cameroon gce A level June 2023 pure maths with mechanics 2

cameroon gce A level June 2023 pure maths with mechanics 2

cameroon gce A level June 2023 pure maths with mechanics 2

(i) Given that (x-1) and (x + 1) are factors of the polynomial f(x) = ax4 + 7×3 + x2 + bx — 3,
find the values of the constants a and b .
Hence, find the set of real values of x for which f(x) > 0.
1.
(3 marks)
(4 marks)
(ii) Given that ?n log3 2 = 101og9 6. Find the values of m. (3 marks)
2. The table below shows the values of two continuous variables x and y obtained experimentally.
x 2 3 4 5 6
y 1 0.75 0.67 0.63 0.60
It is known that x and y have a relation of the form – +- = 1.
x y
By drawing the graph of – against – , estimate, to one decimal place, the values of the conatants
y %
a and b. (8 marks)
3. (i) Given thatzj = 1 + v3i and z2 = —1 — i,
(a) express zr and z2 in the form r(cos 6 + i sin 9 ), where r £ K and —n < 6 n.
Zi 3
(b) evaluate z2 andarg|( )• (7 marks)
(ii) Prove by mathematical induction that 9n — 1 is divisible by 8, V n £ N. (5 marks)
4 . (i) Given that f(x) = sinxln(tanx), find f'(x) when x =
f 10 x
(ii) Evaluate -=
h vx — 1
(5 marks)
dx , using the substitution u 2 = x-1. (5 marks)
5. (i) Solve the equation cosx = sin2x in the interval 0 < x < 7r. (5 marks) (ii) Express 5cosx-12sinx in the form R cos(x + a) , where R >0 , and 0° < a < 9 0°, giving the value
ofa correct to one decimal place.
Hence or otherwise, find the minimum and maximum values of
10
|5 cosx — 12 sinx| + 15’ (6 marks)
dy
6. (i) Solve the differential equation — = xe 2^ given that y = 0 when x = 0, expressing the answer
in the form y = f(x). (5 marks)
(ii) A relation R defined on the set 5 = {1, 2, 3, 4} is given by
R = {(1,1), (1, 3), (2, 2), (3,1), (3, 3), (4, 4)}.
(a) Show that R is an equivalence relation .
(b) Write down the equivalent classes of the elements 1 and 4 in R .
(3 marks)
(2 marks)