AS & A Level Mathematics (9709) — May–June 2025, Paper 13

A curve has equation y = 2x + 12/(x^2). Find the equation of the tangent to the curve at the point (−2, −1). Give your answer in the form y = mx + c.

The first two terms of a geometric progression are 4 sin²θ, 8 sin³θ, where θ is an angle such that 0 < θ < (1/6)π. Given that the sum to infinity of the progression is 1/2, find…

Given that the integral from 1 to 3 of ( a/((4x − 3)^2) + 2 ) dx = 12, find the value of the constant a.

Find the first three terms in the expansion of (2 − (3/2)x)^5 in ascending powers of x.

Solve the equation 4 sin θ tan θ = 1 + 5 cos θ for −180° < θ < 180°.

An arithmetic progression has first term a and common difference 2. The Nth term is 55 and the sum of the first 3N terms is 5760. Find the values of N and a.

A curve is such that dy/dx = 3x² + 10x − 8. Find the set of values of x for which y decreases as x increases.

The diagram shows a square ABCD where each side has length 12 cm. Points E and F lie on the sides BC and CD respectively and are such that BE = (1/3)BC and DF = (1/3)DC. The arc…

Three points P, Q and R have coordinates P(−13, 5), Q(5, 1) and R(2, k), where k is a constant. It is given that the angle PRQ is a right angle. Show that one of the possible…

A curve C has equation y = 9/(2x − 5) + 2x − 5. Find the coordinates of the two stationary points.

The function f is defined by f(x) = x² + 4ax + a for x ∈ ℝ, where a is a constant. The function g is such that g⁻¹(x) = ³√(2x − 4) for x ∈ ℝ. Given that the range of f is f(x) ⩾…