AS & A Level Mathematics (9709) — October–November 2024, Paper 23

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

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Question 1 · 5 marks · Logarithmic and exponential functions (reduction of a relation to linear form)

The variables x and y satisfy the equation a^(2y) = e^(3x+k), where a and k are constants. The graph of y against x is a straight line. Use logarithms to show that the gradient of…

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Question 2 · 4 marks · Algebra (the modulus function — solving inequalities)

Solve the inequality x − 7 4x + 3.

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Question 3 · 7 marks · Differentiation (trigonometric functions; chain rule)

The function f is defined by f(x) = tan²(½x) for 0 ≤ x < π. Find the exact value of f′(⅔π).

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Question 4 · 8 marks · Algebra (factor theorem)

The polynomial p(x) is defined by p(x) = ax³ − ax² − 15x + 18, where a is a constant. It is given that (x + 2) is a factor of p(x). Find the value of a.

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Question 5 · 8 marks · Integration (∫ 1/(ax + b) dx = (1/a) ln|ax + b|)

It is given that ∫ from a to a³ of 10/(2x + 1) dx = 7, where a is a constant greater than 1. Show that a = ³√(0.5 e^(1.4) (2a + 1) − 0.5).

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Question 6 · 7 marks · Differentiation (parametric equations; quotient rule)

A curve has parametric equations x = (e^(2t) − 2)/(e^(2t) + 1), y = e^(3t) + 1. Find an expression for dy/dx in terms of t.

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Question 7 · 11 marks · Trigonometry (compound-angle and double-angle identities)

Prove that cos(θ + 30°) cos(θ + 60°) ≡ ¼√3 − ½ sin 2θ.

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