AS & A Level Mathematics (9709) — October–November 2023, Paper 31

Find the exact coordinates of the points on the curve y = x² / (1 − 3x) at which the gradient of the tangent is equal to 8.

On an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities z − 2i ≤ z + 2 − i and 0 ≤ arg(z + 1) ≤ (1/4)π.

The variables x and y are related by the equation y = ab^x, where a and b are constants. The diagram shows the result of plotting ln y against x for two pairs of values of x and…

The complex number u is defined by u = (3 + 2i) / (a − 5i), where a is real. Express u in the Cartesian form x + iy, where x and y are in terms of a.

Given that sin(x + (1/6)π) − sin(x − (1/6)π) = cos(x + (1/3)π) − cos(x − (1/3)π), find the exact value of tan x.

The parametric equations of a curve are x = √t + 3, y = ln t, for t 0. Obtain a simplified expression for dy/dx in terms of t.

The variables x and θ satisfy the differential equation (x / tan θ) · (dx/dθ) = x² + 3. It is given that x = 1 when θ = 0. Solve the differential equation, obtaining an expression…

By sketching a suitable pair of graphs, show that the equation √x = e^x − 3 has only one root.

The diagram shows the curve y = x·e^(−(1/4)x²), for x ≥ 0, and its maximum point M. Find the exact coordinates of M.

Let f(x) = (24x + 13) / ((1 − 2x)(2 + x)²). Express f(x) in partial fractions.

In the diagram, OABCDEFG is a cuboid in which OA = 3 units, OC = 2 units and OD = 2 units. Unit vectors i, j and k are parallel to OA, OD and OC respectively. M is the midpoint of…