past paper · 2017

TMUA 2017 Paper 1

Applications of Mathematical Knowledge. All 20 questions from the 2017 paper, each with a full worked solution. Topics on this paper: Differentiation and integration (6), Sequences and series (4), Coordinate geometry (3), Algebra and functions (3), Trigonometry (2), Exponentials and logarithms (2).

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Twenty questions, seventy-five minutes, no calculator, in a replica of the real test driver. Reading the solutions before you have tried a paper spends it.

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Questions

1 Given that (dy)/(dx) = 3x² - (2 - 3x)/(x³), x ≠ 0 and y = 5 when x = 1, find y in terms of x. Calculus
2 The function f is given by f(x) = ((2)/(x) - (1)/(2x²))² (x ≠ 0) What is the value of f''(1)? Calculus
3 A line l has equation y = 6 - 2x. A second line is perpendicular to l and passes through the point (-6, 0).… Geometry
4 When (3x² + 8x - 3) is multiplied by (px - 1) and the resulting product is divided by (x + 1), the remainder… Algebra
5 S is the complete set of values of x which satisfy both the inequalities x² - 8x + 12 9 The set S can also… Algebra
6 A tangent to the circle x² + y² = 144 passes through the point (20, 0) and crosses the positive y-axis. What… Geometry
7 The first three terms of an arithmetic progression are p, q and p² respectively, where p The first three… Sequences
8 Find the complete set of values of x, with 0 ≤ x ≤ π, for which (1 - 2sin x)cos x ≥ 0 Trigonometry
9 A circle has equation x² + y² - 18x - 22y + 178 = 0 A regular hexagon is drawn inside this circle so that… Geometry
10 A curve C has equation y = f(x) where f(x) = p³ - 6p²x + 3px² - x³ and p is real. The gradient of the normal… Calculus hard
11 The sequence x_n is defined by the rules x₁ = 7 x_n+1 = (23x_n - 53)/(5x_n + 1) The first three terms in the… Sequences
12 The polynomial function f(x) is such that f(x) > 0 for all values of x. Given ∫₂⁴ f(x),dx = A, which one of… Calculus
13 In the expansion of (a + bx)⁵ the coefficient of x⁴ is 8 times the coefficient of x². Given that a and b are… Sequences
14 The solution of the simultaneous equations 2^x + 3 × 2^y = 3 2^2x - 9 × 2^2y = 6 is x = p, y = q. Find the… Logarithms hard
15 It is given that f(x) = -2x² + 10 Consider the following three curves: (1)  y = f(x) (2)  y =… Calculus
16 The functions f and g are given by f(x) = 3x² + 12x + 4 and g(x) = x³ + 6x² + 9x - 8. What is the complete… Calculus
17 The two functions F(n) and G(n) are defined as follows for positive integers n: F(n) = (1)/(n)∫₀ⁿ (n - x),dx… Sequences
18 The graph of y = log₁₀ x is translated in the positive y-direction by 2 units. This translation is… Logarithms
19 The set of solutions to the inequality x² + bx + c In terms of p, q and c, what is the set of solutions to… Algebra hard
20 The lengths of the sides QR, RP and PQ in triangle PQR are a, a + d and a + 2d respectively, where a and d… Trigonometry hard

Answer key for TMUA 2017 Paper 1

1 C2 C3 A4 B5 C6 B7 B8 A9 F10 E11 A12 B13 C14 F15 B16 E17 D18 A19 D20 E