AS & A Level Mathematics (9709) — May–June 2024, Paper 31

Expand (3 + x)(1 − 2x)^(1/2) in ascending powers of x, up to and including the term in x², simplifying the coefficients.

Solve the equation ln(x − 5) = 7 − ln x. Give your answer correct to 2 decimal places.

The variables x and y satisfy the equation a^y = bx, where a and b are constants. The graph of y against ln x is a straight line passing through the points (0.336, 1.00) and…

The complex number u is given by u = −1 − i√3. Express u in the form r(cos θ + i sin θ), where r 0 and −π < θ ≤ π. Give the exact values of r and θ.

The equation of a curve is y = e^(sin x) / cos²x for 0 ≤ x ≤ 2π. Find dy/dx and hence find the x-coordinates of the stationary points of the curve.

By sketching a suitable pair of graphs, show that the equation cosec(1/2 x) = e^x − 3 has exactly one root, denoted by α, in the interval 0 < x < π.

On a single Argand diagram sketch the loci given by the equations z − 3 + 2i = 2 and w − 3 + 2i = w + 3 − 4i , where z and w are complex numbers.

Use the substitution u = 1 − sin x to find the exact value of ∫ from π to (3/2)π of (sin 2x) / √(1 − sin x) dx. Give your answer in the form a + b√2, where a and b are rational…

The equations of two straight lines l₁ and l₂ are l₁: r = i − 2j + 3k + λ(2i − j + ak) and l₂: r = −i − j − k + μ(3i − 2j − 2k), where a is a constant. The lines l₁ and l₂ are…

Given that 2x = tan y, show that dy/dx = 2 / (1 + 4x²).

In a field there are 300 plants of a certain species, all of which can be infected by a particular disease. At time t after the first plant is infected there are x infected…