AS & A Level Mathematics (9709) — May–June 2024, Paper 21

7 questions · every question answerable online with instant point-by-point AI marking against the official mark scheme.

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Question 1 · 3 marks · Differentiation of trigonometric functions; stationary points

A curve has equation y = 2 tan x − 5 sin x for 0 ≤ x < (1/2)π. Find the x-coordinate of the stationary point of the curve. Give your answer correct to 3 significant figures. [3]

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Question 2 · 5 marks · Implicit differentiation (product rule, logarithmic term)

A curve has equation x² ln y + y² + 4x = 9. Find the gradient of the curve at the point (2, 1). [5]

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Question 3 · 8 marks · The modulus function; graphs of y = |ax + b|

Sketch on the same diagram the graphs of y = 3x − 8 and y = 5 − x. [2]

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Question 4 · 7 marks · Trigonometric identities (double angle, compound angle for tan)

Show that 3 tan 2θ + tan(θ + 45°) ≡ (tan²θ + 8 tan θ + 1)/(1 − tan²θ). [4]

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Question 5 · 9 marks · Differentiation using the quotient rule; forming an equation for a stationary point

A curve has equation y = (1 + e^(2x))/(1 + 3x). The curve has exactly one stationary point P. Find dy/dx and hence show that the x-coordinate of P satisfies the equation x = 1/6 +…

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Question 6 · 9 marks · Numerical integration (trapezium rule)

The diagram shows the curve with equation y = √(sin 2x + sin² 2x) for 0 ≤ x ≤ (1/6)π. The shaded region is bounded by the curve and the straight lines x = (1/6)π and y = 0. Use…

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Question 7 · 9 marks · Polynomial division; remainder theorem

The polynomial p(x) is defined by p(x) = 9x³ + 6x² + 12x + k, where k is a constant. Find the quotient when p(x) is divided by (3x + 2) and show that the remainder is (k − 8). [3]

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