past paper · 2020

TMUA 2020 Paper 2

Mathematical Reasoning. All 20 questions from the 2020 paper, each with a full worked solution. Topics on this paper: Differentiation and integration (4), Trigonometry (3), Proof and counterexample (3), Algebra and functions (2), Coordinate geometry (2), Logic and arguments (2), Graphs and transformations (1), Inequalities and reasoning (1), Sequences and series (1), Number and divisibility (1).

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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 2020 Paper 2 2020 Paper 1

Questions

1 Find the complete set of values of k for which the line y = x - 2 crosses or touches the curve y = x² + kx +… Algebra
2 Given that tanθ = 2 and 180^∘ < θ < 360^∘, find the value of cosθ. Trigonometry
3 A student makes the following claim: For all integers n, the expression 4((9n+1)/(2) - (3n-1)/(2)) is… Proof
4 Consider the following statement: Every positive integer N that is greater than 6 can be written as the sum… Proof
5 Which one of the following shows the graph of y = (2^x)/(1 + 2^x) (Dotted lines indicate asymptotes.) The… Graphs
6 The function f(x) is defined for all real values of x. Which of the following conditions on f(x) is/are… Calculus
7 Consider the following conditions on a parallelogram PQRS, labelled anticlockwise: I   … Geometry
8 A student is asked to prove whether the following statement (*) is true or false: (*) For all real numbers a… Proof
9 A student wishes to evaluate the function f(x) = xsin x, where x is in radians, but has a calculator that… Trigonometry
10 The real numbers a, b, c and d satisfy both 0 must be true? I    a II   b III … Inequalities
11 A spiral line is drawn as shown. Starting from the point (-2, 0) it travels down to (-2, -2), right to (2,… Geometry
12 Which one of A – F correctly completes the following statement? Given that a 0 for all x with a Calculus
13 f(x) is a function for which ∫₀³ (f(x))² dx + ∫₀³ f(x),dx = ∫₀¹ f(x),dx Which of the following claims about… Calculus
14 An arithmetic sequence T has first term a and common difference d, where a and d are non-zero integers.… Sequences
15 Which one of the following is a necessary and sufficient condition for ∑_k=1ⁿsin((kπ)/(3)) = (3)/(2) to be… Trigonometry
16 The Fundamental Theorem of Calculus (FTC) tells us that for any polynomial f: (d)/(dx)(∫₀^x f(t),dt) = f(x)… Calculus
17 A set of six distinct integers is split into two sets of three. The first set of three integers has a mean… Number
18 In this question, f(x) = ax³ + bx² + cx + d and g(x) = px³ + qx² + rx + s are cubic polynomials. If f(x) -… Algebra
19 Nine people are sitting in the squares of a 3 by 3 grid, one in each square. Two people are called… Logic
20 x is a real number and f is a function. Given that exactly one of the following statements is true, which… Logic

Answer key for TMUA 2020 Paper 2

1 E2 F3 C4 G5 A6 A7 H8 E9 F10 F11 G12 D13 D14 A15 D16 D17 E18 G19 E20 C