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Recent questions tagged tifr2018

28 votes

6 answers

1

TIFR CSE 2018 | Part A | Question: 9
How many ways are there to assign colours from range $\left\{1,2,\ldots,r\right\}$ to vertices of the following graph so that adjacent vertices receive distinct colours? $r^{4}$ $r^{4} - 4r^{3}$ $r^{4}-5r^{3}+8r^{2}-4r$ $r^{4}-4r^{3}+9r^{2}-3r$ $r^{4}-5r^{3}+10r^{2}-15r$

Rohit Gupta 8 asked in Graph Theory Dec 10, 2017

by Rohit Gupta 8

3.8k views

8 votes

5 answers

2

TIFR CSE 2018 | Part A | Question: 14
Let $A$ be an $n\times n$ invertible matrix with real entries whose row sums are all equal to $c$. Consider the following statements: Every row in the matrix $2A$ sums to $2c$. Every row in the matrix $A^{2}$ sums to $c^{2}$. Every row in ... and $(2)$ are correct but not necessarily statement $(3)$ all the three statements $(1), (2),$ and $(3)$ are correct

Rohit Gupta 8 asked in Linear Algebra Dec 10, 2017

by Rohit Gupta 8

2.5k views

17 votes

2 answers

3

TIFR CSE 2018 | Part A | Question: 13
A hacker knows that the password to the TIFR server is $10$-letter string consisting of lower-case letters from the English alphabet. He guesses a set of $5$ distinct $10$-letter strings (with lower-case letters) uniformly at random. What is the probability that one of the ... $ \frac{1}{(26)^{10}}$ None of the above

Rohit Gupta 8 asked in Probability Dec 10, 2017

by Rohit Gupta 8

1.6k views

17 votes

3 answers

4

TIFR CSE 2018 | Part A | Question: 15
Suppose a box contains $20$ balls: each ball has a distinct number in $\left\{1,\ldots,20\right\}$ written on it. We pick $10$ balls (without replacement) uniformly at random and throw them out of the box. Then we check if the ball with number $\text{ 1"}$ ... $\text{ 2"}$ on it is present in the box? $9/20$ $9/19$ $1/2$ $10/19$ None of the above

Rohit Gupta 8 asked in Probability Dec 10, 2017

by Rohit Gupta 8

2.0k views

4 votes

1 answer

5

TIFR CSE 2018 | Part B | Question: 15
$G$ respresents an undirected graph and a cycle refers to a simple cycle (no repeated edges or vertices). Define the following two languages. $\text{SCYCLE}=\{(G,k)\mid G \text{ contains a cycle of length at most k}\}$ ... $\text{SCYCLE}$ to $\text{LCYCLE}$).

Arjun asked in Theory of Computation Dec 10, 2017

by Arjun

779 views

12 votes

3 answers

6

TIFR CSE 2018 | Part B | Question: 14
Define the language $\text{INFINITE}_{DFA}\equiv \{(A)\mid A \text{ is a DFA and } L(A) \text{ is an infinite language}\},$ where $(A)$ denotes the description of the deterministic finite automata (DFA).Then which ... It is Turing decidable (recursive). It is Turing recognizable but not decidable. Its complement is Turing recognizable but it is not decidable.

Arjun asked in Theory of Computation Dec 10, 2017

by Arjun

2.9k views

9 votes

1 answer

7

TIFR CSE 2018 | Part B | Question: 12
Consider the following statements: For every positive integer $n,$ let $\#{n}$ be the product of all primes less than or equal to $n.$ Then, $\# {p}+1$ is a prime, for every prime $p.$ $\large\pi$ is a universal constant ... . Only statement (ii) is correct. Only statement (iii) is correct. Only statement (iv) is correct. None of the statements are correct.

Arjun asked in Theory of Computation Dec 10, 2017

by Arjun

1.6k views

11 votes

3 answers

8

TIFR CSE 2018 | Part B | Question: 13
Let $n\geq 3,$ and let $G$ be a simple, connected, undirected graph with the same number $n$ of vertices and edges. Each edge of $G$ has a distinct real weight associated with it. Let $T$ be the minimum weight spanning tree of $G.$ Which of the ... edge of $C$ is not in $T.$ $T$ can be found in $O(n)$ time from the adjacency list representation of $G.$

Arjun asked in Algorithms Dec 10, 2017

by Arjun

2.3k views

12 votes

2 answers

9

TIFR CSE 2018 | Part B | Question: 11
Consider the language $L\subseteq \left \{ a,b,c \right \}^{*}$ defined as $L = \left \{ a^{p}b^{q}c^{r} : p=q\quad or\quad q=r \quad or\quad r=p \right \}.$ Which of the following answer is TRUE about the complexity of this language ... defined as $\overline{L} = \left \{ a,b,c \right \}^{*}\backslash L,$ is regular. $L$ is regular, context-free and decidable

Arjun asked in Theory of Computation Dec 10, 2017

by Arjun

1.9k views

11 votes

1 answer

10

TIFR CSE 2018 | Part B | Question: 10
For two $n$ bit strings $x,y \in\{0,1\}^{n},$ define $z=x\oplus y$ to be the bitwise XOR of the two strings (that is, if $x_{i},y_{i},z_{i}$ denote the $i^{th}$ bits of $x,y,z$ respectively, then $z_{i}=x_{i}+y_{i} \bmod 2$ ... such linear functions for $n \geq 2$ is: $2^{n}$ $2^{n^{2}}$ $\large2^{\frac{n}{2}}$ $2^{4n}$ $2^{n^{2}+n}$

Arjun asked in Set Theory & Algebra Dec 10, 2017

by Arjun

1.2k views

12 votes

1 answer

11

TIFR CSE 2018 | Part B | Question: 9
Let $G=(V,E)$ be a DIRECTED graph, where each edge $\large e$ has a positive weight $\large\omega(e),$ and all vertices can be reached from vertex $\large s.$ For each vertex $\large v,$ let $\large \phi(v)$ be the length of the shortest ... $G,$ then $\omega'(P)<2\times \omega(P).$ All of the above options are necessarily TRUE.

Arjun asked in Algorithms Dec 10, 2017

by Arjun

2.0k views

30 votes

3 answers

12

TIFR CSE 2018 | Part B | Question: 8
In an undirected graph $G$ with $n$ vertices, vertex $1$ has degree $1$, while each vertex $2,\ldots,n-1$ has degree $10$ and the degree of vertex $n$ is unknown, Which of the following statement must be TRUE on the graph $G$? There is a path ... Vertex $n$ has degree $1$. The diameter of the graph is at most $\frac{n}{10}$ All of the above choices must be TRUE

Arjun asked in Graph Theory Dec 10, 2017

by Arjun

4.3k views

13 votes

1 answer

13

TIFR CSE 2018 | Part B | Question: 7
Consider the recursive quicksort algorithm with "random pivoting". That is, in each recursive call, a pivot is chosen uniformly at random from the sub-array being sorted.When this randomized algorithm is applied to an array of size $n$ all whose elements are distinct, ... $\Theta\left(\dfrac{1}{n \log^{2} n}\right)$

Arjun asked in Algorithms Dec 10, 2017

by Arjun

7.5k views

4 votes

2 answers

14

TIFR CSE 2018 | Part B | Question: 6
Consider the following implementation of a binary tree data strucrure. The operator $+$ denotes list-concatenation. That is, $[a,b,c]+[d,e]=[a,b,c,d,e].$ struct TreeNode: int value TreeNode leftChild TreeNode rightChild function preOrder(T): if T == null: ... $\text{Cannot be uniquely determined from given information.}$

Arjun asked in DS Dec 10, 2017

by Arjun

1.1k views

14 votes

1 answer

15

TIFR CSE 2018 | Part B | Question: 5
Which of the following functions, given by there recurrence, grows the fastest asymptotically? $T(n) = 4T\left ( \frac{n}{2} \right ) + 10n$ $T(n) = 8T\left ( \frac{n}{3} \right ) + 24n^{2}$ $T(n) = 16T\left ( \frac{n}{4} \right ) + 10n^{2}$ $T(n) = 25T\left ( \frac{n}{5} \right ) + 20\left ( n \log n \right )^{1.99}$ They all are asymptotically the same

Arjun asked in Algorithms Dec 10, 2017

by Arjun

2.7k views

8 votes

2 answers

16

TIFR CSE 2018 | Part B | Question: 3
How many distinct minimum weight spanning trees does the following undirected, weighted graph have ? $1$ $2$ $4$ $6$ $8$

Arjun asked in Algorithms Dec 10, 2017

by Arjun

2.1k views

15 votes

12 answers

17

TIFR CSE 2018 | Part B | Question: 1
What is the remainder when $4444^{4444}$ is divided by $9?$ $1$ $2$ $5$ $7$ $8$

Arjun asked in Quantitative Aptitude Dec 10, 2017

by Arjun

2.7k views

4 votes

3 answers

18

TIFR CSE 2018 | Part B | Question: 4
The notation "$\Rightarrow$" denotes "implies" and "$\wedge$" denotes "and" in the following formulae. Let $X$ denote the formula: $(b \Rightarrow a ) \Rightarrow ( a \Rightarrow b)$ Let $Y$ denote the formula: ... is not satisfiable. $X$ is not tautology and $Y$ is satisfiable. $X$ is a tautology and $Y$ is satisfiable,

Arjun asked in Mathematical Logic Dec 10, 2017

by Arjun

1.5k views

7 votes

1 answer

19

TIFR CSE 2018 | Part B | Question: 2
Consider the following non-deterministic automation, where $\large s_{1}$ is the start state and $\large s_{4}$ is the final (accepting) state. The alphabet is $\{a,b\}.$ A transition with label $\large\epsilon$ can be taken without consuming any symbol from the input. Which of the ... $aba(a+b)^{*}aba$ $(a+b)aba(b+a)^{*}$ $aba(a+b)^{*}$ $(ab)^{*}aba$

Arjun asked in Theory of Computation Dec 10, 2017

by Arjun

1.2k views

7 votes

2 answers

20

TIFR CSE 2018 | Part A | Question: 10
Let $C$ be a biased coin such that the probability of a head turning up is $p.$ Let $p_n$ denote the probability that an odd number of heads occurs after $n$ tosses for $n \in \{0,1,2,\ldots \},$ ... $p_{n}=1 \text{ if } n \text{ is odd and } 0 \text{ otherwise}.$

Arjun asked in Probability Dec 10, 2017

by Arjun

1.5k views

8 votes

1 answer

21

TIFR CSE 2018 | Part A | Question: 12
An $n \times n$ matrix $M$ with real entries is said to be positive definite if for every non-zero $n$-dimensional vector $x$ with real entries, we have $x^{T}Mx>0.$ Let $A$ and $B$ be symmetric, positive definite matrices of size ... $(3)$ Only $(1)$ and $(3)$ None of the above matrices are positive definite All of the above matrices are positive definite

Arjun asked in Linear Algebra Dec 10, 2017

by Arjun

1.3k views

3 votes

3 answers

22

TIFR CSE 2018 | Part A | Question: 11
We are given a (possibly empty) set of objects. Each object in the set is colored either black or white, is shaped either circular or rectangular, and has a profile that is either fat or thin, Those properties obey the following principles: Each white ... $(i) \text{ and } (iii)$ only None of the statements must be TRUE All of the statements must be TRUE

Arjun asked in Analytical Aptitude Dec 10, 2017

by Arjun

1.0k views

10 votes

2 answers

23

TIFR CSE 2018 | Part A | Question: 8
A crime has been committed with four people at the scene of the crime. You are responsible for finding out who did it. You have recorded the following statements from the four witnesses, and you know one of them has committed the crime. ... $\text{Desmond}$ $\text{Either Anuj or Binky; the information is insufficient to pinpoint the criminal}$

Arjun asked in Analytical Aptitude Dec 10, 2017

by Arjun

1.1k views

15 votes

5 answers

24

TIFR CSE 2018 | Part A | Question: 7
Consider the following function definition. void greet(int n) { if(n>0) { printf("hello"); greet(n-1); } printf("world"); } If you run $\textsf{greet(n)}$ ... "helloworld" $\textsf{n+1}$ times "helloworld" $\textsf{n}$ times "helloworld", followed by "world"

Arjun asked in Programming Dec 10, 2017

by Arjun

2.3k views

15 votes

6 answers

25

TIFR CSE 2018 | Part A | Question: 6
What is the minimum number of students needed in a class to guarantee that there are at least $6$ students whose birthdays fall in the same month ? $6$ $23$ $61$ $72$ $91$

Arjun asked in Combinatory Dec 10, 2017

by Arjun

2.8k views

3 votes

1 answer

26

TIFR CSE 2018 | Part A | Question: 5
Which of the following is the derivative of $f(x)=x^{x}$ when $x>0$ ? $x^{x}$ $x^{x} \ln \;x$ $x^{x}+x^{x}\ln\;x$ $(x^{x}) (x^{x}\ln\;x)$ $\text{None of the above; function is not differentiable for }x>0$

Arjun asked in Calculus Dec 10, 2017

by Arjun

1.1k views

4 votes

1 answer

27

TIFR CSE 2018 | Part A | Question: 1
Consider a point $A$ inside a circle $C$ that is at distance $9$ from the centre of a circle. Suppose you told that there is a chord of length $24$ passing through $A$ with $A$ as its midpoint. How many distinct chords of $C$ have integer length and pass through $A?$ $2$ $6$ $7$ $12$ $14$

Arjun asked in Quantitative Aptitude Dec 10, 2017

by Arjun

1.2k views

2 votes

2 answers

28

TIFR CSE 2018 | Part A | Question: 4
The distance from your home to your office is $4$ kilometers and your normal walking speed is $4$ Km/hr. On the first day, you walk at your normal walking speed and take time $T_1$ to reach office. On the second day, you walk at a speed of $3$ Km/hr from $2$ Kilometers, and at ... $T_{1}=T_{2}$ and $T_{1}<T_{3}$ $T_{1}<T_{2}$ and $T_{1}=T_{3}$

Arjun asked in Quantitative Aptitude Dec 10, 2017

by Arjun

477 views

21 votes

3 answers

29

TIFR CSE 2018 | Part A | Question: 3
Which of the following statements is TRUE for all sufficiently large integers n ? $2^{2^{\sqrt{\log \log n}}} <2^{\sqrt{\log n}} < n$ $2^{\sqrt{\log n}}<n<2^{2^{\sqrt{\log \log n}}}$ $n<2^{\sqrt{\log n}}<2^{2^{\sqrt{\log \log n}}}$ $n<2^{2^{\sqrt{\log \log n}}} <2^{\sqrt{\log n}}$ $2^{\sqrt{\log n}}<2^{2^{\sqrt{\log \log n}}} < n$

Arjun asked in Algorithms Dec 10, 2017

by Arjun

2.4k views

2 votes

1 answer

30

TIFR CSE 2018 | Part A | Question: 2
Consider the following subset of $\mathbb{R} ^{3}$ (the first two are cylinder, the third is a plane): $C_{1}=\left \{ \left ( x,y,z \right ): y^{2}+z^{2}\leq 1 \right \};$ ... $A?$ Circle Ellipse Triangle Square An octagonal convex figure with curved sides

Arjun asked in Quantitative Aptitude Dec 10, 2017

by Arjun

625 views

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