AS & A Level Mathematics (9709) — October–November 2024, Paper 31

The polynomial 4x³ + ax² + 5x + b, where a and b are constants, is denoted by p(x). It is given that (2x + 1) is a factor of p(x). When p(x) is divided by (x − 4) the remainder is…

Find the exact value of ∫₁³ x² ln 3x dx. Give your answer in the form a ln b + c, where a and c are rational and b is an integer.

The equation of a curve is ln(x + y) = 3x²y. Find the gradient of the curve at the point (1, 0).

Show that sec⁴θ − tan⁴θ ≡ 1 + 2 tan²θ.

By sketching a suitable pair of graphs, show that the equation 2 + e^(−0.2x) = ln(1 + x) has only one root.

The diagram shows the curve y = sin 2x(1 + sin 2x), for 0 ≤ x ≤ (3/4)π, and its minimum point M. The shaded region bounded by the curve that lies above the x-axis and the x-axis…

Let f(x) = (5x² + 8x + 5) / ((1 + 2x)(2 + x²)). Express f(x) in partial fractions.

Given that z = 1 + yi and that y is a real number, express 1/z in the form a + bi, where a and b are functions of y.

The position vector of point A relative to the origin O is OA = 8i − 5j + 6k. The line l passes through A and is parallel to the vector 2i + j + 4k. State a vector equation for l.

A large cylindrical tank is used to store water. The base of the tank is a circle of radius 4 metres. At time t minutes, the depth of the water in the tank is h metres. There is a…