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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
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2014
ugmath 9
2013
pgmath 11 ugmath 12
2012
pgmath 24 ugmath 14
2011
pgmath 15 ugmath 12
2010
pgmath 4 ugmath 19
2019 ugmath

15 maths questions

QA1 4 marks Number Theory Divisibility and Divisor Analysis View
For a natural number $m$, define $\Phi_{1}(m)$ to be the number of divisors of $m$ and for $k \geq 2$ define $\Phi_{k}(m) := \Phi_{1}\left(\Phi_{k-1}(m)\right)$. For example, $\Phi_{2}(12) = \Phi_{1}(6) = 4$. Find the minimum $k$ such that $$\Phi_{k}\left(2019^{2019}\right) = 2.$$
QA3 4 marks Stationary points and optimisation Geometric or applied optimisation problem View
You have a piece of land close to a river, running straight. You are required to cut off a rectangular portion of the land, with the river forming one of the sides of the rectangle so, your fence will have three sides to it. You only have 60 meters of fencing. The maximum area that you can enclose is \_\_\_\_
QA4 4 marks Arithmetic Sequences and Series Telescoping or Non-Standard Summation Involving an AP View
The sum $$S = 1 + 111 + 11111 + \cdots + \underbrace{11\cdots1}_{2k+1}$$ is equal to . . . . . . .
QA5 4 marks Probability Definitions Finite Equally-Likely Probability Computation View
You are given an $8 \times 8$ chessboard. If two distinct squares are chosen uniformly at random find the probability that two rooks placed on these squares attack each other. Recall that a rook can move either horizontally or vertically, in a straight line.
QA6 4 marks Number Theory Quadratic Diophantine Equations and Perfect Squares View
For how many natural numbers $n$ is $n^{6} + n^{4} + 1$ a square of a natural number?
QA7 4 marks Number Theory Combinatorial Number Theory and Counting View
A broken calculator has all its 10 digit keys and two operation keys intact. Let us call these operation keys A and B. When the calculator displays a number $n$ pressing A changes the display to $n+1$. When the calculator displays a number $n$ pressing $B$ changes the display to $2n$. For example, if the number 3 is displayed then the key strokes ABBA changes the display in the following steps $3 \rightarrow 4 \rightarrow 8 \rightarrow 16 \rightarrow 17$.
If 1 is on the display what is the least number of key strokes needed to get 260 on the display?
QA8 4 marks Permutations & Arrangements Permutation Properties and Enumeration (Abstract) View
Let $\pi = \pi_{1}\pi_{2}\ldots\ldots\pi_{n}$ be a permutation of the numbers $1,2,3,\ldots,n$. We say $\pi$ has its first ascent at position $k < n$ if $\pi_{1} > \pi_{2} \ldots > \pi_{k}$ and $\pi_{k} < \pi_{k+1}$. If $\pi_{1} > \pi_{2} > \ldots > \pi_{n-1} > \pi_{n}$ we say $\pi$ has its first ascent in position $n$. For example when $n = 4$ the permutation 2134 has its first ascent at position 2.
The number of permutations which have their first ascent at position $k$ is ......
QA9 4 marks Indefinite & Definite Integrals Definite Integral as a Limit of Riemann Sums View
Consider $f : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ defined as follows: $$f(a,b) := \lim_{n \rightarrow \infty} \frac{1}{n} \log_{e}\left[e^{na} + e^{nb}\right]$$
For each statement, state if it is true or false.
(a) $f$ is not onto i.e. the range of $f$ is not all of $\mathbb{R}$.
(b) For every $a$ the function $x \mapsto f(a,x)$ is continuous everywhere.
(c) For every $b$ the function $x \mapsto f(x,b)$ is differentiable everywhere.
(d) We have $f(0,x) = x$ for all $x \geq 0$.
QA10 4 marks Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View
Let $f : \mathbb{R} \rightarrow \mathbb{R}$. For each statement, state if it is true or false.
(a) There is no continuous function $f$ for which $\int_{0}^{1} f(x)(1 - f(x))\,dx < \frac{1}{4}$.
(b) There is only one continuous function $f$ for which $\int_{0}^{1} f(x)(1 - f(x))\,dx = \frac{1}{4}$.
(c) There are infinitely many continuous functions $f$ for which $\int_{0}^{1} f(x)(1 - f(x))\,dx = \frac{1}{4}$.
QB1 10 marks Permutations & Arrangements Counting Functions with Constraints View
For a natural number $n$ denote by $\operatorname{Map}(n)$ the set of all functions $f : \{1,2,\ldots,n\} \rightarrow \{1,2,\ldots,n\}$. For $f, g \in \operatorname{Map}(n)$, $f \circ g$ denotes the function in $\operatorname{Map}(n)$ that sends $x$ to $f(g(x))$.
(a) Let $f \in \operatorname{Map}(n)$. If for all $x \in \{1,\ldots,n\}$ $f(x) \neq x$, show that $f \circ f \neq f$.
(b) Count the number of functions $f \in \operatorname{Map}(n)$ such that $f \circ f = f$.
QB2 10 marks Complex numbers 2 Roots of Unity and Cyclotomic Properties View
(a) Count the number of roots $w$ of the equation $z^{2019} - 1 = 0$ over complex numbers that satisfy $|w + 1| \geq \sqrt{2 + \sqrt{2}}$.
(b) Find all real numbers $x$ that satisfy the following equation: $$\frac{8^{x} + 27^{x}}{12^{x} + 18^{x}} = \frac{7}{6}$$
QB3 10 marks Reduction Formulae Evaluate a Closed-Form Expression Using the Reduction Formula View
Evaluate $\int_{0}^{\infty} \left(1 + x^{2}\right)^{-(m+1)} dx$, where $m$ is a natural number.
QB4 10 marks Proof Direct Proof of a Stated Identity or Equality View
Let $ABCD$ be a parallelogram. Let $O$ be a point in its interior such that $\angle AOB + \angle DOC = 180^{\circ}$. Show that $\angle ODC = \angle OBC$.
QB5 10 marks Sine and Cosine Rules Multi-step composite figure problem View
Three positive real numbers $x, y, z$ satisfy $$\begin{aligned} x^{2} + y^{2} &= 3^{2} \\ y^{2} + yz + z^{2} &= 4^{2} \\ x^{2} + \sqrt{3}\,xz + z^{2} &= 5^{2} \end{aligned}$$ Find the value of $2xy + xz + \sqrt{3}\,yz$.
QB6 10 marks Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View
(a) Compute $\dfrac{d}{dx}\left[\int_{0}^{e^{x}} \log(t)\cos^{4}(t)\,dt\right]$.
(b) For $x > 0$ define $F(x) = \int_{1}^{x} t\log(t)\,dt$.
i. Determine the open interval(s) (if any) where $F(x)$ is decreasing and the open interval(s) (if any) where $F(x)$ is increasing.
ii. Determine all the local minima of $F(x)$ (if any) and the local maxima of $F(x)$ (if any).