tmua

Papers (18)
2023
paper1 20 paper2 20
2022
paper1 20 paper2 20
2021
paper1 20 paper2 20
2020
paper1 20 paper2 20
2019
paper1 20 paper2 20
2018
paper1 20 paper2 20
2017
paper1 20 paper2 20
2016
paper1 20 paper2 20
Unknown
specimen_paper1 18 specimen_paper2 18
2019 paper2

20 maths questions

Q1 1 marks Binomial Theorem (positive integer n) Find a Specific Coefficient in a Product of Binomial/Polynomial Expressions View
Find the coefficient of the $x ^ { 4 }$ term in the expansion of
$$x ^ { 2 } \left( 2 x + \frac { 1 } { x } \right) ^ { 6 }$$
Q2 1 marks Factor & Remainder Theorem Divisibility and Factor Determination View
( 2 x + 1 )$ and $( x - 2 )$ are factors of $2 x ^ { 3 } + p x ^ { 2 } + q$.
What is the value of $2 p + q$ ?
Q3 1 marks Proof True/False Justification View
$a , b$ and $c$ are real numbers.
Given that $a b = a c$, which of the following statements must be true?
I $\quad a = 0$
II $b = 0$ or $c = 0$
III $b = c$
Q4 1 marks Proof True/False Justification View
Consider the following conjecture:
If $N$ is a positive integer that consists of the digit 1 followed by an odd number of 0 digits and then a final digit 1 , then $N$ is a prime number.
Here are three numbers:
I $\quad N = 101$ (which is a prime number)
II $\quad N = 1001$ (which equals $7 \times 11 \times 13$ )
III $N = 10001$ (which equals $73 \times 137$ )
Which of these provide(s) a counterexample to the conjecture?
Q5 1 marks Proof Proof of Equivalence or Logical Relationship Between Conditions View
Consider the following statement about the positive integers $a , b$ and $n$ :
(*): $a b$ is divisible by $n$
The condition 'either $a$ or $b$ is divisible by $n$ ' is:
Q6 1 marks Proof True/False Justification View
A student attempts to solve the equation
$$\cos x + \sin x \tan x = 2 \sin x - 1$$
in the range $0 \leq x \leq 2 \pi$.
The student's attempt is as follows:
$\cos x + \sin x \tan x = 2 \sin x - 1$
So $\quad \cos x - \sin x + \sin x \tan x - \sin x = - 1$
So $\quad ( \sin x - \cos x ) ( \tan x - 1 ) = - 1$
So $\quad \sin x - \cos x = - 1$ or $\tan x - 1 = - 1$
So $\quad ( \sin x - \cos x ) ^ { 2 } = 1$ or $\tan x = 0$
So $\quad 2 \sin x \cos x = 0$ or tan $x = 0$
So $x = 0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 } , 2 \pi$
Which of the following best describes this attempt?
Q7 1 marks Proof True/False Justification View
For which one of the following statements can the fact that $12 ^ { 2 } + 16 ^ { 2 } = 20 ^ { 2 }$ be used to produce a counterexample?
A If $a , b$ and $c$ are positive integers which satisfy the equation $a ^ { 2 } + b ^ { 2 } = c ^ { 2 }$, and the three numbers have no common divisor, then two of them are odd and the other is even.
B The equation $a ^ { 4 } + b ^ { 2 } = c ^ { 2 }$ has no solutions for which $a , b$ and $c$ are positive integers.
C The equation $a ^ { 4 } + b ^ { 4 } = c ^ { 4 }$ has no solutions for which $a , b$ and $c$ are positive integers.
D If $a , b$ and $c$ are positive integers which satisfy the equation $a ^ { 2 } + b ^ { 2 } = c ^ { 2 }$, then one is the arithmetic mean of the other two.
Q8 1 marks Inequalities Ordering and Sign Analysis from Inequality Constraints View
$a , b$ and $c$ are real numbers with $a < b < c < 0$
Which of the following statements must be true?
I $a c < a b < a ^ { 2 }$
II $b ( c + a ) > 0$
III $\frac { c } { b } > \frac { a } { b }$
Q9 1 marks Combinations & Selection View
A large circular table has 40 chairs round it.
What is the smallest number of people who can be sitting at the table already such that the next person to sit down must sit next to someone?
Q10 1 marks Proof View
$P Q R S$ is a quadrilateral, labelled anticlockwise.
Which one of the following is a necessary but not sufficient condition for $P Q R S$ to be a parallelogram?
Q11 1 marks Arithmetic Sequences and Series Properties of AP Terms under Transformation View
An arithmetic series has $n$ terms, all of which are integers.
The sum of the series is 20 .
Which of the following statements must be true?
I The first term of the series is even.
II $n$ is even.
III The common difference is even.
Q12 1 marks Probability Definitions View
Most students in a large college study Mathematics. A teacher chooses three different students at random, one after the other.
Consider these three probabilities:
$R = \mathrm { P }$ (At least one of the students chosen studies Mathematics)
$S = \mathrm { P }$ (The second student chosen studies Mathematics)
$T = \mathrm { P }$ (All three of the students chosen study Mathematics)
Which of the following is true?
Q13 1 marks Numerical integration Quadrature Error Bound Derivation View
A student approximates the integral $\int _ { a } ^ { b } \sin ^ { 2 } x \mathrm {~d} x$ using the trapezium rule with 4 strips. The resulting approximation is an overestimate.
Which of the following is/are necessarily true?
I If the student approximates $\int _ { - b } ^ { - a } \sin ^ { 2 } x \mathrm {~d} x$ in the same way, the result will be an overestimate.
II If the student approximates $\int _ { a } ^ { b } \cos ^ { 2 } x \mathrm {~d} x$ in the same way, the result will be an underestimate.
Q14 1 marks Stationary points and optimisation View
Consider the following statements about the polynomial $\mathrm { p } ( x )$, where $a < b$ :
I $\quad \mathrm { p } ( a ) \leq \mathrm { p } ( b )$
II $\quad \mathrm { p } ^ { \prime } ( a ) \leq \mathrm { p } ^ { \prime } ( b )$
III $\mathrm { p } ^ { \prime \prime } ( a ) \leq \mathrm { p } ^ { \prime \prime } ( b )$
Which of these statements is a necessary condition for $\mathrm { p } ( x )$ to be increasing for $a \leq x \leq b$ ?
Q15 1 marks Laws of Logarithms Solve a Logarithmic Equation View
The numbers $a , b$ and $c$ are each greater than 1 .
The following logarithms are all to the same base:
$$\begin{aligned} \log \left( a b ^ { 2 } c \right) & = 7 \\ \log \left( a ^ { 2 } b c ^ { 2 } \right) & = 11 \\ \log \left( a ^ { 2 } b ^ { 2 } c ^ { 3 } \right) & = 15 \end{aligned}$$
What is this base?
Q16 1 marks Discriminant and conditions for roots Parameter range for no real roots (positive definite) View
The graph of the quadratic
$$y = p x ^ { 2 } + q x + p$$
where $p > 0$, intersects the $x$-axis at two distinct points.
In which one of the following graphs does the shaded region show the complete set of possible values that $p$ and $q$ could take?
Q17 1 marks Proof View
A multiple-choice test question offered the following four options relating to a certain statement:
A The statement is true if and only if $x > 1$
B The statement is true if $x > 1$
C The statement is true if and only if $x > 2$
D The statement is true if $x > 2$
Given that exactly one of these options was correct, which one was it?
Q18 1 marks Inequalities Quadratic Inequality Holding for All x (or a Restricted Domain) View
Consider the following inequality:
$( * ) : \quad a | x | + 1 \leq | x - 2 |$
where $a$ is a real constant.
Which one of the following describes the complete set of values of $a$ such that (*) is true for all real $x$ ?
Q19 1 marks Indices and Surds Simplifying Surd Expressions View
Find the value of the expression
$$\sqrt { 8 - 4 \sqrt { 2 } + 1 } + \sqrt { 9 - 12 \sqrt { 2 } + 8 }$$
Q20 1 marks Function Transformations View
When the graph of the function $y = \mathrm { f } ( x )$, defined on the real numbers, is reflected in the $y$-axis and then translated by 2 units in the negative $x$-direction, the result is the graph of the function $y = \mathrm { g } ( x )$.
When the graph of the same function $y = \mathrm { f } ( x )$ is translated by 2 units in the negative $x$-direction and then reflected in the $y$-axis, the result is the graph of the function $y = \mathrm { h } ( x )$.
Which one of the following conditions on $y = \mathrm { f } ( x )$ is necessary and sufficient for the functions $\mathrm { g } ( x )$ and $\mathrm { h } ( x )$ to be identical?