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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
2012 pgmath

24 maths questions

QA2 5 marks Sequences and series, recurrence and convergence True/false or conceptual reasoning about sequences View
Let $x _ { n }$ be a sequence with the following property: Every subsequence of $x _ { n }$ has a further subsequence which converges to $x$. Then the sequence $x _ { n }$ converges to $x$.
QA3 5 marks Sequences and series, recurrence and convergence True/false or conceptual reasoning about sequences View
Let $f : ( 0 , \infty ) \longrightarrow \mathbb { R }$ be a continuous function. Then $f$ maps any Cauchy sequence to a Cauchy sequence.
QA4 5 marks Sequences and series, recurrence and convergence Sequence of functions convergence View
Let $\left\{ f _ { n } : \mathbb { R } \longrightarrow \mathbb { R } \right\}$ be a sequence of continuous functions. Let $x _ { n } \longrightarrow x$ be a convergent sequence of reals. If $f _ { n } \longrightarrow f$ uniformly then $f _ { n } \left( x _ { n } \right) \longrightarrow f ( x )$.
QA5 5 marks Sequences and series, recurrence and convergence True/false or conceptual reasoning about sequences View
Let $K \subset \mathbb { R } ^ { n }$ such that every real valued continuous function on $K$ is bounded. Then $K$ is compact (i.e closed and bounded).
QA6 5 marks Sequences and series, recurrence and convergence True/false or conceptual reasoning about sequences View
If $A \subset \mathbb { R } ^ { 2 }$ is a countable set, then $\mathbb { R } ^ { 2 } \backslash A$ is connected.
QA7 5 marks Sequences and series, recurrence and convergence True/false or conceptual reasoning about sequences View
The set $A = \left\{ ( z , w ) \in \mathbb { C } ^ { 2 } \mid z ^ { 2 } + w ^ { 2 } = 1 \right\}$ is bounded in $\mathbb { C } ^ { 2 }$.
QA8 5 marks Sequences and series, recurrence and convergence True/false or conceptual reasoning about sequences View
Let $f , g : \mathbb { C } \longrightarrow \mathbb { C }$ be complex analytic, and let $h : [ 0,1 ] \longrightarrow \mathbb { C }$ be a non-constant continuous map. Suppose $f ( z ) = g ( z )$ for every $z \in \operatorname { Im } h$, then $f = g$. (Here $\operatorname { Im } h$ denotes the image of the function $h$.)
QA9 5 marks Groups True/False with Justification View
There is a field with 121 elements.
QA10 5 marks Matrices Diagonalizability and Similarity View
The matrix $\left( \begin{array} { c c } \pi & \pi \\ 0 & \frac { 22 } { 7 } \end{array} \right)$ is diagonalizable over $\mathbb { C }$.
QA11 5 marks Groups True/False with Justification View
There are no infinite group with subgroups of index 5.
QA12 5 marks Groups True/False with Justification View
Every finite group of odd order is isomorphic to a subgroup of $A _ { n }$, the group of all even permutations.
QA13 5 marks Groups True/False with Justification View
Every group of order 6 abelian.
QA14 5 marks Groups True/False with Justification View
Two abelian groups of the same order are isomorphic.
QA15 5 marks Sequences and series, recurrence and convergence True/false or conceptual reasoning about sequences View
There is a non-constant continuous function $f : \mathbb { R } \rightarrow \mathbb { R }$ whose image is contained in $\mathbb { Q }$.
QB1 10 marks Sequences and series, recurrence and convergence True/false or conceptual reasoning about sequences View
Suppose $f : \mathbb { R } \mapsto \mathbb { R } ^ { n }$ be a differentiable mapping satisfying $\| f ( t ) \| = 1$ for all $t \in \mathbb { R }$. Show that $\left\langle f ^ { \prime } ( t ) , f ( t ) \right\rangle = 0$ for all $t \in \mathbb { R }$. (Here $\|$.$\|$ denotes standard norm or length of a vector in $\mathbb { R } ^ { n }$, and $\langle . , .\rangle$ denotes the standard inner product (or scalar product) in $\mathbb { R } ^ { n }$.)
QB2 10 marks Sequences and series, recurrence and convergence True/false or conceptual reasoning about sequences View
Let $A , B \subset \mathbb { R } ^ { n }$ and define $A + B = \{ a + b ; a \in A , b \in B \}$. If $A$ and $B$ are open, is $A + B$ open? If $A$ and $B$ are closed, is $A + B$ closed? Justify your answers.
QB3 10 marks Sequences and series, recurrence and convergence True/false or conceptual reasoning about sequences View
Let $f : X \mapsto Y$ be continuous map onto $Y$, and let $X$ be compact. Also $g : Y \mapsto Z$ is such that $g \circ f$ is continuous. Show $g$ is continuous.
QB4 10 marks Sequences and series, recurrence and convergence Convergence proof and limit determination View
Let $A$ be a $n \times m$ matrix with real entries, and let $B = A A ^ { t }$ and let $\alpha$ be the supremum of $x ^ { t } B x$ where supremum is taken over all vectors $x \in \mathbb { R } ^ { n }$ with norm less than or equal to 1. Consider $$C _ { k } = I + \sum _ { j = 1 } ^ { k } B ^ { j }$$ Show that the sequence of matrices $C _ { k }$ converges if and only if $\alpha < 1$.
QB5 10 marks Sequences and series, recurrence and convergence Series convergence and power series analysis View
Show that a power series $\sum _ { n \geq 0 } a _ { n } z ^ { n }$ where $a _ { n } \rightarrow 0$ as $n \rightarrow \infty$ cannot have a pole on the unit circle. Is the statement true with the hypothesis that $\left( a _ { n } \right)$ is a bounded sequence?
QB6 10 marks Sequences and series, recurrence and convergence True/false or conceptual reasoning about sequences View
Show that a biholomorphic map of the unit ball onto itself which fixes the origin is necessarily a rotation.
QB7 10 marks Groups Subgroup and Normal Subgroup Properties View
(i) Let $G = G L \left( 2 , \mathbb { F } _ { p } \right)$. Prove that there is a Sylow $p$-subgroup $H$ of $G$ whose normalizer $N _ { G } ( H )$ is the group of all upper triangular matrices in $G$.
(ii) Hence prove that the number of Sylow subgroups of $G$ is $1 + p$.
QB8 10 marks Groups Ring and Field Structure View
Calculate the minimal polynomial of $\sqrt { 2 } e ^ { \frac { 2 \pi i } { 3 } }$ over $\mathbb { Q }$.
QB9 10 marks Groups Group Homomorphisms and Isomorphisms View
Let $G$ be a group $\mathbb { F }$ a field and $n$ a positive integer. A linear action of $G$ on $\mathbb { F } ^ { n }$ is a map $\alpha : G \times \mathbb { F } ^ { n } \rightarrow \mathbb { F } ^ { n }$ such that $\alpha ( g , v ) = \rho ( g ) v$ for some group homomorphism $\rho : G \rightarrow \mathrm { GL } _ { n } ( \mathbb { F } )$. Show that for every finite group $G$, there is an $n$ such that there is a linear action $\alpha$ of $G$ on $\mathbb { F } ^ { n }$ and such that there is a nonzero vector $v \in \mathbb { F } ^ { n }$ such that $\alpha ( g , v ) = v$ for all $g \in G$.
QB10 10 marks Groups Ring and Field Structure View
Let $R$ be an integral domain containing a field $F$ as a subring. Show that if $R$ is a finite-dimensional vector space over $F$, then $R$ is a field.