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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. $430$ $440$ $460$ $470$ $480$

Lakshman Patel RJIT asked in Others Feb 11, 2020

by Lakshman Patel RJIT

638 views

2 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? $20$ $30$ $32$ $160$ $1024$

Lakshman Patel RJIT asked in Digital Logic Feb 11, 2020

by Lakshman Patel RJIT

519 views

1 vote

1 answer

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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

Lakshman Patel RJIT asked in Others Feb 11, 2020

by Lakshman Patel RJIT

463 views

1 vote

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

Lakshman Patel RJIT asked in Others Feb 11, 2020

by Lakshman Patel RJIT

501 views

0 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 $(1),(2),(3)$

Lakshman Patel RJIT asked in Graph Theory Feb 11, 2020

by Lakshman Patel RJIT

1.7k views

8 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? $2^{\log n}$ $n^{10}$ $(\sqrt{\log n})^{\log ^{2} n}$ $(\log n)^{\sqrt{\log n}}$ $2^{2^{\sqrt{\log\log n}}}$

Lakshman Patel RJIT asked in Algorithms Feb 11, 2020

by Lakshman Patel RJIT

2.6k views

0 votes

2 answers

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

Lakshman Patel RJIT asked in Others Feb 11, 2020

by Lakshman Patel RJIT

468 views

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

Lakshman Patel RJIT asked in Operating System Feb 11, 2020

by Lakshman Patel RJIT

615 views

0 votes

0 answers

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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 ... $\Theta(\log K)$ $\Theta({K})$ $\Theta({K\log K})$ $\Theta({K^{2}})$ $\Theta({2^{K}})$

Lakshman Patel RJIT asked in Others Feb 11, 2020

by Lakshman Patel RJIT

291 views

0 votes

1 answer

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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 $b$

Lakshman Patel RJIT asked in Others Feb 11, 2020

by Lakshman Patel RJIT

278 views

1 vote

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? $50^{\circ}$ $85^{\circ}$ $90^{\circ}$ $145^{\circ}$ $165^{\circ}$

Lakshman Patel RJIT asked in Others Feb 11, 2020

by Lakshman Patel RJIT

269 views

0 votes

2 answers

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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

Lakshman Patel RJIT asked in Others Feb 11, 2020

by Lakshman Patel RJIT

442 views

4 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?$\{n\in \mathbb{N}: n\leq 182 \text{ and n has } \textit{exactly four} \text{ ones in its binary representation}\}$ $91$ $70$ $54$ $35$ $27$

Lakshman Patel RJIT asked in Digital Logic Feb 11, 2020

by Lakshman Patel RJIT

1.2k views

2 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.

Lakshman Patel RJIT asked in Theory of Computation Feb 11, 2020

by Lakshman Patel RJIT

1.0k views

0 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

Lakshman Patel RJIT asked in Digital Logic Feb 11, 2020

by Lakshman Patel RJIT

392 views

2 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 $s_{0}$? $81$ $95$ $100$ $121$ $190$

Lakshman Patel RJIT asked in Quantitative Aptitude Feb 11, 2020

by Lakshman Patel RJIT

521 views

1 vote

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

Lakshman Patel RJIT asked in Quantitative Aptitude Feb 11, 2020

by Lakshman Patel RJIT

373 views

1 vote

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

Lakshman Patel RJIT asked in Calculus Feb 11, 2020

by Lakshman Patel RJIT

327 views

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

Lakshman Patel RJIT asked in Linear Algebra Feb 11, 2020

by Lakshman Patel RJIT

383 views

3 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

Lakshman Patel RJIT asked in Probability Feb 11, 2020

by Lakshman Patel RJIT

538 views

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 ... the front and one at the back. How many pages are missing from Tharoor's book? $45$ $135$ $136$ $198$ $450$

Lakshman Patel RJIT asked in Verbal Aptitude Feb 11, 2020

by Lakshman Patel RJIT

608 views

1 vote

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

Lakshman Patel RJIT asked in Probability Feb 10, 2020

by Lakshman Patel RJIT

1.3k views

0 votes

1 answer

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

Lakshman Patel RJIT asked in Calculus Feb 10, 2020

by Lakshman Patel RJIT

551 views

4 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 ... $\dfrac{48}{2401} \\$ $\dfrac{105}{2401} \\$ $\dfrac{175}{2401} \\$ $\dfrac{294}{2401}$

Lakshman Patel RJIT asked in Probability Feb 10, 2020

by Lakshman Patel RJIT

756 views

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 $A(n)$.

Lakshman Patel RJIT asked in Quantitative Aptitude Feb 10, 2020

by Lakshman Patel RJIT

516 views

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 $i\neq n$

Lakshman Patel RJIT asked in Probability Feb 10, 2020

by Lakshman Patel RJIT

464 views

1 vote

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

Lakshman Patel RJIT asked in Linear Algebra Feb 10, 2020

by Lakshman Patel RJIT

615 views

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

Lakshman Patel RJIT asked in Linear Algebra Feb 10, 2020

by Lakshman Patel RJIT

427 views

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

Lakshman Patel RJIT asked in Linear Algebra Feb 10, 2020

by Lakshman Patel RJIT

905 views

5 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

Lakshman Patel RJIT asked in Probability Feb 10, 2020

by Lakshman Patel RJIT

1.5k views

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