past paper · 2023

TMUA 2023 Paper 2

Mathematical Reasoning. All 20 questions from the 2023 paper, each with a full worked solution. Topics on this paper: Logic and arguments (4), Differentiation and integration (3), Proof and counterexample (3), Coordinate geometry (3), Algebra and functions (2), Number and divisibility (2), Exponentials and logarithms (1), Inequalities and reasoning (1), Trigonometry (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 2023 Paper 2 2023 Paper 1

Questions

1 Given that (1)/(√(x) - 6) - (1)/(√(x) + 6) = (3)/(11) what is the value of x? Algebra
2 Evaluate ∫₉¹⁶((1)/(√(x)) + √(x))² dx - ∫₉¹⁶((1)/(√(x)) - √(x))² dx Calculus
3 Consider the claim: For all positive real numbers x and y, √(x^y) = x^√(y) Which of the following is/are a… Proof
4 A student attempts to answer the following question. What is the largest number of consecutive odd integers… Proof
5 Consider the two statements R:   k is an integer multiple of π S:   ∫₀^k sin 2x,dx = 0… Logic
6 Consider the following equation where a is a real number and a > 1: (*) a^x = x Which of the following… Logarithms
7 The graph of the line ax + by = c is drawn, where a, b and c are real non-zero constants. Which one of the… Geometry
8 A student draws a triangle that is acute-angled or obtuse-angled but not right-angled. The student counts… Geometry hard
9 Consider the following statement about a pentagon P: (*) If at least one of the interior angles in P is… Logic
10 Here is an attempt to solve the inequality x⁴ - 2x² - 3 x⁴ - 2x² - 3 I     if and only… Inequalities
11 In this question, k is a positive integer. Consider the following theorem: If 2^k + 1 is a prime, then k is… Logic
12 In this question, p is a real constant. The equation sin x cos² x = p² sin x has n distinct solutions in the… Trigonometry hard
13 Let x be a real number. Which one of the following statements is a sufficient condition for exactly three of… Logic
14 Three lines are given by the equations: ax + by + c = 0 bx + cy + a = 0 cx + ay + b = 0 where a, b and c are… Geometry hard
15 The base 10 number 0.03841 has the value 0× 10⁻¹ + 3× 10⁻² + 8× 10⁻³ + 4× 10⁻⁴ + 1× 10⁻⁵ = 0.03841… Number
16 A sequence is defined by u₁ = a u₂ = b u_n+2 = u_n + u_n+1 for n ≥ 1 where a and b are positive integers.… Number hard
17 The ceiling of x, written ⌈ x ⌉, is defined to be the value of x rounded up to the nearest integer. For… Calculus
18 The equation x⁴ + bx² + c = 0 has four distinct real roots if and only if which of the following conditions… Algebra
19 In this question, f(x) is a non-constant polynomial, and g(x) = x,f'(x). f(x) = 0 for exactly M real values… Proof
20 Let f be a polynomial with real coefficients. The integral I_p,q where p Which of the following statements… Calculus hard

Answer key for TMUA 2023 Paper 2

1 H2 F3 C4 G5 A6 F7 E8 D9 D10 A11 G12 C13 C14 F15 B16 B17 F18 D19 H20 D