AS & A Level Mathematics (9709) — May–June 2025, Paper 12

The diagram shows the graphs with equations y = f(x) and y = g(x). The graph of y = f(x): starts at (−2, 4), rises to a maximum at (0, 6), then falls linearly to (3, 3) and…

Find the coordinates of the points of intersection of the curve and the line with equations 2xy + 5y^2 = 24 and 2x + y + 4 = 0.

The coefficient of x^7 in the expansion of (px^2 + (4/p) x)^5 is 1280. Find the value of the constant p.

A point P is moving along the curve with equation y = a x^(3/2) − 12x in such a way that the x-coordinate of P is increasing at a constant rate of 5 units per second. Find the…

The equation of a curve is y = 4 cos 2x + 3 for 0 ≤ x ≤ 2π. State the greatest and least possible values of y.

The diagram shows the curve with equation y = 9 / (5x + 4)^(1/2) and the line y = 6 − 3x. The line and the curve intersect at the point P which has y-coordinate 3. The shaded…

Prove the identity (tan θ + 7) / (tan^2 θ − 3) ≡ (sin θ cos θ + 7 cos^2 θ) / (1 − 4 cos^2 θ).

The diagram shows the circle with equation x^2 + y^2 − 14x + 8y + 36 = 0 and the line y = −2. The line intersects the circle at the points A and B. The centre of the circle is C.…

The equation of a curve is such that d^2y/dx^2 = −24/x^3. It is given that the curve has a stationary point at (−2, 19). Find an expression for dy/dx.

The first, second and third terms of an arithmetic progression are 4k, k^2 and 8k respectively, where k is a non-zero constant. Find the value of k.

Express x^2 + 4x + 2 in the form (x + a)^2 + b, where a and b are integers.