Recent questions tagged tifr2020

2 votes

1 answer

1

TIFR2020-B-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 $s$ and, after visiting ... $s, t, b$ and $f$ 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

455 views

0 votes

1 answer

2

TIFR2020-B-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 $X\oplus Y:=(X\wedge \neg Y)\vee (\neg X \wedge Y)$ ... $H$ have? $20$ $30$ $32$ $160$ $1024$

Lakshman Patel RJIT asked in Digital Logic Feb 11, 2020

by Lakshman Patel RJIT

319 views

0 votes

1 answer

3

TIFR2020-B-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 must be true? ... then it 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

255 views

0 votes

2 answers

4

TIFR2020-B-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 ; } return x; } ... is always $1$ 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

266 views

0 votes

3 answers

5

TIFR2020-B-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

851 views

7 votes

4 answers

6

TIFR2020-B-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

1.5k views

0 votes

2 answers

7

TIFR2020-B-9
A particular Panini-Backus-Naur Form definition for a $<\textsf{word}>$ is given by the following rules: $<\textsf{word}>:: = \:<\text{letter}> \mid <\text{letter}>\:<\text{pairlet}>\mid <\text{letter}>\:<\text{pairdig}>$ ... $I,II\:\text{or}\:III$ $I$ 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

229 views

1 vote

3 answers

8

TIFR2020-B-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 ... following 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

361 views

0 votes

0 answers

9

TIFR2020-B-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 integers $g.$ ... case? $\Theta(\log K)$ $\Theta({K})$ $\Theta({K\log K})$ $\Theta({K^{2}})$ $\Theta({2^{K}})$

Lakshman Patel RJIT asked in Others Feb 11, 2020

by Lakshman Patel RJIT

153 views

0 votes

1 answer

10

TIFR2020-B-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? $(ab)^{\ast} + (ba)^{\ast}$ $(abba) {\ast} + (baab)^{\ast}$ ... $a^{n}b^{n}$ 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

163 views

1 vote

1 answer

11

TIFR2020-B-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 $u$. Consider the following ... $w$ 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

153 views

0 votes

1 answer

12

TIFR2020-B-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 $y$ ... 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

221 views

4 votes

1 answer

13

TIFR2020-B-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

836 views

2 votes

3 answers

14

TIFR2020-B-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 all strings in which ... $(1),(2)$ 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

645 views

0 votes

0 answers

15

TIFR2020-B-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 (with no feedback) ... $c$ $n^{\omega(1)}$, 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

249 views

1 vote

2 answers

16

TIFR2020-A-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

365 views

0 votes

1 answer

17

TIFR2020-A-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 up The ball travels ... $3$ is correct None of the Statements $1,2$ 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

252 views

0 votes

1 answer

18

TIFR2020-A-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

218 views

0 votes

1 answer

19

TIFR2020-A-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 time at $12$ ... $\dfrac{11}{12}$ 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

234 views

2 votes

1 answer

20

TIFR2020-A-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

346 views

2 votes

2 answers

21

TIFR2020-A-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 order. Note that every ... at 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

409 views

0 votes

2 answers

22

TIFR2020-A-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

785 views

0 votes

1 answer

23

TIFR2020-A-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 $f'$ ... is zero 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

349 views

3 votes

2 answers

24

TIFR2020-A-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 of the other winners ... $\dfrac{48}{2401} \\$ $\dfrac{105}{2401} \\$ $\dfrac{175}{2401} \\$ $\dfrac{294}{2401}$

Lakshman Patel RJIT asked in Probability Feb 10, 2020

by Lakshman Patel RJIT

517 views

1 vote

1 answer

25

TIFR2020-A-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

281 views

0 votes

0 answers

26

TIFR2020-A-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 position $i,$ it ... $i\neq 1$ $\overrightarrow{\pi}(n) = 1$ and $\overrightarrow{\pi}(i) = 0$ for $i\neq n$

Lakshman Patel RJIT asked in Probability Feb 10, 2020

by Lakshman Patel RJIT

285 views

1 vote

1 answer

27

TIFR2020-A-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 $A^{-1}$ ... statement $(3)$ is correct but not statements $(1)$ 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

402 views

0 votes

0 answers

28

TIFR2020-A-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 $f_{a}(x)$ ... is a $d$-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

270 views

2 votes

1 answer

29

TIFR2020-A-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

536 views

4 votes

2 answers

30

TIFR2020-A-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

1k views

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