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Papers (31)
2025
mscds 11
2024
mscds 10 pgmath 1 ugmath 20
2023
pgmath 3 ugmath 16
2022
pgmath 2 ugmath_22may 16 ugmath_23may 16
2021
pgmath 1 ugmath 15
2020
ugmath 15
2019
pgmath 2 ugmath 10
2018
pgmath 2 ugmath 14
2017
pgmath 1 ugmath 13
2016
pgmath 2 ugmath 11
2015
pgmath 1 ugmath 13
2014
ugmath 9
2013
pgmath 8 ugmath 10
2012
pgmath 2 ugmath 14
2011
pgmath 1 ugmath 12
2010
pgmath 4 ugmath 19
2019 ugmath

10 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 $$\Phi_{k}\left(2019^{2019}\right) = 2.$$
QA2 4 marks Differentiating Transcendental Functions Evaluate derivative at a point or find tangent slope View
Let $f$ be a real valued continuous function defined on $\mathbb{R}$ satisfying $$f'\left(\tan^{2}\theta\right) = \cos 2\theta + \tan\theta \sin 2\theta, \text{ for all real numbers } \theta.$$ If $f'(0) = -\cos\frac{\pi}{12}$ then find $f(1)$.
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 Proof 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 Permutations & Arrangements 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 ......
QB3 10 marks Standard Integrals and Reverse Chain Rule 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.
QB6 10 marks Connected Rates of Change 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).