past paper · 2019

TMUA 2019 Paper 2

Mathematical Reasoning. All 20 questions from the 2019 paper, each with a full worked solution. Topics on this paper: Algebra and functions (4), Proof and counterexample (3), Sequences and series (2), Number and divisibility (2), Inequalities and reasoning (2), Logic and arguments (2), Differentiation and integration (2), Coordinate geometry (1), Exponentials and logarithms (1), Graphs and transformations (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 2019 Paper 2 2019 Paper 1

Questions

1 Find the coefficient of the x⁴ term in the expansion of x²(2x + (1)/(x))⁶ Sequences
2 (2x + 1) and (x - 2) are factors of 2x³ + px² + q. What is the value of 2p + q? Algebra
3 a, b and c are real numbers. Given that ab = ac, which of the following statements must be true?… Algebra
4 Consider the following conjecture: If N is a positive integer that consists of the digit 1 followed by an… Proof
5 Consider the following statement about the positive integers a, b and n: (*):  ab is divisible by n The… Number
6 A student attempts to solve the equation cos x + sin x tan x = 2sin x - 1 in the range 0 ≤ x ≤ 2π. The… Proof
7 For which one of the following statements can the fact that 12² + 16² = 20² be used to produce a… Proof
8 a, b and c are real numbers with a Which of the following statements must be true? I    ac… Inequalities
9 A large circular table has 40 chairs round it. What is the smallest number of people who can be sitting at… Number
10 PQRS is a quadrilateral, labelled anticlockwise. Which one of the following is a necessary but not… Geometry
11 An arithmetic series has n terms, all of which are integers . The sum of the series is 20. Which of the… Sequences
12 Most students in a large college study Mathematics. A teacher chooses three different students at random,… Logic
13 A student approximates the integral ∫_a^b sin² x,dx using the trapezium rule with 4 strips. The resulting… Calculus
14 Consider the following statements about the polynomial p(x), where a I    p(a) ≤ p(b)… Calculus
15 The numbers a, b and c are each greater than 1. The following logarithms are all to the same base: log(ab²c)… Logarithms
16 The graph of the quadratic y = px² + qx + p where p > 0, intersects the x-axis at two distinct points. In… Algebra
17 A multiple-choice test question offered the following four options relating to a certain statement:… Logic
18 Consider the following inequality: (*):   a|x| + 1 ≤ |x - 2| where a is a real constant. Which one… Inequalities hard
19 Find the value of the expression √(8 - 4√(2) + 1) + √(9 - 12√(2) + 8) Algebra
20 When the graph of the function y = f(x), defined on the real numbers, is reflected in the y-axis and then… Graphs

Answer key for TMUA 2019 Paper 2

1 E2 C3 A4 D5 B6 D7 B8 E9 D10 A11 A12 F13 D14 B15 B16 F17 D18 E19 E20 B