P425/2

APPLIED
MATHEMATIC
PAPER 42
JULY/AUG 2024
3 hours

ASK INTEGRATED TEACHER’S MOCK

EXAMINATIONS BURREAU

AITEL JOINT MOCK EXAMINATIONS 2024.

UGANDA ADVANCED CERTIFICATE OF EDUCATION
APPLIED MATHEMATICS
PAPER 2
3 HOURS

INSTRUCTIONS TO CANDIDATES:

Answer all the eight questions in section A and any five questions from section B
Any additional question(s) answered will not be marked.
All necessary working must be shown clearly
Silent, non-programmable scientific calculators and mathematical tables with a list
of formulae may be used.
In numerical work, take g to be 9.8ms-2.
Graph paper is provided

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 SECTION A (40 Marks)

(Attempt all questions in this section)

5 7 1
1. Two events A and B are such that 𝑃(𝐴) = , 𝑃(𝐴/𝐵′ ) = , 𝑃(𝐴 ∩ 𝐵) = , find;
12 12 8

(i) 𝑃(𝐴 ∪ 𝐵) (03 marks)

(ii) 𝑃(𝐵/𝐴′ ) (02 marks)
2. A train being brought to rest with uniform retardation travels 30m in 2 seconds
and then a further 30m in 4 seconds. Find the retardation of the train.
(05 marks)
3. The tables of the normal probability distribution give f(1.36) = 0.9131,
f(1.37) = 0.9147. Find by linear extrapolation the value of x when f(x) = 0.9129
correct to three decimal places. (05 marks)
4. The table below shows the age groups of a random sample of 200 people that
attend a music festival.
Age (years) 15 - < 20 < 25 < 30 < 35 < 40 < 50

No. of people 22 42 70 38 16 12

Construct a histogram and use it to estimate the mode. (05 marks)

5. A body of mass 8kg in contact with a rough plane inclined at 50 to the
0

horizontal is just prevented from sliding down the plane by a horizontal force
P. if the angle of friction between the plane and the body is 25 0, calculate the
magnitude of P. (05 marks)
6
6. Use the trapezium rule with 7 ordinates to estimate ∫1 𝑥 ln 𝑥 𝑑𝑥 giving your
answer correct to 3 decimal places. (05 marks)
7. A coin is biased so that it is twice as likely to show a head as a tail. If it is tossed
5 times, find the probability that at least 4 tails show up. ( 05 marks)
8. A piston moves with simple harmonic motion performing 3 oscillations per
minute. Given that the maximum speed of the piston is 0.5ms-1, find the
(i) amplitude of the motion. (03 marks)
(ii) maximum acceleration. (02 marks)
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 SECTION B. (60 MARKS)
(Attempt any five questions from this section. All questions carry equal marks)
9. A random variable 𝑥 has probability density fuction 𝑓(𝑥) given by
𝑎𝑥 2 (𝑑 − 𝑥), 0≤𝑥≤1
𝑓 (𝑥 ) = {
0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒.
Given that the mean of 𝑥 𝑖𝑠 0.6, determine;
(i) the value of 𝑎 and 𝑑.
(ii) 𝑃(0.9 ≤ 𝑥 ≤ 1)
(iii) the cumulative distribution of 𝑥. (12 marks)
10. Two particles of mass 2kg and 3kg are connected by a light inextensible string
passing over a fixed smooth pulley. Initially, the system is at rest with the string
taut and vertical with both particles at a height of 2m above the ground. When
the system is released, find the;
(i) time that elapses before the 3kg mass hits the ground.
(ii) maximum height reached by the 2kg mass. (12 marks)
11. Given the equation 𝑒 𝑥 + 1 = −2𝑥;
(a) Show graphically that the equation has a root between 0 and -1.
(b)(i) Show that the Newton Raphson formula for approximating the root of the
𝑥𝑛 𝑒 𝑥𝑛 −𝑒 𝑥𝑛 − 1
equation is given by 𝑥𝑛+1 =
𝑒 𝑥𝑛 +2

(ii) Use the formula in b(i) above and the initial approximation 𝑥0 in (a) above
to find the root of the given equation to two decimal places. (12 marks)
12. The weights of a certain type of cows are normally distributed. Out of 8000
cows selected at random, 440 weigh below 130kg while 340 weigh above
200kg. find the;
(i) mean weight and standard deviation. (08 marks)
(ii) probability that a cow chosen at random weighs at least 120kg. (04 marks)
13. A ship A moving with a constant velocity (4𝒊 + 3𝒋) passes through a point with
position vector (3𝒊 + 2𝒋). At the same instant, a ship B passes through the point
with position vector (3𝒊 + 4𝒋) moving with a constant velocity of (𝒊 − 2𝒋).
Find the (i) position vector of A relative to B at any time 𝑡.
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 (ii) Shortest distance between P and Q in the subsequent motion.
(iii) time that elapses before the particles are nearest to one another.
(iv) the position vector of ship A after 3 hours. (12 marks)
14. Two numbers X and Y are approximated by 𝑥 and 𝑦 with errors 𝑒1 and 𝑒2
respectively.
𝑒 𝑒
(a) Show that the maximum relative error in 𝑥 2 𝑦 is given by 2 | 1| + | 2|
𝑥 𝑦
(05 marks)
(b) If 𝑥 = 2.23 and 𝑦 = 2.013 are each rounded off to the given number of
decimal places, calculate the;
(i) percentage error in 𝑥𝑦.
(ii) limits with in which 𝑥𝑦 is expected to lie. Given your answer correct to 3
decimal places. (07 marks)
15. Two adjudicators at a music competition award marks to 10 pianists as
follows;
Adjudicator 1 78 66 73 73 84 66 89 87 67 77
Adjudicator 2 81 68 81 75 80 67 85 63 66 78
(a) (i) Draw a scatter diagram to show the awards of the two adjudicators.
(ii) Draw a line of best fit on the scatter diagram and estimate a mark
adjudicator 2 would give if adjudicators 1 gave 75 marks.
(b) Calculate the rank correlation coefficient and comment on your result.
(12 marks)
16. A non-uniform beam AB of length 4m rests in a horizontal position on vertical
support at A and B. the centre of gravity is at a point 1.5m from A. if the
reaction at B is 37.5N,
(a)Find the;(i) weight of the beam.
(ii) reaction at A.
(b) The beam is made to lean against a smooth vertical wall with A on a rough
horizontal ground. Find the coefficient of friction necessary to maintain the
beam inclined at 300 to the horizontal. (12 marks)
END
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