past paper · 2016

TMUA 2016 Paper 2

Mathematical Reasoning. All 20 questions from the 2016 paper, each with a full worked solution. Topics on this paper: Differentiation and integration (4), Coordinate geometry (4), Trigonometry (2), Proof and counterexample (2), Sequences and series (2), Inequalities and reasoning (2), Logic and arguments (1), Exponentials and logarithms (1), Graphs and transformations (1), Algebra and functions (1).

Sit it properly first

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.

Sit 2016 Paper 2 2016 Paper 1

Questions

1 Find the value of ∫₁² (x² - (4)/(x²))² dx Calculus
2 f(x) = (x²+5)(2x)[4]x³, x > 0 Which one of the following is equal to f'(x)? Calculus
3 What is the value, in radians, of the largest angle x in the range 0 ≤ x ≤ 2π that satisfies the equation… Trigonometry
4 Five sealed urns, labelled P, Q, R, S, and T, each contain the same (non-zero) number of balls. The… Logic
5 Consider the statement: (*) A whole number n is prime if it is 1 less or 5 less than a multiple of 6. How… Proof
6 The sequence of functions f₁(x), f₂(x), f₃(x), … is defined as follows: f₁(x) = x¹⁰ f_n+1(x) = x f_n'(x) for… Sequences
7 The four real numbers a, b, c, and d are all greater than 1. Suppose that they satisfy the equation log_c d… Logarithms
8 A region is defined by the inequalities x + y > 6 and x - y > -4. Consider the three statements:… Inequalities
9 Triangles ABC and XYZ have the same area . Which of these extra conditions, taken independently, would imply… Geometry
10 In this question x and y are non-zero real numbers. Which one of the following is sufficient to conclude… Inequalities
11 f(x) is a polynomial with real coefficients. The equation f(x) = 0 has exactly two real roots, x = -p and x… Calculus
12 The first term of an arithmetic sequence is a and the common difference is d. The sum of the first n terms… Sequences
13 In this question, a, b, and c are positive integers. The following is an attempted proof of the false… Proof
14 f(x) = ax⁴ + bx³ + cx² + dx + e, where a, b, c, d, and e are real numbers. Suppose f(x) = 1 has p distinct… Graphs hard
15 Consider the quadratic f(x) = x² - 2px + q and the statement: (*) f(x) = 0 has two real roots whose… Algebra
16 In the figure, PQRS is a trapezium with PQ parallel to SR. The diagonals of the trapezium meet at X. U lies… Geometry
17 Consider these simultaneous equations, where c is a constant: y = 3sin x + 2 y = x + c Which of the… Trigonometry hard
18 Consider this statement about a function f(x): (*) If (f(x))² ≤ 1 for all -1 ≤ x ≤ 1 then ∫_-1¹(f(x))² dx ≤… Calculus
19 Some identical unit cubes are used to construct a three-dimensional object by gluing them together face to… Geometry
20 Each interior angle of a regular polygon with n sides is (3)/(4) of each interior angle of a second regular… Geometry

Answer key for TMUA 2016 Paper 2

1 A2 B3 D4 C5 C6 C7 C8 B9 D10 E11 D12 F13 E14 B15 D16 C17 H18 D19 F20 E