AS & A Level Mathematics (9709) — May–June 2025, Paper 31

Sketch the graph of y = 2x − 3 .

It is given that 2 ln p + ln(p − 1) − (1/2) ln(q + 1) = 3. Find q in terms of p.

Find the complex numbers z for which (z + 5i)/(z − 5) is real and z = √17. Give your answers in the form z = x + iy, where x and y are real.

The parametric equations of a curve are x = e^(tan t), y = 3 tan² t. Find the equation of the tangent to the curve at the point (e, 3). Give your answer in the form y = mx + c,…

The polynomial 3x³ + pax² + 7a²x + qa³ is denoted by f(x), where p, q and a are constants and a ≠ 0. When f(x) is divided by (x + 2a) the remainder is −22a³. When f(x) is divided…

It is given that z₁ = 3 e^((1/4)πi), z₂ = (3/2) e^((1/6)πi) and ω = 2 e^((1/2)πi). State the values of ωz₁ and ωz₂. Give your answers in the form r e^(iθ), where r 0 and −π < θ ⩽…

Express 5 sin(x + (1/6)π) − 4 cos x in the form R sin(x − α), where R 0 and 0 < α < (1/2)π. State the exact value of R and give the value of α correct to 3 decimal places.

With respect to the origin O, the points A and B have position vectors 2i + 4k and 5i + j + 6k respectively. The line l₁ passes through the points A and B. Find a vector equation…

The constant a is such that ∫ from 1 to a of 6x ln x dx = 4. Show that a = exp((1/6)(5/a² + 3)), where exp(x) denotes e^x.

Find the quotient and remainder when x³ + 5x² − 2x − 15 is divided by x² − 3.

The diagram shows the curve y = cos x √(sin 2x) for 0 ⩽ x ⩽ (1/2)π. The curve has a maximum point at M, where x = a. Find the exact value of a.