Recent questions tagged tifr2021 / GATE Overflow for GATE CSE

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1

TIFR CSE 2021 | Part A | Question: 1
A box contains $5$ red marbles, $8$ green marbles, $11$ blue marbles, and $15$ yellow marbles. We draw marbles uniformly at random without replacement from the box. What is the minimum number of marbles to be drawn to ensure that out of the marbles drawn, at least $7$ are of the same colour? $7$ $8$ $23$ $24$ $39$

A box contains $5$ red marbles, $8$ green marbles, $11$ blue marbles, and $15$ yellow marbles. We draw marbles uniformly at random without replacement from the box. What ...

soujanyareddy13

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soujanyareddy13 asked Mar 25, 2021

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2

TIFR CSE 2021 | Part A | Question: 2
What is the area of a rectangle with the largest perimeter that can be inscribed in the unit circle (i.e., all the vertices of the rectangle are on the circle with radius $1$)? $1$ $2$ $3$ $4$ $5$

What is the area of a rectangle with the largest perimeter that can be inscribed in the unit circle (i.e., all the vertices of the rectangle are on the circle with radius...

soujanyareddy13

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soujanyareddy13 asked Mar 25, 2021

2 votes

1 answer

3

TIFR CSE 2021 | Part A | Question: 3
Let $M$ be an $n \times m$ real matrix. Consider the following: Let $k_{1}$ be the smallest number such that $M$ can be factorized as $A \cdot B$, where $A$ is an $n \times k_{1}$ and $B$ is a $k_{1} \times m$ matrix. Let $k_{2}$ be the smallest number ... $k_{1}= k_{2}= k_{3}$ No general relationship exists among $k_{1}, k_{2}$ and $k_{3}$

Let $M$ be an $n \times m$ real matrix. Consider the following:Let $k_{1}$ be the smallest number such that $M$ can be factorized as $A \cdot B$, where $A$ is an $n \time...

soujanyareddy13

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soujanyareddy13 asked Mar 25, 2021

1 votes

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4

TIFR CSE 2021 | Part A | Question: 4
What is the probability that at least two out of four people have their birthdays in the same month, assuming their birthdays are uniformly distributed over the twelve months? $\frac{25}{48}$ $\frac{5}{8}$ $\frac{5}{12}$ $\frac{41}{96}$ $\frac{55}{96}$

What is the probability that at least two out of four people have their birthdays in the same month, assuming their birthdays are uniformly distributed over the twelve mo...

soujanyareddy13

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soujanyareddy13 asked Mar 25, 2021

1 votes

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5

TIFR CSE 2021 | Part A | Question: 5
Let $n, m$ and $k$ be three positive integers such that $n \geq m \geq k$. Let $S$ be a subset of $\left \{ 1, 2,\dots, n \right \}$ of size $k$. Consider sampling a function $f$ uniformly at random from the set of all functions mapping $\left \{ 1,\dots, n \right \}$ ... $1-\frac{k!\binom{n}{k}}{n^{k}}$ $1-\frac{k!\binom{n}{k}}{m^{k}}$

Let $n, m$ and $k$ be three positive integers such that $n \geq m \geq k$. Let $S$ be a subset of $\left \{ 1, 2,\dots, n \right \}$ of size $k$. Consider sampling a func...

soujanyareddy13

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soujanyareddy13 asked Mar 25, 2021

1 votes

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6

TIFR CSE 2021 | Part A | Question: 6
A matching in a graph is a set of edges such that no two edges in the set share a common vertex. Let $G$ be a graph on $n$ $\textit{vertices}$ in which there is a subset $M$ of $m$ $\textit{edges}$ which is a matching. Consider a random process where each vertex in the ... $\left ( 1-p^{2} \right )^{m}$ $1-\left ( 1-p\left ( 1-p \right ) \right )^{m}$

A matching in a graph is a set of edges such that no two edges in the set share a common vertex. Let $G$ be a graph on $n$ $\textit{vertices}$ in which there is a subset ...

soujanyareddy13

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soujanyareddy13 asked Mar 25, 2021

1 votes

1 answer

7

TIFR CSE 2021 | Part A | Question: 7
Let $d$ be the positive square integers (that is, it is a square of some integer) that are factors of $20^{5} \times 21^{5}$. Which of the following is true about $d$? $50\leq d< 100$ $100\leq d< 150$ $150\leq d< 200$ $200\leq d< 300$ $300\leq d$

Let $d$ be the positive square integers (that is, it is a square of some integer) that are factors of $20^{5} \times 21^{5}$. Which of the following is true about $d$?$50...

soujanyareddy13

524 views

soujanyareddy13 asked Mar 25, 2021

1 votes

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8

TIFR CSE 2021 | Part A | Question: 8
Consider the sequence $y_{n}=\frac{1}{\int_{1}^{n}\frac{1}{\left ( 1+x/n \right )^{3}}dx}$ for $\text{n} = 2,3,4, \dots$ Which of the following is $\text{TRUE}$? The sequence $\{y_{n}\}$ does not have a limit as $n\rightarrow \infty$ ... and is equal to $0$. The sequence $\{y_{n}\}$ first increases and then decreases as $\text{n}$ takes values $2, 3, 4, \dots$

Consider the sequence$$y_{n}=\frac{1}{\int_{1}^{n}\frac{1}{\left ( 1+x/n \right )^{3}}dx}$$for $\text{n} = 2,3,4, \dots$ Which of the following is $\text{TRUE}$?The seque...

soujanyareddy13

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soujanyareddy13 asked Mar 25, 2021

2 votes

2 answers

9

TIFR CSE 2021 | Part A | Question: 9
Fix $n\geq 6$. Consider the set $\mathcal{C}$ of binary strings $x_{1}, x_{2} \dots x_{n}$ of length $n$ such that the bits satisfy the following set of equalities, all modulo $2$: $x_{i}+x_{i+1}+x_{i+2}=0$ ... $3$ $\left | \mathcal{C} \right |=4$. If $n\geq 6$ is not divisible by $3$ then $\left | \mathcal{C} \right |=1$.

Fix $n\geq 6$. Consider the set $\mathcal{C}$ of binary strings $x_{1}, x_{2} \dots x_{n}$ of length $n$ such that the bits satisfy the following set of equalities, all m...

soujanyareddy13

498 views

soujanyareddy13 asked Mar 25, 2021

1 votes

1 answer

10

TIFR CSE 2021 | Part A | Question: 10
Lavanya and Ketak each flip a fair coin (i.e., both heads and tails have equal probability of appearing) $n$ times. What is the probability that Lavanya sees more heads than ketak? In the following, the binomial coefficient $\binom{n}{k}$ counts the number of $k$-element subsets of ... $\sum_{i=0}^{n}\frac{\binom{n}{i}}{2^{n}}$

Lavanya and Ketak each flip a fair coin (i.e., both heads and tails have equal probability of appearing) $n$ times. What is the probability that Lavanya sees more heads t...

soujanyareddy13

556 views

soujanyareddy13 asked Mar 25, 2021

1 votes

1 answer

11

TIFR CSE 2021 | Part A | Question: 11
Find the following sum. $\frac{1}{2^{2}-1}+\frac{1}{4^{2}-1}+\frac{1}{6^{2}-1}+\cdots+\frac{1}{40^{2}-1}$ $\frac{20}{41}$ $\frac{10}{41}$ $\frac{10}{21}$ $\frac{20}{21}$ $1$

Find the following sum.$$\frac{1}{2^{2}-1}+\frac{1}{4^{2}-1}+\frac{1}{6^{2}-1}+\cdots+\frac{1}{40^{2}-1}$$$\frac{20}{41}$$\frac{10}{41}$$\frac{10}{21}$$\frac{20}{21}$$1$

soujanyareddy13

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soujanyareddy13 asked Mar 25, 2021

1 votes

1 answer

12

TIFR CSE 2021 | Part A | Question: 12
How many numbers in the range ${0, 1, \dots , 1365}$ have exactly four $1$'s in their binary representation? (Hint: $1365_{10}$ is $10101010101_{2}$, that is, $1365=2^{10} + 2^{8}+2^{6}+2^{4}+2^{2}+2^{0}.)$ ... $\binom{11}{4}+\binom{9}{3}+\binom{7}{2}+\binom{5}{1}$ $1024$

How many numbers in the range ${0, 1, \dots , 1365}$ have exactly four $1$’s in their binary representation? (Hint: $1365_{10}$ is $10101010101_{2}$, that is, $$1365=2^...

soujanyareddy13

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soujanyareddy13 asked Mar 25, 2021

1 votes

3 answers

13

TIFR CSE 2021 | Part A | Question: 13
What are the last two digits of $7^{2021}$? $67$ $07$ $27$ $01$ $77$

What are the last two digits of $7^{2021}$?$67$$07$$27$$01$$77$

soujanyareddy13

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soujanyareddy13 asked Mar 25, 2021

3 votes

1 answer

14

TIFR CSE 2021 | Part A | Question: 14
Five married couples attended a party. In the party, each person shook hands with those they did not know. Everyone knows his or her spouse. At the end of the party, Shyamal, one of the attendees, listed the number of hands that other attendees ... in the list. How many persons shook hands with Shyamal at the party? $2$ $4$ $6$ $8$ Insufficient information

Five married couples attended a party. In the party, each person shook hands with those they did not know. Everyone knows his or her spouse. At the end of the party, Shya...

soujanyareddy13

711 views

soujanyareddy13 asked Mar 25, 2021

2 votes

0 answers

15

TIFR CSE 2021 | Part A | Question: 15
Let $P$ be a convex polygon with sides $5, 4, 4, 3$. For example, the following: Consider the shape in the plane that consists of all points within distance $1$ from some point in $P$. If $\ell$ is the perimeter of the shape, which of the following ... the given information. $20\leq \ell < 21$ $21\leq \ell< 22$ $22\leq \ell< 23$ $23\leq \ell< 24$

Let $P$ be a convex polygon with sides $5, 4, 4, 3$. For example, the following:Consider the shape in the plane that consists of all points within distance $1$ from some ...

soujanyareddy13

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soujanyareddy13 asked Mar 25, 2021

2 votes

2 answers

16

TIFR CSE 2021 | Part B | Question: 1
Consider the following statements about propositional formulas. $\left ( p\wedge q \right )\rightarrow r$ and $\left ( p \rightarrow r \right )\wedge \left ( q\rightarrow r \right )$ are $\textit{not }$ ... values $p$ and $q$, $\text{(i)}$ can be either true or false, while $\text{(ii)}$ is always false.

Consider the following statements about propositional formulas.$\left ( p\wedge q \right )\rightarrow r$ and $\left ( p \rightarrow r \right )\wedge \left ( q\rightarrow ...

soujanyareddy13

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soujanyareddy13 asked Mar 25, 2021

1 votes

1 answer

17

TIFR CSE 2021 | Part B | Question: 2
Let $L$ be a singly-linked list $X$ and $Y$ be additional pointer variables such that $X$ points to the first element of $L$ and $Y$ points to the last element of $L$. Which of the following operations cannot be done in time that is ... after the last element of $L$. Add an element before the first element of $L$. Interchange the first two elements of $L$.

Let $L$ be a singly-linked list $X$ and $Y$ be additional pointer variables such that $X$ points to the first element of $L$ and $Y$ points to the last element of $L$. Wh...

soujanyareddy13

788 views

soujanyareddy13 asked Mar 25, 2021

2 votes

3 answers

18

TIFR CSE 2021 | Part B | Question: 3
What is the prefix expression corresponding to the expression: $\left ( \left ( 9+8 \right ) \ast 7+\left ( 6\ast \left ( 5+4 \right ) \right )\ast 3\right )+2?$ You may assume that $\ast$ has precedence over $+$? $\ast + +\: 987 \ast \ast \: 6 + + \:5432$ ... $+ + \ast +\: 987 \ast \ast \: 6 + \:5432$ $+ \ast + \ast \: 987+ + \: 6 \ast \:5432$

What is the prefix expression corresponding to the expression:$\left ( \left ( 9+8 \right ) \ast 7+\left ( 6\ast \left ( 5+4 \right ) \right )\ast 3\right )+2?$You may as...

soujanyareddy13

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soujanyareddy13 asked Mar 25, 2021

2 votes

1 answer

19

TIFR CSE 2021 | Part B | Question: 4
Consider the following two languages. ... $\text{NP}$ vs. $\text{P}$ question.

Consider the following two languages.$\begin{array}{rcl} \text{PRIME} & = & \{ 1^{n} \mid n \text{ is a prime number} \}, \\ \text{FACTOR} & = & \{ 1^{n}0 1^{a} 01^{b} ...

soujanyareddy13

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soujanyareddy13 asked Mar 25, 2021

1 votes

1 answer

20

TIFR CSE 2021 | Part B | Question: 5
For a language $L$ over the alphabet $\{a, b\}$, let $\overline{L}$ denote the complement of $L$ and let $L^{\ast}$ denote the Kleene-closure of $L$. Consider the following sentences. $\overline{L}$ and $L^{\ast}$ are both context-free. $\overline{L}$ is not ... ? Both (i) and (iii) Only (i) Only (iii) Only (ii) None of the above

For a language $L$ over the alphabet $\{a, b\}$, let $\overline{L}$ denote the complement of $L$ and let $L^{\ast}$ denote the Kleene-closure of $L$. Consider the followi...

soujanyareddy13

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soujanyareddy13 asked Mar 25, 2021

2 votes

2 answers

21

TIFR CSE 2021 | Part B | Question: 6
Consider the following pseudocode: procedure HowManyDash(n) if n=0 then print '-' else if n=1 then print '-' else HowManyDash(n-1) HowManyDash(n-2) end if end procedure How many ‘-’ does HowManyDash$(10)$ print? $9$ $10$ $55$ $89$ $1024$

Consider the following pseudocode:procedure HowManyDash(n) if n=0 then print '-' else if n=1 then print '-' else HowManyDash(n-1) HowManyDash(n-2) end if end procedureHow...

soujanyareddy13

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soujanyareddy13 asked Mar 25, 2021

1 votes

2 answers

22

TIFR CSE 2021 | Part B | Question: 7
Which of the following regular expressions defines a language that is different from the other choices? $b^{\ast }\left ( a+b \right )^\ast a\left ( a+b \right )^ \ast ab^\ast \left ( a+b \right )^{\ast }$ ...

Which of the following regular expressions defines a language that is different from the other choices?$b^{\ast }\left ( a+b \right )^\ast a\left ( a+b \right )^ \ast ab^...

soujanyareddy13

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soujanyareddy13 asked Mar 25, 2021

3 votes

1 answer

23

TIFR CSE 2021 | Part B | Question: 8
Let $A$ and $B$ be two matrices of size $n \times n$ and with real-valued entries. Consider the following statements. If $AB = B$, then $A$ must be the identity matrix. If $A$ is an idempotent (i.e. $A^{2} = A$) nonsingular matrix, then $A$ must be the identity matrix. ... true of $A$? $1, 2 $ and $3$ Only $2$ and $3$ Only $1$ and $2$ Only $1$ and $3$ Only $2$

Let $A$ and $B$ be two matrices of size $n \times n$ and with real-valued entries. Consider the following statements.If $AB = B$, then $A$ must be the identity matrix.If ...

soujanyareddy13

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soujanyareddy13 asked Mar 25, 2021

2 votes

1 answer

24

TIFR CSE 2021 | Part B | Question: 9
Let $L$ be a context-free language generated by the context-free grammar $G = (V, \Sigma, R, S)$ where $V$ is the finite set of variables, $\Sigma$ the finite set of terminals (disjoints from $V$), $R$ the finite set of rules and $S \in V$ the start variable. ... ${L}'=L^{\ast }$ ${L}'=\left \{ xx \mid x \in L \right \}$ None of the above

Let $L$ be a context-free language generated by the context-free grammar $G = (V, \Sigma, R, S)$ where $V$ is the finite set of variables, $\Sigma$ the finite set of term...

soujanyareddy13

671 views

soujanyareddy13 asked Mar 25, 2021

2 votes

1 answer

25

TIFR CSE 2021 | Part B | Question: 10
Let $G$ be a connected bipartite simple graph (i.e., no parallel edges) with distinct edge weights. Which of the following statements on $\text{MST}$ (minimum spanning tree) need $\text{NOT}$ be true? $G$ has a unique $\text{MST}$ ... edge. Every $\text{MST}$ in $G$ contains the third lightest edge. No $\text{MST}$ in $G$ contains the heaviest edge.

Let $G$ be a connected bipartite simple graph (i.e., no parallel edges) with distinct edge weights. Which of the following statements on $\text{MST}$ (minimum spanning tr...

soujanyareddy13

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soujanyareddy13 asked Mar 25, 2021

2 votes

2 answers

26

TIFR CSE 2021 | Part B | Question: 11
Suppose we toss a fair coin (i.e., both beads and tails have equal probability of appearing) repeatedly until the first time by which at least $\textit{two}$ heads and at least $\textit{two}$ tails have appeared in the sequence of tosses made. What is the expected number of coin tosses that we would have to make? $8$ $4$ $5.5$ $7.5$ $4.5$

Suppose we toss a fair coin (i.e., both beads and tails have equal probability of appearing) repeatedly until the first time by which at least $\textit{two}$ heads and at...

soujanyareddy13

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soujanyareddy13 asked Mar 25, 2021

3 votes

1 answer

27

TIFR CSE 2021 | Part B | Question: 12
Let $G$ be an undirected graph. For any two vertices $u, v$ in $G$, let $\textrm{cut} (u, v)$ be the minimum number of edges that should be deleted from $G$ so that there is no path between $u$ and $v$ in the resulting graph. Let $a, b, c, d$ be $4$ ... $\textrm{cut} (b,d) = 2$, $\textrm{cut} (b,c) = 2$ and $\textrm{cut} (c,d) = 1$

Let $G$ be an undirected graph. For any two vertices $u, v$ in $G$, let $\textrm{cut} (u, v)$ be the minimum number of edges that should be deleted from $G$ so that there...

soujanyareddy13

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soujanyareddy13 asked Mar 25, 2021

2 votes

1 answer

28

TIFR CSE 2021 | Part B | Question: 13
Let $A$ be a $3 \times 6$ matrix with real-valued entries. Matrix $A$ has rank $3$. We construct a graph with $6$ vertices where each vertex represents distinct column in $A$, and there is an edge between two vertices if the two columns represented ... is connected. There is a clique of size $3$. The graph has a cycle of length $4$. The graph is $3$-colourable.

Let $A$ be a $3 \times 6$ matrix with real-valued entries. Matrix $A$ has rank $3$. We construct a graph with $6$ vertices where each vertex represents distinct column in...

soujanyareddy13

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soujanyareddy13 asked Mar 25, 2021

3 votes

2 answers

29

TIFR CSE 2021 | Part B | Question: 14
Consider the following greedy algorithm for colouring an $n$-vertex undirected graph $G$ with colours $c_{1}, c_{2}, \dots:$ consider the vertices of $G$ ... $m\left ( n, r \right ) = nr$ $m\left ( n, r \right ) = n\binom{r}{2}$

Consider the following greedy algorithm for colouring an $n$-vertex undirected graph $G$ with colours $c_{1}, c_{2}, \dots:$ consider the vertices of $G$ in any sequence ...

soujanyareddy13

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soujanyareddy13 asked Mar 25, 2021

2 votes

1 answer

30

TIFR CSE 2021 | Part B | Question: 15
Let $A\left [ i \right ] : i=0, 1, 2, \dots , n-1$ be an array of $n$ distinct integers. We wish to sort $A$ in ascending order. We are given that each element in the array is at a position that is at most $k$ away from its position in the sorted array, ... $t\left ( n \right ) = \Theta \left ( nk \right )$

Let $A\left [ i \right ] : i=0, 1, 2, \dots , n-1$ be an array of $n$ distinct integers. We wish to sort $A$ in ascending order. We are given that each element in the arr...

soujanyareddy13

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soujanyareddy13 asked Mar 25, 2021

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force match	+apple
views	views:100
score	score:10
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