cmi-entrance

Q1 4 marks Matrices True/False or Multiple-Select Conceptual Reasoning View

Let $T : \mathbb { R } ^ { 3 } \longrightarrow \mathbb { R } ^ { 3 }$ be a linear transformation such that $T \neq 0$ and $T ^ { 4 } = 0$. Pick the correct statement(s) from below.
(A) $T ^ { 3 } = 0$.
(B) $\operatorname { Image } ( T ) \neq \operatorname { Image } \left( T ^ { 2 } \right)$.
(C) $\operatorname { rank } \left( T ^ { 2 } \right) \leq 1$.
(D) $\operatorname { rank } ( T ) = 2$.

Q2 4 marks Matrices Projection and Orthogonality View

Let $W = \left\{ ( a , b , c , d ) \in \mathbb { R } ^ { 4 } \mid 3 a - b + 6 c = 0 \right\}$ and $T : \mathbb { R } ^ { 4 } \longrightarrow W$ be a linear map with $T ^ { 2 } = T$. Suppose $T$ is onto. Pick the correct statement(s) from below.
(A) $T ( u + v ) = T ( u ) + v$ for all $u \in \mathbb { R } ^ { 4 } , v \in W$.
(B) $\operatorname { ker } ( T - I )$ contains three linearly independent vectors.
(C) $( 1,3,0,2 ) \in \operatorname { ker } ( T )$.
(D) If $v _ { 1 } , v _ { 2 } \in \operatorname { ker } ( T )$ are nonzero, then $v _ { 1 } = c v _ { 2 }$ for some $c \in \mathbb { R }$.

Q4 4 marks Groups True/False with Justification View

Let $k$ be a finite field of characteristic $p > 2$ and $G$ the subgroup of $\mathrm { GL } _ { 2 } ( k )$ consisting of all matrices whose first column is $\left[ \begin{array} { l } 1 \\ 0 \end{array} \right]$. Pick the correct statement(s) from below.
(A) $G$ is a normal subgroup of $\mathrm { GL } _ { 2 } ( k )$.
(B) $G$ is a $p$-group.
(C) $\left\{ \left[ \begin{array} { l l } 1 & a \\ 0 & 1 \end{array} \right] : a \in k \right\}$ is a normal subgroup of $G$.
(D) $G$ is abelian.

Q6 4 marks Sequences and Series Convergence/Divergence Determination of Numerical Series View

Let $\mathbb { R } ^ { + } = \{ x \in \mathbb { R } : x \geq 0 \}$. For $x \in \mathbb { R } ^ { + }$, denote by $\operatorname{FRAC}( x )$ the fractional part of $x$, i.e., $x - n$ where $n$ is the largest integer that is less than or equal to $x$. Consider the series $\sum _ { n = 1 } ^ { \infty } \frac { \operatorname { FRAC } ( x / n ) } { n }$. Pick the correct statement(s) from below.
(A) The above series converges for all $x \in \mathbb { R } ^ { + } - \mathbb { Z }$.
(B) The above series diverges for some non-negative integer $x$.
(C) The above series defines a continuous function in a neighbourhood of $\frac { 1 } { 2 }$.
(D) The above series defines a continuous function in a neighbourhood of 1.

Q7 4 marks Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View

Let $f _ { n } ( x ) = \frac { 1 } { 1 + x ^ { n } }$. Pick the correct statement(s) from below.
(A) $f _ { n }$ converges uniformly on $[ 0,1 / 2 ]$.
(B) $f _ { n }$ converges uniformly on $[ 0,1 )$.
(C) $f _ { n }$ converges uniformly on $[ 0,2 ]$.
(D) $f _ { n }$ converges pointwise on $[ 0 , \infty )$.

Q8 4 marks Complex numbers 2 Properties of Analytic/Entire Functions View

Pick the correct statement(s) from below.
(A) If $f ( z )$ is a function defined on $\mathbb { C }$ that satisfies the Cauchy-Riemann equations at $z = 0$, then $f ( z )$ is complex-differentiable at $z = 0$.
(B) The function $\frac { ( \sin z - z ) \bar { z } ^ { 3 } } { | z | ^ { 6 } }$ is holomorphic on $\{ z \in \mathbb{C} : 0 < | z | < 1 \}$ and has a removable singularity at $z = 0$.
(C) There exists a holomorphic function on $\{ z \in \mathbb { C } : | z | > 3 \}$ whose derivative is $\frac { z } { ( z - 2 ) ^ { 2 } }$.
(D) There exists a holomorphic function on the upper half plane $\{ z \in \mathbb { C } : \mathfrak { I } z > 0 \}$ whose derivative is $\frac { z } { ( z - 2 ) ^ { 2 } \left( z ^ { 2 } + 4 \right) }$.

Q9 4 marks Groups Ring and Field Structure View

Pick the correct statement(s) from below.
(A) There exists a maximal ideal $M$ of $\mathbb { Z } [ x ]$ such that $M \cap \mathbb { Z } = ( 0 )$.
(B) If $M$ is a maximal ideal of $\mathbb { Z } [ x ]$, then $\mathbb { Z } [ x ] / M$ is finite.
(C) If $I$ is an ideal of $\mathbb { Z } [ x ]$ such that $\mathbb { Z } [ x ] / I$ is finite, then $I$ is maximal.
(D) The ideal $\left( 7 , x ^ { 2 } - 14 x - 2 \right)$ in $\mathbb { Z } [ x ]$ is maximal.

Q10 4 marks Groups True/False with Justification View

Let $M _ { n } ( \mathbb { R } )$ be the space of $n \times n$ real matrices. View $M _ { n } ( \mathbb { R } )$ as a metric space with $$d \left( \left[ a _ { i , j } \right] , \left[ b _ { i , j } \right] \right) : = \max _ { i , j } \left| a _ { i , j } - b _ { i , j } \right|$$ Let $U \subset M _ { n } ( \mathbb { R } )$ be the subset of matrices $M \in M _ { n } ( \mathbb { R } )$ such that $\left( M - I _ { n } \right) ^ { n } = 0$.
(A) $U$ is closed.
(B) $U$ is open.
(C) $U$ is compact.
(D) $U$ is neither closed or open.

Q11 10 marks Groups Subgroup and Normal Subgroup Properties View

Let $G$ be an abelian group and let $H$ be a nontrivial subgroup of $G$, that is, $H$ is a subgroup containing at least two elements. Show that the following two statements are equivalent.
(A) For every nontrivial subgroup $K$ of $G$, the subgroup $K \cap H$ is also nontrivial.
(B) $H$ contains every nontrivial minimal subgroup of $G$ and every element of the quotient group $G / H$ has finite order.

Q12 10 marks Groups Ring and Field Structure View

Consider the ring $\mathcal { C } ( \mathbb { R } )$ of continuous real-valued functions on $\mathbb { R }$, with pointwise addition and multiplication. For $A \subset \mathbb { R }$, the ideal of $A$ is $I ( A ) = \{ f \in \mathcal { C } ( \mathbb { R } ) \mid f ( a ) = 0$ for all $a \in A \}$. For a subset $I$ of $\mathcal { C } ( \mathbb { R } )$, the zero-set of $I$ is $Z ( I ) = \{ a \in \mathbb { R } \mid f ( a ) = 0$ for all $f \in I \}$. Prove the following:
(A) (3 marks) $Z ( I \cap J ) = Z ( I J )$ for ideals $I$ and $J$ of $\mathcal { C } ( \mathbb { R } )$.
(B) (2 marks) For each $a \in \mathbb { R } , I ( a )$ is a maximal ideal.
(C) (3 marks) The set $\{ f \in \mathcal { C } ( \mathbb { R } ) \mid f$ has compact support $\}$ is a proper ideal, and its zero set is empty.
(D) (2 marks) True/False: For each prime ideal $\mathfrak { p }$ of $\mathcal { C } ( \mathbb { R } ) , Z ( \mathfrak { p } )$ is a singleton set. (Justify your answer.)

Q13 10 marks Sequences and Series Functional Equations and Identities via Series View

Let $f , g , h$ be functions from $\mathbb { R }$ to $\mathbb { R }$ such that $$h ( f ( x ) + g ( y ) ) = x y$$ for all $x , y \in \mathbb { R }$. Show the following:
(A) $(2$ marks$)$ $h$ is surjective.
(B) $(3$ marks$)$ If $f$ is continuous then $f$ is strictly monotone.
(C) $(5$ marks$)$ There do not exist continuous functions $f , g , h$ satisfying $(*)$.

Q14 10 marks Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View

Let $f : [ 0,1 ] \longrightarrow \mathbb { R }$ and $g : \mathbb { R } \longrightarrow \mathbb { R }$ be continuous functions. Assume that $g$ is periodic with period 1. Show that $$\lim _ { n \mapsto \infty } \int _ { 0 } ^ { 1 } f ( x ) g ( n x ) d x = \left( \int _ { 0 } ^ { 1 } f ( x ) d x \right) \left( \int _ { 0 } ^ { 1 } g ( x ) d x \right)$$

Q15 10 marks Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

Prove or disprove each of the statements below.
(A) (4 marks) Let $f : \mathbb { R } ^ { 2 } \longrightarrow \mathbb { R }$ be a continuous function that takes both positive and negative values. Then $f$ has infinitely many zeros.
(B) (6 marks) Let $f : \mathbb { R } \longrightarrow \mathbb { R } ^ { 2 }$ be a continuous function. Then $f$ is not open.

Q16 10 marks Complex numbers 2 Properties of Analytic/Entire Functions View

Prove or disprove the following statements:
(A) (5 marks) Suppose that $f ( z )$ is a complex analytic function in the punctured unit disk $0 < | z | < 1$ such that $\lim _ { n \longrightarrow \infty } f \left( \frac { 1 } { n } \right) = 0$ and $\lim _ { n \longrightarrow \infty } f \left( \frac { 2 } { 2 n - 1 } \right) = 1$, then there exists a positive integer $N > 0$ such that $\lim _ { z \rightarrow 0 } \left| z ^ { - N } f ( z ) \right| = \infty$.
(B) (5 marks) There exists a non-zero entire function $f$ such that $f \left( e ^ { 2 \pi i e n ! } \right) = 0$ for all $n \geq 2025$.

Q17 10 marks Groups Symplectic and Orthogonal Group Properties View

Let $X \subseteq \mathbb { R } ^ { n }$ and $p \in X$. By a tangent vector of $X$ at $p$, we mean $\gamma ^ { \prime } ( 0 )$, where $\gamma : ( - \epsilon , \epsilon ) \longrightarrow X$ is a differentiable function with $\gamma ( 0 ) = p$. ($\epsilon \in \mathbb { R } , \epsilon > 0$.) The tangent space of $X$ at $p$ is the $\mathbb { R }$-vector space of all the tangent vectors at $p$. Think of $\mathrm { GL } _ { n } ( \mathbb { C } )$ as a subspace of $\mathbb { R } ^ { 2 n ^ { 2 } }$, with the euclidean topology.
Let $G : = \left\{ A \in \mathrm { GL } _ { 2 } ( \mathbb { C } ) \mid A ^ { * } A = A A ^ { * } = I _ { 2 } , \operatorname { det } A = 1 \right\}$.
(A) (2 marks) Show that every tangent vector of $\mathrm { GL } _ { n } ( \mathbb { C } )$ at $I _ { n }$ is of the form $\gamma _ { A } ^ { \prime } ( 0 )$ where $A$ is a $n \times n$ complex matrix and $\gamma _ { A } : \mathbb { R } \longrightarrow \mathrm { GL } _ { n } ( \mathbb { C } )$ is the function $t \mapsto e ^ { t A }$.
(B) (3 marks) Show that the tangent space of $G$ at $I _ { 2 }$ is $V : = \left\{ \left. \left[ \begin{array} { c c } i a & z \\ - \bar { z } & - i a \end{array} \right] \right\rvert \, a \in \mathbb { R } , z \in \mathbb { C } \right\}$.
(C) (5 marks) Consider the homeomorphism $\Phi : G \longrightarrow \mathbb { S } ^ { 3 }$ (where $\mathbb { S } ^ { 3 }$ denotes the unit sphere in $\mathbb { R } ^ { 4 }$) given by $$\left[ \begin{array} { c c } \alpha & \beta \\ \bar { \beta } & \bar { \alpha } \end{array} \right] \mapsto ( \Re ( \alpha ) , \Im ( \alpha ) , \Re ( \beta ) , \Im ( \beta ) )$$ Define a 'multiplication' on $V$ by $[ A , B ] = \frac { A B - B A } { 2 }$. Determine the multiplication on the tangent space at $\Phi \left( I _ { 2 } \right)$ induced by the derivative $D \Phi$. (Hint: The map $( A , B ) \longrightarrow [ A , B ]$ is $\mathbb { R }$-bilinear.)

Q18 10 marks Groups Ring and Field Structure View

Let $\mathbb { F } _ { q }$ be the finite field with $q$ elements and $P \in \mathbb { F } _ { q } [ x ]$ be a monic irreducible polynomial of even degree $2 d$. Then show that $P$, when considered as a polynomial in $\mathbb { F } _ { q ^ { 2 } } [ x ]$, decomposes into a product $P = Q _ { 1 } Q _ { 2 }$ of irreducible polynomials $Q _ { i }$ in $\mathbb { F } _ { q ^ { 2 } } [ x ]$ with $\operatorname { deg } \left( Q _ { i } \right) = d$.

Q19 10 marks Complex numbers 2 Properties of Analytic/Entire Functions View

Show that the power series $\sum _ { n = 1 } ^ { \infty } z ^ { n ! }$ represents an analytic function $f ( z )$ in the open unit disk $\Delta$ centred at 0. Show that $f ( z )$ cannot be extended to a continuous function on any connected open set $U$ such that $U$ is strictly larger than $\Delta$.

Q20 10 marks Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

It is known that there exist surjective continuous maps $I \longrightarrow I ^ { 2 }$ where $I = [ 0,1 ]$ is the unit interval.
(A) (4 marks) Using the above result or otherwise, show that there exists a surjective continuous map $f : \mathbb { R } \longrightarrow \mathbb { R } ^ { 2 }$.
(B) (6 marks) Let $f : \mathbb { R } \longrightarrow \mathbb { R } ^ { 2 }$ be a surjective continuous map. Let $\Gamma = \{ ( x , f ( x ) ) \mid x \in \mathbb { R } \} \subset \mathbb { R } ^ { 3 }$. Show that $\mathbb { R } ^ { 3 } \setminus \Gamma$ is path connected.

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