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11 maths questions

QA1 4 marks Groups Group Order and Structure Theorems View
Pick the correct statement(s) below.
(a) There exists a group of order 44 with a subgroup isomorphic to $\mathbb { Z } / 2 \oplus \mathbb { Z } / 2$.
(b) There exists a group of order 44 with a subgroup isomorphic to $\mathbb { Z } / 4$.
(c) There exists a group of order 44 with a subgroup isomorphic to $\mathbb { Z } / 2 \oplus \mathbb { Z } / 2$ and a subgroup isomorphic to $\mathbb { Z } / 4$.
(d) There exists a group of order 44 without any subgroup isomorphic to $\mathbb { Z } / 2 \oplus \mathbb { Z } / 2$ or to $\mathbb { Z } / 4$.
QA2 4 marks Groups True/False with Justification View
Let $G$ be a group. The following statements hold.
(a) If $G$ has nontrivial centre $C$, then $G / C$ has trivial centre.
(b) If $G \neq 1$, there exists a nontrivial homomorphism $h : \mathbb { Z } \rightarrow G$.
(c) If $| G | = p ^ { 3 }$, for $p$ a prime, then $G$ is abelian.
(d) If $G$ is nonabelian, then it has a nontrivial automorphism.
QA3 4 marks Groups Ring and Field Structure View
Let $C [ 0,1 ]$ be the space of continuous real-valued functions on the interval $[ 0,1 ]$. This is a ring under point-wise addition and multiplication. The following are true.
(a) For any $x \in [ 0,1 ]$, the ideal $M ( x ) = \{ f \in C [ 0,1 ] \mid f ( x ) = 0 \}$ is maximal.
(b) $C [ 0,1 ]$ is an integral domain.
(c) The group of units of $C [ 0,1 ]$ is cyclic.
(d) The linear functions form a vector-space basis of $C [ 0,1 ]$ over $\mathbb { R }$.
QA4 4 marks Invariant lines and eigenvalues and vectors True/false or multiple-choice on spectral properties View
Let $A : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ be a linear transformation with eigenvalues $\frac { 2 } { 3 }$ and $\frac { 9 } { 5 }$. Then, there exists a non-zero vector $v \in \mathbb { R } ^ { 2 }$ such that
(a) $\| A v \| > 2 \| v \|$;
(b) $\| A v \| < \frac { 1 } { 2 } \| v \|$;
(c) $\| A v \| = \| v \|$;
(d) $A v = 0$;
QA5 4 marks Groups Ring and Field Structure View
Let $F$ be a field with 256 elements, and $f \in F [ x ]$ a polynomial with all its roots in $F$. Then,
(a) $f \neq x ^ { 15 } - 1$;
(b) $f \neq x ^ { 63 } - 1$;
(c) $f \neq x ^ { 2 } + x + 1$;
(d) if $f$ has no multiple roots, then $f$ is a factor of $x ^ { 256 } - x$.
QA6 4 marks Groups True/False with Justification View
Let $h : \mathbb { C } \rightarrow \mathbb { C }$ be an analytic function such that $h ( 0 ) = 0 ; h \left( \frac { 1 } { 2 } \right) = 5$, and $| h ( z ) | < 10$ for $| z | < 1$. Then,
(a) the set $\{ z : | h ( z ) | = 5 \}$ is unbounded by the Maximum Principle;
(b) the set $\left\{ z : \left| h ^ { \prime } ( z ) \right| = 5 \right\}$ is a circle of strictly positive radius;
(c) $h ( 1 ) = 10$;
(d) regardless of what $h ^ { \prime }$ is, $h ^ { \prime \prime } \equiv 0$.
QA12 4 marks Sequences and Series Convergence/Divergence Determination of Numerical Series View
The series $\sum _ { n = 1 } ^ { \infty } a _ { n }$ where $a _ { n } = ( - 1 ) ^ { n + 1 } n ^ { 4 } e ^ { - n ^ { 2 } }$
(a) has unbounded partial sums;
(b) is absolutely convergent;
(c) is convergent but not absolutely convergent;
(d) is not convergent, but partial sums oscillate between $-1$ and $+1$.
QA13 4 marks Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
Let $f$ be continuously differentiable on $\mathbb { R }$. Let $f _ { n } ( x ) = n \left( f \left( x + \frac { 1 } { n } \right) - f ( x ) \right)$. Then,
(a) $f _ { n }$ converges uniformly on $\mathbb { R }$;
(b) $f _ { n }$ converges on $\mathbb { R }$, but not necessarily uniformly;
(c) $f _ { n }$ converges to the derivative of $f$ uniformly on $[ 0,1 ]$;
(d) there is no guarantee that $f _ { n }$ converges on any open interval.
QB2 10 marks Matrices Matrix Group and Subgroup Structure View
(a) Show that there exists a $3 \times 3$ invertible matrix $M \neq I _ { 3 }$ with entries in the field $\mathbb { F } _ { 2 }$ such that $M ^ { 7 } = I _ { 3 }$.
(b) Let $A$ be an $m \times n$ matrix, and $\mathbf { b }$ an $m \times 1$ vector, both with integer entries.
  1. Suppose that there exists a prime number $p$ such that the equation $A \mathbf { x } = \mathbf { b }$ seen as an equation over the finite field $\mathbb { F } _ { p }$ has a solution. Then does there exist a solution to $A \mathbf { x } = \mathbf { b }$ over the real numbers?
  2. If $A \mathbf { x } = \mathbf { b }$ has a solution over $\mathbb { F } _ { p }$ for every prime $p$, is a real solution guaranteed?
QB3 10 marks Matrices Diagonalizability and Similarity View
Let $M _ { n } ( \mathbb { C } )$ denote the set of $n \times n$ matrices over $\mathbb { C }$. Think of $M _ { n } ( \mathbb { C } )$ as the $2 n ^ { 2 }$-dimensional Euclidean space $\mathbb { R } ^ { 2 n ^ { 2 } }$. Show that the set of all diagonalizable $n \times n$ matrices is dense in $M _ { n } ( \mathbb { C } )$.
QB8 10 marks Number Theory Prime Number Properties and Identification View
(a) Let $f \in \mathbb { Z } [ x ]$ be a non-constant polynomial with integer coefficients. Show that as $a$ varies over the integers, the set of divisors of $f ( a )$ includes infinitely many different primes.
(b) Assume known the following result: If $G$ is a finite group of order $n$ such that for integer $d > 0$, $d \mid n$, there is no more than one subgroup of $G$ of order $d$, then $G$ is cyclic. Using this (or otherwise) prove that the multiplicative group of units in any finite field is cyclic.