(a) Given y = sin2 3x, show that dy/dx = 3sin 6x.
(b) The curve C has equation y = ln(x2 + 1) - x. Show that C has exactly one stationary point, find its coordinates, and show that it is a point of inflection.
From Paper 1: Pure Mathematics
1111 marks
The number of fish, P, in a lake t years after stocking is modelled by P = 2000/(1 + 9e-0.4t).
(a) Find the initial number of fish.
(b) State the long-term number of fish predicted by the model.
(c) Find the time taken for the population to reach 1000.
(d) Show that dP/dt = 0.4P (1 - P/2000 ).
(e) Hence find the greatest rate of increase of the population.
From Paper 1: Pure Mathematics
1212 marks
A curve has parametric equations x = t + 1/t, y = t - 1/t, for t > 0.
(a) Find a Cartesian equation of the curve.
(b) Find dy/dx in terms of t.
(c) Find the exact coordinates of the point where the gradient of the curve is 2.
(d) Find the equation of the tangent at this point.
From Paper 2: Pure Mathematics and Statistics
38 marks
A curve has equation y3 + 3xy - x2 = 3.
(a) Find dy/dx in terms of x and y.
(b) Verify that (1, 1) lies on the curve and find the equation of the tangent there.
From Paper 3: Pure Mathematics and Mechanics
38 marks
The curve C has equation y = exsin x, for 0 ≤ x ≤ 2π.
(a) Find the exact x-coordinates of the stationary points of C.
(b) Determine the nature of each stationary point.
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This paper is written by Plasmo in the style of the OCR A-Level Maths exam. It is not a past paper and is not produced or endorsed by the exam board.