past paper · 2019

TMUA 2019 Paper 1

Applications of Mathematical Knowledge. All 20 questions from the 2019 paper, each with a full worked solution. Topics on this paper: Differentiation and integration (6), Sequences and series (3), Exponentials and logarithms (3), Trigonometry (3), Algebra and functions (2), Coordinate geometry (2), Graphs and transformations (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.

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Questions

1 f(x) is a quadratic function in x. The graph of y = f(x) passes through the point (1, -1) and has a turning… Algebra
2 Find the complete set of values of the real constant k for which the expression x² + kx + 2x + 1 - 2k is… Algebra
3 Find the coefficient of x in the expression (1+x)⁰ + (1+x)¹ + (1+x)² + (1+x)³ + ⋯ + (1+x)⁷⁹ + (1+x)⁸⁰ Sequences
4 The sequence x_n is given by x₁ = 10 x_n+1 = √(x_n) for n ≥ 1 What is the value of x₁₀₀? [Note that a^b^c… Sequences
5 S is a geometric sequence. The sum of the first 6 terms of S is equal to 9 times the sum of the first 3… Sequences
6 The circles with equations (x+4)² + (y+1)² = 64 and (x-8)² + (y-4)² = r², where r > 0 have exactly one point… Geometry
7 A curve has equation y = (2q - x²)(2qx + 3) The gradient of the curve at x = -1 is a function of q. Find the… Calculus
8 The function f is such that 0 The trapezium rule with n equal intervals is used to estimate ∫₀¹ f(x),dx and… Calculus
9 p is a positive constant. Find the area enclosed between the curves y = p√(x) and x = p√(y). Calculus
10 Evaluate ∫_-1³ |x|(1-x),dx Calculus
11 Find the sum of the real values of x that satisfy the simultaneous equations log₃(xy²) = 1 (log₃ x)(log₃ y)… Logarithms
12 It is given that (dV)/(dt) = (24π(t-1))/(1 + √(t)) for t ≥ 1 and V = 7 when t = 1. Find the value of V when… Calculus
13 Find the maximum value of 4^sin x - 4 × 2^sin x + (17)/(4) for real x. Logarithms
14 x satisfies the simultaneous equations sin 2x + √(3)cos 2x = -1 and √(3)sin 2x - cos 2x = √(3) where 0^∘ ≤ x… Trigonometry
15 Find the real non-zero solution to the equation 2^(9^x)8^(3^x) = 14 Logarithms
16 Given that 2∫₀¹ f(x),dx + 5∫₁² f(x),dx = 14 and ∫₀¹ f(x+1),dx = 6 find the value of ∫₀² f(x),dx Calculus
17 Find the fraction of the interval 0 ≤ θ ≤ π for which the inequality (sin 2θ - 12)(sinθ - cosθ) ≥ 0 is… Trigonometry hard
18 Find the shortest distance between the curve y = x² + 4 and the line y = 2x - 2. Geometry
19 Find the value of ∑_k=0⁹⁰ sin(10 + 90k)^∘ Trigonometry
20 What is the complete range of values of k for which the curves with equations y = x³ - 12x and y = k -… Graphs hard

Answer key for TMUA 2019 Paper 1

1 A2 A3 E4 C5 E6 C7 F8 E9 D10 F11 H12 C13 B14 B15 A16 C17 C18 B19 C20 E