AS & A Level Mathematics (9709) — May–June 2025, Paper 35

Solve the equation 3^(4−2x) = 5(6^(x−1)). Give your answer correct to 3 significant figures. [4]

Solve the equation 3 cot θ − 4 cosec^2 θ + 5 = 0 for −π <= θ <= π. [5]

The complex numbers s and t are given by s = 5(cos 0.25 + i sin 0.25) and t = 6e^(3i). Express s/t in the form r e^(iθ), where −π < θ <= π and r 0. [2]

Find the exact coordinates of the stationary point of the curve with equation y = 3x^3 ln(x^4), for x 0. [5]

The diagram shows the locus of points representing the complex numbers, z, satisfying z + 5 − 4i = 3. The locus is a circle of radius 3 centred at the point representing −5 + 4i…

The parametric equations of a curve are x = 2/(cos 3t) and y = tan 3t, for 0 <= t <= 2π. Show that dy/dx can be written as A cosec 3t, where A is a constant to be found. [5]

The equation of a curve is y = tan^(−1)(4x). Find the exact values of x when the gradient of the curve is 1/4. [3]

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

Express (12x^2 + 55x − 2) / ((3x − 2)(x + 6)) in partial fractions. [5]

With respect to the origin O, the points A, B and C have position vectors given by OA = 2i − j − 6k, OB = bi − 2j + 3k and OC = −4i + 5j − 2k. It is given that AB = BC . Find the…

The variables x and y satisfy the differential equation (x^2 + 3)(dy/dx) = e^(3y)(x − 2). It is given that y = 0 when x = 0. Solve the differential equation, and find the value of…