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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
2016 pgmath

10 maths questions

Q1 4 marks Number Theory Combinatorial Number Theory and Counting View
We say that two subsets $X$ and $Y$ of $\mathbb{R}$ are order-isomorphic if there is a bijective map $\phi : X \longrightarrow Y$ such that for every $x_1 \leq x_2 \in X$, $\phi(x_1) \leq \phi(x_2)$, where '$\leq$' denotes the usual order on $\mathbb{R}$. Choose the correct statement(s) from below:
(A) $\mathbb{N}$ and $\mathbb{Z}$ are not order-isomorphic;
(B) $\mathbb{N}$ and $\mathbb{Q}$ are order-isomorphic;
(C) $\mathbb{Z}$ and $\mathbb{Q}$ are order-isomorphic;
(D) The sets $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{Q}$ are pairwise non-order-isomorphic.
Q2 4 marks Sequences and series, recurrence and convergence Multiple-choice on sequence properties View
Let $x_n = \left(1 - \frac{1}{n}\right) \sin \frac{n\pi}{3}$, $n \geq 1$. Write $l = \liminf x_n$ and $s = \limsup x_n$. Choose the correct statement(s) from below:
(A) $-\frac{\sqrt{3}}{2} \leq l < s \leq \frac{\sqrt{3}}{2}$;
(B) $-\frac{1}{2} \leq l < s \leq \frac{1}{2}$;
(C) $l = -1$ and $s = 1$;
(D) $l = s = 0$.
Q3 4 marks Differentiating Transcendental Functions Regularity and smoothness of transcendental functions View
Let $f : \mathbb{R} \longrightarrow \mathbb{R}$ be defined as $$f(x) = \begin{cases} x^2 \sin\left(\frac{1}{x^2}\right), & \text{if } x \neq 0 \\ 0, & \text{otherwise} \end{cases}$$ Choose the correct statement(s) from below:
(A) $f$ is continuous;
(B) $f$ is discontinuous at 0;
(C) $f$ is differentiable;
(D) $f$ is continuously differentiable.
Q4 4 marks Matrices True/False or Multiple-Select Conceptual Reasoning View
Let $A \in M_{m \times n}(\mathbb{R})$ be of rank $m$. Choose the correct statement(s) from below:
(A) The map $\mathbb{R}^n \longrightarrow \mathbb{R}^m$ given by $v \mapsto Av$ is injective;
(B) There exist matrices $B \in M_m(\mathbb{R})$ and $C \in M_n(\mathbb{R})$ such that $BAC = \left[I_m \mid \mathbf{0}_{n-m}\right]$;
(C) There exist matrices $B \in \mathrm{GL}_m(\mathbb{R})$ and $C \in \mathrm{GL}_n(\mathbb{R})$ such that $BAC = \left[I_m \mid \mathbf{0}_{n-m}\right]$;
(D) For every $(B, C) \in M_m(\mathbb{R}) \times M_n(\mathbb{R})$ such that $BAC = \left[I_m \mid \mathbf{0}_{n-m}\right]$, $C$ is uniquely determined by $B$.
Q5 4 marks Number Theory Algebraic Number Theory and Minimal Polynomials View
Let $f : \mathbb{C} \longrightarrow \mathbb{C}$ be an entire function such that $f(z+1) = f(z+\imath) = f(z)$ for every $z \in \mathbb{C}$. Choose the correct statement(s) from below:
(A) $f$ is constant;
(B) $f(z) = 0$ for every $z \in \mathbb{C}$;
(C) There exist complex numbers $a, b$ such that for every $x, y \in \mathbb{R}$, $f(x + \imath y) = a\sin(x) + \imath b\cos(y)$;
(D) $f$ is not necessarily constant but $|f(z)|$ is constant.
Q6 4 marks Groups Centre and Commutant Computation View
What is the cardinality of the centre of $O_2(\mathbb{R})$? (Definition: The centre of a group $G$ is $\{g \in G \mid gh = hg \text{ for every } h \in G\}$. Hint: Reflection matrices and permutation matrices are orthogonal.)
(A) 1;
(B) 2;
(C) The cardinality of $\mathbb{N}$;
(D) The cardinality of $\mathbb{R}$.
Q7 4 marks Number Theory Combinatorial Number Theory and Counting View
Let $U \subseteq \mathbb{R}$ be a non-empty open subset. Choose the correct statement(s) from below:
(A) $U$ is uncountable;
(B) $U$ contains a closed interval as a proper subset;
(C) $U$ is a countable union of disjoint open intervals;
(D) $U$ contains a convergent sequence of real numbers.
Q8 4 marks Number Theory Algebraic Structures in Number Theory View
Let $R$ be a commutative ring. The characteristic of $R$ is the smallest positive integer $n$ such that $a + a + \cdots + a$ ($n$ times) is zero for every $a \in R$, if such an integer exists, and zero, otherwise. Choose the correct statement(s) from below:
(A) For every $n \in \mathbb{N}$, there exists a commutative ring whose characteristic is $n$;
(B) There exists an integral domain with characteristic 57;
(C) The characteristic of a field is either 0 or a prime number;
(D) For every prime number $p$, every commutative ring of characteristic $p$ contains $\mathbb{F}_p$ as a subring.
Q9 4 marks Number Theory Algebraic Number Theory and Minimal Polynomials View
Consider the $\mathbb{Q}$-vector-space $$\{ f : \mathbb{R} \longrightarrow \mathbb{R} \mid f \text{ is continuous and } \mathrm{Image}(f) \subseteq \mathbb{Q} \}$$ What is its dimension?
Q10 4 marks Number Theory Algebraic Number Theory and Minimal Polynomials View
Let $p$ be a prime number and $F$ a field of $p^{23}$ elements. Let $\phi : F \longrightarrow F$ be the field automorphism of $F$ sending $a$ to $a^p$. Let $K := \{ a \in F \mid \phi(a) = a \}$. What is the value of $[K : \mathbb{F}_p]$?