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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
2015 pgmath

5 maths questions

Q3 4 marks Groups Group Homomorphisms and Isomorphisms View
Let $$G = \left\{\left(\begin{array}{cc}a & b \\ 0 & a^{-1}\end{array}\right) : a, b \in \mathbb{R}, a > 0\right\}, \quad N = \left\{\left(\begin{array}{cc}1 & b \\ 0 & 1\end{array}\right) : b \in \mathbb{R}\right\}.$$ Which of the following are true?
(A) $G/N$ is isomorphic to $\mathbb{R}$ under addition.
(B) $G/N$ is isomorphic to $\{a \in \mathbb{R} : a > 0\}$ under multiplication.
(C) There is a proper normal subgroup $N'$ of $G$ which properly contains $N$.
(D) $N$ is isomorphic to $\mathbb{R}$ under addition.
Q17* 10 marks Number Theory Algebraic Structures in Number Theory View
Determine the cardinality of set of subrings of $\mathbb{Q}$. (Hint: For a set $P$ of positive prime numbers, consider the smallest subring of $\mathbb{Q}$ that contains $\left\{\left.\frac{1}{p}\right\rvert\, p \in P\right\}$.)
Q18* 10 marks Number Theory Irrationality and Transcendence Proofs View
Let $$f(x) = \sum_{n \geq 1} \frac{\sin\left(\frac{x}{n}\right)}{n}$$ Show that $f$ is continuous. Determine (with justification) whether $f$ is differentiable.
Q19* 10 marks Groups Subgroup and Normal Subgroup Properties View
Let $m$ and $n$ be positive integers and $p$ a prime number. Let $G \subseteq \mathrm{GL}_{m}(\mathbb{F}_{p})$ be a subgroup of order $p^{n}$. Let $U \subseteq \mathrm{GL}_{m}(\mathbb{F}_{p})$ be the subgroup that consists of all the matrices with 1's on the diagonal and 0's below the diagonal. Show that there exists $A \in \mathrm{GL}_{m}(\mathbb{F}_{p})$ such that $AGA^{-1} \subseteq U$.
Q20* 10 marks Matrices Determinant and Rank Computation View
Let $m$ and $n$ be positive integers and $0 \leq k \leq \min\{m,n\}$ an integer. Prove or disprove: The subspace of $M_{m \times n}(\mathbb{C})$ consisting of all matrices of rank equal to $k$ is connected. (You may use the following fact: For $t \geq 2$, $\mathrm{GL}_{t}(\mathbb{C})$ is connected.)