past paper · 2022

TMUA 2022 Paper 1

Applications of Mathematical Knowledge. All 20 questions from the 2022 paper, each with a full worked solution. Topics on this paper: Trigonometry (4), Differentiation and integration (4), Coordinate geometry (3), Sequences and series (3), Graphs and transformations (3), Algebra and functions (3).

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 2022 Paper 1 2022 Paper 2

Questions

1 How many real solutions are there to the equation 2cos⁴θ - 5cos²θ + 3 = 0 in the interval 0 ≤ θ ≤ 2π? Trigonometry
2 Find the complete set of values of p for which the equation x² - 2px + y² - 6y - p² + 8p + 9 = 0 describes a… Geometry
3 Given the following statements about a function f f''(x) = a for all x f(0) = 1, f(1) = 2 ∫₀¹ f(x),dx = 1… Calculus
4 Two sectors of circles are similar. The smaller has radius r and the larger has radius r + 3. The arc length… Trigonometry
5 The terms x_n of a sequence follow the rule x_n+1 = (x_n + p)/(x_n + q) where p and q are real numbers.… Sequences
6 Given that ∫_log_2 5^log₂ 20 x,dx = log₂ M what is the value of M? Calculus
7 Find the finite area enclosed between the line y = 0 and the curve y = x² - 4|x| - 12. Calculus
8 A geometric sequence has first term a and common ratio r, where a and r are positive integers and r is… Sequences
9 This question is about pairs of functions f and g that satisfy f(x) - g(x) = 2sin x f(x),g(x) = cos² x for… Trigonometry hard
10 A sequence of translations is applied to the graph of y = x³. Which of the following graphs could be the… Graphs
11 Evaluate ∑_n=1¹⁰⁰log₁₀(3^,1-n) Sequences
12 A family of quadratic curves is given by y_k = 2(x - (k)/(2))² + (k²)/(2) + 4k + 3 where k is any real… Algebra
13 Given that (a³ + (2)/(b³))((2)/(a³) - b³) = √(2) where a and b are real numbers, what is the least value of… Algebra hard
14 A circle has centre O and radius 6. P, Q and R are points on the circumference with angle POQ ≥ (π)/(2). The… Geometry hard
15 A rectangle is drawn in the region enclosed by the curves p and q, where p(x) = 8 - 2x² q(x) = x² - 2 such… Calculus hard
16 The solutions to 7x⁴ - 6x² + 1 = 0 are ±cosθ and ±cosβ. Which one of the following equations has solutions… Algebra
17 A triangle has a side of length x - 1 opposite an angle of 30^∘, and a side of length -x² + 6x - 5 adjacent… Trigonometry hard
18 It is given that f(x) = x²(x-1)²(x-2) g(x) = -p(x-q)²(x-r)² where p, q and r are positive and q Find the set… Graphs hard
19 Circle C₁ is defined as x² + y² = 25. A second circle C₂ has radius 4 and centre (a, b) where -2 ≤ a ≤ 2 -3… Geometry
20 n is the number of points of intersection of the graphs y = |x² - a²| y = a²|x - 1| where a is a real… Graphs hard

Answer key for TMUA 2022 Paper 1

1 C2 D3 F4 C5 H6 F7 E8 B9 E10 C11 A12 D13 A14 D15 H16 B17 D18 B19 F20 B