past paper · 2021

TMUA 2021 Paper 2

Mathematical Reasoning. All 20 questions from the 2021 paper, each with a full worked solution. Topics on this paper: Differentiation and integration (4), Coordinate geometry (3), Logic and arguments (3), Proof and counterexample (3), Exponentials and logarithms (2), Trigonometry (2), Number and divisibility (1), Inequalities and reasoning (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 2021 Paper 2 2021 Paper 1

Questions

1 Find the value of ∫₁⁴(3√(x) + (4)/(x²))dx Calculus
2 A(0,2) and C(4,0) are opposite vertices of the square ABCD. What is the equation of the straight line… Geometry
3 A student is chosen at random from a class. Each student is equally likely to be chosen. Which of the… Logic
4 Consider the following claim about positive integers a, b and c: if a is a factor of bc, then a is a factor… Number
5 On which line is the first error in the following argument? Proof
6 Consider the following two statements about the polynomial f(x): P:   f(x) = 0 for exactly three… Logic
7 A circle has equation (x-9)² + (y+2)² = 4 A square has vertices at (1,0), (1,2), (-1,2) and (-1,0). A… Geometry
8 Consider the following statement about the polynomial p(x), where a and b are real numbers with a… Calculus
9 Consider the following statements about a polynomial f(x): I    f(x) = px³ + qx² + rx + s,… Logic
10 The first seven terms of a sequence of positive integers are: u₁ = 15, u₂ = 21, u₃ = 30, u₄ = 37, u₅ = 44,… Proof
11 A student attempts to solve the following problem, where a and b are non-zero real numbers: Show that if a²… Proof
12 Which of the following statements about polynomials f and g is/are true? I    If f(x) ≥ g(x)… Calculus
13 A region R in the (x,y)-plane is defined by the simultaneous inequalities y - x every point in R?… Inequalities
14 Consider the following simultaneous equations, where p is a real number: p,2^x + log₂ y = 2 2^x + log₂ y = 1… Logarithms
15 A circle has equation x² + ax + y² + by + c = 0 where a, b and c are non-zero real constants. Which one of… Geometry
16 p and q are real numbers, and the equation x|x| = px + q has exactly k distinct real solutions for x. Which… Algebra hard
17 Consider the following functions defined for x > 1: f(x) = log₂(log₂ √(x)) g(x) = log₂(√(log₂ x)) Which one… Logarithms hard
18 A student chooses two distinct real numbers x and y with 0 The student then attempts to draw a triangle ABC… Trigonometry hard
19 The angle θ can take any of the values 1^∘, 2^∘, 3^∘, …, 359^∘, 360^∘. For how many of these values of θ is… Trigonometry hard
20 A sequence of functions f₁, f₂, f₃, … is defined by f₁(x) = |x| f_n+1(x) = |f_n(x) + x| for n ≥ 1 Find the… Calculus hard

Answer key for TMUA 2021 Paper 2

1 D2 E3 C4 C5 B6 D7 B8 C9 C10 E11 C12 B13 A14 C15 B16 E17 F18 C19 F20 E