past paper · 2021

TMUA 2021 Paper 1

Applications of Mathematical Knowledge. All 20 questions from the 2021 paper, each with a full worked solution. Topics on this paper: Differentiation and integration (6), Algebra and functions (3), Trigonometry (3), Graphs and transformations (3), Coordinate geometry (2), Sequences and series (2), Exponentials and logarithms (1).

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Questions

1 Two circles have the same radius. The centre of one circle is (-2, 1). The centre of the other circle is (3,… Geometry
2 The curve y = x³ - 6x + 3 has turning points at x = α and x = β, where β > α. Find ∫_α^β x³ - 6x + 3,dx Calculus
3 An arithmetic progression and a convergent geometric progression each have first term (1)/(2) The sum of the… Sequences
4 Find the minimum value of the function 2^2x - 2^x+3 + 4 Logarithms
5 The function f is such that, for all positive integers m and n: f(mn) = f(m)f(n) if mn is a multiple of 3… Algebra
6 The function f is given by f(x) = (cos x + 3)/(7 + 5cos x - sin² x) Find the positive difference between the… Trigonometry
7 The function f is such that f(0) = 0, and x,f(x) > 0 for all non-zero values of x. It is given that ∫_-2²… Calculus hard
8 The line y = 2x + 3 meets the curve y = x² + bx + c at exactly one point. The line y = 4x - 2 also meets the… Algebra
9 Find the area enclosed by the graph of |x| + |y| = 1 Graphs
10 Use the trapezium rule with 3 strips to estimate ∫_(1)/(2)² 2log₁₀ x,dx Calculus
11 The function f is given by f(x) = x^(1)/(7)(x² - x + 1) Find the fraction of the interval 0 < x < 1 for… Calculus
12 The minimum value of the function x⁴ - p²x² is -9, where p is a real number. Find the minimum value of the… Algebra
13 The function f is such that, for every integer n ∫_nⁿ⁺¹ f(x),dx = n + 1 Evaluate ∑_r=1⁸(∫₀^r f(x),dx) Calculus
14 This question uses radians. Find the number of distinct values of x that satisfy the equation (x+1)(3-x) =… Trigonometry hard
15 The diagram shows the graph of y = f(x). It consists of alternating straight-line segments of gradient 1 and… Calculus hard
16 Consider the expansion of (a + bx)ⁿ The third term, in ascending powers of x, is 105x² The fourth term, in… Sequences hard
17 Which of the following sketches shows the graph of sin(x² + y²) = (1)/(2) where x² + y² ≤ 8π? All five… Graphs
18 The curve with equation x = y² - 6y + 11 is rotated 90^∘ clockwise about the point P to give the curve C. P… Graphs hard
19 The equation sin²(4^cosθ × 60^∘) = (3)/(4) has exactly three solutions in the range 0^∘ ≤ θ ≤ x^∘ What is… Trigonometry hard
20 Find the length of the curve with equation 2log₁₀(x - y) = log₁₀(2 - 2x) + log₁₀(y + 5) Geometry hard

Answer key for TMUA 2021 Paper 1

1 F2 F3 G4 B5 F6 D7 G8 A9 C10 B11 A12 E13 C14 B15 C16 B17 A18 B19 B20 D