P425/1

PURE MATHEMATICS
Paper 1
3 hrs.

STANDARD HIGH SCHOOL ZZANA

Uganda Advanced Certificate of Education
Pure Mathematics
Paper 1
3 hours

INSTRUCTIONS TO CANDIDATES:
Answer all the eight questions in Section A and five questions from Section B.
Any additional question(s) answered will not be marked.
All working must be shown clearly.
Begin each answer on a fresh sheet of paper.
Graph paper is provided.
Silent non – programmable scientific calculators and Mathematical tables with a list of
formulae may be used.

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 SECTION A: (40 marks)

Attempt all questions in this section.

1. Solve the equations;

√𝑥 + 2 − √𝑥 − 3 = √𝑥 − 6 and hence verify your answer. (05 marks)

𝑆𝑖𝑛6𝜃 − 𝐶𝑜𝑠6𝜃 + 1
2. Show that = 𝑡𝑎𝑛3𝜃 . (05 marks)
𝑆𝑖𝑛6𝜃 + 𝐶𝑜𝑠6𝜃 + 1

3. The sides of an equilateral triangle increases at a rate of 3𝑐𝑚𝑠 −1 without the

angles changing. Find the rate of increase of area when the side is 10𝑐𝑚.
(05 marks)

4. Determine the value of 𝑟 if;

8𝐶𝑟 = 70 and 8𝑃𝑟 = 1680. (05 marks)

5. Prove that;
(05 marks)
1
−1 (3𝑥) 2
3 6𝑡𝑎𝑛 𝜋
∫ 𝑑𝑥 =
0 5 + 45𝑥 2 80

6𝑡𝑎𝑛−1 (3𝑥) 𝜋2
6. Find the Cartesian equation of a
𝑑𝑥 = plane passing through the midpoint of 𝑃(−1, 0, 5)
5 + 45𝑥 2 80 𝑥−6 𝑧−5
and 𝑄(7, −4, 1) which is perpendicular to the line = −2 − 𝑦 = .
7 2
(05 marks)

7. Determine the length of tangent drawn from a point (5, −3) to the circle
𝑥 2 + 𝑦 2 − 4𝑦 = 12. (05 marks)

𝑑𝑦
8. Solve the differential equation = 𝑒 2𝑥 + 5𝑦 given that y = −3 𝑎𝑡 𝑥 = 0.
𝑑𝑥
(05 marks)

SECTION B: (60 marks)

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 Answer any five questions from this section. All questions carry equal marks.

9. (a) Show that 𝑖 – 1 is a root to the equation

𝑍 4 − 2𝑍 3 − 𝑍 2 + 2𝑍 + 10 = 0 and hence find all other roots to the
equation; . 𝑍 4 − 2𝑍 3 − 𝑍 2 + 2𝑍 + 10 = 0 (06 marks)

(b) Given that the complex numbers 𝑍 and its conjugate 𝑍̅ satisfies the
𝜋
equations; 𝑍 + 3𝑖 − 2 = 2, 𝐴𝑟𝑔 (𝑍 + 2𝑍̅ + 5𝑖) =
4
Find the complex number 𝑍. (06 marks)

10. (a) Solve the equation; 8 𝐶𝑜𝑠 4 𝜃 − 5𝐶𝑜𝑠 2𝜃 = 2 for 0𝑜 ≤ 𝜃 ≤ 360𝑜 .

(07 marks)
(b) If 𝐴, 𝐵 and 𝐶 are angles of a triangle, prove that;
𝑆𝑖𝑛2 𝐴 + 𝑆𝑖𝑛2 𝐵 + 𝑆𝑖𝑛2 𝐶 − 2 = 2𝐶𝑜𝑠𝐴𝐶𝑜𝑠𝐵𝐶𝑜𝑠𝐶 (05 marks)

𝑥+1 𝑦+2 𝑥−6 𝑍

11. Two lines 𝐿1 and 𝐿2 are given by = = 𝑍 − 6 and =𝑦−3=
5 −1 2 5

respectively. 𝐿1 meets 𝑦 = 0 at 𝐴, while 𝐿2 meets 𝑍 = −5 at 𝐵.

(a) Find the;
(i) coordinates of 𝐴 and 𝐵,
(ii) Cartesian equation of line 𝐴𝐵.
(07 marks)

(b) Determine the angle between the plane 5𝑥 + 7𝑦 − 𝑍 = 11 and line in (a) (ii)
above; (05 marks)

𝑑𝑥
12. (a) Find ∫
1−𝑠𝑖𝑛𝑥
(06 marks)
(b) Given that 𝑦 = 𝑒 −𝑥 𝐶𝑜𝑠(𝑥 + 𝜃) where 𝜃 is a constant, show that;
𝑑2𝑦 𝑑𝑦
+2 + 2𝑦 = 0 . (06 marks)
𝑑𝑥 2 𝑑𝑥

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13. (a) Find the equation of normal chord to the rectangular hyperbola 𝑥𝑦 = 4 at a
2
point (2𝑡, ) . (05 marks)
𝑡

(b) If the normal chord to the hyperbola above drawn from a point 𝐴(4, 1)
meets the hyperbola again at point 𝐵, find length of the chord 𝐴𝐵.
(07 marks)

𝑥−1
14. (a) Prove that the curve 𝑦 = has no turning points. (04 marks)
2𝑥 2 − 5𝑥 −3

𝑥−1
(b) Sketch the curve 𝑦 = . (08 marks)
2𝑥 2 − 5𝑥 −3

3 1−2𝑥 1
15. (a) Expand √(
1+𝑥
) in ascending powers of 𝑥 upto 𝑥 3 and hence using 𝑥 = 7
3
estimate √5 correct to three significant figures. (08 marks)

(b) Find the term independent of x in the expansion;

1 6
(3𝑥 2 − 2𝑥) . (04 marks)

16. The temperature of a hot liquid drops at a rate which is directly proportional to the
difference between its temperature 𝜃 and that of the surrounding. At 4: 35𝑃𝑚, the
temperature of a hot liquid was 95𝑜 𝐶 and that of the surrounding was 30𝑜 𝐶. At
4: 37𝑃𝑚 the temperature of the liquid droped to 89𝑜 𝐶. The temperature of the
surrounding is constant;

(a) Show that 𝜃 = 30 + 65𝑒 −𝐾𝑡 where 𝐾 is constant and 𝜃 is temperature after
time 𝑡. (05 marks)
(b) Find the;
(i) temperature of the liquid in the next 4 minutes,
(ii) time when the temperature of the liquid is 60𝑜 𝐶.
(07 marks)

END