past paper · 2016

TMUA 2016 Paper 1

Applications of Mathematical Knowledge. All 20 questions from the 2016 paper, each with a full worked solution. Topics on this paper: Differentiation and integration (6), Algebra and functions (4), Sequences and series (3), Trigonometry (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 It is given that the expansion of (ax + b)³ is 8x³ - px² + 18x - 3√(3), where a, b and p are real constants.… Algebra
2 The expression 3x³ + 13x² + 8x + a, where a is a constant, has (x + 2) as a factor. Which one of the… Algebra
3 A line is drawn normal to the curve y = (2)/(x²) at the point on the curve where x = 1. This line cuts the… Calculus
4 The sequence a_n is defined by the rule a_n = (-1)ⁿ - (-1)ⁿ⁻¹ + (-1)ⁿ⁺² for n ≥ 1. Find the value of ∑_n=1³⁹… Sequences
5 What is the total area enclosed between the curve y = x² - 1, the x-axis and the lines x = -2 and x = 2? Calculus
6 P, Q, and R are each mixtures of red and white paint. The percentage by volume of red paint in P is 30%. The… Algebra
7 60% of a sports club's members are women and the remainder are men. This sports club offers the opportunity… Algebra
8 Find the maximum angle x in the range 0^∘ ≤ x ≤ 360^∘ which satisfies the equation cos²(2x) + √(3)sin(2x) -… Trigonometry
9 The line segment joining the points (3, 3) and (7, 5) is a diameter of a circle. This circle is translated… Geometry
10 How many solutions does the equation xtan x = 1 have in the interval -2π ≤ x ≤ 2π? Trigonometry
11 The real roots of the equation 4^2x + 12 = 2^2x+3 are p and q, where p > q. The value of p - q can be… Logarithms
12 A right circular cylinder is contained within a sphere of radius 5 cm in such a way that the whole of the… Calculus
13 How many real roots does the equation 3x⁵ - 10x³ - 120x + 30 = 0 have? Calculus
14 The terms of an infinite series S are formed by adding together the corresponding terms in two infinite… Sequences hard
15 The least possible value of the gradient of the curve y = (2x + a)(x - 2a)² at the point where x = 1, as a… Calculus
16 Given the simultaneous equations log₁₀ 2 + log₁₀(y-1) = 2log₁₀ x log₁₀(y + 3 - 3x) = 0 the values of y are Logarithms
17 It is given that y = (1 + 2cos x)cos 2x for 0 The complete set of values of x for which y is negative is Trigonometry
18 The function 1-x[3]x² is defined for all x ≠ 0. The complete set of values of x for which the function is… Calculus
19 The coefficient of x³ in the expansion of (1 + 2x + 3x²)⁶ is equal to twice the coefficient of x⁴ in the… Sequences hard
20 The diagram shows a square-based pyramid with base PQRS and vertex O. All the edges of the pyramid are of… Geometry hard

Answer key for TMUA 2016 Paper 1

1 H2 E3 C4 B5 C6 C7 B8 F9 D10 E11 E12 E13 C14 D15 C16 C17 D18 A19 B20 D