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2025
pgmath 18
2024
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2023
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2020
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ugmath 15
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2015
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ugmath 9
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2012
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2011
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2020 pgmath

18 maths questions

Q1 4 marks Groups True/False with Justification View
Let $G$ be a group and $N$ be a proper normal subgroup. Pick the true statement(s) from below.
(A) If $N$ and the quotient $G / N$ is finite, then $G$ is finite.
(B) If the complement $G \backslash N$ of $N$ in $G$ is finite, then $G$ is finite.
(C) If both $N$ and the quotient $G / N$ are cyclic, then $G$ is cyclic.
(D) $G$ is isomorphic to $N \times G / N$.
Q2 4 marks Groups Ring and Field Structure View
Let $R$ denote the ring of all continuous functions from $\mathbb{R}$ to $\mathbb{R}$, where addition and multiplication are given, respectively, by $(f + g)(x) = f(x) + g(x)$ and $(fg)(x) = f(x)g(x)$ for every $f, g \in R$ and $x \in \mathbb{R}$. A zero-divisor in $R$ is a non-zero $f \in R$ such that $fg = 0$ for some non-zero $g \in R$. Pick the true statement(s) from below:
(A) $R$ has zero-divisors.
(B) If $f$ is a zero-divisor, then $f^{2} = 0$.
(C) If $f$ is a non-constant function and $f^{-1}(0)$ contains a non-empty open set, then $f$ is a zero-divisor.
(D) $R$ is an integral domain.
Q3 4 marks Proof True/False Justification View
Let $U = \left\{(x, y) \in \mathbb{R}^{2} \mid x < y^{2} < 4\right\}$ and $V = \left\{(x, y) \in \mathbb{R}^{2} \mid 0 < xy < 4\right\}$, both taken with the subspace topology from $\mathbb{R}^{2}$. Which of the following statement(s) is/are true?
(A) There exists a non-constant continuous map $V \longrightarrow \mathbb{R}$ whose image is not an interval.
(B) Image of $U$ under any continuous map $U \longrightarrow \mathbb{R}$ is bounded.
(C) There exists an $\epsilon > 0$ such that given any $p \in V$ the open ball $B_{\epsilon}(p)$ with centre $p$ and radius $\epsilon$ is contained in $V$.
(D) If $C$ is a closed subset of $\mathbb{R}^{2}$ which is contained in $U$, then $C$ is compact.
Q4 4 marks Matrices True/False or Multiple-Select Conceptual Reasoning View
Let $A$ and $B$ be $5 \times 5$ real matrices with $A^{2} = B^{2}$. Which of the following statements is/are correct?
(A) Either $A = B$ or $A = -B$.
(B) $A$ and $B$ have the same eigen spaces.
(C) $A$ and $B$ have the same eigen values.
(D) $A^{13} B^{3} = A^{3} B^{13}$.
Q5 4 marks Proof True/False Justification View
Consider the function $f : \mathbb{R}^{2} \longrightarrow \mathbb{R}$ given by
$$f(x, y) = \left(1 - \cos \frac{x^{2}}{y}\right) \sqrt{x^{2} + y^{2}}$$
for $y \neq 0$ and $f(x, 0) = 0$. (The square root is chosen to be non-negative). Pick the correct statement(s) from below:
(A) $f$ is continuous at $(0,0)$.
(B) $f$ is an open map.
(C) $f$ is differentiable at $(0,0)$.
(D) $f$ is a bounded function.
Q6 4 marks Sequences and Series Convergence/Divergence Determination of Numerical Series View
Which of the following is/are true for a series of real numbers $\sum a_{n}$?
(A) If $\sum a_{n}$ converges then $\sum a_{n}^{2}$ converges;
(B) If $\sum a_{n}^{2}$ converges then $\sum a_{n}$ converges;
(C) if $\sum a_{n}^{2}$ converges then $\sum \frac{1}{n} a_{n}$ converges;
(D) If $\sum |a_{n}|$ converges then $\sum \frac{1}{n} a_{n}$ converges;
Q7 4 marks Proof True/False Justification View
Which of the following functions are uniformly continuous on $\mathbb{R}$?
(A) $f(x) = x$;
(B) $f(x) = x^{2}$;
(C) $f(x) = (\sin x)^{2}$;
(D) $f(x) = e^{-|x|}$.
Q8 4 marks Proof True/False Justification View
Let $U$ and $V$ be non-empty open connected subsets of $\mathbb{C}$ and $f : U \longrightarrow V$ an analytic function. Which of the following statement(s) is/are true?
(A) $f^{\prime}(z) \neq 0$ for every $z \in U$.
(B) If $f$ is bijective, then $f^{\prime}(z) \neq 0$ for every $z \in U$.
(C) If $f^{\prime}(z) \neq 0$ for every $z \in U$, then $f$ is bijective.
(D) If $f^{\prime}(z) \neq 0$ for every $z \in U$, then $f$ is injective.
Q9 4 marks Proof True/False Justification View
Let $U$ denote the unit open disc centred at 0. Let $f : U \backslash \{0\} \longrightarrow \mathbb{C}$ be an analytic function. Assume that $\lim_{z \longrightarrow 0} z f(z) = 0$.
(A) $\lim_{z \longrightarrow 0} |f(z)|$ exists and is in $\mathbb{R}$.
(B) $f$ has a pole of order 1 at 0.
(C) $zf(z)$ has a zero of order 1 at 0.
(D) There exists an analytic function $g : U \longrightarrow \mathbb{C}$ such that $g(z) = f(z)$ for every $z \in U \backslash \{0\}$.
Q10 4 marks Groups Ring and Field Structure View
Let $f(x) = x^{2} + ax + b \in \mathbb{F}_{3}[X]$. What is the number of non-isomorphic quotient rings $\mathbb{F}_{3}[X] / (f(X))$?
Q11 10 marks Proof Proof That a Map Has a Specific Property View
Let $(X, d)$ be a compact metric space. For $x \in X$ and $\epsilon > 0$, define $B_{\epsilon}(x) := \{y \in X \mid d(x, y) < \epsilon\}$. For $C \subseteq X$ and $\epsilon > 0$, define $B_{\epsilon}(C) := \cup_{x \in C} B_{\epsilon}(x)$. Let $\mathcal{K}$ be the set of non-empty compact subsets of $X$. For $C, C^{\prime} \in \mathcal{K}$, define $\delta\left(C, C^{\prime}\right) = \inf\{\epsilon \mid C \subseteq B_{\epsilon}\left(C^{\prime}\right)$ and $C^{\prime} \subseteq B_{\epsilon}(C)\}$. Show that $(\mathcal{K}, \delta)$ is a compact metric space.
Q12 10 marks Proof Proof of Set Membership, Containment, or Structural Property View
Let $f$ be a non-constant entire function with $f(z) \neq 0$ for all $z \in \mathbb{C}$. Consider the set $U = \{z : |f(z)| < 1\}$. Show that all connected components of $U$ are unbounded.
Q13 10 marks Proof Proof That a Map Has a Specific Property View
Let $F \subseteq \mathbb{R}^{3}$ be a non-empty finite set, and $X = \mathbb{R}^{3} \backslash F$, taken with the subspace topology of $\mathbb{R}^{3}$. Show that $X$ is homeomorphic to a complete metric space. (Hint: Look for a suitable continuous function from $X$ to $\mathbb{R}$.)
Q14 10 marks Proof Existence Proof View
Show that there is no differentiable function $f : \mathbb{R} \longrightarrow \mathbb{R}$ such that $f(0) = 1$ and $f^{\prime}(x) \geq (f(x))^{2}$ for every $x \in \mathbb{R}$.
Q15 10 marks Proof Direct Proof of a Stated Identity or Equality View
Let $a_{1}, \ldots, a_{n}$ be distinct complex numbers. Show that the functions $e^{a_{1} z}, \ldots, e^{a_{n} z}$ are linearly independent over $\mathbb{C}$.
Q16 10 marks Proof Direct Proof of a Stated Identity or Equality View
The Frattini subgroup of a finite group $G$ is the intersection of all its proper maximal subgroups. Let $p$ be a prime number. Show that the Frattini subgroup of $\mathbb{Z} / p^{n}$, $n \geq 2$, is generated by $p$.
Q17* 10 marks Groups Decomposition and Basis Construction View
Let $M \in M_{n}(\mathbb{C})$. Show that $M$ is diagonalizable if and only if for every polynomial $P(X) \in \mathbb{C}[X]$ such that $P(M)$ is nilpotent, $P(M) = 0$.
Q20* 10 marks Sequences and series, recurrence and convergence Convergence proof and limit determination View
Let $a_{n}$, $n \geq 1$ be a sequence of real numbers. If $a_{n} \rightarrow a$, show that
$$b_{n} = \frac{a_{1} + 2a_{2} + 3a_{3} + \cdots + na_{n}}{n^{2}} \rightarrow \frac{a}{2}.$$