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
2011 pgmath

15 maths questions

QA1 5 marks Proof True/False Justification View
There is a sequence of open intervals $I _ { n } \subset \mathbb { R }$ such that $\bigcap _ { n = 1 } ^ { \infty } I _ { n } = [ 0,1 ]$.
QA2 5 marks Proof True/False Justification View
The set $S$ of real numbers of the form $\frac { m } { 10 ^ { n } }$ with $m , n \in \mathbb { Z }$ and $n \geq 0$ is a dense subset of $\mathbb { R }$.
QA3 5 marks Proof True/False Justification View
There is a continuous bijection from $\mathbb { R } ^ { 2 } \rightarrow \mathbb { R }$.
QA4 5 marks Proof True/False Justification View
There is a bijection between $\mathbb { Q }$ and $\mathbb { Q } \times \mathbb { Q }$.
QA5 5 marks Proof True/False Justification View
If $\left\{ a _ { n } \right\} _ { n = 1 } ^ { \infty } , \left\{ b _ { n } \right\} _ { n = 1 } ^ { \infty }$ are two sequences of positive real numbers with the first converging to zero, and the second diverging to $\infty$, then the sequence of complex numbers $c _ { n } = a _ { n } e ^ { i b _ { n } }$ also converges to zero.
QA6 5 marks Proof Existence Proof View
For any polynomial $f ( x )$ with real coefficients and of degree 2011 , there is a real number $b$ such that $f ( b ) = f ^ { \prime } ( b )$.
QA7 5 marks Proof True/False Justification View
If $f : [ 0,1 ] \rightarrow [ - \pi , \pi ]$ is a continuous bijection then it is a homeomorphism.
QA8 5 marks Matrices True/False or Multiple-Select Conceptual Reasoning View
For any $n \geq 2$ there is an $n \times n$ matrix $A$ with real entries such that $A ^ { 2 } = A$ and trace $( A ) = n + 1$.
QA9 5 marks Invariant lines and eigenvalues and vectors Compute or factor the characteristic polynomial View
There is $2 \times 2$ real matrix with characteristic polynomial $x ^ { 2 } + 1$.
QA10 5 marks Groups True/False with Justification View
There is a field with 10 elements.
QA11 5 marks Groups True/False with Justification View
There are at least three non-isomorphic rings with 4 elements.
QA12 5 marks Groups True/False with Justification View
The group $( \mathbb { Q } , + )$ is a finitely generated abelian group.
QA13 5 marks Groups True/False with Justification View
$\mathbb { Q } ( \sqrt { 7 } )$ and $\mathbb { Q } ( \sqrt { 17 } )$ are isomorphic as fields.
QA14 5 marks Proof True/False Justification View
A vector space of dimension $\geq 2$ can be expressed as a union of two proper subspaces.
QA15 5 marks Proof True/False Justification View
There is a bijective analytic function from the complex plane to the upper half-plane.