past paper · 2018

TMUA 2018 Paper 1

Applications of Mathematical Knowledge. All 20 questions from the 2018 paper, each with a full worked solution. Topics on this paper: Differentiation and integration (5), Algebra and functions (4), Trigonometry (4), Sequences and series (3), Coordinate geometry (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 Find the value of ∫₁⁴ (3 - 2x)/(x√(x)),dx Calculus
2 An arithmetic progression has first term a and common difference d. The sum of the first 5 terms is equal to… Sequences
3 Find the shortest distance between the two circles with equations: (x + 2)² + (y - 3)² = 18 (x - 7)² + (y +… Geometry
4 Consider the simultaneous equations 3x² + 2xy = 4 x + y = a where a is a real constant. Find the complete… Algebra
5 The function f is defined by f(x) = x³ + ax² + bx + c. a, b and c take the values 1, 2 and 3 with no two of… Algebra
6 Find the number of solutions of the equation xsin 2x = cos 2x with 0 ≤ x ≤ 2π. Trigonometry hard
7 The non-zero constant k is chosen so that the coefficients of x⁶ in the expansions of (1 + kx²)⁷ and (k +… Sequences
8 The sum to infinity of a geometric progression is 6. The sum to infinity of the squares of each term in the… Sequences
9 Find the complete set of values of the constant c for which the cubic equation 2x³ - 3x² - 12x + c = 0 has… Calculus
10 x and y satisfy |2 - x| ≤ 6 and |y + 2| ≤ 4. What is the greatest possible value of |xy|? Algebra
11 The line y = mx + 5, where m > 0, is normal to the curve y = 10 - x² at the point (p, q). What is the value… Calculus
12 A curve has equation y = f(x), where f(x) = x(x - p)(x - q)(r - x) with 0 You are given that: ∫₀^r f(x),dx =… Calculus
13 The function f(x) has derivative f'(x). The diagram shows the graph of y = f'(x). Coming in from the top… Calculus
14 The line y = mx + 4 passes through the points (3, log₂ p) and (log₂ p, 4). What are the possible values of p? Logarithms
15 Find the sum of the real solutions of the equation: 3^x - (√(3))^x+4 + 20 = 0 Logarithms
16 The curve C has equation y = x² + bx + 2, where b ≥ 0. Find the value of b that minimises the distance… Geometry
17 There are two sets of data: the mean of the first set is 15, and the mean of the second set is 20. One of… Algebra
18 What is the smallest positive value of a for which the line x = a is a line of symmetry of the graph of y =… Trigonometry
19 A triangle ABC is to be drawn with AB = 10 cm, BC = 7 cm and the angle at A equal to θ, where θ is a certain… Trigonometry
20 Find the value of sin² 0^∘ + sin² 1^∘ + sin² 2^∘ + sin² 3^∘ + ⋯ + sin² 87^∘ + sin² 88^∘ + sin² 89^∘ + sin²… Trigonometry

Answer key for TMUA 2018 Paper 1

1 D2 C3 E4 G5 D6 E7 A8 D9 B10 E11 C12 F13 C14 B15 E16 F17 A18 B19 D20 E