cmi-entrance

Q1 4 marks Groups Group Order and Structure Theorems View

By a simple group, we mean a group $G$ in which the only normal subgroups are $\left\{ 1 _ { G } \right\}$ and $G$. Pick the correct statement(s) from below.
(A) No group of order 625 is simple.
(B) $\mathrm { GL } ( 2 , \mathbb { R } )$ is simple.
(C) Let $G$ be a simple group of order 60. Then $G$ has exactly six subgroups of order 5 .
(D) Let $G$ be a group of order 60. Then $G$ has exactly seven subgroups of order 3 .

Q2 4 marks Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View

Let $f : \mathbb { R } \longrightarrow ( 0 , \infty )$ be an infinitely differentiable function with $\int _ { - \infty } ^ { \infty } f ( t ) d t = 1$. Pick the correct statement(s) from below.
(A) $f ( t )$ is bounded.
(B) $\lim _ { | t | \rightarrow \infty } f ^ { \prime } ( t ) = 0$.
(C) There exists $t _ { 0 } \in \mathbb { R }$ such that $f \left( t _ { 0 } \right) \geq f ( t )$ for all $t \in \mathbb { R }$.
(D) $f ^ { \prime \prime } ( a ) = 0$ for some $a \in \mathbb { R }$.

Q3 4 marks Matrices True/False or Multiple-Select Conceptual Reasoning View

Let $\mathcal { P } _ { n } = \{ f ( x ) \in \mathbb { R } [ x ] \mid \operatorname { deg } f ( x ) \leq n \}$, considered as an ($n + 1$)-dimensional real vector space. Let $T$ be the linear operator $f \mapsto f + \frac { \mathrm { d } f } { \mathrm {~d} x }$ on $\mathcal { P } _ { n }$. Pick the correct statement(s) from below.
(A) $T$ is invertible.
(B) $T$ is diagonalizable.
(C) $T$ is nilpotent.
(D) $( T - I ) ^ { 2 } = ( T - I )$ where $I$ is the identity map.

Q4 4 marks Groups True/False with Justification View

Pick the correct statement(s) from below.
(A) There exists a finite commutative ring $R$ of cardinality 100 such that $r ^ { 2 } = r$ for all $r \in R$.
(B) There is a field $K$ such that the additive group ( $K , +$ ) is isomorphic to the multiplicative group ( $K ^ { \times } , \cdot$ ).
(C) An irreducible polynomial in $\mathbb { Q } [ x ]$ is irreducible in $\mathbb { Z } [ x ]$.
(D) A monic polynomial of degree $n$ over a commutative ring $R$ has at most $n$ roots in $R$.

Q5 4 marks Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View

Pick the correct statement(s) from below.
(A) If $f$ is continuous and bounded on $( 0,1 )$, then $f$ is uniformly continuous on $( 0,1 )$.
(B) If $f$ is uniformly continuous on $( 0,1 )$, then $f$ is bounded on $( 0,1 )$.
(C) If $f$ is continuous on $( 0,1 )$ and $\lim _ { x \rightarrow 0 ^ { + } } f ( x )$ and $\lim _ { x \rightarrow 1 ^ { - } } f ( x )$ exists, then $f$ is uniformly continuous on $( 0,1 )$.
(D) Product of a continuous and a uniformly continuous function on $[ 0,1 ]$ is uniformly continuous.

Q6 4 marks Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View

Let $X$ be the metric space of real-valued continuous functions on the interval $[ 0,1 ]$ with the ``supremum distance'': $$d ( f , g ) = \sup \{ | f ( x ) - g ( x ) | : x \in [ 0,1 ] \} \text { for all } f , g \in X$$ Let $Y = \{ f \in X : f ( [ 0,1 ] ) \subset [ 0,1 ] \}$ and $Z = \left\{ f \in X : f ( [ 0,1 ] ) \subset \left[ 0 , \frac { 1 } { 2 } \right) \cup \left( \frac { 1 } { 2 } , 1 \right] \right\}$. Pick the correct statement(s) from below.
(A) $Y$ is compact.
(B) $X$ and $Y$ are connected.
(C) $Z$ is not compact.
(D) $Z$ is path-connected.

Q7 4 marks Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View

Let $X : = \left\{ ( x , y , z ) \in \mathbb { R } ^ { 3 } \mid z \leq 0 \right.$, or $\left. x , y \in \mathbb { Q } \right\}$ with subspace topology. Pick the correct statement(s) from below.
(A) $X$ is not locally connected but path connected.
(B) There exists a surjective continuous function $X \longrightarrow \mathbb { Q } _ { \geq 0 }$ (the set of non-negative rational numbers, with the subspace topology of $\mathbb { R }$ ).
(C) Let $S$ be the set of all points $p \in X$ having a compact neighbourhood (i.e. there exists a compact $K \subset X$ containing $p$ in its interior). Then $S$ is open.
(D) The closed and bounded subsets of $X$ are compact.

Q8 4 marks Complex numbers 2 Solving Polynomial Equations in C View

Consider the complex polynomial $P ( x ) = x ^ { 6 } + i x ^ { 4 } + 1$. (Here $i$ denotes a square-root of $-1$.) Pick the correct statement(s) from below.
(A) $P$ has at least one real zero.
(B) $P$ has no real zeros.
(C) $P$ has at least three zeros of the form $\alpha + i \beta$ with $\beta < 0$.
(D) $P$ has exactly three zeros $\alpha + i \beta$ with $\beta > 0$.

Q9 4 marks Invariant lines and eigenvalues and vectors Spectral properties of structured or special matrices View

Let $v$ be a (fixed) unit vector in $\mathbb { R } ^ { 3 }$. (We think of elements of $\mathbb { R } ^ { n }$ as column vectors.) Let $M = I _ { 3 } - 2 v v ^ { t }$. Pick the correct statement(s) from below.
(A) $O$ is an eigenvalue of $M$.
(B) $M ^ { 2 } = I _ { 3 }$.
(C) 1 is an eigenvalue of $M$.
(D) The eigenspace for the eigenvalue $-1$ is 2-dimensional.

Q10 4 marks Complex numbers 2 Properties of Analytic/Entire Functions View

Let $f ( z ) = \sum _ { n \geq 0 } a _ { n } z ^ { n }$ be an analytic function on the open unit disc $D$ around 0 with $a _ { 1 } \neq 0$. Suppose that $\sum _ { n \geq 2 } \left| n a _ { n } \right| < \left| a _ { 1 } \right|$. Then which of the following are true?
(A) There are only finitely many such $f$.
(B) $\left| f ^ { \prime } ( z ) \right| > 0$ for all $z \in D$.
(C) If $z , w \in D$ are such that $z \neq w$ and $f ( z ) = f ( w )$, then $a _ { 1 } = - \sum _ { n \geq 2 } a _ { n } \left( z ^ { n - 1 } + z ^ { n - 2 } w + \cdots + w ^ { n - 1 } \right)$.
(D) $f$ is one-one on $D$.

Q11 10 marks Invariant lines and eigenvalues and vectors Eigenvalue constraints from matrix properties View

Let $A \in \mathrm { GL } ( 3 , \mathbb { Q } )$ with $A ^ { t } A = I _ { 3 }$. Assume that $$A \left[ \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right] = \lambda \left[ \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right]$$ for some $\lambda \in \mathbb { C }$.
(A) Determine the possible values of $\lambda$.
(B) Determine $x + y + z$ where $x , y , z$ is given by $$\left[ \begin{array} { l } x \\ y \\ z \end{array} \right] = A \left[ \begin{array} { c } 1 \\ - 1 \\ 0 \end{array} \right]$$

Q12 10 marks Sequences and Series Limit Evaluation Involving Sequences View

Consider the function $S ( a )$ defined by the limit below: $$S ( a ) : = \lim _ { n \rightarrow \infty } \frac { 1 ^ { a } + 2 ^ { a } + 3 ^ { a } + \cdots + n ^ { a } } { ( n + 1 ) ^ { a - 1 } [ ( n a + 1 ) + ( n a + 2 ) + \cdots + ( n a + n ) ] }$$ Find the sum of all values $a$ such that $S ( a ) = \frac { 1 } { 60 }$.

Q13 10 marks Complex numbers 2 Properties of Analytic/Entire Functions View

Let $U$ and $V$ be non-empty open connected subsets of $\mathbb { C }$ and $f : U \longrightarrow V$ an analytic function. Suppose that for all compact subsets $K$ of $V , f ^ { - 1 } ( K )$ is compact. Show that $f ( U ) = V$.

Q14 10 marks Groups Subgroup and Normal Subgroup Properties View

Let $G$ be a finite group that has a non-trivial subgroup $N$ (i.e. $\left\{ 1 _ { G } \right\} \neq N \neq G$ ) that is contained in every non-trivial subgroup of $G$. Show that
(A) $G$ is a $p$-group for some prime number $p$;
(B) $N$ is a normal subgroup of $G$.

Q15 10 marks Complex numbers 2 Properties of Analytic/Entire Functions View

Let $f$ be an entire function such that $f$ maps the open unit ball $D$ around 0 to itself. Suppose further that $f ( 0 ) = 0$ and $f ( 1 ) = 1$. Show that $f ^ { \prime } ( 1 ) \in \mathbb { R }$ and that $\left| f ^ { \prime } ( 1 ) \right| \geq 1$.

Q16 10 marks Groups Ring and Field Structure View

Let $F$ be a field such that it has a finite non-Galois extension field. Let $V$ be a finite-dimensional vector-space over $F$. Let $V _ { 1 } , \ldots , V _ { r }$ be proper subspaces of $V$. Prove or disprove the following assertion: $V \neq \bigcup _ { i = 1 } ^ { r } V _ { i }$.

Q17 10 marks Groups Ring and Field Structure View

For a ring homomorphism $R \longrightarrow S$ (of commutative rings) and an ideal $I$ of $R$, the fibre over $I$ is the ring $S / I S$, i.e., the quotient of $S$ by the $S$-ideal generated by the image of $I$ in $S$. Let $S = \mathbb { C } [ X , Y ] / ( X Y - 1 )$ and $R = \mathbb { C } [ x + \alpha y ]$ where $\alpha \in \mathbb { C }$ and $x , y$ are the images of $X , Y$ in $S$. Consider the ring homomorphism $R \subseteq S$. Let $I = ( x + \alpha y - \beta ) R$, where $\beta \in \mathbb { C }$. For each nonnegative integer $n$, determine the set of ( $\alpha , \beta$ ) such that the fibre over $I$ has exactly $n$ maximal ideals.

Q18 10 marks Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View

Let $Q$ be the space of infinite sequences $$\mathbf { x } : = \left( x _ { 1 } , x _ { 2 } , \ldots , x _ { n } , \ldots \right)$$ of real numbers $x _ { n } \in [ 0,1 ]$, with the product topology coming from the identification $Q = [ 0,1 ] ^ { \mathbb { N } }$. ($[ 0,1 ]$ has the euclidean topology.) Let $S : Q \longrightarrow \mathbb { R }$ be the map $$S ( \mathbf { x } ) : = \sum _ { n } \frac { x _ { n } } { n ^ { 2 } } .$$ (A) Let $Q _ { 2 } : = \left\{ \left( y _ { 1 } , y _ { 2 } , \ldots , y _ { n } , \ldots \right) \left\lvert \, 0 \leq y _ { n } \leq \frac { 1 } { n } \right. \right\}$. Show that $Q _ { 2 }$ is compact.
(B) Let $D : Q _ { 2 } \longrightarrow Q$ be the map $$\left( y _ { 1 } , y _ { 2 } , \ldots , y _ { n } , \ldots \right) \mapsto \left( y _ { 1 } , 2 y _ { 2 } , \ldots , n y _ { n } , \ldots \right)$$ Show that $D$ is a homeomorphism. (Hint: first show that $Q$ is Hausdorff.)
(C) Show that $S \circ D : Q _ { 2 } \longrightarrow \mathbb { R }$ is continuous. (Hint: Show that there is a suitable inner-product space $( L , \langle - , - \rangle )$ and a vector $\mathbf { a } \in L$ such that $( S \circ D ) ( \mathbf { x } ) = \langle \mathbf { x } , \mathbf { a } \rangle$ for each $\mathbf { x } \in Q _ { 2 }$.)
(D) Show that $S$ is continuous.

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