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Papers (30)
2025
pgmath 18
2024
pgmath 4 ugmath 20
2023
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2022
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2021
pgmath 11 ugmath 15
2020
pgmath 18 ugmath 16
2019
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2018
pgmath 4 ugmath 6
2017
ugmath 15
2016
pgmath 10 ugmath 16
2015
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2014
ugmath 9
2013
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2012
pgmath 24 ugmath 14
2011
pgmath 15 ugmath 12
2010
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2021 pgmath

11 maths questions

Q1 4 marks Groups Group Order and Structure Theorems View
Which of the following can not be the class equation for a group of appropriate order?
(A) $14 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 7$.
(B) $18 = 1 + 1 + 1 + 1 + 2 + 3 + 9$.
(C) $6 = 1 + 2 + 3$.
(D) $31 = 1 + 3 + 6 + 6 + 7 + 8$.
Q2 4 marks Sequences and Series Convergence/Divergence Determination of Numerical Series View
Consider the improper integral $\int _ { 2 } ^ { \infty } \frac { 1 } { x ( \log x ) ^ { 2 } } d x$ and the infinite series $\sum _ { k = 2 } ^ { \infty } \frac { 1 } { k ( \log k ) ^ { 2 } }$. Which of the following is/are true?
(A) The integral converges but the series does not converge.
(B) The integral does not converge but the series converges.
(C) Both the integral and the series converge.
(D) The integral and the series both fail to converge.
Q3 4 marks Matrices True/False or Multiple-Select Conceptual Reasoning View
Let $A \in M _ { 2 } ( \mathbb { R } )$ be a nonzero matrix. Pick the correct statement(s) from below.
(A) If $A ^ { 2 } = 0$, then $\left( I _ { 2 } - A \right) ^ { 5 } = 0$.
(B) If $A ^ { 2 } = 0$, then ( $I _ { 2 } - A$ ) is invertible.
(C) If $A ^ { 3 } = 0$, then $A ^ { 2 } = 0$.
(D) If $A ^ { 2 } = A ^ { 3 } \neq 0$, then $A$ is invertible.
Q9 4 marks Matrices True/False or Multiple-Select Conceptual Reasoning View
For $A \in M _ { 3 } ( \mathbb { C } )$, let $W _ { A } = \left\{ B \in M _ { 3 } ( \mathbb { C } ) \mid A B = B A \right\}$. Which of the following is/are true?
(A) For all diagonal $A \in M _ { 3 } ( \mathbb { C } )$, $W _ { A }$ is a linear subspace of $M _ { 3 } ( \mathbb { C } )$ with $\operatorname { dim } _ { \mathbb { C } } W _ { A } \geq 3$.
(B) For all $A \in M _ { 3 } ( \mathbb { C } ) , W _ { A }$ is a linear subspace of $M _ { 3 } ( \mathbb { C } )$ with $\operatorname { dim } _ { \mathbb { C } } W _ { A } > 3$.
(C) There exists $A \in M _ { 3 } ( \mathbb { C } )$ such that $W _ { A }$ is a linear subspace of $M _ { 3 } ( \mathbb { C } )$ with $\operatorname { dim } _ { \mathbb { C } } W _ { A } = 3$.
(D) If $A \in M _ { 3 } ( \mathbb { C } )$ is diagonalizable, then every element of $W _ { A }$ is diagonalizable.
Q10 4 marks Groups Ring and Field Structure View
Let $K$ be a field of order 243 and let $F$ be a subfield of $K$ of order 3. Pick the correct statement(s) from below.
(A) There exists $\alpha \in K$ such that $K = F ( \alpha )$.
(B) The polynomial $x ^ { 242 } = 1$ has exactly 242 solutions in $K$.
(C) The polynomial $x ^ { 26 } = 1$ has exactly 26 roots in $K$.
(D) Let $f ( x ) \in F [ x ]$ be an irreducible polynomial of degree 5. Then $f ( x )$ has a root in $K$.
Q11 10 marks Groups Subgroup and Normal Subgroup Properties View
Let $G$ be a finite group and $X$ the set of all abelian subgroups $H$ of $G$ such that $H$ is a maximal subgroup of $G$ (under inclusion) and is not normal in $G$. Let $M$ and $N$ be distinct elements of $X$. Show the following:
(A) The subgroup of $G$ generated by $M$ and $N$ is contained in the centralizer of $M \cap N$ in $G$.
(B) $M \cap N$ is the centre of $G$.
Q16 10 marks Groups Ring and Field Structure View
Consider the following statement: Let $F$ be a field and $R = F [ X ]$ the polynomial ring over $F$ in one variable. Let $I _ { 1 }$ and $I _ { 2 }$ be maximal ideals of $R$ such that the fields $R / I _ { 1 } \simeq R / I _ { 2 } \neq F$. Then $I _ { 1 } = I _ { 2 }$.
Prove or find a counterexample to the following claims:
(A) The above statement holds if $F$ is a finite field.
(B) The above statement holds if $F = \mathbb { R }$.
Q17 10 marks Groups Symplectic and Orthogonal Group Properties View
Let $\mathrm { O } ( 2 , \mathbb { R } )$ be the subgroup of $\mathrm { GL } ( 2 , \mathbb { R } )$ consisting of orthogonal matrices, i.e., matrices $A$ satisfying $A ^ { \operatorname { tr } } A = I$. Let $\mathrm { B } _ { + } ( 2 , \mathbb { R } )$ be the subgroup of $\mathrm { GL } ( 2 , \mathbb { R } )$ consisting of upper triangular matrices with positive entries on the diagonal.
(A) Let $A \in \mathrm { GL } ( 2 , \mathbb { R } )$. Show that there exist $A _ { o } \in \mathrm { O } ( 2 , \mathbb { R } )$ and $A _ { b } \in \mathrm { B } _ { + } ( 2 , \mathbb { R } )$ such that $A = A _ { o } A _ { b }$. (Hint: use appropriate elementary column operations.)
(B) Show that the map $$\phi : \mathrm { O } ( 2 , \mathbb { R } ) \times \mathrm { B } _ { + } ( 2 , \mathbb { R } ) \longrightarrow \mathrm { GL } ( 2 , \mathbb { R } ) \quad \left( A ^ { \prime } , A ^ { \prime \prime } \right) \mapsto A ^ { \prime } A ^ { \prime \prime }$$ is injective.
(C) Show that $\mathrm { GL } ( 2 , \mathbb { R } )$ is homeomorphic to $\mathrm { O } ( 2 , \mathbb { R } ) \times \mathrm { B } _ { + } ( 2 , \mathbb { R } )$. (Hint: first show that the map $A \mapsto A _ { b }$ is continuous.)
Q18 10 marks Groups Symplectic and Orthogonal Group Properties View
Let $F$ be a field of characteristic $p > 0$ and $V$ a finite-dimensional $F$-vector-space. Let $\phi \in \mathrm { GL } ( V )$ be an element of order $p ^ { 3 }$. Show that there exists a basis of $V$ with respect to which $\phi$ is given by an upper-triangular matrix with 1's on the diagonal.
Q19 10 marks Groups Ring and Field Structure View
Let $\zeta _ { 5 } \in \mathbb { C }$ be a primitive 5th root of unity; let $\sqrt [ 5 ] { 2 }$ denote a real 5th root of 2, and let $l$ denote a square root of $-1$. Let $K = \mathbb { Q } \left( \zeta _ { 5 } , \sqrt [ 5 ] { 2 } \right)$.
(A) Find the degree $[ K : \mathbb { Q } ]$ of the field $K$ over $\mathbb { Q }$.
(B) Determine if $l \in \mathbb { Q } \left( \zeta _ { 5 } \right)$. (Hint: You may use, without proof, the following fact: if $\zeta _ { 20 } \in \mathbb { C }$ is a primitive 20th root of unity, then $\left[ \mathbb { Q } \left( \zeta _ { 20 } \right) : \mathbb { Q } \right] > 4$.)
(C) Determine if $l \in K$.
Q20 10 marks Sequences and series, recurrence and convergence Closed-form expression derivation View
Let $a _ { 0 }$ and $a _ { 1 }$ be complex numbers and define $a _ { n } = 2 a _ { n - 1 } + a _ { n - 2 }$ for $n \geq 2$.
(A) Show that there are polynomials $p ( z ) , q ( z ) \in \mathbb { C } [ z ]$ such that $q ( 0 ) \neq 0$ and $\sum _ { n \geq 0 } a _ { n } z ^ { n }$ is the Taylor series expansion (around 0) of $\frac { p ( z ) } { q ( z ) }$.
(B) Let $a _ { 0 } = 1$ and $a _ { 1 } = 2$. Show that there exist complex numbers $\beta _ { 1 } , \beta _ { 2 } , \gamma _ { 1 } , \gamma _ { 2 }$ such that $$a _ { n } = \beta _ { 1 } \gamma _ { 1 } ^ { n + 1 } + \beta _ { 2 } \gamma _ { 2 } ^ { n + 1 }$$ for all $n$.