Let $G$ be a finite group of odd order and $1 < d < | G |$ be a divisor of $| G |$. Assume that $G$ has exactly three subgroups $H _ { 1 } , H _ { 2 }$ and $H _ { 3 }$ of order $d$. Suppose that $H _ { 1 }$ is not normal in $G$. For each $i = 1,2,3$, let $N _ { i }$ denote the normalizer of $H _ { i }$. Let $S : = \left\{ H _ { 1 } , H _ { 2 } , H _ { 3 } \right\}$.
(A) (4 marks) For each $g \in G$ let $s _ { g }$ denote the cardinality of the set $\left\{ H \in S \mid g H g ^ { - 1 } = H \right\}$. Show that $\sum _ { g \in G } s _ { g } = \left| N _ { 1 } \right| + \left| N _ { 2 } \right| + \left| N _ { 3 } \right|$.
(B) (6 marks) Show that $G \neq N _ { 1 } \cup N _ { 2 } \cup N _ { 3 }$.

13. In the following code, A is an array indexed from 0 whose elements are all positive integers, and n is the number of elements in A .
\begin{verbatim} function foo(A,n) { max = 0; curr = 0; for i from 1 to (n-1) { if A[i] > A[i-1] { curr = curr + 1; if curr > max { max = curr; } } else { curr = 0; } } return(max+1); } \end{verbatim}
If $A = [ 1,3,5,2,4,7,6,8 ]$, what will $f \circ \circ ( A , 8 )$ return?
(a) 2
(b) 3
(c) 4
(d) 5

(A) (5 marks) Let $X$ be a non-empty finite set and let $R$ be the ring of $\mathbb { Z }$-valued functions on $X$, with pointwise addition and multiplication. Let $S$ be an additive subgroup of $R$ such that the multiplicative identity $1 _ { R } \notin S$ and such that for all $s , s ^ { \prime } \in S , s s ^ { \prime } \in S$. Show that there exists $x \in X$ and a prime number $p$ such that $f ( x )$ is divisible by $p$ for all $f \in S$ (Hint: Consider the sets $\{ f ( x ) : f \in S \}$ for all $x \in X$.)
(B) (5 marks) Let $K$ be a subfield of $\mathbb { C }$ with $[ K : \mathbb { Q } ] = 2$. Let $P \in \mathbb { Q } [ x ]$ be irreducible over $\mathbb { Q }$. Show that $P$ is either irreducible in $K [ x ]$ or splits as product of two irreducible polynomials in $K [ x ]$.

14. A prominent newspaper conducted a survey among a carefully chosen sample of $1,00,000$ voters under the age of 25 . A randomly chosen subset of $60 \%$ of this sample were asked the following question: "Have you ever voted in a general election?" The remaining $40 \%$ were asked the following, different question: "Is it true that you have never voted in a general election?" The editor in charge of collating the data from the survey, accidentally deleted some of the contents of the survey database. The only information that survived this mishap was that a total of 54,000 respondents replied Yes to the question that they got. Specifically, it is not known how many of these Yes answers correspond to the first question, and how many to the second.
Assume that the sampling was done properly and that the respondents answered truthfully. What estimate can we derive from just the total number of Yes answers, about the percentage of voters under the age of 25 that have voted in some general election?
(a) At most $30 \%$.
(b) More than $30 \%$, but less than $60 \%$.
(c) At least $60 \%$, but less than $75 \%$.
(d) No such estimate can be drawn from just the total number of Yes answers.

Let $A ( X ) , B ( X )$ be non-zero polynomials in $\mathbb { C } [ X ]$ such that $0 \leq \operatorname { deg } A \leq \operatorname { deg } B - 2$ and $A ( X )$ and $B ( X )$ do not share any roots. Let $\alpha _ { 1 } , \alpha _ { 2 } , \ldots , \alpha _ { k }$ be the roots of $B ( X )$. Suppose that each of them is a simple root.
Show that
$$\sum _ { j = 1 } ^ { k } \frac { A \left( \alpha _ { j } \right) } { B ^ { \prime } \left( \alpha _ { j } \right) } = 0$$

15. Let $f , g$ be two real valued functions defined on the set of real numbers. Which of the following statements are true?
(a) If $f ( x )$ is continuous and $g ( x )$ discontinuous then the function $f ( x ) + g ( x )$ is necessarily discontinuous.
(b) If $f ( x )$ is continuous and $g ( x )$ discontinuous then the function $f ( x ) \cdot g ( x )$ is necessarily discontinuous.
(c) If the functions $f ( x ) , g ( x )$ are discontinuous at a point $c$ in the domain the function $f ( x ) + g ( x )$ is also discontinuous at $c$.
(d) If the functions $f ( x ) , g ( x )$ are discontinuous at a point $c$ in the domain the function $f ( x ) \cdot g ( x )$ is also discontinuous at $c$.

(A) (3 marks) Let $f : \mathbb { R } \longrightarrow \mathbb { R }$ be a continuous function such that $f ( r ) = f \left( r + \frac { 1 } { n } \right)$ for each $r \in \mathbb { Q }$ and each positive integer $n$. Prove or disprove the following statement: $f$ is a constant function.
(B) (7 marks) Let $a _ { n } , n \geq 1$ be a sequence of non-negative real numbers such that $a _ { m + n } \leq a _ { m } + a _ { n }$ for all $m , n$. Show that
$$\lim _ { n \longrightarrow \infty } \frac { a _ { n } } { n } = \inf \left\{ \left. \frac { a _ { n } } { n } \right\rvert\, n \geq 1 \right\}$$

16. In the following code, A is an array indexed from 0 whose elements are all positive integers, n is the number of elements in A , and x is a positive integer. The call abs( z ) returns the absolute value of integer z .
\begin{verbatim} function foo(A,n,x) { max = 0; for i from 0 to (n-1) { for j from (i+1) to (n-1) { diff = abs(A[i]-A[j]); if (diff >= x) and (diff >= max) { max = diff; } } } return(max); } \end{verbatim}
If $A = [ 10,8,10,4,10,7,1,2,2,9 ]$, what will $f _ { \circ \circ } ( A , 10,5 )$ return?
(a) 2
(b) 4
(c) 6
(d) 9

Let $p$ be a prime number. Let $n \geq 2$ be an integer. Let $V$ be an $n$-dimensional $\mathbb { F } _ { p }$-vector space. Determine, with a proof, the number of two-dimensional $\mathbb { F } _ { p }$-subspaces of $V$.

Suppose $f$ is a function whose domain is $X$ and codomain is $Y$. It is given that $|X|>1$ and $|Y|>1$. No other information is known about $X$, $Y$ and $f$. Instruction: Write the number of a single correct option for the given statement S.
$\mathrm{S} =$ "For each $x$ in $X$, there exists $y$ in $Y$ such that $f(x) = y$." [1 point]
Options:

1. S is always true.
2. S is always false.
3. S is true if and only if $f$ is one-to-one.
4. If S is true then $f$ is one-to-one but the converse is false.
5. If $f$ is one-to-one then S is true but the converse is false.
6. S is true if and only if $f$ is onto.
7. If S is true then $f$ is onto but the converse is false.
8. If $f$ is onto then S is true but the converse is false.
9. S is true if and only if $f$ is a constant function.
10. If S is true then $f$ is a constant function but the converse is false.
11. If $f$ is a constant function then S is true but the converse is false.
12. None of the above.

Let $R$ be the ring of all the real-valued functions on $\mathbb { N } \times \mathbb { N }$. Show that $R$ contains a subring isomorphic to the polynomial ring $\mathbb { R } [ X , Y ]$.

Suppose $f$ is a function whose domain is $X$ and codomain is $Y$. It is given that $|X|>1$ and $|Y|>1$. No other information is known about $X$, $Y$ and $f$. Instruction: Write the number of a single correct option for the given statement S.
$\mathrm{S} =$ "For each $y$ in $Y$, there exists $x$ in $X$ such that $f(x) = y$." [1 point]
Options:

1. S is always true.
2. S is always false.
3. S is true if and only if $f$ is one-to-one.
4. If S is true then $f$ is one-to-one but the converse is false.
5. If $f$ is one-to-one then S is true but the converse is false.
6. S is true if and only if $f$ is onto.
7. If S is true then $f$ is onto but the converse is false.
8. If $f$ is onto then S is true but the converse is false.
9. S is true if and only if $f$ is a constant function.
10. If S is true then $f$ is a constant function but the converse is false.
11. If $f$ is a constant function then S is true but the converse is false.
12. None of the above.

Let $M _ { n } ( \mathbb { R } )$ be the space of $n \times n$ matrices with real entries, identified with the euclidean space $\mathbb { R } ^ { n ^ { 2 } }$. Let $X$ be a compact subset of $M _ { n } ( \mathbb { R } )$, and $S \subset \mathbb { C }$ be the set of all eigenvalues of the matrices in $X$. Show that $S$ is a compact subset of $\mathbb { C }$.

Suppose $f$ is a function whose domain is $X$ and codomain is $Y$. It is given that $|X|>1$ and $|Y|>1$. No other information is known about $X$, $Y$ and $f$. Instruction: Write the number of a single correct option for the given statement S.
$\mathrm{S} =$ "There exists a unique $x$ in $X$ such that for each $y$ in $Y$ it is true that $f(x) = y$." [1 point]
Options:

1. S is always true.
2. S is always false.
3. S is true if and only if $f$ is one-to-one.
4. If S is true then $f$ is one-to-one but the converse is false.
5. If $f$ is one-to-one then S is true but the converse is false.
6. S is true if and only if $f$ is onto.
7. If S is true then $f$ is onto but the converse is false.
8. If $f$ is onto then S is true but the converse is false.
9. S is true if and only if $f$ is a constant function.
10. If S is true then $f$ is a constant function but the converse is false.
11. If $f$ is a constant function then S is true but the converse is false.
12. None of the above.

19. We shuffle the deck $[ 3,6,11,4,7,9,2,8,5,10,12,1 ]$. What are the neighbours of 4 after the shuffle?
(a) 1
(b) 5
(c) 10
(d) 12

Fix $0 < \lambda < 1$. Choose $\epsilon > 0$ such that $\epsilon + \frac { \lambda ^ { 2 } } { 4 } \leq \frac { \lambda } { 2 }$. Consider the metric space
$$X : = \left\{ \psi \in \mathcal { C } ^ { 1 } ( [ - \epsilon , \epsilon ] ) : | y + \psi ( y ) | \leq \frac { \lambda } { 2 } \text { for all } y \in [ - \epsilon , \epsilon ] \right\}$$
with the induced supremum metric from $\mathcal { C } ^ { 1 } ( [ - \epsilon , \epsilon ] )$, which we denote by $d$. (Recall: $\mathcal { C } ^ { 1 } ( [ - \epsilon , \epsilon ] )$ is the set of real-valued differentiable functions on $[ - \epsilon , \epsilon ]$ whose derivative is continuous.)
(A) (1 mark) Show that there is a function $A : X \longrightarrow X$ given by
$$( A \psi ) ( y ) = - ( y + \psi ( y ) ) ^ { 2 } .$$
(B) (2 marks) Show that if $\psi \in X$ is such that $A \psi = \psi$, then the function $x = y + \psi ( y )$ is an inverse to the function $y = x + x ^ { 2 }$, locally near the origin. In the next few steps, we show that such a $\psi$ exists.
(C) (2 marks) Show that $d \left( A \psi _ { 1 } , A \psi _ { 2 } \right) \leq \lambda d \left( \psi _ { 1 } , \psi _ { 2 } \right)$.
(D) (4 marks) Let $\phi \in X$. Show that the sequence $A ^ { n } \phi , n \geq 1$ is a Cauchy sequence, and it has a limit. By $A ^ { n }$, we mean the $n$-fold composition $A \circ A \circ \cdots \circ A$. (You may use the fact that $X$ is complete with respect to d.)
(E) (1 mark) Show that there exists $\psi \in X$ such that $A \psi = \psi$.

Suppose $f$ is a function whose domain is $X$ and codomain is $Y$. It is given that $|X|>1$ and $|Y|>1$. No other information is known about $X$, $Y$ and $f$. Instruction: Write the number of a single correct option for the given statement S.
$\mathrm{S} =$ "There exists a unique $y$ in $Y$ such that for each $x$ in $X$ it is true that $f(x) = y$." [1 point]
Options:

1. S is always true.
2. S is always false.
3. S is true if and only if $f$ is one-to-one.
4. If S is true then $f$ is one-to-one but the converse is false.
5. If $f$ is one-to-one then S is true but the converse is false.
6. S is true if and only if $f$ is onto.
7. If S is true then $f$ is onto but the converse is false.
8. If $f$ is onto then S is true but the converse is false.
9. S is true if and only if $f$ is a constant function.
10. If S is true then $f$ is a constant function but the converse is false.
11. If $f$ is a constant function then S is true but the converse is false.
12. None of the above.

20. After shuffling once we obtain a deck with cards $[ 15,6,4,1,13,9,8,7,12,11,5,10,3,14,2,16 ]$. What were the neighbours of 2 before the deck was shuffled?
(a) 3
(b) 6
(c) 7
(d) 14

Part (B) - Short-answer questions

For questions in part ( $B$ ), you have to write your answer with a short explanation in the space provided for the question in your answer sheet. If you need more space, you may continue on the pages provided for rough work. Any such overflows must be clearly labeled.

Questions 1 and 2 are based on the following description.

Each round of a TV game show consists of ten questions. Before each round the host takes ten boxes and places prizes in nine of them, leaving one empty. The host then shuffles these boxes and labels them from 1 to 10. When the guest answers a question correctly, the host opens the corresponding box. If the box has a prize, the guest earns the prize. If the box is empty, the round ends and the guest gets to keep their earnings so far.

1. In the Easy Round, all questions are easy, and each prize is ₹1000. Guest Chatur knows all the answers. What is the expected earnings for Chatur in this round? Explain your answer.
2. To proceed to the next round, a guest must earn at least ₹7000 in the Easy Round. What is the probability that guest Chatur, who knows all the answers of the Easy Round, progresses to the second round? Explain your answer.
3. Find values of $x$ and $y$ that satisfy both the following equations:

$$\begin{aligned} & \sqrt { \frac { x } { y } } + \sqrt { \frac { y } { x } } = \frac { 5 } { 2 } \\ & \frac { x } { \sqrt { y } } + \frac { y } { \sqrt { x } } = \frac { 9 } { 2 } \end{aligned}$$

Questions 4 and 5 are based on the following description.

The following question appeared in a quiz:
\begin{displayquote} "Write the code for a function SecondBest $( A , n )$ that takes an array $A$ and a positive integer $n$ as arguments. The elements of $A$ are all integers, and $n$ is the number of elements in $A$. The call SecondBest $( A , n )$ should return the second largest element in $A$. If $A$ has no second largest element, then the function should return the special value None." \end{displayquote}
A student submitted the code below as the answer to this question. In the code the array A is indexed from 0 .
\begin{verbatim} function SecondBest(A, n) { if n == 1 { return(None); } first = A[0]; second = A[1]; for i from 2 to (n-1) { if (A[i] >= first) and (A[i] >= second) { second = first; first = A[i]; } else { if A[i] >= second { second = A[i]; } } } if first != second { return(second); } else { return(None); } } \end{verbatim}
This answer turned out to be wrong; this function gives the correct answer for some valid inputs, and wrong answers for other valid inputs. Answer the next two questions about this function.

Suppose a differentiable function $f$ from $\mathbb{R}$ to $\mathbb{R}$ has a local maximum at $(a, f(a))$ (This means there are numbers $m$ and $M$ such that (i) $m < a < M$ and (ii) $f(a) \geq f(x)$ for any $x \in [m,M]$.) The proof of a standard result is sketched below. Complete it as instructed.
Proof: For sufficiently $\_\_\_\_$ 1 $h > 0$, it is given that $f(a+h)$ $\_\_\_\_$ 2 3. Therefore for such $h$ the quantity $\_\_\_\_$ 4 must be $\_\_\_\_$ 5. By taking the limit of this quantity as $h \rightarrow 0$ from the right, we get that $\_\_\_\_$ 7 must be $\_\_\_\_$ 8. A parallel argument for suitable negative values of $h$ gives that $\_\_\_\_$ 10 must be $\_\_\_\_$ 11. Combining both conclusions gives the desired result: $\_\_\_\_$ 13 $\_\_\_\_$ 14. Note that the mentioned limits exist because $\_\_\_\_$ 16.
Write a sequence of 9 letters indicating the correct options to fill in the numbered blanks 1 to 9. Do not use any spaces, full stop or other punctuation. E.g., ABACDIJKB is in the correct format. [3 points]
Options: A. small B. large C. $\geq$ D. $>$ E. $\leq$ F. $<$ G. $=$ H. $\neq$ I. 0 J. $f(a)$ K. $\frac{f(a+h)-f(a)}{h}$ L. $f'(a)$ M. $f$ is differentiable N. $f$ is continuous

Suppose a differentiable function $f$ from $\mathbb{R}$ to $\mathbb{R}$ has a local maximum at $(a, f(a))$ (This means there are numbers $m$ and $M$ such that (i) $m < a < M$ and (ii) $f(a) \geq f(x)$ for any $x \in [m,M]$.) The proof of a standard result is sketched below. Complete it as instructed.
Proof: For sufficiently $\_\_\_\_$ 1 $h > 0$, it is given that $f(a+h)$ $\_\_\_\_$ 2 3. Therefore for such $h$ the quantity $\_\_\_\_$ 4 must be $\_\_\_\_$ 5. By taking the limit of this quantity as $h \rightarrow 0$ from the right, we get that $\_\_\_\_$ 7 must be $\_\_\_\_$ 8. A parallel argument for suitable negative values of $h$ gives that $\_\_\_\_$ 10 must be $\_\_\_\_$ 11. Combining both conclusions gives the desired result: $\_\_\_\_$ 13 $\_\_\_\_$ 14. Note that the mentioned limits exist because $\_\_\_\_$ 16.
Write a sequence of 7 letters indicating the correct options to fill in the numbered blanks 10 to 16. [2 points]
Options: A. small B. large C. $\geq$ D. $>$ E. $\leq$ F. $<$ G. $=$ H. $\neq$ I. 0 J. $f(a)$ K. $\frac{f(a+h)-f(a)}{h}$ L. $f'(a)$ M. $f$ is differentiable N. $f$ is continuous

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) $\operatorname { rank } ( T ) = 2$.

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 $c \in \mathbb { R }$.

Let $K$ be the splitting field of $X ^ { n } - 1$ over $\mathbb { F } _ { p }$, where $n$ is a positive integer. Pick the correct statement(s) from below.
(A) $K$ has $p ^ { n }$ elements.
(B) If $p \nmid n$, then the group of field automorphisms of $K$ is isomorphic to the multiplicative group $( \mathbb { Z } / n \mathbb { Z } ) ^ { \times }$.
(C) $K$ is a separable extension of $\mathbb { F } _ { p }$.
(D) There exists $m > n$ such that $K$ is the splitting field of $X ^ { m } - 1$ over $\mathbb { F } _ { p }$.

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.

Consider the map $f : \mathbb { R } ^ { 2 } \longrightarrow \mathbb { R } ^ { 2 } , ( x , y ) \mapsto \left( - x - y ^ { 2 } , y + x ^ { 2 } \right)$. Pick the correct statement(s) from below.
(A) There exist infinitely many $( a , b ) \in \mathbb { R } ^ { 2 }$ such that there is an open neighbourhood $U$ of $( a , b )$ such that $\left. f \right| _ { U }$ is a homeomorphism from $U$ to $f ( U )$.
(B) The derivative $D f$ maps some non-zero tangent vector to $\mathbb { R } ^ { 2 }$ at $\left( \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$, to the zero tangent vector at $\left( - \frac { 3 } { 4 } , \frac { 3 } { 4 } \right)$.
(C) There exist infinitely many $( a , b ) \in \mathbb { R } ^ { 2 }$ such that for every open neighbourhood $U$ of $( a , b ) , \left. f \right| _ { U }$ is not a homeomorphism from $U$ to $f ( U )$.
(D) For every differentiable curve $\gamma$ through $( 0,0 ) , f \circ \gamma$ is differentiable curve.