cameroon gce advanced level June 2026 pure mathematics with statistics 3

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1. (i) The polynomial f(x), where

f(x) = 2x³ + ax² + bx – 3,

is exactly divisible by (x – 1) and has remainder 9 when divided by (x + 2).

Find the values of the constants a and b.

Hence solve the equation f(x) = 0. (7 marks)

(ii) Find the set of real values of x for which

2 / (x – 2) < 1 / (x + 1) (4 marks)

2. (i) If the roots of the quadratic equation ax² + 2bx + c = 0 are α and α + 4, prove that

4a² + ca – b² = 0.

Hence show also that 8a = -c ± √(c² + 16b²) (6 marks)

(ii) Given that β = -1 + 2i, express β² and β³ in the form a + bi, where a, b ∈ ℝ.

Hence show that β is a root of the equation z³ + 7z² + 15z + 25 = 0 and state a second complex root of this equation. (6 marks)

3. (i) The first three terms in the expansion of (1 + bx)ⁿ in ascending powers of x are

1 – (3/5)x – (27/100)x².

Find the values of the constants b and n. (5 marks)

(ii) Consider the relation ℛ on a non-empty set A.

State the conditions that would make ℛ a partial order on A.

Given that A = {1, 2, 3, 4, 6, 12} and that ℛ is the relation defined by:

aℛb ⟺ a divides b,

show that ℛ defines a partial order on A. (6 marks)

4. The table below shows the population P, in thousands to one decimal place, of a certain town T years after 1980.

T	1	3	5	7	9
P	17.8	63.1	223.9	707.9	2511.9

It is believed that P and T are connected by a law of the form

P = abᵀ

By plotting log P against T, estimate to one decimal place, the values of the constants a and b.

Hence, state the approximate population of this town in 1980. (9 marks)

5. (i) The vector equations of two lines L₁ and L₂ are

• L₁ : r = (j – k) + λ(i + 2j + k),

• L₂ : r = (i + 7j – 4k) + μ(i + 3k), where λ and μ are scalar variables.

Find:

• (a) the values of λ and μ for which L₁ and L₂ intersect, stating the position vector of the point of intersection.

• (b) the Cartesian equation of the plane containing L₁ and L₂. (4, 4 marks)

(ii) A bus driver has 8 passengers to carry but he only has room for 4.

In how many ways can he choose the 4 passengers, if two of them are sisters who must not be separated? (3 marks)