AS & A Level Mathematics (9709) — May–June 2024, Paper 12

10 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 · Series (binomial expansion)

The coefficient of x² in the expansion of (1 − 4x)⁶ is 12 times the coefficient of x² in the expansion of (2 + ax)⁵. Find the value of the positive constant a.

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Question 2 · 5 marks · Functions - transformations of graphs

The curve y = x² is transformed to the curve y = 4(x − 3)² − 8. Describe fully a sequence of transformations that have been combined, making clear the order in which the…

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Question 3 · 6 marks · Trigonometry - identities and equations

Show that the equation (7 tan θ)/(cos θ) + 12 = 0 can be expressed as 12 sin²θ − 7 sin θ − 12 = 0.

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Question 4 · 7 marks · Functions - inverse functions

The function f is defined by f(x) = √x − 1 for x 1. Find an expression for f⁻¹(x).

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Question 5 · 9 marks · Series - arithmetic progression

The first and second terms of an arithmetic progression are tan θ and sin θ respectively, where 0 < θ < ½π. Given that θ = ¼π, find the exact sum of the first 40 terms of the…

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Question 6 · 9 marks · Differentiation - stationary points

The curve with equation y = 2x − 8x^(1/2) has a minimum point at A and intersects the positive x-axis at B. Find the coordinates of A and B.

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Question 7 · 8 marks · Coordinate geometry - circles

The equation of a circle is (x − 6)² + (y + a)² = 18. The line with equation y = 2a − x is a tangent to the circle. Find the two possible values of the constant a.

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Question 8 · 10 marks · Circular measure

It is given that r = 0.4 cm. Show that the length EF = 2.4 cm.

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Question 9 · 8 marks · Differentiation - increasing and decreasing functions

A function f is such that f′(x) = 6(2x − 3)² − 6x for x ∈ ℝ. Determine the set of values of x for which f(x) is decreasing.

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Question 10 · 10 marks · Differentiation - connected rates of change

The equation of a curve is y = (5 − 2x)^(3/2) + 5 for x < 5/2. A point P is moving along the curve in such a way that the y-coordinate of point P is decreasing at 5 units per…

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