past paper · 2022

TMUA 2022 Paper 2

Mathematical Reasoning. All 20 questions from the 2022 paper, each with a full worked solution. Topics on this paper: Logic and arguments (5), Proof and counterexample (3), Coordinate geometry (3), Differentiation and integration (2), Sequences and series (2), Inequalities and reasoning (2), Exponentials and logarithms (1), Graphs and transformations (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 2022 Paper 2 2022 Paper 1

Questions

1 Determine the number of stationary points on the curve with equation y = 3x⁴ + 4x³ + 6x² - 5 Calculus
2 Find the coefficient of the x⁵ term in the expansion of (1+x)⁵ × ∑_i=0⁵ x^i Sequences
3 Consider the following statement about the positive integer n: if n is prime, then n² + 2 is not prime Which… Proof
4 The point P has coordinates (p, q), and the equation of a circle is x² + 2fx + y² + 2gy + h = 0 where f, g,… Geometry
5 A straight line L passes through (1, 2). Let P be the statement if the y-intercept of L is negative, then… Logic
6 A list consists of n integers. Consider the following statements: P:   n is odd. Q:  … Logic
7 Consider the following claim: The difference between two consecutive positive cube numbers is always prime.… Proof
8 A selection, S, of n terms is taken from the arithmetic sequence 1, 4, 7, 10, …, 70. Consider the following… Sequences hard
9 Consider the following statement: (*) For all real numbers x, if x then x² What is the complete set of… Logic
10 Which of the following statements is/are true? I   For all real numbers x and for all positive… Logic
11 A kite PQRS has diagonals meeting at O, with OP = x, OQ = y, OR = x, OS = z Which of the following is… Geometry
12 Place the following integrals in order of size, starting with the smallest. P = ∫₀¹ 2^√(x),dx Q = ∫₀¹ 2^x,dx… Calculus
13 Consider the statement (*) about a real number x: (*) There exists a real number y such that x - xy + y is… Logic
14 Consider the two inequalities: |x + 5| < |x + 11| |x + 11| < |x + 1| Which one of the following is correct? Inequalities
15 The real numbers x, y and z are all greater than 1, and satisfy the equations log_x y = z log_y z = x Which… Logarithms
16 In this question, a₁, …, a₁₀₀ and b₁, …, b₁₀₀ and c₁, …, c₁₀₀ are three sequences of integers such that a_n… Inequalities
17 A student answered the following question. a and b are non-zero real numbers. Prove that the equation x³ +… Proof hard
18 Four graphs P, Q, R and S show y = (cos x)^cos x, y = (sin x)^sin x, y = (cos x)^sin x, y = (sin x)^cos x… Graphs hard
19 A polygon has n vertices, where n ≥ 3. It has the following properties: Every vertex of the polygon lies on… Geometry hard
20 The functions f₁ to f₅ are defined on the real numbers by f₁(x) = cos x f₂(x) = sin(cos x) f₃(x) =… Trigonometry hard

Answer key for TMUA 2022 Paper 2

1 B2 E3 C4 B5 F6 C7 F8 C9 A10 G11 C12 F13 E14 D15 F16 D17 E18 E19 B20 E