Indian National Astronomy Olympiad – 2009

Senior Category Roll Number:

Question Paper

INAO – 2009 Date: 31st January 2009

Duration: Three Hours Maximum Marks: 100

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Please Note:

• Please write your roll number on top of this page in the space provided.

• Before starting, please ensure that you have received a copy of the question paper
containing total 3 pages (6 sides).

• In Section A, there are 10 multiple choice questions with 4 alternatives out of which

wrong answer. CS
only 1 is correct. You get 3 marks for each correct answer and -1 mark for each

• In Section B, there are 4 multiple choice questions with 4 alternatives each, out of
which any number of alternatives may be correct. You get 5 marks for each correct
answer. No marks are deducted for any wrong answers. You will get credit for
the question if and only if you mark all correct choices and no wrong
choices. There is no partial credit.

• For both these sections, you have to indicate the answers on the page 2 of the
answersheet by putting a × in the appropriate box against the relevant question
number, like this:
Q.NO. (a) (b) (c) (d) Q.NO. (a) (b) (c) (d)
22  ⊠   OR 35 ⊠ ⊠  
Marking a cross (×) means affirmative response (selecting the particular choice).
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Do not use ticks or any other signs to mark the correct answers.

• In Section C, there are 5 analytical questions totalling 50 marks.

• Blank spaces are provided in the question paper for the rough work. No rough work
should be done on the answer-sheet.

• No calculators are allowed.

• The answer-sheet must be returned to the invigilator. You can take this
question booklet back with you.

HOMI BHABHA CENTRE FOR SCIENCE EDUCATION

Tata Institute of Fundamental Research
V. N. Purav Marg, Mankhurd, Mumbai, 400 088
 Useful Physical Constants

Mass of the Earth ME ≈ 5.97 × 1024 kg

Radius of the Earth RE ≈ 6.4 × 106 m
Mass of the Sun M⊙ ≈ 1.99 × 1030 kg
Radius of the Sun R⊙ ≈ 7 × 108 m
Radius of the Moon Rm ≈ 1.7 × 106 m
3 × 108 m/s

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Speed of Light c ≈
Astronomical Unit 1 A. U. ≈ 1.5 × 1011 m
Gravitational Constant G ≈ 6.67 × 10−11 m3 /(Kg s2 )
Gravitational Acceleration on the Earth g ≈ 9.8 m/s2
Gravitational Acceleration on the Moon gm ≈ 1.6 m/s2

Space for Rough Work

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HB

HOMI BHABHA CENTRE FOR SCIENCE EDUCATION

Tata Institute of Fundamental Research
V. N. Purav Marg, Mankhurd, Mumbai, 400 088
 INAO – Sr – 2009

Part A: (10 Q × 3 marks each)

1. If ax = by = cz and b2 = ac, then y = ?
2xz xz
√ √
(a) x+z (b) x+z (c) 2xz (d) xz

2. Each of the figures below depict a constellation. Find the odd one out.

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(a) (b) (c) (d)

3. Gravitational force between two identical uniform solid gold spheres of radius r each

(a) r 4 (b) r 2 CS
in contact is proportional to
(c) 1
r2
(d) 1
r3

4. A copper cube and a wooden cube of volume 10−3 m3 each are initially at room
temperature. They are then moved to an enclosure of ambient temperature 50◦ C.
What can we conclude about the temperatures attained by both cubes after 5 hours?
(a) Tcopper > Twood as thermal conductivity of copper is greater than that of
wood.
(b) Twood > Tcopper as specific heat capacity of wood is greater than that of
copper.
(c) The temperatures will depend on the interplay between specific heat ca-
pacity and thermal conductivity of the materials.
(d) Both temperatures will be practically the same, as they are in the enclosure
for 5 hours.
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5. If the product of all the numbers from 1 to 100 is divisible by 2n , then what is the
maximum possible value for n?
(a) 128 (b) 97 (c) 64 (d) 87
−
→ −
→
6. Two vectors P and Q are acting at a point such that their resultant is perpendicular
−
→ −
→ −
→
to Q . If θ is the angle between P and Q then |P
|Q|
|
is given by,
(a) cos θ (b) sec θ (c) − cos θ (d) − sec θ

7. What will be the approximate period of Chandrayaan moving in an orbit 100 km

above the moon’s surface?
(a) 57 min (b) 30 min (c) 118 min (d) 79 min

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 INAO – Sr – 2009

8. In the following figure, A, B, C are three light source positions with respect to the ob-
stacle and the screen. Which of these light source positions will result in the longest
shadow of the obstacle on the screen?

11
00
A

E
B00
11 Screen

11
00
C

Obstacle

(a) A
(b) C
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(c) A and C form shadows of same length, while B forms a smaller shadow.
(d) All the three light sources will form shadows of same length.

9. The following figure shows skeleton chart of the Orion constellation. Approximate
“North” is indicated by the letter ....
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(a) A (b) B (c) C (d) D

10. Find the resultant focal length for following system where the radius of curvature is
15cm.

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glass air

(a) 40 cm (b) 60 cm (c) 120 cm (d) ∞

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 INAO – Sr – 2009

Section B: (4 Q × 5 marks each)

11. Which of the following observations support the statement that “Every system tends
to adjust by itself to have minimum Potential Energy”.
(a) Andromeda galaxy and Milky Way are approaching each other.
(b) Two unlike, free charges move towards each other.
(c) External work is required to compress a spring.

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(d) A powerful magnet can deflect a compass needle from equilibrium position.
12. Consider a sealed cylinder with frictionless, free-moving piston of fixed mass attached
on the top, kept in an environment of constant atmospheric pressure. The ideal gas
in the cylinder is slowly heated and it expands. Which of the following is / are true?
(a) The density of the gas will increase.
(b) The pressure of the gas will decrease.

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(c) The internal energy of the system will remain the same.
(d) In this process work is done by the gas.
13. In one of the truly revolutionary finds of the 20th century, Howard Carter discovered
the tomb of the Egyptian Pharaoh (emperor) Tutankhamen in 1922. Following items
were removed from the tomb, along with the mummy of the Pharaoh. Which of these
items could have been carbon dated to fix the period of the Pharaoh?
(a) Fragments of glass
(b) Golden Bracelets
(c) Dried Fruits
(d) Leather Shoe
14. Which of the following phenomena is / are useful, in estimating distances in the
Universe?
(a) Some time Venus can be seen transiting over the solar disc.
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(b) Stars with no proper motion appear to change their position in the sky
when viewed six months apart.
(c) Stars exhibit Doppler shift.
(d) All supernovae of Type Ia have same absolute brightness.

Section C: Analytical Questions

α. (8 marks) What will be area of the largest cyclic quadrilateral that can be inscribed
in a given circle? Justify your answer qualitatively (formal proof not necessary).
β. (12 marks) Jayshree claimed that she saw a solar eclipse when the size of the solar
disk was 26′ and that of the lunar disk was 30′ . She also claimed that at the time of the
maximum eclipse, distance between the centres of the two disks was 7′ . Qualitatively
show that she could not have observed a total eclipse. Find the percentage
of the
−1 1
solar disk covered at the time of the maximum eclipse. (Given: cos 26
≈ 0.49π
rad).

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 INAO – Sr – 2009

γ. (8 marks) The famous Indian astronomer, Aryabhata, expressed the value of π in

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what we now know as continuing fractions i.e..π = 3.1416 = a + where a,
1
b+
1
c+
d
b, c, d are positive integers. Find a, b, c, d.

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δ. (10 marks) Kedar sent a container of marbles by road from Mumbai to Parag in
Pune. The container was 2m × 2m × 2.5m in size, with height being the larger
dimension. Marbles of 2 cm diameter were arranged to fill the entire base and then
additional layers of marbles were arranged with each upper marble exactly on top
of corresponding marble in the previous layer (see figure). The marbles were thus
placed upto 2m height to complete the cubical structure. However, on reaching Pune,
when Parag opened the container, he found the height of structure was not 2m as
promised by Kedar, but something else. Kedar defended saying that marbles may

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have readjusted due to jiggling. Can you find new height of this marble pile?

ǫ. Chiraag performed an experiment using a simple pendulum to find value of g. He

measured time taken for 30 oscillations of the pendulum for various values of length
(repeated thrice for each value of length), following readings were obtained.
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L Readings
cm t1 (Sec) t2 (Sec) t3 (Sec)
20.0 26.9 26.9 27.0
25.0 30.1 30.1 30.1
30.0 32.9 32.8 32.7
35.0 35.6 35.8 35.7
40.0 38.0 38.1 38.1
45.0 40.4 40.4 40.5

Mass of the bob of the pendulum was known to be 50 gm.

(a) (9 marks) Plot appropriate graph to represent the data.
(b) (2 marks) Find the value of g.
(c) (1 mark) Using the graph, find out required length of the pendulum to get a
time period of 1.24 sec.

prepared using LATEX2ǫ