y2 – 3x = 0

, x2 + y2 – 2

3x = 0

S( –

2, 1) = 4 + 1 + 3 = 8 > 0Hence ( –2, 1) lies out side the circle

distance of center " #

$%&

' 0,

4

3 from point ( –2, 1) is

=2

2

)10(24

32)"

#

$%&

' ) = 1

4

112

)" #

$%&

' =

16

137 >

4

11

Hence there are no point on the circle(A) is correct.

22. (D)

23. (A)R = 6 , H = 8

V = )8()6(3

1 2( sin + =R8

R

2 =

5

3

(since tan + =4

3)

5R = 24 – 3R

R = 3

V =8

3

24. (B)

25. (C)








26. (D)

DExlim

x

2x

3x" #

$%&

' )2

= DExlim

x

x

21

x

31

""""

#

$

%%%%

&

'

)

2

; 1D form

= x.12x

3xlim

x

e

" #

$%&

' 2

)2

DE

= " #

$%&

' )DE 2x

x5 –

xlim

e = e –5

Hence (D) is correct.

27. (A)

28. (A)AM F GMNow a2 + b2 F 2abb2 + c2 F 2bcc2 + d2 F 2cdd2 + a2 F 2da

On addition2(a2 + b2 + c2 + d2) F 2(ab + bc + cd + da)

, a2 + b2 + c2 + d2 F ab + bc + cd + da

, ab + bc + cd + da / 45

(A) is correct.

29. (A)foci (3, 1) and (1, 1)

, major axes y = 1, center (2, 1)minor axies x = 2Let equation of ellipse

2

2

a

)2x( 2

+ 2

2

b

)1y( 2

= 1

it passes through P(1, 3) so that

22 b

4

a

1) = 1 ....(2)

distance between foci is = 2

, 2ae = 2

Now e2 = 1 – 2

2

a

b

, a2e2 = a2 – b2

, 1 = a2 – b2

Put in (2)

2b1

1

) + 2b

4 = 1 , b4 – 4b2 = 4

, b2 = 2 + 2 2

, a2 = 3 + 2 2

, e2 = 1 – 223

222

)

) =

223

1

)

, e =

)12(

1

)

= )12( 2

(A) is correct.

30. (C)

31. (C)Let Roots are 0 & 02

, 0 + 02 = –

3

p& 0.02 = 1

, (0 + 02)3 = –

27

p3

, 03 + 06 + 30.02(0 + 02) = – 27

p3

, 1 + 1 + 3 " #

$%&

' 2

3

p = –

27

p3

, 54 – 27p = –p3

, p3 – 27p + 54 = 0

, (p – 3) (p2 + 3p – 18) = 0

, p = 3 or p2 + 3p – 18 = 0

, p = 3 or p2 + 3p – 18 = 0

, p = 3 or (p + 6) (p – 3) = 0

, p = 3 or p = –

6

but p > 0

, p = 3(C) is correct.

32. (B)

cos + =ab

ba""

G

)b4 –a5).(b2a(""""

) = 0,

0b8 –b.a10b.a4 –a5 22 :)""""

cos + =63

cos + =2

1

+ = 60º

(B) is correct

33. (A)

34. (C)

35.

36. (D)

A =333

4

5

666

7

8

111

111

111

A – HI =H

H

H

–111

1 –11

11 –1

= 0

, (1 – H) (H2 – 2H) – 1 (1 – H – 1) + 1 (1 – 1 + H)= 0= (1 – H) (H2 – 2H) + H + H = 0= H2 – 2H – H3 + 2H2 + 2H = 0

= –

H3

+ 3H2

= 0H3 = 3H2 ; A3 = 3A2

(d) is correct








37. (D)

Z =emit

emitobs

H

H2H ; Z =

1216

12162432 2=1

Z > 0 is known as red shift.

38. (B)Cube root of 64 isx = (64)1/3 (1)1/3

x = 4, 4!, 4!2

x = 4, "" #

$%%&

' )22

i314 , "

" #

$%%&

' 222

i314

x = 4, – 2 + 2 3 i, –2 – 2 3 i

so x = 2 3 i – 2 (b) is correct

39. (A)n

1 = 2 n

1 = 2

n2 = 3 n

2 = 4

Z = 1 Z = 2

H1 = 6561 H2 = ?

" #

$%&

' 2:

H 9

1

4

1R

1

1

" #

$%&

' 2:

H 16

1

4

12Rx

1 2

2 3494

165

1

2

<<<<

:HH

H2 = 243 × 5

H2 = 1215 Aº

40. (B)

xxx – x)xx( = 0

xxx = x)xx(

x x log x = x log x x

x3/2 log x = xlog2

x3

logx " #

$%&

' 2

x3 –x 2 / 3

= 0

logx = 0, x3/2 – 2

x3= 0

x = 1, " # $%& ' 23

–xx = 0

, x = 0 and4

9x : = 2.25 ; (B) is correct

41. (B)

10 MeV =r

qKq 21

10 × 1.6 × 10 –19 + 6 = 12

389

106.1

2956.221092

2

<

<<<<

= 18 × 1.6 ×29 × 1021 – 38

18 × 1.6 × 2y × 10 –17 = 8.3 × 10 –

15 m

42. (B)

E = 0V I 0B I 0

43. (D)

irms

=60

120=

45

vC

vC = 901502 = 1202 + (v

L – 90)2

)120150( 22 2 = |(vL – 90)|

|VL – 90| = 10 × 9

vL = 180 or v

L = 0 (not possible)

44. (B)425 : 595 : 76585 : 119 : 153

595 – 425 =!4

v2

! = 1 m45. (D)

m = 6 + 35

n = 6 – 35

m · n = 36 – 35 = 1

Now J K2n –m = m + n – mn2

= 12 – 2.1 = 10

, n –m = 10 ; (D) is correct

46. (D)

(h – h´) Lg = 02 pv

2

1)L

(h –

h´) L g = 2

1

L v2

+ h L g

v2 = [(h – h´)g – h] × 2

v = 0

47. (D)

AB = 34

567

8

00

11 3

4

567

8

1

10

1 =

334

5

667

8

010)0

1)11)0

22

2

0 = !1 = !2

1 + ! + !2 = 0

!2 = 1

So AB = 34

567

8

21 –

1 –1 –

(D) is correct








48. (D)

9 = 4a)a(23

)a2)(a2( 22

<H)<H

= " #

$%&

' )H

3

164a3

=3

28a

a6

m 3 < =

2

6m

9

14" #

$%&

' <

!

=2

m369

14!< ; 9 =

2

m162

7!

49. (C)

" #

$%&

' 22;)"

#

$%&

' 2;:

R

1)1(

R

1)1(

f

121

= )11(R

121 ;2)2; ; f =

21

R

;2;

50. (C)

EqL =

40

1

20

140

6

20

10

)

2 =

40

340

620 2

i =

3

4010

3

14

) =

10

2 = 0.2 AA

i = 200 mAB E A

51. (A)

x1 + x = 2 cos +

3

xx

1" #

$%&

' ) = 8 cos3 +

x3 + 3x

1 + 3 "

#

$%&

' ) x

x

1 = 8 cos3+

3

6

x

1x ) + 3 (2cos +) = 8 cos3 +

3

6

x1x ) + 3 (2 cos +) = 8 cos3 +

3

6

x

1x ) = 2(4cos3 + – 3cos +) = 2 cos 3+

Alter :x

1 + x = 2 cos +

, x = cos + + i sin +

Now 3

6

x

1x ) = x3 + 3x

1

= (cos+ + isin+)2 + (cos + – i sin +)3 = 2 cos3+

(A) is correct

52. (B)30s

< vL

W1 = W

0 – 11

1 = v L g

1 < 10

1 M , W NW

2 > W

1

53. (B)

!:H2

)x(2

):H

!

!2

Vf1 :

)x(2

Vf2

)

:!

> f = f1 – f

2

= 34

567

8) x

1 –

1

2

V

!!

=)x(2

Vx

)!! x << l

>f = 22

Vx

!(B)

54. (C)

55. (A)

Angle are12

7,

6,

4

((( ; a = BC = 1 + 3

Csin

c

Bsin

b

Asin

a:: ;

O)105sin

31 =

6 / sin

b

(

, b =

"" #

$%%&

' )

)

22

13

)31(

2

1= 2

Area of >ABC =2

1ab sin C

= J K J K " #

$%&

' ()

4sin231

2

1 =

2

31) ; =

13

1

2

(A) is correct








56. (B)

>u = n CV >T

= )T(1 –

nR>

P =1 –

V)1 –5.2(P

P =)1 –(

PV

2

3

P

Correct option is (B)

57. (B)Energy ELiner momentum pAngular frequency wQuntam no. n

w

Ep indendent of n

(B) option is correct.

58. (B) Standard Result

59. (D)

A"

+B"

= C"

and A" – B

" = D"

Then C2 + D2 = 2(A2 + B2)(D) is correct

60. (B)(1 + cot + + cosec + ) ( 1 + cot + – cosec+)= (1 + cot +)2 – (cosec2+)= 1 + cot2+ + 2 cot+ – cosec2+= 2 cot +(B) is correct

61. (C)

62. (A)C

1 (0, 0), C

2 = (3, 4)

r1 = 2, r

2 = 24169 )) = 7

C1C

2 = 5, r

1 + r

2 = 7

|r1 – r

2| = C

1C

2 < r

1 + r

2

Circles are internally touching so one tangent canbe drawn.

63. (C)

k, ! ;

" #

$%&

' )

(:

3

1

2

1k5

m2T =

k25

m62( = T

5

6

64. (A)

r

8 = cos + – 1

8 = r cos + – r

8 = x – 22 yx )

, (x – 8)2 = x2 + y2

, x2 – 16x + 64 = x2 + y2

, y2 + 16x – 64 = 0 (Parabola) (A) is correct

65. (B)

H = (l – x) ;

QQ

dm

xdm

= 0.3

2

6

2 –

3 –

2

dx)x –(

dxx)x –(

2

3

22

32

0

0 !

!

!

!!

!!!

!

!

!

!

:::

Q

Q

66. (A)

a)e –1(ae2

1 –

2

a)e –1(2a( = Area of shaded part

a 34

567

8(

2

e –)e –1(a2

= Area ;m2

L

dt

dA:

T = " #

$%&

' L

m2 (a (1 – e) a × 2

t ="

#

$%&

'

L

m2

a

2

(1 –

e)"

#

$%&

' (

2

e –

" #

$%&

' (

(:

2

e –

2

t

T

; " #

$%&

' (

(:

2

e –

2

Tt ; "

#

$%&

' (

:2

e –1

2

Tt

67. (D)

Area of >ABC =2

1

1bac

1acb

1cba

)

)

)

C1 E C

1 + C

2

=2

1(a + b + c)

1ba1

1ac11cb1

)

)) = 0

(D) is correct

68. (B)

n J KEH ? = 52% ...(1)

, 100% – n(H @ E) = 52%n(H) = 35% , n (H @ E) = 48%n(E) = 23% , n(H @ E) = n(H) + n(E) – n(H ? E)

, 48% = 35% + 23% – n(H ? E)

, n(H ? E) = 10%

So total = 1500 ×10

100 = 15000 ; (B) is correct








69. (C)

Tn =

nn

32 )

850

1 =

nn

32 )

, n = 50

Sn

=

R ) nn

32 =

R ""

#

$%%

&

'

)

)

)1n(n

)n( –)1n(3

Sn = 3 3

4

567

8) 1n

1 –

n

1

S50

= 3 34

567

8"

#

$%&

' )"

#

$%&

' )"

#

$%&

' )"

#

$%&

' 51

1 –

50

1....

4

1 –

3

1

3

1 –

2

1

2

1 –

1

1

= 3 34

567

851

1 –1 =

17

50 =

17

162

(C) is correct

70. (C)

!12

1 –

!14

1 =

!15

x

!12

1 ""

#

$%%&

'

)13()14(

1 –1 =

!12)13()14()15(

x

, )13()14(

181 =

)13()14(15

x

, x = 181 × 15

x = 2715 ; (C) is correct

71. (A)

dx

dy = 4x3 + 24x2 + 52x + 32 = 0 (for tangent paral-

lel to x-axis)x = –1 satisfy

(x + 1) (4x2 + 20x + 32) = 0

, 4(x + 1) (x2 + 5x + 8) = 0x = –1 only one point

(1, –6)

(A) is correct

72. (A)AM F GM. equality holds only if 53x –4 = 54 –3x

, 4 – 3x = 0 , x =3

4

73. (D)

f(x) =2

1sin 2x

f '(x) = cos2x > 0 (for increasing)

x . " # $%

& ' (

4,0 @ "

# $%

& ' (( ,

4

3 ; (D) is correct

74. (C)(a + b)n

Tn+1

= nCn a°bn = bn = 16807

...(1)T

n = nC

n –1a1bn –1 = nabn –1 = 48020 ..(2)

Tn –1

= nCn –2

a2bn –2 =2

)1 –n(n a2bn –2 = 54880...(3)

by (2) / (1) b

na = 16807

48020

...(4)

by (3)/(2)b

a

2

)1 –n( =

48020

54880...(5)

by (4) & (5)n2

1 –n =

48020

16807

48020

54880<

, n

1 –n =

2305920400

1844736320 ,

n

1 = 1 –

2305920400

1844736320

,n

1 =

2305920400

461184080 , n = 5

by (1) b5 = 16807, b = 7 and a = 4(C) is correct

75. (C)

5

1 –

2

1

r

1: ; mm

3

10

3

25r :

<:

(C) Concave towards A

76. (D)

2

1 kA2 = 2 ;

2

1 k (.2)2 = 2 ; k =

04.

4 = 100

T =k

m2(

100

52T

(:

=100

05.2( = 14.04 × 10 –2 = 140 × 10 –3 = 140 ms

77. (A)

9 = Q(

)

6 /

0

44

4

dx)x3(sinx3cos

)x3(cos...(1)

using properties Qa

0

dx)x(f = Q 2a

0

dx)xa(f

9 = Q(

)

6 /

0

44

4

dx)x3(sin)x3(cos

)x3(sin...(2)

adding

29 = Q( 6 /

0

dx.1 ; 29 =6

(

9 = 12

((A) is correct








78. (B)

Y = B.)BA( )Y = (1 + 0) . 1 = 1Y = (0 + 1) . 0 = 0

79. (D)

Hd =

mV

h =

34567

8 < nZ1018.2.m

h

6

; =

S T6

1018.2m

3h

<

<

= 1 nm Ans. (D)

80. (D)

9 = Q )

) 2

dxx5

)5(5logx5x

1x4

Let 5x + x5 = t(5x log 5 + 5x4) dx = dt

, 9 = Q t

dt

5

1 = ctlog

5

1)

= c)x5log(5

1 5x ))

(D) is correct

Physical constants you many need :

(1) Charge on electron e = 1.6 × 10-19 C

(2) Mass of an electron me = 9.1 × 10 –31 kg

(3) Universal gravitational constant G = 6.67 × 10 –11 Nm2 / kg2

(4) Permittivity of free space =0 = 8.85 × 10 –12 C2 / Nm2

(5) Gas constant R = 8.31 J / K mol

(6) Planck constant h = 6.62 × 10 –34 J s

(7) Stefan constant U = 5.67 × 10 –8 W/m2 k4

(8) Boltzmann constant k = 1.38 × 10 –23 J/K

ANSWER KEY

1. (A) 2. (B) 3. (D) 4. (D) 5. (D) 6. (D) 7. (B)

8. (B) 9. (B) 10. (C) 11. (C) 12. (A) 13. (C) 14. (C)

15. (A) 16. (B) 17. (A) 18. (B) 19. (C) 20. (A) 21. (A)

22. (D) 23. (A) 24. (B) 25. (C) 26. (D) 27. (A) 28. (A)

29. (A) 30. (C) 31. (C) 32. (B) 33. (A) 34. (C) 35.

36. (D) 37. (D) 38. (B) 39. (A) 40. (B) 41. (B) 42. (B)

43. (D) 44. (B) 45. (D) 46. (D) 47. (D) 48. (D) 49. (C)

50. (C) 51. (A) 52. (B) 53. (B) 54. (C) 55. (A) 56. (B)

57. (B) 58. (B) 59. (D) 60. (B) 61. (C) 62. (A) 63. (C)

64. (A) 65. (B) 66. (A) 67. (D) 68. (B) 69. (C) 70. (C)

71. (A) 72. (A) 73. (D) 74. (C) 75. (C) 76. (D) 77. (A)

78. (B) 79. (D) 80. (D)



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