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University Admissions Tests UK (Pearson VUE) · United Kingdom

TMUA 2023 Paper 1

2023

1h15

20 questions. Each part is annotated: check the hint if you are stuck, or reveal the worked solution once you have committed to an approach.

Given that $\int_{0}^{1} (a x + b) d x = 1$ and $\int_{0}^{1} x (a x + b) d x = 1$ find the value of $a + b$ .
Integration
The graphs of $y = x^{2} + 5 x + 6$ and $y = m x - 3$ , where $m$ is a constant, are plotted on the same set of axes.

Given that the graphs do not meet, what is the complete range of possible values of $m$ ?
A $- 1 < m < 11$
Algebra & Inequalities
For any integer $n \geq 0$ , $\int_{n}^{n + 1} f (x) d x = n + 1$ Evaluate $\int_{0}^{3} f (x) d x + \int_{1}^{3} f (x) d x + \int_{2}^{3} f (x) d x + \int_{4}^{3} f (x) d x + \int_{5}^{3} f (x) d x$
Integration
Evaluate $n = 0 \sum \infty \frac{sin ( nπ + \frac{π}{3} )}{2 ^{n}}$
Sequences & Series
The following shape has two lines of reflectional symmetry.

$M N O P$ is a square of perimeter 40 cm. The vertices of rectangle $R S T U$ lie on the edge of square $M N O P$ . $M R$ has length $x$ cm.

What is the largest possible value of $x$ such that $R S T U$ has area $20 cm^{2}$ ?
Coordinate Geometry & Circles
In the simplified expansion of $(2 + 3 x)^{12}$ , how many of the terms have a coefficient that is divisible by 12?
A $0$
B $2$
C $5$
D $10$
E $11$
F $12$
Combinatorics & Binomial
$P (x)$ and $Q (x)$ are defined as follows: $P (x) = 2^{x} + 4$ $Q (x) = 2^{(2 x - 2)} - 2^{(x + 2)} + 16$ Find the largest value of $x$ such that $P (x)$ and $Q (x)$ are in the ratio $4 : 1$ , respectively.
Exponentials & Logarithms
A triangle $X Y Z$ is called *fun* if it has the following properties: $angle Y X Z = 3 0^{\circ}$ $X Y = 3 a$ $Y Z = a$ where $a$ is a constant.

For a given value of $a$ , there are two distinct *fun* triangles $S$ and $T$ , where the area of $S$ is greater than the area of $T$ .

Find the ratio $area of S : area of T$
Trigonometry
How many solutions are there to $(1 + 3 cos 3 θ)^{2} = 4$ in the interval $0^{\circ} \leq θ \leq 18 0^{\circ}$ ?
Trigonometry
The trapezium rule with 4 strips is used to estimate the integral $\int_{- 2}^{2} 4 - x^{2} d x$ What is the positive difference between the estimate and the exact value of the integral?
Integration
It is given that $f (x) = x^{2} - 6 x$ .

The curves $y = f (k x)$ and $y = f (x - c)$ have the same minimum point, where $k > 0$ and $c > 0$ .

Which of the following is a correct expression for $k$ in terms of $c$ ?
Functions & Graphs
How many solutions are there to the equation $\frac{2 ^{t a n^{2} x}}{4 ^{s i n^{2} x}} = 1$ in the range $0 \leq x \leq 2 π$ ?
Trigonometry
Point $P$ lies on the circle with equation $(x - 2)^{2} + (y - 1)^{2} = 16$ .

Point $Q$ lies on the circle with equation $(x - 4)^{2} + (y + 5)^{2} = 16$ .

What is the maximum possible length of $P Q$ ?
Coordinate Geometry & Circles
The function $f (x) = \frac{2}{3} x^{3} + 2 m x^{2} + n, m > 0$ has three distinct real roots.

What is the complete range of possible values of $n$ , in terms of $m$ ?
Differentiation
The difference between the maximum and minimum values of the function $f (x) = a^{c o s x}$ , where $a > 0$ and $x$ is real, is 3.

Find the sum of the possible values of $a$ .
A $0$
B $\frac{3}{2}$
Exponentials & Logarithms
A right-angled triangle has vertices at $(2, 3)$ , $(9, - 1)$ and $(5, k)$ .

Find the sum of all the possible values of $k$ .
A $- 8$
B $- 6$
C $0.25$
D $2$
Coordinate Geometry & Circles
A circle $C_{n}$ is defined by $x^{2} + y^{2} = 2 n (x + y)$ where $n$ is a positive integer.

$C_{1}$ and $C_{2}$ are drawn and the area between them is shaded. Next, $C_{3}$ and $C_{4}$ are drawn and the area between them is shaded. This process continues until 100 circles have been drawn.

What is the total shaded area?
Sequences & Series
You are given that $S = 4 + \frac{8 k}{7} + \frac{16 k ^{2}}{49} + \frac{32 k ^{3}}{343} + \dots + 4 (\frac{2 k}{7})^{n} + \dots$ The value for $k$ is chosen as an integer in the range $- 5 \leq k \leq 5$ .

All possible values for $k$ are equally likely to be chosen.

What is the probability that the value of $S$ is a finite number greater than 3?
Sequences & Series
The solution to the differential equation $\frac{d y}{d x} = ∣ - 6 x ∣ for all x$ is $y = f (x) + c$ , where $c$ is a constant.

Which one of the following is a correct expression for $f (x)$ ?
Differentiation
The diagram shows the graph of $y = f (x)$ .

The function $f$ attains its maximum value of 2 at $x = 1$ , and its minimum value of $- 2$ at $x = - 1$ .

Find the difference between the maximum and minimum values of $(f (x))^{2} - f (x)$ .
Functions & Graphs