AS & A Level Mathematics (9709) — October–November 2023, Paper 32

Sketch the graph of y = 4x − 2 .

The parametric equations of a curve are x = (ln t)², y = e^(2 − t²), for t 0. Find the gradient of the curve at the point where t = e, simplifying your answer.

The polynomial 2x³ + ax² − 11x + b is denoted by p(x). It is given that p(x) is divisible by (2x − 1) and that when p(x) is divided by (x + 1) the remainder is 12. Find the values…

On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities z − 4 − 3i ≤ 2 and Re z ≤ 3.

Find the exact value of ∫₀⁶ [x(x + 1) / (x² + 4)] dx.

By sketching a suitable pair of graphs, show that the equation cot x = 2 − cos x has one root in the interval 0 < x ≤ (1/2)π.

By expressing 3θ as 2θ + θ, prove the identity cos 3θ ≡ 4 cos³θ − 3 cos θ.

It is given that (2 + 3ai) / (a + 2i) = λ(2 − i), where a and λ are real constants. Show that 3a² + 4a − 4 = 0.

The diagram shows the curve y = sin x cos 2x, for 0 ≤ x ≤ π, and a maximum point M, where x = a. The shaded region between the curve and the x-axis is denoted by R. Find the value…

The equations of the lines l and m are given by l: r = (3, −2, 1) + λ(1, 1, 2) and m: r = (6, −3, 6) + μ(−2, 4, c), where the position and direction vectors are column vectors and…

The variables x and y satisfy the differential equation x² (dy/dx) + y² + y = 0. It is given that x = 1 when y = 1. Solve the differential equation to obtain an expression for y…