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41
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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Others
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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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Others
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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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Others
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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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TIFR CSE 2024 | Part B | Question: 11
Let $\mathbb{C}$ denote the set of complex numbers and let $k$ be a positive integer. Given a non-zero univariate polynomial $f(x)$ with coefficients in $\mathbb{C}$ and an $a \in \mathbb{C}$, we say that $a$ is a zero of $f$ ... larger than $d$ as well. The number of distinct zeroes in $\mathbb{C}$ of $f$ of multiplicity $k$ is equal to $d$.
Let $\mathbb{C}$ denote the set of complex numbers and let $k$ be a positive integer. Given a non-zero univariate polynomial $f(x)$ with coefficients in $\mathbb{C}$ and ...
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TIFR CSE 2024 | Part B | Question: 12
In the $n$-queens completion problem, the input is an $n \times n$ chess board with queens on some squares, and the goal is to determine if there is a way to place more queens so that the total number of queens is $n$ and no two queens attack each other (two queens are ... $\text{(iii),(iv) and (v)}$. Only $\text{(i), (iii) and (iv)}$.
In the $n$-queens completion problem, the input is an $n \times n$ chess board with queens on some squares, and the goal is to determine if there is a way to place more q...
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TIFR CSE 2024 | Part B | Question: 13
Suppose we are given a graph $\text{G=(V, E)}$ with non-negative edge weights $\left\{w_{e}\right\}_{e \in E}$. Consider the following problems: P1: Finding a minimum spanning tree of $\text{G}$. P2: Finding a maximum spanning tree of $\text{G}$. P3: Finding a ... $\text{P1 but not P2,P3, P4}$. $\text{P1,P2,P3 but not P4}$. $\text{P1, P4 but not P2, P3}$.
Suppose we are given a graph $\text{G=(V, E)}$ with non-negative edge weights $\left\{w_{e}\right\}_{e \in E}$.Consider the following problems:P1: Finding a minimum spann...
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TIFR CSE 2024 | Part B | Question: 14
For an undirected graph $G$, let $\bar{G}$ refer to the complement (a graph on the same vertex set as $G$, with $(i, j)$ as an edge in $\bar{G}$ if and only if it is not an edge in $G$ ). Consider the following statements. $G$ has ... (i) is equivalent to (ii) and (iv). (i) is equivalent to (ii) and (v) None of the five statements are equivalent to each other.
For an undirected graph $G$, let $\bar{G}$ refer to the complement (a graph on the same vertex set as $G$, with $(i, j)$ as an edge in $\bar{G}$ if and only if it is not ...
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TIFR CSE 2024 | Part B | Question: 15
Consider the following automata: Let $N$ be the number of $0 / 1$-strings of length exactly $6$ accepted by this automata. Which of the following is true about $\text{N}$? $\text{N} \leq 4$. $4$ $8$ $16$ $32$
Consider the following automata:Let $N$ be the number of $0 / 1$-strings of length exactly $6$ accepted by this automata. Which of the following is true about $\text{N}$?...
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TIFR CSE 2024 | Part A | Question: 1
If $\text{O}$ is the center of the circle, what is the area of the shaded portion in square cm? $4 \pi-4 \sqrt{2}$ $\frac{7}{2} \pi-4 \sqrt{2}$ $\frac{7}{2} \pi-4 \sqrt{3}$ $\frac{8}{3} \pi-4 \sqrt{3}$ $\frac{8}{3} \pi-4 \sqrt{2}$
If $\text{O}$ is the center of the circle, what is the area of the shaded portion in square cm?$4 \pi-4 \sqrt{2}$$\frac{7}{2} \pi-4 \sqrt{2}$$\frac{7}{2} \pi-4 \sqrt{3}$$...
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TIFR CSE 2024 | Part A | Question: 2
Let $\sigma$ be a uniform random permutation of $\{1, \ldots, 100\}$. What is the probability that $\sigma(1)<\sigma(2)<\sigma(3)$ ... $\frac{3}{100 !}$ $\frac{3 !}{100 !}$ $\frac{6}{100}$ $\frac{1}{6}$ $\frac{1}{3}$
Let $\sigma$ be a uniform random permutation of $\{1, \ldots, 100\}$. What is the probability that $\sigma(1)<\sigma(2)<\sigma(3)$ (i.e., what is the probability that the...
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TIFR CSE 2024 | Part A | Question: 3
There is a $100 \mathrm{~cm}$ long ruler that has 11 ants on positions $0 \mathrm{~cm}, 10 \mathrm{~cm}, 20 \mathrm{~cm}, 30 \mathrm{~cm}$, ..., $100 \mathrm{~cm}$. The ant at the $0 \mathrm{~cm}$ mark ... without knowing the directions of all ants. $100$ seconds. More than $100$ seconds, but cannot be determined without knowing the directions of all ants.
There is a $100 \mathrm{~cm}$ long ruler that has 11 ants on positions $0 \mathrm{~cm}, 10 \mathrm{~cm}, 20 \mathrm{~cm}, 30 \mathrm{~cm}$, ..., $100 \mathrm{~cm}$. The a...
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TIFR CSE 2024 | Part A | Question: 4
Let $z_{1}, z_{2}, z_{3}, \ldots, z_{2023}$ be a permutation of the numbers $1,2,3, \ldots, 2023$. Which of the following is true about the product $\prod_{i=1}^{2023}\left(z_{i}-i\right)$ ? Note: The parity of an ... such that swapping their values does not change the parity of the above product. None of the above statements is true.
Let $z_{1}, z_{2}, z_{3}, \ldots, z_{2023}$ be a permutation of the numbers $1,2,3, \ldots, 2023$. Which of the following is true about the product $\prod_{i=1}^{2023}\le...
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TIFR CSE 2024 | Part A | Question: 5
Let $p(x)$ be a polynomial with real coefficients which satisfies $p(r)=p(-r)$ for every real number $r$. Let $n \geq 5$ be a positive integer. Suppose that $p(i)=i$ for all $1 \leq i \leq n$. What is the maximum possible value of the absolute value of the coefficient of ${x^{5}}$ in $p(x)$ ? $0$ $5$ $10$ $n$ $n+1$
Let $p(x)$ be a polynomial with real coefficients which satisfies $p(r)=p(-r)$ for every real number $r$. Let $n \geq 5$ be a positive integer. Suppose that $p(i)=i$ for ...
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