A curve has equation y = x e-2x.
(a) Show that dy/dx = e-2x(1 - 2x).
(b) Find the exact coordinates of the stationary point and determine its nature.
(c) Find the equation of the tangent to the curve at the origin.
From Paper 1: Pure Mathematics
106 marks
A curve is defined by the parametric equations x = t2, y = 2t - t3, for t ≠ 0.
(a) Find dy/dx in terms of t.
(b) Find the exact coordinates of the points where the tangent to the curve is horizontal.
From Paper 1: Pure Mathematics
117 marks
The temperature T °C of a cup of tea t minutes after it is made is modelled by T = 20 + 70e-kt. After 10 minutes the temperature is 60°C.
(a) Show that k = 1/10 ln 7/4.
(b) Find the time taken for the tea to cool to 30°C.
(c) Find the rate at which the tea is cooling when t = 10.
From Paper 2: Pure and Mechanics
11 mark
Find d/dx(tan 3x).
From Paper 2: Pure and Mechanics
45 marks
A curve has equation x2 + xy + y2 = 7.
Find the equation of the normal to the curve at the point (1, 2), giving your answer in the form ax + by + c = 0 where a, b, c are integers.
From Paper 3: Pure and Statistics
34 marks
Given y = (x2 + 1)/(2x - 1), find dy/dx in its simplest form and its value when x = 1.
From Paper 3: Pure and Statistics
78 marks
A spherical balloon is inflated so that its volume increases at a constant rate of 10 cm3 s-1.
(a) Find the rate of increase of the radius when the radius is 5 cm.
(b) Find the rate of increase of the surface area at this instant.
[V = 4/3π r3, S = 4π r2]
Score 0 / 39 · 0 of 7 marked
This paper is written by Plasmo in the style of the AQA A-Level Maths exam. It is not a past paper and is not produced or endorsed by the exam board.