AS & A Level Further Mathematics (9231) — May–June 2024, Paper 23

8 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 · Integration (inverse trigonometric form, completing the square)

Find the exact value of the integral from x = 2 to x = 7/2 of 1/sqrt(4x - x^2 - 1) dx.

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Question 2 · 9 marks · Arc length (parametric form, hyperbolic functions)

The curve C has parametric equations x = cosh t, y = sinh t, for 0 < t <= 3/5. The length of C is denoted by s. Show that s = the integral from t = 0 to t = 3/5 of sqrt(cosh 2t)…

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Question 3 · 8 marks · Implicit differentiation

The curve C has equation x^3 + 2xy + 8y^3 = -12. Show that, at the point (-2, -1) on C, dy/dx = -1/2.

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Question 4 · 10 marks · Summation of series (integral comparison, lower bound)

The diagram shows the curve with equation y = x^(-2) for 2 <= x <= N together with a set of (N - 2) rectangles of unit width. By considering the sum of the areas of these…

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Question 5 · 10 marks · Second-order linear differential equations (constant coefficients)

Find the general solution of the differential equation d^2x/dt^2 + 10 dx/dt + 25x = 338 sin t.

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Question 6 · 7 marks · Complex numbers / geometric series

Show that the sum from r = 1 to n of z^(4r) = (z^(4n+2) - z^2) / (z^2 - z^(-2)), for z^2 not equal to z^(-2).

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Question 7 · 12 marks · Differentiation (inverse hyperbolic functions, product rule)

Show that d/dx[ (x/2) sqrt(x^2 - 9) - (9/2) cosh^(-1)(x/3) ] = sqrt(x^2 - 9).

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Question 8 · 14 marks · Vectors (Cartesian equations of planes)

The planes Pi1 and Pi2 do not intersect and are both perpendicular to the vector i + 2j + 3k. The line l intersects Pi1 at the point (1, 6, 0) and intersects Pi2 at the point (3,…

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