AS & A Level Mathematics (9709) — October–November 2024, Paper 22

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 · Logarithms (laws of logarithms, exponential equations)

Use logarithms to show that the equation 5^(8y) = 6^(7x) can be expressed in the form y = kx. Give the value of the constant k correct to 3 significant figures.

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Question 2 · 6 marks · Differentiation (chain rule, trigonometric functions)

Let f(x) = 4 sin²3x. Find the value of f'(¼π).

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Question 3 · 6 marks · Differentiation (implicit differentiation)

A curve has equation 6e^(−x)y² + e^(2x) − 12y + 7 = 0. Find the gradient of the curve at the point (ln 3, 2).

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Question 4 · 7 marks · Graphs (exponential and modulus functions)

Sketch the graphs of y = 1 + e^(2x) and y = x − 4 on the same diagram.

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Question 5 · 10 marks · Algebra (factor theorem, remainder theorem)

The polynomial p(x) is defined by p(x) = ax³ + bx² − ax + 8, where a and b are constants. It is given that (x + 2) is a factor of p(x), and that the remainder is 24 when p(x) is…

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

The diagram shows the curves with equations y = ∛(5x² + 7) and y = 27/(2x + 5) for x ≥ 0. The curves meet at the point (2, 3). Region A is bounded by the curve y = ∛(5x² + 7) and…

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Question 7 · 9 marks · Trigonometry (double-angle formulae, R sin(θ − α) form)

Express 4 sin θ sin(θ + 60°) in the form a + R sin(2θ − α), where a and R are positive integers and 0° < α < 90°.

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