AS & A Level Mathematics (9709) — October–November 2023, Paper 33

Find the set of values of x satisfying the inequality 2^(x+1) − 2 < 0.5, giving your answer to 3 significant figures.

On an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities z − 1 + 2i ⩽ z and z − 2 ⩽ 1.

The polynomial 2x³ + ax² + bx + 6, where a and b are constants, is denoted by p(x). When p(x) is divided by (x + 2) the remainder is −38 and when p(x) is divided by (2x − 1) the…

Solve the quadratic equation (3 + i)w² − 2w + 3 − i = 0, giving your answers in the form x + iy, where x and y are real.

Find the exact coordinates of the stationary points of the curve y = e^(3x²−1) / (1 − x²).

Show that the equation cot²θ + 2cos2θ = 4 can be written in the form 4sin⁴θ + 3sin²θ − 1 = 0.

The equation of a curve is x³ + y² + 3x² + 3y = 4. Show that dy/dx = −(3x² + 6x) / (2y + 3).

The variables x and y satisfy the differential equation e^(4x) · dy/dx = cos²3y. It is given that y = 0 when x = 2. Solve the differential equation, obtaining an expression for y…

Let f(x) = (17x² − 7x + 16) / ((2 + 3x²)(2 − x)). Express f(x) in partial fractions.

The diagram shows the curve y = x cos 2x, for x ⩾ 0. Find the equation of the tangent to the curve at the point where x = ½π.

The line l has equation r = i − 2j − 3k + λ(−i + j + 2k). The points A and B have position vectors −2i + 2j − k and 3i − j + k respectively. Find a unit vector in the direction of…