AS & A Level Mathematics (9709) — October–November 2024, Paper 33

The complex number z satisfies z = 2 and 0 ≤ arg z ≤ (1/4)π. On the Argand diagram below, sketch the locus of the points representing z.

Let f(x) = 2x³ − 5x² + 4. Show that if a sequence of values given by the iterative formula x{n+1} = √(4 / (5 − 2xn)) converges, then it converges to a root of the equation f(x) =…

The number of bacteria in a population, P, at time t hours is modelled by the equation P = ae^{kt}, where a and k are constants. The graph of ln P against t, shown in the diagram,…

Find the complex number z satisfying the equation (z − 3i) / (z + 3i) = (2 − 9i) / 5. Give your answer in the form x + iy, where x and y are real.

Show that cos⁴θ − sin⁴θ − 4 sin²θ cos²θ ≡ cos²2θ + cos2θ − 1.

The lines l and m have vector equations l: r = 2i + j − 3k + λ(−i + 2k) and m: r = 2i + j − 3k + μ(2i − j + 5k). Lines l and m intersect at the point P. State the coordinates of P.

The parametric equations of a curve are x = 3 sin 2t, y = tan t + cot t, for 0 < t < (1/2)π. Show that dy/dx = −2 / (3 sin²2t).

Let f(x) = 7a² / ((a − 2x)(3a + x)), where a is a positive constant. Express f(x) in partial fractions.

Find the quotient and remainder when x⁴ + 16 is divided by x² + 4.

A water tank is in the shape of a cuboid with base area 40 000 cm². At time t minutes the depth of water in the tank is h cm. Water is pumped into the tank at a rate of 50 000 cm³…

The diagram shows the curve y = 2 sin x √(2 + cos x), for 0 ≤ x ≤ 2π, and its minimum point M, where x = a. Find the value of a correct to 2 decimal places.