cameroon gce O level June 2023 additional mathematics paper 2

cameroon gce O level June 2023 additional mathematics paper 2

cameroon gce O level June 2023 additional mathematics paper 2

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SECTION A: PURE MATHEMATICS
THIS SECTION IS COMPULSORY TO ALL CANDIDATES
(ANSWER ALL QUESTIONS)
1. (i) Given that (x — 1) is a factor off (x), where f (x) = x3 — 2×2 — kx + 6.
a) Find the value of k,
With this value of k,
b) Factorise f (x) completely.
(ii) Given that a and fl are the roots of the equation 2×2 — x — 1 = 0.
a) Find the values of a + /I and a/3
b) Hence, find a quadratic equation with integral coefficient whose roots are a2 and /32 . (2 marks)
(2 marks)
(1 mark)
(3 marks)

2. (i) In how many ways can 3boys and 2 girls be chosen from a class of 6 boys and 4 girls. (3 marks)
(ii) Expand (1 — 2x)-1 in ascending powers of x, up to and including the term in x3 (4 marks)
3. Chia, Mbako and Fru started a business with total amount of 350,000 FCFA. Of this amount, Chia contributed
50,000 FCFA, Mbako contributed 100,000 FCFA and Fru contributed 200,000 FCFA. The monthly profit they made was shared amongst the three of them in the ratio of the amount they contributed. Given that they made a total profit of 21,000 FCFA in the first month, 28,000 FCFA in the second month and 35,000 FCFA in the third month and that the profit kept increasing in this manner for the first 12 months,
4. (i) Given that a binary operation * defined over the set G = (0, 1, 2, 3,4) by x * y = (x + y)modulo 5 forms a group
(a) Copy and complete the table.

* 0 1 2 3 4
0 1 3 4
1 1 2 3
2 2
3 3 4 0
4 4 1 2 3

From the table,
(b) State the identity element,
(c) State the inverse of each element.
(ii) The transformation T is defined by the matrix M,where M = (2
1
(a) Find the inverse of the matrix M
Hence or otherwise,
(b) find the point whose image is (6 , —5)
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5. Eposi has only 6,000 FCFA to buy her books and pens. She buys x books for 600 FCFA each and y pens for 300 FCFA.
(a) Write down the total cost in terms ofx and y of these materials and hence
show that 2x + y < 20. (2 marks)
She intended to buy at least 4 books and decided to buy more than twice as many books as pens.
(b) Write down further two inequalities involving x and y. (2 marks)
On a graph paper, using the scale 2 cm to represent 2 units on the x- axis and 2 cm to represent 4 units on they- axis:
(c) Shade, so as to leave unshaded , the region satisfied by these three inequalities and hence determine the minimum number of books and pens she buys. (4 marks)

6. (i) Solve, for x, in the range, 0° S x 180 0, the equation cosx sin2x = 0.
(ii) A function f, of a real variable, 0, is defined by f (0) = 2cosO + 3sin0 , for 0 S 9 < 27r. (a) Copy and complete the table below.
0 0 77. 27r g 47r 57r 27r
3 3 3 3
f(0) 4 -2 2′

Taking 2cm to represent 23 on the x-axis and 2cm to represent 1 unit on they-axis,
(b) Draw the graph of y = f (0).
(c) From your graph, state the maximum value of y = f (0).

7. Given that the vector equations of the lines 11 and 12 are given by:
11: r = + 3j + s(3i – 2j),
12: r = 5i – j + t(2i + j) , where s and t are constants respectively, intersect,
(a) Find the value of and t . (4 marks)
(b) Write down the position vector of the point of intersection of 11 and 12 (1 mark)
(c) Find the cosine of the angle between 11 and 12. (3 marks)

dy
8. (i) Given that y = X2 , x # -1 , find — simplifying your answer as far as possible. i+x dx
ir
(ii) Evaluate j 02 (2x – sin 2x)cbc (3 marks) (5 marks)

SECTION B: MECHANICS
IF THIS SECTION IS CHOSEN, THEN SECTION C MAY NOT BE CHOSEN
(ANSWER ANY TWO QUESTIONS)
9. (i) Two particles A and B are moving on the coordinates axes OX and OY. At time t seconds, particle A is at the point with position vector r = [(1 + 3t)i + 2t2j]m and particle B is at point with
position vector r = [2t2i – 3j]m.
Find,
(a) the magnitude of the velocity of particle A relative to particle B when t = 2. (4 marks)

(b) Show that the acceleration of B is constant (2 marks)
(ii) Two particles, of masses 2kg and 3kg are moving towards each other with speeds 6ms-1 and 2ms-1 respectively. After collision, they coalesce.
Find;
(a) their common velocity after collision (3 marks)
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(b) the loss in kinetic energy due to the collision (3 marks)
(iii) Two particles M and N of masses 4 kg and 5kg are connected by a light inextensible sting passing over a smooth fixed pulley. The system is released from rest with the strings taut and hanging freely.
(a) draw a diagram showing all the forces acting on the system (1 mark)
(b) find the acceleration of the particles and the tension in the string. (4 marks)
10. (i) The volume of a sphere is increasing at the rate of 25n-cm3s-1. Given that the radius of the sphere
is 5cm, find the rate at which;
(a) the radius is increasing, (3 marks)
(b) the surface area is increasing. (3 marks)
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[The volume of the sphere,V = —3 nr3 and the surface area of the sphere, A = 47r.r2]
(ii) The area bounded by the curve y2 = x3 and the x-axis and the lines x = 2 to x = 4 is rotated completely about the x-axis. Find the volume of the solid generated. (5 marks)
(iii) The position vectors of the centre of gravity of three particles of mass mkg, 5kg and 3kg which are at the point with position vectors 31 — j, —6i + 2j and —2i + 4j respectively is —2i + aj. Given that m and a are constants, find the values of m and a. (6 marks)