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31
TIFR Mathematics 2024 | Part A | Question: 11
What is the least positive integer $n>1$ such that $x^{n}$ and $x$ are conjugate, for every $x \in S_{11}$? Here, $S_{11}$ denotes the symmetric group on $11$ letters. $10$ $11$ $12$ $13$
What is the least positive integer $n>1$ such that $x^{n}$ and $x$ are conjugate, for every $x \in S_{11}$? Here, $S_{11}$ denotes the symmetric group on $11$ letters.$10...
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TIFR Mathematics 2024 | Part A | Question: 12
Consider the following statements: $\text{(A)}$ Let $G$ be a group and let $H \subset G$ be a subgroup of index 2 . Then $[G, G] \subseteq H$. $\text{(B)}$ Let $G$ be a group and let $H \subset G$ be a subgroup that contains the commutator subgroup ... false $\text{(A)}$ is true and $\text{(B)}$ is false $\text{(A)}$ is false and $\text{(B)}$ is true
Consider the following statements:$\text{(A)}$ Let $G$ be a group and let $H \subset G$ be a subgroup of index 2 . Then $[G, G] \subseteq H$.$\text{(B)}$ Let $G$ be a gro...
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TIFR Mathematics 2024 | Part A | Question: 13
For any symmetric real matrix $A$, let $\lambda(A)$ denote the largest eigenvalue of $A$. Let $S$ be the set of positive definite symmetric $3 \times 3$ real matrices. Which of the following assertions is correct? There exist $A, B \in S$ ... $\lambda(A+B)=\max (\lambda(A), \lambda(B))$ None of the remaining three assertions is correct
For any symmetric real matrix $A$, let $\lambda(A)$ denote the largest eigenvalue of $A$. Let $S$ be the set of positive definite symmetric $3 \times 3$ real matrices. Wh...
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TIFR Mathematics 2024 | Part A | Question: 14
Let $\theta \in(0, \pi / 2)$. Let $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ be the linear map which sends a vector $v$ to its reflection with respect to the line through $(0,0)$ and $(\cos \theta, \sin \theta)$. Then the ... $\left(\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right)$
Let $\theta \in(0, \pi / 2)$. Let $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ be the linear map which sends a vector $v$ to its reflection with respect to the line thr...
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TIFR Mathematics 2024 | Part A | Question: 15
For a polynomial $f(x, y) \in \mathbb{R}[x, y]$, let $X_{f}=\left\{(a, b) \in \mathbb{R}^{2} \mid f(a, b)=1\right\} \subset \mathbb{R}^{2}$. Which of the following statements is correct? If $f(x, y)=x^{2}+4 x y+3 y^{2}$, ... then $X_{f}$ is compact If $f(x, y)=x^{2}-4 x y-y^{2}$, then $X_{f}$ is compact None of the remaining three statements is correct
For a polynomial $f(x, y) \in \mathbb{R}[x, y]$, let $X_{f}=\left\{(a, b) \in \mathbb{R}^{2} \mid f(a, b)=1\right\} \subset \mathbb{R}^{2}$. Which of the following statem...
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TIFR Mathematics 2024 | Part A | Question: 16
What is the number of distinct subfields of $\mathbb{C}$ isomorphic to $\mathbb{Q}[\sqrt[3]{2}]$? $1$ $2$ $3$ Infinite
What is the number of distinct subfields of $\mathbb{C}$ isomorphic to $\mathbb{Q}[\sqrt[3]{2}]$?$1$$2$$3$Infinite
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TIFR Mathematics 2024 | Part A | Question: 17
Let $\mathbb{F}_{3}$ denote the finite field with 3 elements. What is the number of one dimensional vector subspaces of the vector space $\mathbb{F}_{3}^{5}$ over $\mathbb{F}_{3}$? $5$ $121$ $81$ None of the remaining three options
Let $\mathbb{F}_{3}$ denote the finite field with 3 elements. What is the number of one dimensional vector subspaces of the vector space $\mathbb{F}_{3}^{5}$ over $\mathb...
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TIFR Mathematics 2024 | Part A | Question: 18
For a positive integer $n$, let $a_{n}, b_{n}, c_{n}, d_{n}$ be the real numbers such that \[ \left(\begin{array}{ll} 1 & 1 \\ 1 & 0 \end{array}\right)^{n}=\left(\begin{array}{ll} a_{n} & b_{n} \\ c_{n} & ... the following numbers equals $\lim _{n \rightarrow \infty} a_{n} / b_{n}$ ? $1$ $e$ $3 / 2$ None of the remaining three options
For a positive integer $n$, let $a_{n}, b_{n}, c_{n}, d_{n}$ be the real numbers such that\[\left(\begin{array}{ll}1 & 1 \\1 & 0\end{array}\right)^{n}=\left(\begin{array}...
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TIFR Mathematics 2024 | Part A | Question: 19
Consider the complex vector space $V=\{f \in \mathbb{C}[x] \mid f$ has degree at most 50 , and $f(i x)=-f(x)$ for all $x \in \mathbb{C}\}$. Then the dimension of $V$ equals $50$ $25$ $13$ $47$
Consider the complex vector space$V=\{f \in \mathbb{C}[x] \mid f$ has degree at most 50 , and $f(i x)=-f(x)$ for all $x \in \mathbb{C}\}$.Then the dimension of $V$ equals...
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TIFR Mathematics 2024 | Part A | Question: 20
Let $S$ denote the set of sequences $a=\left(a_{1}, a_{2}, \ldots\right)$ of real numbers such that $a_{k}$ equals 0 or 1 for each $k$. Then the function $f: S \rightarrow \mathbb{R}$ defined by \[ f\left(\ ... }}{10}+\frac{a_{2}}{10^{2}}+\ldots \] is injective but not surjective surjective but not injective bijective neither injective nor surjective
Let $S$ denote the set of sequences $a=\left(a_{1}, a_{2}, \ldots\right)$ of real numbers such that $a_{k}$ equals 0 or 1 for each $k$. Then the function $f: S \rightarro...
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TIFR CSE 2024 | Part B | Question: 1
Consider the following three functions defined for all positive integers $n \geq 0$ ... . Only $\text{(ii)}$ and $\text{(iii)}$ are true All of $\text{(i), (ii)}$, and $\text{(iii)}$ are true.
Consider the following three functions defined for all positive integers $n \geq 0$.\[\begin{array}{l}f(n)=|\sin (n)+n|, \\g(n)=n, \\h(n)=|\sin (n)| .\end{array}\]Which o...
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Algorithms
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TIFR CSE 2024 | Part B | Question: 2
Let $\text{S}$ be the set of all $4$ -digit numbers created using just the digits $1,2,3,4,5$ such that no two successive digits are the same. If the numbers in $\text{S}$ are arranged in ascending order, what is the $100$ th number in this sequence? $2135$ $2324$ $2315$ $2352$ $2415$
Let $\text{S}$ be the set of all $4$ -digit numbers created using just the digits $1,2,3,4,5$ such that no two successive digits are the same. If the numbers in $\text{S}...
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TIFR CSE 2024 | Part B | Question: 3
For any positive integer $\text{N}$, let $\text{p(N)}$ be the probability that a uniformly random number $a \in\{1, \ldots, N\}$ ... $p(N)=\Theta\left(\frac{1}{\sqrt{N}}\right)$. $p(N)=\Theta\left(\frac{1}{\log N}\right)$.
For any positive integer $\text{N}$, let $\text{p(N)}$ be the probability that a uniformly random number $a \in\{1, \ldots, N\}$ has an odd number of factors (including 1...
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TIFR CSE 2024 | Part B | Question: 4
Consider functions $f$ and $g$ from the set of positive real numbers to itself, recursively defined as follows. \[ \begin{array}{rrl} \forall n \leq 1 & f(n), g(n) & =1, \\ \forall n>1 & f(n) & =3 f(n / 3)+n^{2}, \\ \forall n>1 & g( ... $g(n)=\Theta(n \log \log n)$ $f(n)=\Theta\left(n^{2} \log n\right)$ and $g(n)=\Theta(n \log n)$
Consider functions $f$ and $g$ from the set of positive real numbers to itself, recursively defined as follows.\[\begin{array}{rrl}\forall n \leq 1 & f(n), g(n) & =1, \\\...
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Algorithms
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TIFR CSE 2024 | Part B | Question: 5
For two languages $\text{A, B}$ over the alphabet $\Sigma$, let the perfect shuffle of $\text{A}$ and $\text{B}$ be the language \begin{Bmatrix} w=a_1 b_1 a_2 b_2 \cdots a_k b_k \text{where} a_1 a_2 \cdots a_k \in \text{A} and b_1 b_2 \cdots b_k \in B.& \\ ... $\text{(ii)}$. Only $\text{(ii) and (iii)}$. None of $\text{(i), (ii), (iii)}$ is true.
For two languages $\text{A, B}$ over the alphabet $\Sigma$, let the perfect shuffle of $\text{A}$ and $\text{B}$ be the language\begin{Bmatrix}w=a_1 b_1 a_2 b_2 \cdots a_...
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TIFR CSE 2024 | Part B | Question: 6
The four nucleotides in $\text{DNA}$ are called $\text{A, C, G}$, and $\text{T}$. Consider the following languages over the alphabet $\{\mathrm{A}, \mathrm{C}, \mathrm{G}$, and $\mathrm{T}\}$. \[ \begin{array}{l} L_{1}=\left\{(\mathrm{AC})^{n}(\mathrm{GT})^{n} ... $L_{1}$ and $L_{3} \cdot$ Only $L_{1}$ and $L_{2}$. All three of $L_{1}, L_{2}, L_{3}$.
The four nucleotides in $\text{DNA}$ are called $\text{A, C, G}$, and $\text{T}$. Consider the following languages over the alphabet $\{\mathrm{A}, \mathrm{C}, \mathrm{G}...
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TIFR CSE 2024 | Part B | Question: 7
Consider the following algorithm that takes as input a positive integer $n$. if (n == 1) { return "Neither prime nor composite." } m=2 while (m < n) { if (m divides n ){ return "Composite." } m=m+1 } return "Prime. ... at most $\left\lceil n^{1 / 9}\right\rceil$ times only if $p, q, r$ are distinct primes or distinct prime powers.
Consider the following algorithm that takes as input a positive integer $n$.if (n == 1) { return "Neither prime nor composite." } m=2 while (m < n) { if (m divides n ){ r...
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TIFR CSE 2024 | Part B | Question: 8
In the following pseudocode, assume that for any pair of integers $x \leq y$, the function random ( $\mathrm{x}, \mathrm{y})$ produces an integer uniformly chosen from the set $\{x, x+1, \ldots, y\}$. n=9 for (i=1 to ... equal probability, and does not print any other output. The output is always $987654321$. The output may not be a permutation of $123456789$.
In the following pseudocode, assume that for any pair of integers $x \leq y$, the function random ( $\mathrm{x}, \mathrm{y})$ produces an integer uniformly chosen from th...
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TIFR CSE 2024 | Part B | Question: 9
Given $m$ vectors $\vec{x}_{1}, \vec{x}_{2}, \ldots, \vec{x}_{m}$ in $\mathbb{R}^{d}$, we construct an undirected graph $G=(V, E)$ as follows. Each vector $\vec{x}_{i}$ is represented by a vertex $v_{i}$. We add an edge between ... size at most $d$ Any clique has size at most $m / 2$ The maximum degree of any vertex in $G$ is at most $d$ None of the above.
Given $m$ vectors $\vec{x}_{1}, \vec{x}_{2}, \ldots, \vec{x}_{m}$ in $\mathbb{R}^{d}$, we construct an undirected graph $G=(V, E)$ as follows. Each vector $\vec{x}_{i}$ i...
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TIFR CSE 2024 | Part B | Question: 10
Arun has a non-empty subset $\text{S}$ of the numbers $\{1,2,3, \ldots, 1000\}$. Bela wants to find any number $\text{x}$ in Arun's set $\text{S}$. To do this, Arun and Bela decide to play a game which proceeds in rounds. In each round, Bela ... rounds will Bela need to find out some $\text{x}$ in Arun's set $\text{S}$? $9$ $10$ $11$ $1023$ $1024$
Arun has a non-empty subset $\text{S}$ of the numbers $\{1,2,3, \ldots, 1000\}$. Bela wants to find any number $\text{x}$ in Arun's set $\text{S}$.To do this, Arun and Be...
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