PHYSICS

Wednesday 5 November 2008

Time allowed: 2 hours

For candidates applying for Physics, and Physics and Philosophy

There are two parts (A and B) to this test, carrying equal weight. Attempt as many questions as you can from each part. Write on white A4 lined paper. At the end of the test put your answer sheets in order and collate with a treasury tag. Attach a cover sheet to your answers. DO NOT write your name anywhere except on the cover sheet.

Marks for each question are indicated in the right hand margin. There are a total of 100 marks available and total marks for each section are indicated at the start of a section. You are advised to divide your time according to the marks available, and to spend equal eﬀort on parts A and B. No calculators, tables or formula sheets may be used.

Answers in Part A should be given exactly unless indicated otherwise. Numeric answers in Part B should be calculated to 2 signiﬁcant ﬁgures. Use g = 10 m s−2 .

Do NOT turn over until told that you may do so.

Part A: Mathematics for Physics [50 Marks]

Answers in Part A should be given exactly unless indicated otherwise. 1. Evaluate the sum of integers 1 + 2 + 3 + · · · + 99 + 100.

[3]

2. Evaluate (0.25)−1/2 and (0.09)3/2

[4]

3. The ﬁrst three terms of the series expansion of (1 + x)m are: 1 + mx +

m(m − 1)x2

.

2

Find the ﬁrst three terms in the series expansion of (1+ x)m+1 (1 − 2x)m . [5]

4. Find the set of values of x for which

x2 + 2

< 3.

1 − x2

[3]

5. Given that x = log9 2, ﬁnd, in terms of x, (i) log2 9; (ii) log8 3.

[4]

6. Find the two values of x for which 1, x2 , x are successive terms of an arithmetic progression.

[3]

2

3

4

7. Determine the value of a such that the curve y = x + x + x + x + · · · 2

3

4

and the line y = ax have the same gradient at x = 0. What value will a have if instead they have the same gradient at x = 1 ?.

[4]

4

8. The points (5,2) and (−3, 8) are at opposite ends of the diameter of a circle. Determine the equation of the circle.

[4]

9. A die is biased so that the numbers 5 and 6 are obtained three times as often as 2, 3 and 4, and the number 1 is never obtained. Calculate the probability that (i) a two is thrown; (ii) two consecutive throws give a total ≥ 10.

[5]

10. A cube has side a. Find the length of its body diagonal.

[3]

11. Evaluate

(i)

(ii)

1

−1

1

0

x + x3 + x5 + x7 dx

x9 + x99

dx.

11

[4]

12. In the ﬁgure below, the shaded area ADEC is deﬁned by concentric circles which share a common centre O with the shaded triangle ABC. The straight lines AD and CE, if extended, pass through O. The lengths AD=AB=BC=CA=CE. Find the ratio of the two shaded areas ADEC

and ABC.

[8]

Part B: Physics [50 Marks]

Numeric answers in Part B should be calculated to 2 signiﬁcant ﬁgures. Use g = 10 m s−2 .

Multiple choice (10 marks)

13. A symmetric seesaw is 3 m long from end to end. If a boy of mass 20 kg sits on one end, how far away from him should a girl of mass 30 kg sit to balance the seesaw?

A 0.5 m

C 2.0 m

B 1.0 m

D 2.5 m

[1]

14. When nuclear ﬁssion occurs in a commercial nuclear reactor the mass of the products compared with the mass of the reactants is

A

B

C

D

increased

decreased

stays the same

it depends on the reaction

[1]

15. The visible universe contains about 400 billion galaxies (where 1 billion equals 109 ). Our galaxy contains about 250 billion stars. The mass of our sun is about 2 × 1030 kg. NASA estimates that dark matter out-masses stars by about 20:1. Use this data to estimate the total mass of the visible universe.

A 4.2 × 1036 kg

C 2.0 × 1053 kg

B 9.5 × 1051 kg

D 4.2 × 1054 kg

[1]

16. A solar eclipse can only occur when the moon’s phase is A new moon

C waning

B full moon

D waxing

[1]

17. When an ideal gas is heated in a container of ﬁxed volume then A

B

C

D

the pressure and density both rise

the...