past paper · 2018

TMUA 2018 Paper 2

Mathematical Reasoning. All 20 questions from the 2018 paper, each with a full worked solution. Topics on this paper: Proof and counterexample (5), Logic and arguments (3), Differentiation and integration (2), Sequences and series (2), Trigonometry (2), Number and divisibility (2), Coordinate geometry (1), Graphs and transformations (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 2018 Paper 2 2018 Paper 1

Questions

1 The function f is given, for x > 0, by f(x) = (x³ - 4x)/(2√(x)) Find the value of f'(4). Calculus
2 Find the value of the constant term in the expansion of (x⁶ - (1)/(x²))¹² Sequences
3 Consider the following statement: A car journey consists of two parts. In the first part, the average speed… Proof
4 The non-zero real number c is such that the equation cos x = c has two solutions for 0 How many solutions of… Trigonometry
5 The two diagonals of the quadrilateral Q are perpendicular. Consider the following statements: I  … Geometry
6 Which one of the following functions provides a counterexample to the statement: if f'(x) > 0 for all real… Proof
7 Sequence 1 is an arithmetic progression with first term 11 and common difference 3. Sequence 2 is an… Number
8 The diagram shows an example of a mountain profile : a zig-zag path made of strokes above a dashed… Proof
9 Consider the following attempt to solve the equation 4x√(2x-1) = 10x - 5: 4x√(2x-1) = 10x - 5… Proof
10 The function f(x) is defined for all real numbers. Consider the following three conditions, where a is a… Logic
11 Consider the equation 2^x = mx + c, where m and c are real constants. Which of the following statements… Logic
12 Consider the following statement: For any positive integer N there is a positive integer K such that N(Km+1)… Logic
13 The following is an attempted proof of the conjecture: if tanθ > 0, then sinθ + cosθ > 1. Suppose tanθ > 0,… Proof
14 In the triangle PQR, PR = 2, QR = p and ∠ RPQ = 30^∘. What is the set of all the values of p for which this… Trigonometry hard
15 It is given that f(x) = x³ + 3qx² + 2, where q is a real constant. The equation f(x) = 0 has 3 distinct real… Graphs hard
16 In this question, x₁, x₂, x₃, … is an arithmetic progression , all of whose terms are integers. Let n be a… Sequences
17 A positive integer is called a squaresum if and only if it can be written as the sum of the squares of two… Number
18 f(x) is a polynomial function defined for all real x. Which of the following is a necessary condition for… Calculus
19 Three real numbers x, y and z satisfy x > y > z > 1. Which one of the following statements must be true? Inequalities
20 It is given that the equation √(x+p) + √(x) = p has at least one real solution for x, where p is a real… Algebra

Answer key for TMUA 2018 Paper 2

1 C2 B3 F4 D5 A6 C7 B8 D9 F10 B11 A12 F13 F14 E15 G16 F17 C18 C19 C20 B