Recent questions tagged tifr2020 / GATE Overflow for GATE CSE

6 votes

1 answer

1

TIFR CSE 2020 | Part B | Question: 14
The figure below describes the network of streets in a city where Motabhai sells $\text{pakoras}$ from his cart. The number next to an edge is the time (in minutes) taken to traverse the corresponding street. At present, the cart is required to start at point $s$ ... $f$ are the only odd degree nodes in the figure above. $430$ $440$ $460$ $470$ $480$

The figure below describes the network of streets in a city where Motabhai sells $\text{pakoras}$ from his cart. The number next to an edge is the time (in minutes) taken...

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admin asked Feb 10, 2020

3 votes

1 answer

2

TIFR CSE 2020 | Part B | Question: 15
Suppose $X_{1a}, X_{1b},X_{2a},X_{2b},\dots , X_{5a},X_{5b}$ are ten Boolean variables each of which can take the value TRUE or FLASE. Recall the Boolean XOR $X\oplus Y:=(X\wedge \neg Y)\vee (\neg X \wedge Y)$. Define the Boolean logic ... $H$ have? $20$ $30$ $32$ $160$ $1024$

Suppose $X_{1a}, X_{1b},X_{2a},X_{2b},\dots , X_{5a},X_{5b}$ are ten Boolean variables each of which can take the value TRUE or FLASE. Recall the Boolean XOR $X\oplus Y:=...

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admin asked Feb 10, 2020

3 votes

1 answer

3

TIFR CSE 2020 | Part B | Question: 13
Let $G$ be an undirected graph. An Eulerian cycle of $G$ is a cycle that traverses each edge of $G$ exactly once. A Hamiltonian cycle of $G$ is a cycle that traverses each vertex of $G$ exactly once. Which of the following ... has a Hamiltonian cycle A complete graph always has both an Eulerian cycle and a Hamiltonian cycle All of the other statements are true

Let $G$ be an undirected graph. An Eulerian cycle of $G$ is a cycle that traverses each edge of $G$ exactly once. A Hamiltonian cycle of $G$ is a cycle that traverses ea...

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803 views

admin asked Feb 10, 2020

1 votes

2 answers

4

TIFR CSE 2020 | Part B | Question: 12
Given the pseudocode below for the function $\textbf{remains()}$, which of the following statements is true about the output, if we pass it a positive integer $n>2$? int remains(int n) { int x = n; for (i=(n-1);i>1;i--) { x = x % i ; } ... $1$ Output is $0$ only if $n$ is NOT a prime number Output is $1$ only if $n$ is a prime number None of the above

Given the pseudocode below for the function $\textbf{remains()}$, which of the following statements is true about the output, if we pass it a positive integer $n>2$?int r...

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969 views

admin asked Feb 10, 2020

1 votes

3 answers

5

TIFR CSE 2020 | Part B | Question: 11
Which of the following graphs are bipartite? Only $(1)$ Only $(2)$ Only $(2)$ and $(3)$ None of $(1),(2),(3)$ All of $(1),(2),(3)$

Which of the following graphs are bipartite?Only $(1)$Only $(2)$Only $(2)$ and $(3)$None of $(1),(2),(3)$All of $(1),(2),(3)$

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3.5k views

admin asked Feb 10, 2020

9 votes

4 answers

6

TIFR CSE 2020 | Part B | Question: 10
Among the following asymptotic expressions, which of these functions grows the slowest (as a function of $n$) asymptotically? $2^{\log n}$ $n^{10}$ $(\sqrt{\log n})^{\log ^{2} n}$ $(\log n)^{\sqrt{\log n}}$ $2^{2^{\sqrt{\log\log n}}}$

Among the following asymptotic expressions, which of these functions grows the slowest (as a function of $n$) asymptotically?$2^{\log n}$$n^{10}$$(\sqrt{\log n})^{\log ^{...

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4.6k views

admin asked Feb 10, 2020

0 votes

2 answers

7

TIFR CSE 2020 | Part B | Question: 9
A particular Panini-Backus-Naur Form definition for a $<\textsf{word}>$ is given by the following rules: $<\textsf{word}>:: = \:<\text{letter}> \mid <\text{letter}>\:<\text{pairlet}>\mid <\text{letter}>\:<\text{pairdig}>$ ... $\text{III}$ only $\text{II}$ and $\text{III}$ only $\text{I},\text{II}$ and $\text{III}$

A particular Panini-Backus-Naur Form definition for a $<\textsf{word}>$ is given by the following rules:$<\textsf{word}>:: = \:<\text{letter} \mid <\text{letter}>\:<\text...

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1.3k views

admin asked Feb 10, 2020

3 votes

3 answers

8

TIFR CSE 2020 | Part B | Question: 8
Jobs keep arriving at a processor. A job can have an associated time length as well as a priority tag. New jobs may arrive while some earlier jobs are running. Some jobs may keep running indefinitely. A $\textsf{starvation free}$ ... job-scheduling policies is starvation free? Round - robin Shortest job first Priority queuing Latest job first None of the others

Jobs keep arriving at a processor. A job can have an associated time length as well as a priority tag. New jobs may arrive while some earlier jobs are running. Some jobs ...

admin

1.2k views

admin asked Feb 10, 2020

0 votes

0 answers

9

TIFR CSE 2020 | Part B | Question: 7
Consider the following algorithm (Note: For positive integers, $p,q,p/q$ denotes the floor of the rational number $\dfrac{p}{q}$, assume that given $p,q,p/q$ can be computed in one step): $\textbf{Input:}$ Two positive integers $a,b,a\geq b.$ $\textbf{Output:}$ A positive ... $\Theta(\log K)$ $\Theta({K})$ $\Theta({K\log K})$ $\Theta({K^{2}})$ $\Theta({2^{K}})$

Consider the following algorithm (Note: For positive integers, $p,q,p/q$ denotes the floor of the rational number $\dfrac{p}{q}$, assume that given $p,q,p/q$ can be compu...

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663 views

admin asked Feb 10, 2020

0 votes

1 answer

10

TIFR CSE 2020 | Part B | Question: 6
Consider the context-free grammar below ($\epsilon$ denotes the empty string, alphabet is $\{a,b\}$): $S\rightarrow \epsilon \mid aSb \mid bSa \mid SS.$ What language does it generate? $(ab)^{\ast} + (ba)^{\ast}$ $(abba) {\ast} + (baab)^{\ast}$ ... of the form $a^{n}b^{n}$ or $b^{n}a^{n},n$ any positive integer Strings with equal numbers of $a$ and $b$

Consider the context-free grammar below ($\epsilon$ denotes the empty string, alphabet is $\{a,b\}$):$$S\rightarrow \epsilon \mid aSb \mid bSa \mid SS.$$What language doe...

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506 views

admin asked Feb 10, 2020

1 votes

1 answer

11

TIFR CSE 2020 | Part B | Question: 5
Let $u$ be a point on the unit circle in the first quadrant (i.e., both coordinates of $u$ are positive). Let $\theta$ be the angle subtended by $u$ and the $x$ axis at the origin. Let $\ell _{u}$ denote the infinite line passing through the origin and $u$. ... and the $x$-axis at the origin? $50^{\circ}$ $85^{\circ}$ $90^{\circ}$ $145^{\circ}$ $165^{\circ}$

Let $u$ be a point on the unit circle in the first quadrant (i.e., both coordinates of $u$ are positive). Let $\theta$ be the angle subtended by $u$ and the $x$ axis at t...

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506 views

admin asked Feb 10, 2020

0 votes

2 answers

12

TIFR CSE 2020 | Part B | Question: 4
A $\textit{clamp}$ gate is an analog gate parametrized by two real numbers $a$ and $b$, and denoted as $\text{clamp}_{a,b}$. It takes as input two non-negative real numbers $x$ and $y$ ... outputs the maximum of $x$ and $y?$ $1$ $2$ $3$ $4$ No circuit composed only of clamp gates can compute the max function

A $\textit{clamp}$ gate is an analog gate parametrized by two real numbers $a$ and $b$, and denoted as $\text{clamp}_{a,b}$. It takes as input two non-negative real numbe...

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803 views

admin asked Feb 10, 2020

6 votes

1 answer

13

TIFR CSE 2020 | Part B | Question: 3
Consider the (decimal) number $182$, whose binary representation is $10110110$. How many positive integers are there in the following set?$\{n\in \mathbb{N}: n\leq 182 \text{ and n has } \textit{exactly four} \text{ ones in its binary representation}\}$ $91$ $70$ $54$ $35$ $27$

Consider the (decimal) number $182$, whose binary representation is $10110110$. How many positive integers are there in the following set?$$\{n\in \mathbb{N}: n\leq 182 \...

admin

1.8k views

admin asked Feb 10, 2020

3 votes

4 answers

14

TIFR CSE 2020 | Part B | Question: 2
Consider the following statements. The intersection of two context-free languages is always context-free The super-set of a context-free languages is never regular The subset of a decidable language is always decidable Let $\Sigma = \{a,b,c\}.$ Let $L\subseteq \Sigma$ be the language of ... Only $(1),(2)$ and $(3)$ Only $(4)$ None of $(1),(2),(3),(4)$ are true.

Consider the following statements.The intersection of two context-free languages is always context-freeThe super-set of a context-free languages is never regularThe subse...

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1.6k views

admin asked Feb 10, 2020

1 votes

0 answers

15

TIFR CSE 2020 | Part B | Question: 1
Consider the following Boolean valued function on $n$ Boolean variables: $f(x_{1},\dots,x_{n}) = x_{1} + \dots + x_{n}(\text{mod } 2)$, where addition is over integers, mapping $\textbf{ FALSE'}$ to $0$ and $\textbf{ TRUE'}$ to $1.$ Consider Boolean circuits ... , for some fixed constant $c$ $n^{\omega(1)}$, but $n^{O(\log n)}$ $2^{\Theta(n)}$ None of the others

Consider the following Boolean valued function on $n$ Boolean variables: $f(x_{1},\dots,x_{n}) = x_{1} + \dots + x_{n}(\text{mod } 2)$, where addition is over integers, m...

admin

678 views

admin asked Feb 10, 2020

3 votes

2 answers

16

TIFR CSE 2020 | Part A | Question: 15
The sequence $s_{0},s_{1},\dots , s_{9}$ is defined as follows: $s_{0} = s_{1} + 1$ $2s_{i} = s_{i-1} + s_{i+1} + 2 \text{ for } 1 \leq i \leq 8$ $2s_{9} = s_{8} + 2$ What is $s_{0}$? $81$ $95$ $100$ $121$ $190$

The sequence $s_{0},s_{1},\dots , s_{9}$ is defined as follows:$s_{0} = s_{1} + 1$$2s_{i} = s_{i-1} + s_{i+1} + 2 \text{ for } 1 \leq i \leq 8$$2s_{9} = s_{8} + 2$What is...

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797 views

admin asked Feb 10, 2020

1 votes

1 answer

17

TIFR CSE 2020 | Part A | Question: 14
A ball is thrown directly upwards from the ground at a speed of $10\: ms^{-1}$, on a planet where the gravitational acceleration is $10\: ms^{-2}$. Consider the following statements: The ball reaches the ground exactly $2$ seconds after it is thrown ... $1,2$ or $3$ is correct All of the Statements $1,2$ and $3$ are correct

A ball is thrown directly upwards from the ground at a speed of $10\: ms^{-1}$, on a planet where the gravitational acceleration is $10\: ms^{-2}$. Consider the following...

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596 views

admin asked Feb 10, 2020

1 votes

1 answer

18

TIFR CSE 2020 | Part A | Question: 13
What is the area of the largest rectangle that can be inscribed in a circle of radius $R$? $R^{2}/2$ $\pi \times R^{2}/2$ $R^{2}$ $2R^{2}$ None of the above

What is the area of the largest rectangle that can be inscribed in a circle of radius $R$?$R^{2}/2$$\pi \times R^{2}/2$$R^{2}$$2R^{2}$None of the above

admin

675 views

admin asked Feb 10, 2020

0 votes

1 answer

19

TIFR CSE 2020 | Part A | Question: 12
The hour needle of a clock is malfunctioning and travels in the anti-clockwise direction, i.e., opposite to the usual direction, at the same speed it would have if it was working correctly. The minute needle is working correctly. Suppose the two needles show the correct ... $\dfrac{11}{12}$ hour $\dfrac{12}{13}$ hour $\dfrac{19}{22}$ hour One hour

The hour needle of a clock is malfunctioning and travels in the anti-clockwise direction, i.e., opposite to the usual direction, at the same speed it would have if it was...

admin

690 views

admin asked Feb 10, 2020

4 votes

1 answer

20

TIFR CSE 2020 | Part A | Question: 11
Suppose we toss $m=3$ labelled balls into $n=3$ numbered bins. Let $A$ be the event that the first bin is empty while $B$ be the event that the second bin is empty. $P(A)$ and $P(B)$ denote their respective probabilities. Which of the following is true? $P(A)>P(B)$ $P(A) = \dfrac{1}{27}$ $P(A)>P(A\mid B)$ $P(A)<P(A\mid B)$ None of the above

Suppose we toss $m=3$ labelled balls into $n=3$ numbered bins. Let $A$ be the event that the first bin is empty while $B$ be the event that the second bin is empty. $P(A)...

admin

858 views

admin asked Feb 10, 2020

3 votes

2 answers

21

TIFR CSE 2020 | Part A | Question: 9
A contiguous part, i.e., a set of adjacent sheets, is missing from Tharoor's GRE preparation book. The number on the first missing page is $183$, and it is known that the number on the last missing page has the same three digits, but in a different ... front and one at the back. How many pages are missing from Tharoor's book? $45$ $135$ $136$ $198$ $450$

A contiguous part, i.e., a set of adjacent sheets, is missing from Tharoor’s GRE preparation book. The number on the first missing page is $183$, and it is known that t...

admin

901 views

admin asked Feb 10, 2020

1 votes

2 answers

22

TIFR CSE 2020 | Part A | Question: 10
In a certain year, there were exactly four Fridays and exactly four Mondays in January. On what day of the week did the $20^{th}$ of the January fall that year (recall that January has $31$ days)? Sunday Monday Wednesday Friday None of the others

In a certain year, there were exactly four Fridays and exactly four Mondays in January. On what day of the week did the $20^{th}$ of the January fall that year (recall th...

admin

2.2k views

admin asked Feb 10, 2020

0 votes

2 answers

23

TIFR CSE 2020 | Part A | Question: 8
Consider a function $f:[0,1]\rightarrow [0,1]$ which is twice differentiable in $(0,1).$ Suppose it has exactly one global maximum and exactly one global minimum inside $(0,1)$. What can you say about the behaviour of the first derivative $f'$ and and second ... at at least one point $f'$ is zero at at least two points, $f''$ is zero at at least two points

Consider a function $f:[0,1]\rightarrow [0,1]$ which is twice differentiable in $(0,1).$ Suppose it has exactly one global maximum and exactly one global minimum inside $...

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1.3k views

admin asked Feb 10, 2020

5 votes

2 answers

24

TIFR CSE 2020 | Part A | Question: 7
A lottery chooses four random winners. What is the probability that at least three of them are born on the same day of the week? Assume that the pool of candidates is so large that each winner is equally likely to be born on any of the seven days of the week independent ... $\dfrac{48}{2401} \\$ $\dfrac{105}{2401} \\$ $\dfrac{175}{2401} \\$ $\dfrac{294}{2401}$

A lottery chooses four random winners. What is the probability that at least three of them are born on the same day of the week? Assume that the pool of candidates is so ...

admin

1.3k views

admin asked Feb 10, 2020

2 votes

1 answer

25

TIFR CSE 2020 | Part A | Question: 6
What is the maximum number of regions that the plane $\mathbb{R}^{2}$ can be partitioned into using $10$ lines? $25$ $50$ $55$ $56$ $1024$ Hint: Let $A(n)$ be the maximum number of partitions that can be made by $n$ lines. Observe that $A(0) = 1, A(2) = 2, A(2) = 4$ etc. Come up with a recurrence equation for $A(n)$.

What is the maximum number of regions that the plane $\mathbb{R}^{2}$ can be partitioned into using $10$ lines?$25$$50$$55$$56$$1024$Hint: Let $A(n)$ be the maximum numbe...

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834 views

admin asked Feb 10, 2020

0 votes

0 answers

26

TIFR CSE 2020 | Part A | Question: 4
Fix $n\geq 4.$ Suppose there is a particle that moves randomly on the number line, but never leaves the set $\{1,2,\dots,n\}.$ Let the initial probability distribution of the particle be denoted by $\overrightarrow{\pi}.$ In the first step, if the particle is at ... $i\neq 1$ $\overrightarrow{\pi}(n) = 1$ and $\overrightarrow{\pi}(i) = 0$ for $i\neq n$

Fix $n\geq 4.$ Suppose there is a particle that moves randomly on the number line, but never leaves the set $\{1,2,\dots,n\}.$ Let the initial probability distribution of...

admin

764 views

admin asked Feb 10, 2020

1 votes

1 answer

27

TIFR CSE 2020 | Part A | Question: 5
Let $A$ be am $n\times n$ invertible matrix with real entries whose column sums are all equal to $1$. Consider the following statements: Every column in the matrix $A^{2}$ sums to $2$ Every column in the matrix $A^{3}$ sums to $3$ Every column in the matrix ... $(1)$ or $(2)$ all the $3$ statements $(1),(2),$ and $(3)$ are correct

Let $A$ be am $n\times n$ invertible matrix with real entries whose column sums are all equal to $1$. Consider the following statements:Every column in the matrix $A^{2}$...

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1.2k views

admin asked Feb 10, 2020

0 votes

0 answers

28

TIFR CSE 2020 | Part A | Question: 3
Let $d\geq 4$ and fix $w\in \mathbb{R}.$ Let $S = \{a = (a_{0},a_{1},\dots ,a_{d})\in \mathbb{R}^{d+1}\mid f_{a}(w) = 0\: \text{and}\: f'_{a}(w) = 0\},$ where the polynomial function $f_{a}(x)$ ... $d$-dimensional vector subspace of $\mathbb{R}^{d+1}$ $S$ is a $(d-1)$-dimensional vector subspace of $\mathbb{R}^{d+1}$ None of the other options

Let $d\geq 4$ and fix $w\in \mathbb{R}.$ Let $$S = \{a = (a_{0},a_{1},\dots ,a_{d})\in \mathbb{R}^{d+1}\mid f_{a}(w) = 0\: \text{and}\: f’_{a}(w) = 0\},$$ where the pol...

admin

710 views

admin asked Feb 10, 2020

2 votes

1 answer

29

TIFR CSE 2020 | Part A | Question: 2
Let $M$ be a real $n\times n$ matrix such that for$ every$ non-zero vector $x\in \mathbb{R}^{n},$ we have $x^{T}M x> 0.$ Then Such an $M$ cannot exist Such $Ms$ exist and their rank is always $n$ Such $Ms$ exist, but their eigenvalues are always real No eigenvalue of any such $M$ can be real None of the above

Let $M$ be a real $n\times n$ matrix such that for$ every$ non-zero vector $x\in \mathbb{R}^{n},$ we have $x^{T}M x 0.$ ThenSuch an $M$ cannot existSuch $Ms$ exist and th...

admin

1.4k views

admin asked Feb 10, 2020

6 votes

2 answers

30

TIFR CSE 2020 | Part A | Question: 1
Two balls are drawn uniformly at random without replacement from a set of five balls numbered $1,2,3,4,5.$ What is the expected value of the larger number on the balls drawn? $2.5$ $3$ $3.5$ $4$ None of the above

Two balls are drawn uniformly at random without replacement from a set of five balls numbered $1,2,3,4,5.$ What is the expected value of the larger number on the balls dr...

admin

2.4k views

admin asked Feb 10, 2020

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