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2025
pgmath 18
2024
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2023
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2022
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2021
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2020
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2019
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2018
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2017
ugmath 15
2016
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2015
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2014
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2013
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2012
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2011
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2010
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2024 pgmath

4 maths questions

Q4 Groups Group Order and Structure Theorems View
Let $G$ (respectively, $H$ ) be a Sylow 2-subgroup (respectively, Sylow 7-subgroup) of the symmetric group $S _ { 17 }$. Pick the correct statement(s) from below.
(A) The order of $G$ is $2 ^ { 15 }$.
(B) $H$ is abelian.
(C) $G$ has a subgroup isomorphic to $\mathbb { Z } / 8 \mathbb { Z } \times \mathbb { Z } / 8 \mathbb { Z }$.
(D) If $\sigma \in S _ { 17 }$ has order 4 , then $\sigma$ is a 4-cycle.
Q5 Matrices True/False or Multiple-Select Conceptual Reasoning View
Let $p \geq 3$ be a prime number and $V$ be an $n$-dimensional vector space over $\mathbb { F } _ { p }$. Let $T : V \rightarrow V$ be a linear transformation. Select all the true statement(s) from below.
(A) $T$ has an eigenvalue in $\mathbb { F } _ { p }$.
(B) If $T ^ { p - 1 } = I$, then the minimal polynomial of $T$ has distinct roots in $\mathbb { F } _ { p }$.
(C) If $T \neq I$ and $T ^ { p - 1 } = I$, then the characteristic polynomial of $T$ has distinct roots in $\mathbb { F } _ { p }$.
(D) If $T ^ { p - 1 } = I$, then $T$ is diagonalizable over $\mathbb { F } _ { p }$.
Q7 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View
Let $f$ be a continuous real-valued function on $[ 0,1 ]$ such that
$$\int _ { 0 } ^ { 1 } f ( x ) d x = \int _ { 0 } ^ { 1 } x f ( x ) d x = 0$$
Pick the correct statement(s) from below.
(A) $f$ must have a zero in $[ 0,1 ]$.
(B) $f$ has at least two zeros, counted with multiplicity, in $[ 0,1 ]$.
(C) If $f \not\equiv 0$, then $f$ has exactly two zeros in $[ 0,1 ]$.
(D) $f \equiv 0$.
Q8 Number Theory Algebraic Structures in Number Theory View
Which of the following statement(s) are true?
(A) Every prime ideal of a finite commutative ring with unity is maximal.
(B) A commutative ring with unity whose set of all ideals is countably infinite is necessarily a countable ring.
(C) Let $R$ be a unique factorisation domain and $K$ be its field of fractions. There exists an irreducible element $\alpha \in R$ and an element $\beta \in K$ such that $\beta ^ { 2 } = \alpha$.
(D) Every subring $R$ (with unity) of $\mathbb { Q }$ with $\mathbb { Z } \varsubsetneqq R \varsubsetneqq \mathbb { Q }$ has infinitely many prime ideals.