AS & A Level Mathematics (9709) — May–June 2023, Paper 33

Solve the equation ln(x + 5) = 5 + ln x. Give your answer correct to 3 decimal places.

Find the quotient and remainder when 2x⁴ − 27 is divided by x² + x + 3.

On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities z − 3 − i ⩽ 3 and z ⩾ z − 4i .

The parametric equations of a curve are x = cos θ / (2 − sin θ), y = θ + 2 cos θ. Show that dy/dx = (2 − sin θ)².

The diagram shows the part of the curve y = x² cos 3x for 0 ⩽ x ⩽ ⅙π, and its maximum point M, where x = a. Show that a satisfies the equation a = ⅓ tan⁻¹(2/(3a)).

Express 3 cos x + 2 cos(x − 60°) in the form R cos(x − α), where R 0 and 0° < α < 90°. State the exact value of R and give α correct to 2 decimal places.

Use the substitution u = cos x to show that ∫₀^π sin 2x e^(2 cos x) dx = ∫₋₁¹ 2u e^(2u) du.

The variables x and y satisfy the differential equation dy/dx = (y² + 4) / (x(y + 4)) for x 0. It is given that x = 4 when y = 2√3. Solve the differential equation to obtain the…

The lines l and m have equations l: r = a i + 3 j + b k + λ(c i − 2 j + 4 k), m: r = i + 2 j + 3 k + μ(2 i − 3 j + k). Relative to the origin O, the position vector of the point P…

Let f(x) = (21 − 8x − 2x²) / ((1 + 2x)(3 − x)²). Express f(x) in partial fractions.

The complex number z is defined by z = (5a − 2i) / (3 + ai), where a is an integer. It is given that arg z = −¼π. Find the value of a and hence express z in the form x + iy, where…