AS & A Level Mathematics (9709) — October–November 2024, Paper 32

Expand (9 − 3x)^(1/2) in ascending powers of x, up to and including the term in x², simplifying the coefficients.

By sketching a suitable pair of graphs, show that the equation cot 2x = sec x has exactly one root in the interval 0 < x < ½π.

The square roots of 6 − 8i can be expressed in the Cartesian form x + iy, where x and y are real and exact. By first forming a quartic equation in x or y, find the square roots of…

Solve the equation 5^x = 5^(x+2) − 10. Give your answer correct to 3 decimal places.

The complex number u is given by u = (cos ⅐π + i sin ⅐π)⁴ / (cos ⅐π − i sin ⅐π). Find the exact value of arg u.

The variables x and y satisfy the equation ay = b^x, where a and b are constants. The graph of ln y against x is a straight line passing through the points (0.50, 2.24) and (3.40,…

Show that the equation tan³x + 2 tan 2x − tan x = 0 may be expressed as tan⁴x − 2 tan²x − 3 = 0 for tan x ≠ 0.

The parametric equations of a curve are x = tan²2t, y = cos 2t, for 0 < t < ¼π. Show that dy/dx = −½ cos³2t.

With respect to the origin O, the points A, B and C have position vectors given by OA = (2, 1, −3), OB = (0, 4, 1) and OC = (−3, −2, 2) (each written as a column vector). The…

A balloon in the shape of a sphere has volume V and radius r. Air is pumped into the balloon at a constant rate of 40π starting when time t = 0 and r = 0. At the same time, air…

Let f(x) = 2e^(2x) / (e^(2x) − 3e^x + 2). Find f′(x) and hence find the exact coordinates of the stationary point of the curve with equation y = f(x).