Paper snapshot

Questions

Worked solutions

Chapters covered

University Admissions Tests UK (Pearson VUE) · United Kingdom

TMUA Specimen Paper 2

specimen

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.

The radius of the circle $2 x^{2} + 2 y^{2} - 8 x + 12 y + 15 = 0$ is
A $\frac{5}{2}$
Coordinate Geometry & Circles
The gradient of the curve $y = \frac{( 3 x - 2 ) ^{2}}{x x}$ at the point where $x = 2$ is
Differentiation
Consider the following attempt to solve an equation. The steps have been numbered for reference. $x + 5 = x + 3$ $(1) x + 5 = x^{2} + 6 x + 9$ $(2) x^{2} + 5 x + 4 = 0$ $(3) (x + 4) (x + 1) = 0$ $x = - 4 or x = - 1$ Which one of the following statements is true?
Logic & Proof
A set of five cards each have a letter printed on their front and a number printed on their back, as follows: Card A has front A and back 3; Card B has front B and back 4; Card C has front C and back 1; Card D has front D and back 7; Card E has front E and back 8. Which one of the five cards (A, B, C, D or E) provides a counterexample to the following statement? "Every card that has a vowel on its front has an even number on its back."
ACard A
BCard B
CCard C
DCard D
ECard E
Logic & Proof
Using the observation that $2^{5} \approx 3^{3}$ , it is possible to deduce that $lo g_{3} 2$ is approximately
A $\frac{3}{5}$
Exponentials & Logarithms
The area of a rectangle is measured to be $5600 cm^{2}$ correct to 2 significant figures. The width of the rectangle is measured to be $80 cm$ correct to the nearest centimetre. Which one of the following expressions gives the greatest possible height of the rectangle?
A $70.5 cm$
B $75 cm$
C $\frac{5650}{85} cm$
Logic & Proof
Which one of the following is a sketch of the graph $(x + y) (x^{2} - x y + y^{2}) = 1$ ?
$Four graphs labelled A, B, C, D, each plotted on axes from approximately $-4$ to $6$ on the $x$-axis and $-4$ to $4$ on the $y$-axis. Graph A: a curve that passes through the second quadrant (top-left) and fourth quadrant (bottom-right), passing near $(-1,0)$ and $(0,-1)$, resembling a curve symmetric about $y=-x$. Graph B: a curve that passes through $(1,0)$ and $(0,1)$ broadly, with branches going to top-right and bottom-left. Graph C: a curve passing through $(1,0)$ and $(0,1)$; for large positive $x$ the curve goes to large positive $y$ in the first quadrant, and for large negative $x$ the curve asymptotes along $y \approx -x$ (lower-left). Graph D: a curve with a steep section near the origin, passing through positive $y$ for positive $x$ near zero but going sharply negative for negative $x$ near zero.$
Functions & Graphs
Consider the following statement about the positive integer $n$ : Statement (*): The sum of the four consecutive integers, the smallest of which is $n$ , is a multiple of 6. Which one of the following is true?
AStatement (*) is true for all values of $n$ .
BStatement (*) is true for all values of $n$ which are odd, but not for any other values of $n$ .
CStatement (*) is true for all values of $n$ which are multiples of 3, but not for any other values of $n$ .
DStatement (*) is true for all values of $n$ which are multiples of 6, but not for any other values of $n$ .
Proof & Number Theory
Consider the statement about Fred: (*) Every day next week, Fred will do at least one maths problem. If statement (*) is NOT true, which of the following is certainly true?
AEvery day next week, Fred will do more than one maths problem.
BSome day next week, Fred will do more than one maths problem.
COn no day next week will Fred do more than one maths problem.
DEvery day next week, Fred will do no maths problems.
ESome day next week, Fred will do no maths problems.
FOn no day next week will Fred do no maths problems.
Logic & Proof
Which one of the following is a sketch of the graph of $y = lo g_{x} 2$ for $x > 1$ ?
$Six graphs (A through F) each plotted for $x > 0$ (domain $0$ to $10$) with $y$-axis ranging approximately from $-5$ to $5$. Graph A: an increasing curve in the region $x \in (0,3)$, $y \in (0,5)$, steeply rising. Graph B: a decreasing curve below the $x$-axis for $x > 1$, going from near 0 down to about $-5$ as $x$ increases. Graph C: a slowly increasing concave-down curve above the $x$-axis for $x > 1$, starting near $y=0$ and rising gently toward about $y=3$. Graph D: a curve starting above the $x$-axis near $x=1$, crossing and becoming negative, decreasing gradually toward $-4$. Graph E: a curve that starts very large (around $y=5$) very close to $x=1$ and rapidly decreases, approaching $y=0$ asymptotically from above. Graph F: a curve that starts very negative (around $y=-5$) near $x=1$ and increases, approaching $y=0$ asymptotically from below.$
AGraph A: an increasing curve for $x > 0$ , with $y > 0$ and growing, starting near $(1, 2)$ and increasing steeply.
Exponentials & Logarithms
Which one of the following numbers is largest in value? (All angles are given in radians.)
A $tan (\frac{3 π}{4})$
B $lo g_{10} 100$
C $sin^{10} (\frac{π}{2})$
Exponentials & Logarithms
A polynomial $p (x)$ has the property that $p (1) = 2$ . Which one of the following can be deduced from this?
A $p (x) = (x - 1) q (x) + 2$ for some polynomial $q (x)$ .
B $p (x) = (x + 1) q (x) + 2$ for some polynomial $q (x)$ .
Algebra & Inequalities
Five runners competed in a race: Fred, George, Hermione, Lavender, and Ron. The following results are known: Fred beat George. Hermione beat Lavender. Lavender beat George. Ron beat George. Assuming there were no ties, how many possible finishing orders could there have been, given only this information?
A1
B6
C12
D18
E24
F120
Combinatorics & Binomial
The graph of the polynomial function $y = a x^{5} + b x^{4} + c x^{3} + d x^{2} + e x + f$ , is sketched, where $a, b, c, d, e$ , and $f$ are real constants with $a \neq = 0$ . Which one of the following is NOT possible?
Functions & Graphs
For any real numbers $a$ , $b$ , and $c$ where $a \geq b$ , consider these three statements: 1. $- b \geq - a$ . 2. $a^{2} + b^{2} \geq 2 ab$ . 3. $a c \geq b c$ . Which of the statements 1, 2, and 3 must be true?
Logic & Proof
The sequence $a_{n}$ is given by the rule: $a_{1} = 2$ and $a_{n + 1} = a_{n} + (- 1)^{n}$ for $n \geq 1$ . What is $n = 1 \sum 100 a_{n}$ ?
Sequences & Series
Let $S$ be a set of positive integers (for example $S$ could consist of 3, 4, and 8). A positive integer $n$ is called an $S$ -number if and only if for every factor $m$ of $n$ with $m > 1$ , the number $m$ is a multiple of some number in $S$ . (So in the given example, 9 is an $S$ -number since its factors greater than 1 are 3 and 9, and each is a multiple of 3.) Positive integer $n$ is therefore NOT an $S$ -number if and only if
Logic & Proof
A group of five numbers are such that their mean is 0 and their range is 20. What is the largest possible median of the five numbers?
A0
B4
C $4 \frac{1}{2}$
D $6 \frac{1}{2}$
E
Probability & Statistics
The positive real numbers $a$ , $b$ , and $c$ are such that the equation $x^{3} + a x^{2} = b x + c$ has three real roots, one positive and two negative. Which one of the following correctly describes the real roots of the equation $x^{3} + c = a x^{2} + b x$ ?
Algebra & Inequalities
Five logicians each make a statement, as follows. Mr P: Of these five statements, an odd number are true. Ms Q: Both statements made by women are true. Mr R: My first name is Robert and Mr P's statement is true. Ms S: Exactly one statement made by a man is true. Mr T: Neither statement made by a woman is true. How many of the five statements can be simultaneously true?
Anone
B1 only
C2 only
D3 only
E4 only
Fnone or 1 only
G1 or 2 only
H2 or 3 only
Logic & Proof