cmi-entrance

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
2018 pgmath

4 maths questions

Q15 10 marks Linear transformations View
Let $A$ be a $2 \times 2$ orthogonal matrix such that $\operatorname{det}(A) = -1$. Show that $A$ represents reflection about a line in $\mathbb{R}^2$.
Q16 10 marks Groups Subgroup and Normal Subgroup Properties View
A subgroup $H$ of a group $G$ is said to be a characteristic subgroup if $\sigma(H) = H$ for every group isomorphism $\sigma : G \longrightarrow G$ of $G$.
(A) Determine all the characteristic subgroups of $(\mathbb{Q}, +)$ (the additive group).
(B) Show that every characteristic subgroup of $G$ is normal in $G$. Determine whether the converse is true.
Q17* 10 marks Groups Group Homomorphisms and Isomorphisms View
Write $V$ for the space of $3 \times 3$ skew-symmetric real matrices.
(A) Show that for $A \in SO_3(\mathbb{R})$ and $M \in V$, $AMA^t \in V$. Write $A \cdot M$ for this action.
(B) Let $\Phi : \mathbb{R}^3 \longrightarrow V$ be the map $$\begin{bmatrix} u \\ v \\ w \end{bmatrix} \mapsto \begin{bmatrix} 0 & w & -v \\ -w & 0 & u \\ v & -u & 0 \end{bmatrix}$$ With the usual action of $SO_3(\mathbb{R})$ on $\mathbb{R}^3$ and the above action on $V$, show that $\Phi(Av) = A \cdot \Phi(v)$ for every $A \in SO_3(\mathbb{R})$ and $v \in \mathbb{R}^3$.
(C) Show that there does not exist $M \in V$, $M \neq 0$ such that for every $A \in SO_3(\mathbb{R})$, $A \cdot M$ belongs to the span of $M$.
Q19* 10 marks Groups Centre and Commutant Computation View
Let $\mathbb{k}$ be a field, $n$ a positive integer and $G$ a finite subgroup of $\mathrm{GL}_n(\mathbb{k})$ such that $|G| > 1$. Further assume that every $g \in G$ is upper-triangular and all the diagonal entries of $g$ are 1.
(A) Show that $\operatorname{char}\,\mathbb{k} > 0$. (Hint: consider the minimal polynomials of elements of $G$.)
(B) Show that the order of $g$ is a power of $\operatorname{char}\,\mathbb{k}$, for every $g \in G$.
(C) Show that the centre of $G$ has at least two elements.