past paper · 2023

TMUA 2023 Paper 1

Applications of Mathematical Knowledge. All 20 questions from the 2023 paper, each with a full worked solution. Topics on this paper: Differentiation and integration (5), Sequences and series (4), Coordinate geometry (3), Trigonometry (3), Exponentials and logarithms (2), Graphs and transformations (2), 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 2023 Paper 1 2023 Paper 2

Questions

1 Given that ∫₀¹ (ax + b),dx = 1 and ∫₀¹ x(ax + b),dx = 1 find the value of a + b. Calculus
2 The graphs of y = x² + 5x + 6 and y = mx - 3, where m is a constant, are plotted on the same set of axes.… Algebra
3 For any integer n ≥ 0, ∫_nⁿ⁺¹ f(x),dx = n + 1 Evaluate ∫₀³ f(x),dx + ∫₁³ f(x),dx + ∫₂³ f(x),dx + ∫₄³ f(x),dx… Calculus
4 Evaluate ∑_n=0^∞ (sin(nπ + (π)/(3)))/(2ⁿ) Sequences
5 The following shape has two lines of reflectional symmetry. MNOP is a square of perimeter 40 cm. The… Geometry
6 In the simplified expansion of (2 + 3x)¹², how many of the terms have a coefficient that is divisible by 12? Sequences hard
7 P(x) and Q(x) are defined as follows: P(x) = 2^x + 4 Q(x) = 2^(2x-2) - 2^(x+2) + 16 Find the largest value… Logarithms
8 A triangle XYZ is called fun if it has the following properties: angle YXZ = 30^∘ XY = √(3),a YZ = a where a… Trigonometry
9 How many solutions are there to (1 + 3cos 3θ)² = 4 in the interval 0^∘ ≤ θ ≤ 180^∘? Trigonometry
10 The trapezium rule with 4 strips is used to estimate the integral ∫_-2²√(4 - x²),dx What is the positive… Calculus
11 It is given that f(x) = x² - 6x. The curves y = f(kx) and y = f(x - c) have the same minimum point, where k… Graphs
12 How many solutions are there to the equation 2^tan^2 x4^sin^2 x = 1 in the range 0 ≤ x ≤ 2π? Trigonometry
13 Point P lies on the circle with equation (x-2)² + (y-1)² = 16. Point Q lies on the circle with equation… Geometry
14 The function f(x) = (2)/(3)x³ + 2mx² + n, m > 0 has three distinct real roots. What is the complete range of… Calculus hard
15 The difference between the maximum and minimum values of the function f(x) = a^cos x, where a > 0 and x is… Logarithms hard
16 A right-angled triangle has vertices at (2, 3), (9, -1) and (5, k). Find the sum of all the possible values… Geometry
17 A circle C_n is defined by x² + y² = 2n(x + y) where n is a positive integer. C₁ and C₂ are drawn and the… Sequences hard
18 You are given that S = 4 + (8k)/(7) + (16k²)/(49) + (32k³)/(343) + ⋯ + 4((2k)/(7))ⁿ + ⋯ The value of k is… Sequences hard
19 The solution to the differential equation (dy)/(dx) = |-6x| for all x is y = f(x) + c, where c is a… Calculus
20 The diagram shows the graph of y = f(x). The function f attains its maximum value of 2 at x = 1, and its… Graphs hard

Answer key for TMUA 2023 Paper 1

1 F2 A3 C4 C5 F6 E7 F8 B9 E10 B11 B12 F13 F14 A15 F16 E17 E18 E19 D20 F