past paper · 2020

TMUA 2020 Paper 1

Applications of Mathematical Knowledge. All 20 questions from the 2020 paper, each with a full worked solution. Topics on this paper: Algebra and functions (8), Differentiation and integration (3), Exponentials and logarithms (3), Sequences and series (2), Trigonometry (2), Graphs and transformations (1), Coordinate geometry (1).

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Questions

1 Which of the following is an expression for the first derivative with respect to x of (x³ - 5x²)/(2x√(x)) Calculus
2 (2x + 1) and (x - 2) are factors of 2x³ + px² + q. What is the value of 2p + q? Algebra
3 Find the complete set of values of x for which (x+4)(x+3)(1-x) > 0 and (x+2)(x-2) < 0 Algebra
4 The 1 st , 2 nd and 3 rd terms of a geometric progression are also the 1 st , 4 th and 6 th terms,… Sequences
5 The curve S has equation y = px² + 6x - q where p and q are constants. S has a line of symmetry at x = -14… Algebra
6 Find the maximum value of the function f(x) = 15^2x - 4(5^x) + 7 Logarithms
7 Given that 2^3x = 8^(y+3) and 4^(x+1) = 16^(y+1)8^(y+3) what is the value of x + y? Logarithms
8 The function f is defined for all real x as f(x) = (p - x)(x + 2) Find the complete set of values of p for… Algebra
9 The quadratic expression x² - 14x + 9 factorises as (x - α)(x - β), where α and β are positive real numbers.… Algebra
10 The following sequence of transformations is applied to the curve y = 4x²: 1.  Translation by pmatrix 3… Graphs
11 The diagram shows a quadratic function passing through (2, 0) and (q, 0), where q > 2. The curve opens… Calculus
12 How many real solutions are there to the equation 3cos x = √(x) where x is in radians? Trigonometry
13 Find the coefficient of x²y⁴ in the expansion of (1 + x + y²)⁷. Sequences
14 The area enclosed between the line y = mx and the curve y = x³ is 6. What is the value of m? Calculus
15 Find the positive difference between the two real values of x for which (log₂ x)⁴ + 12(log₂((1)/(x)))² - 2⁶… Logarithms
16 The circle C₁ has equation (x+2)² + (y-1)² = 3. The circle C₂ has equation (x-4)² + (y-1)² = 3. The straight… Geometry
17 Find the complete set of values of m in terms of c such that the graphs of y = mx + c and y = √(x) have two… Algebra
18 Find the number of solutions and the sum of the solutions of the equation 1 - 2cos² x = |cos x| where 0 ≤ x… Trigonometry
19 Find the lowest positive integer for which x² - 52x - 52 is positive. Algebra
20 For how many values of a is the equation (x - a)(x² - x + a) = 0 satisfied by exactly two distinct values of… Algebra hard

Answer key for TMUA 2020 Paper 1

1 C2 C3 B4 D5 A6 C7 A8 D9 C10 A11 E12 D13 F14 E15 C16 C17 A18 A19 E20 C