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Papers (30)
2025
pgmath 18
2024
pgmath 4 ugmath 20
2023
pgmath 12 ugmath 16
2022
pgmath 18 ugmath_22may 16 ugmath_23may 16
2021
pgmath 11 ugmath 15
2020
pgmath 18 ugmath 16
2019
pgmath 20 ugmath 15
2018
pgmath 4 ugmath 6
2017
ugmath 15
2016
pgmath 10 ugmath 16
2015
pgmath 5 ugmath 13
2014
ugmath 9
2013
pgmath 11 ugmath 12
2012
pgmath 24 ugmath 14
2011
pgmath 15 ugmath 12
2010
pgmath 4 ugmath 19
2016 ugmath

16 maths questions

QB1 14 marks Conditional Probability Conditional Probability with Normal Distribution View
Out of the 14 students taking a test, 5 are well prepared, 6 are adequately prepared and 3 are poorly prepared. There are 10 questions on the test paper. A well prepared student can answer 9 questions correctly, an adequately prepared student can answer 6 questions correctly and a poorly prepared student can answer only 3 questions correctly.
For each probability below, write your final answer as a rational number in lowest form.
(a) If a randomly chosen student is asked two distinct randomly chosen questions from the test, what is the probability that the student will answer both questions correctly?
Note: The student and the questions are chosen independently of each other. "Random" means that each individual student/each pair of questions is equally likely to be chosen.
(b) Now suppose that a student was chosen at random and asked two randomly chosen questions from the exam, and moreover did answer both questions correctly. Find the probability that the chosen student was well prepared.
QB2 14 marks Stationary points and optimisation Geometric or applied optimisation problem View
By definition the region inside the parabola $y = x^{2}$ is the set of points $(a,b)$ such that $b \geq a^{2}$. We are interested in those circles all of whose points are in this region. A bubble at a point $P$ on the graph of $y = x^{2}$ is the largest such circle that contains $P$. (You may assume the fact that there is a unique such circle at any given point on the parabola.)
(a) A bubble at some point on the parabola has radius 1. Find the center of this bubble.
(b) Find the radius of the smallest possible bubble at some point on the parabola. Justify.
QB3 14 marks Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View
Consider the function $f(x) = x^{\cos(x) + \sin(x)}$ defined for $x \geq 0$.
(a) Prove that
$$0.4 \leq \int_{0}^{1} f(x)\, dx \leq 0.5$$
(b) Suppose the graph of $f(x)$ is being traced on a computer screen with the uniform speed of 1 cm per second (i.e., this is how fast the length of the curve is increasing). Show that at the moment the point corresponding to $x = 1$ is being drawn, the $x$ coordinate is increasing at the rate of
$$\frac{1}{\sqrt{2 + \sin(2)}} \text{ cm per second.}$$
QB4 14 marks Number Theory Combinatorial Number Theory and Counting View
Let $A$ be a non-empty finite sequence of $n$ distinct integers $a_{1} < a_{2} < \cdots < a_{n}$. Define
$$A + A = \left\{ a_{i} + a_{j} \mid 1 \leq i, j \leq n \right\}$$
i.e, the set of all pairwise sums of numbers from $A$. E.g., for $A = \{1,4\}$, $A + A = \{2,5,8\}$.
(a) Show that $|A + A| \geq 2n - 1$. Here $|A + A|$ means the number of elements in $A + A$.
(b) Prove that $|A + A| = 2n - 1$ if and only if the sequence $A$ is an arithmetic progression.
(c) Find a sequence $A$ of the form $0 < 1 < a_{3} < \cdots < a_{10}$ such that $|A + A| = 20$.
QB5 14 marks Factor & Remainder Theorem Polynomial Construction from Root/Value Conditions View
Find a polynomial $p(x)$ that simultaneously has both the following properties.
(i) When $p(x)$ is divided by $x^{100}$ the remainder is the constant polynomial 1.
(ii) When $p(x)$ is divided by $(x-2)^{3}$ the remainder is the constant polynomial 2.
QB6 14 marks Number Theory Prime Number Properties and Identification View
Find all pairs $(p, n)$ of positive integers where $p$ is a prime number and $p^{3} - p = n^{7} - n^{3}$.
Q1 4 marks Number Theory Combinatorial Number Theory and Counting View
Four children $\mathrm{K}$, $\mathrm{L}$, $\mathrm{M}$ and R are about to run a race. They make some predictions as follows.
K says: M will win. Myself will come second. R says: M will come second. L will be third. M says: L will be last. R will be second. After the race, it turns out that each person has made exactly one correct and one incorrect prediction. Write the result of the race in the order from first to the last.
Q2 4 marks Number Theory Combinatorial Number Theory and Counting View
A country's GDP grew by $7.8\%$ within a period. During the same period the country's per-capita-GDP ($=$ ratio of GDP to the total population) increased by $10\%$. During this period, the total population of the country increased/decreased by $\_\_\_\_$ \%. (Choose the correct option and fill in the blank if possible.)
Q3 4 marks Number Theory Arithmetic Functions and Multiplicative Number Theory View
You are told that $n = 110179$ is the product of two primes $p$ and $q$. The number of positive integers less than $n$ that are relatively prime to $n$ (i.e. those $m$ such that $\operatorname{gcd}(m,n) = 1$) is 109480. Write the value of $p + q$. Then write the values of $p$ and $q$.
Q4 4 marks Permutations & Arrangements Lattice Path / Grid Route Counting View
A step starting at a point $P$ in the $XY$-plane consists of moving by one unit from $P$ in one of three directions: directly to the right or in the direction of one of the two rays that make the angle of $\pm 120^{\circ}$ with positive $X$-axis. (An opposite move, i.e. to the left/southeast/northeast, is not allowed.) A path consists of a number of such steps, each new step starting where the previous step ended. Points and steps in a path may repeat.
Find the number of paths starting at $(1,0)$ and ending at $(2,0)$ that consist of
(i) exactly 6 steps
(ii) exactly 7 steps.
Q5 4 marks Trig Graphs & Exact Values View
Find the value of the following sum of 100 terms. (Possible hint: also consider the same sum with $\sin^{2}$ instead of $\cos^{2}$.)
$$\cos^{2}\left(\frac{\pi}{101}\right) + \cos^{2}\left(\frac{2\pi}{101}\right) + \cos^{2}\left(\frac{3\pi}{101}\right) + \cdots + \cos^{2}\left(\frac{99\pi}{101}\right) + \cos^{2}\left(\frac{100\pi}{101}\right)$$
Q6 4 marks Number Theory Combinatorial Number Theory and Counting View
A function $f(x)$ is defined by the following formulas
$$f(x) = \begin{cases} x^{2} + 1 & \text{when } x \text{ is irrational} \\ \tan(x) & \text{when } x \text{ is rational} \end{cases}$$
At how many $x$ in the interval $[0, 4\pi]$ is $f(x)$ continuous?
Q7 4 marks Number Theory Combinatorial Number Theory and Counting View
We want to construct a nonempty and proper subset $S$ of the set of non-negative integers. This set must have the following properties. For any $m$ and any $n$,
if $m \in S$ and $n \in S$ then $m + n \in S$ \quad and \quad if $m \in S$ and $m + n \in S$ then $n \in S$.
For each statement below, state if it is true or false.
(i) 0 must be in $S$.
(ii) 1 cannot be in $S$.
(iii) There are only finitely many ways to construct such a subset $S$.
(iv) There is such a subset $S$ that contains both $2015^{2016}$ and $2016^{2015}$.
Q8 4 marks Factor & Remainder Theorem True/False or Multiple-Statement Evaluation View
A function $g$ satisfies the property that $g(k) = 3k + 5$ for each of the 15 integer values of $k$ in $[1,15]$.
For each statement below, state if it is true or false.
(i) If $g(x)$ is a linear polynomial, then $g(x) = 3x + 5$.
(ii) $g$ cannot be a polynomial of degree 10.
(iii) $g$ cannot be a polynomial of degree 20.
(iv) If $g$ is differentiable, then $g$ must be a polynomial.
Q9 4 marks Tangents, normals and gradients Existence or count of tangent lines with given properties View
Given a continuous function $f$, define the following subsets of the set $\mathbb{R}$ of real numbers.
$T =$ set of slopes of all possible tangents to the graph of $f$.
$S =$ set of slopes of all possible secants, i.e. lines joining two points on the graph of $f$.
For each statement below, state if it is true or false.
(i) If $f$ is differentiable, then $S \subset T$.
(ii) If $f$ is differentiable, then $T \subset S$.
(iii) If $T = S = \mathbb{R}$, then $f$ must be differentiable everywhere.
(iv) Suppose 0 and 1 are in $S$. Then every number between 0 and 1 must also be in $S$.
Q10 4 marks Straight Lines & Coordinate Geometry Collinearity and Concurrency View
You are given a triangle ABC, a point D on segment AC, a point E on segment AB and a point F on segment BC. Let BD and CE intersect in point P. Join P with F. Suppose that $\angle\mathrm{EPB} = \angle\mathrm{BPF} = \angle\mathrm{FPC} = \angle\mathrm{CPD}$ and $\mathrm{PD} = \mathrm{PE} = \mathrm{PF}$.
For each statement below, state if it is true or false.
(i) AP must bisect $\angle\mathrm{BAC}$.
(ii) $\triangle\mathrm{ABC}$ must be isosceles.
(iii) $\mathrm{A}$, $\mathrm{P}$, $\mathrm{F}$ must be collinear.
(iv) $\angle\mathrm{BAC}$ must be $60^{\circ}$.