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Recent questions tagged tifr2019

8 votes

6 answers

1

TIFR CSE 2019 | Part A | Question: 1
Let $X$ be a set with $n$ elements. How many subsets of $X$ have odd cardinality? $n$ $2^n$ $2^{n/2}$ $2^{n-1}$ Can not be determined without knowing whether $n$ is odd or even

Arjun asked in Set Theory & Algebra Dec 18, 2018

by Arjun

2.8k views

5 votes

2 answers

2

TIFR CSE 2019 | Part A | Question: 2
How many proper divisors (that is, divisors other than $1$ or $7200$) does $7200$ have ? $18$ $20$ $52$ $54$ $60$

Arjun asked in Quantitative Aptitude Dec 18, 2018

by Arjun

1.2k views

8 votes

2 answers

3

TIFR CSE 2019 | Part A | Question: 3
$A$ is $n \times n$ square matrix for which the entries in every row sum to $1$. Consider the following statements: The column vector $[1,1,\ldots,1]^T$ is an eigen vector of $A.$ $ \text{det}(A-I) = 0.$ $\text{det}(A) = 0.$ Which of the above statements must be ... Only $(i)$ Only $(ii)$ Only $(i)$ and $(ii)$ Only $(i)$ and $(iii)$ $(i),(ii) \text{ and }(iii)$

Arjun asked in Linear Algebra Dec 18, 2018

by Arjun

2.6k views

7 votes

1 answer

4

TIFR CSE 2019 | Part A | Question: 4
What is the probability that a point $P=(\alpha,\beta)$ picked uniformly at random from the disk $x^2 +y^2 \leq 1$ satisfies $\alpha + \beta \leq 1$? $\frac{1}{\pi}$ $\frac{3}{4} + \frac{1}{4} \cdot \frac{1}{\pi}$ $\frac{3}{4}+ \frac{1}{4} \cdot \frac{2}{\pi}$ $1$ $\frac{2}{\pi}$

Arjun asked in Probability Dec 18, 2018

by Arjun

1.5k views

10 votes

6 answers

5

TIFR CSE 2019 | Part A | Question: 5
Asha and Lata play a game in which Lata first thinks of a natural number between $1$ and $1000$. Asha must find out that number by asking Lata questions, but Lata can only reply by saying Yes or no . Assume that Lata always tells the truth. What is ... she can always find out the number Lata has thought of? $10$ $32$ $100$ $999$ $\text{None of the above}$

Arjun asked in Algorithms Dec 18, 2018

by Arjun

3.3k views

1 vote

1 answer

6

TIFR CSE 2019 | Part A | Question: 6
A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is said to be $\textit{convex}$ if for all $x,y \in \mathbb{R}$ and $\lambda$ such that $0 \leq \lambda \leq1,$ $f(\lambda x+ (1-\lambda)y) \leq \lambda f (x) + (1-\lambda) f(y)$. Let $f:$\ ... . Which of the functions $p,q$ and $r$ must be convex? Only $p$ Only $q$ Only $r$ Only $p$ and $r$ Only $q$ and $r$

Arjun asked in Set Theory & Algebra Dec 18, 2018

by Arjun

853 views

12 votes

2 answers

7

TIFR CSE 2019 | Part A | Question: 7
What are the last two digits of $1! + 2! + \dots +100!$? $00$ $13$ $30$ $33$ $73$

Arjun asked in Quantitative Aptitude Dec 18, 2018

by Arjun

1.1k views

1 vote

1 answer

8

TIFR CSE 2019 | Part A | Question: 8
Consider the following toy model of traffic on a straight , single lane, highway. We think of cars as points, which move at the maximum speed $v$ ... the following graphs most accurately captures the relationship between the speed $v$ and the density $\rho$ in this model ?

Arjun asked in Quantitative Aptitude Dec 18, 2018

by Arjun

901 views

3 votes

1 answer

9

TIFR CSE 2019 | Part A | Question: 9
Let $A$ and $B$ be two containers. Container $A$ contains $50$ litres of liquid $X$ and container $B$ contains $100$ litres of liquid $Y$. Liquids $X$ and $Y$ are soluble in each other. We now take $30$ ml of liquid $X$ from container $A$ and put it into container $B$. ... $V_{AY} > V_{BX}$ $V_{AY} = V_{BX}$ $V_{AY} + V_{BX}=30$ $V_{AY} + V_{BX}=20$

Arjun asked in Quantitative Aptitude Dec 18, 2018

by Arjun

772 views

3 votes

1 answer

10

TIFR CSE 2019 | Part A | Question: 10
Avni and Badal alternately choose numbers from the set $\{1,2,3,4,5,6,7,8,9\}$ without replacement (starting with Avni). The first person to choose numbers of which any $3$ sum to $15$ wins the game (for example, Avni wins ... strategy Both of them have a winning strategy Neither of them has a winning strategy The Player that picks $9$ has a winning strategy

Arjun asked in Analytical Aptitude Dec 18, 2018

by Arjun

937 views

7 votes

5 answers

11

TIFR CSE 2019 | Part A | Question: 11
Suppose there are $n$ guests at a party (and no hosts). As the night progresses, the guests meet each other and shake hands. The same pair of guests might shake hands multiple times. for some parties stretch late into the night , and it is hard to keep track.Still, ... $2 \mid \text{Odd} \mid - \mid \text{Even} \mid$

Arjun asked in Analytical Aptitude Dec 18, 2018

by Arjun

1.4k views

3 votes

1 answer

12

TIFR CSE 2019 | Part A | Question: 12
Let $f$ be a function with both input and output in the set $\{0,1,2, \dots ,9\}$, and let the function $g$ be defined as $g(x) = f(9-x)$. The function $f$ is non-decreasing, so that $f(x) \geq f(y)$ for $x \geq y$. Consider the following statements ... and $g$ ? Only $\text{(i)}$ Only $\text{(i)}$ and $\text{(ii)}$ Only $\text{(iii)}$ None of them All of them

Arjun asked in Set Theory & Algebra Dec 18, 2018

by Arjun

1.4k views

7 votes

2 answers

13

TIFR CSE 2019 | Part A | Question: 13
Consider the integral $\int^{1}_{0} \frac{x^{300}}{1+x^2+x^3} dx$ What is the value of this integral correct up to two decimal places? $0.00$ $0.02$ $0.10$ $0.33$ $1.00$

Arjun asked in Calculus Dec 18, 2018

by Arjun

2.0k views

8 votes

1 answer

14

TIFR CSE 2019 | Part A | Question: 14
A drawer contains $9$ pens, of which $3$ are red, $3$ are blue, and $3$ are green. The nine pens are drawn from the drawer one at at time (without replacement) such that each pen is drawn with equal probability from the remaining pens in the drawer. What is ... that two red pens are drawn in succession ? $7/12$ $1/6$ $1/12$ $1/81$ $\text{None of the above}$

Arjun asked in Probability Dec 18, 2018

by Arjun

1.8k views

6 votes

5 answers

15

TIFR CSE 2019 | Part A | Question: 15
Consider the matrix $A = \begin{bmatrix} \frac{1}{2} &\frac{1}{2} & 0\\ 0& \frac{3}{4} & \frac{1}{4}\\ 0& \frac{1}{4} & \frac{3}{4} \end{bmatrix}$ What is $\displaystyle \lim_{n→\infty}$A^n$ ? $\begin{bmatrix} \ 0 ... $\text{The limit exists, but it is none of the above}$

Arjun asked in Calculus Dec 18, 2018

by Arjun

2.1k views

3 votes

2 answers

16

TIFR CSE 2019 | Part B | Question: 1
Which of the following decimal numbers can be exactly represented in binary notation with a finite number of bits ? $0.1$ $0.2$ $0.4$ $0.5$ All the above

Arjun asked in Digital Logic Dec 18, 2018

by Arjun

3.5k views

11 votes

5 answers

17

TIFR CSE 2019 | Part B | Question: 2
How many distinct minimum weight spanning trees does the following undirected, weighted graph have ? $8$ $16$ $32$ $64$ None of the above

Arjun asked in Algorithms Dec 18, 2018

by Arjun

3.5k views

3 votes

1 answer

18

TIFR CSE 2019 | Part B | Question: 3
A graph is $d$ – regular if every vertex has degree $d$. For a $d$ – regular graph on $n$ vertices, which of the following must be TRUE? $d$ divides $n$ Both $d$ and $n$ are even Both $d$ and $n$ are odd At least one of $d$ and $n$ is odd At least one of $d$ and $n$ is even

Arjun asked in Graph Theory Dec 18, 2018

by Arjun

1.3k views

6 votes

1 answer

19

TIFR CSE 2019 | Part B | Question: 4
Let $\varphi$ be a propositional formula on a set of variables $A$ and $\psi$ be a propositional formula on a set of variables $B$ , such that $\varphi \Rightarrow \psi$ . A $\textit{Craig interpolant}$ of $\varphi$ and $\psi$ is a propositional formula $\mu$ ... interpolant for $\varphi$ and $\psi$ ? $q$ $\varphi$ itself $q \vee s$ $q \vee r$ $\neg q \wedge s$

Arjun asked in Mathematical Logic Dec 18, 2018

by Arjun

1.0k views

9 votes

2 answers

20

TIFR CSE 2019 | Part B | Question: 5
Stirling's approximation for $n!$ states for some constants $c_1,c_2$ $c_1 n^{n+\frac{1}{2}}e^{-n} \leq n! \leq c_2 n^{n+\frac{1}{2}}e^{-n}.$ What are the tightest asymptotic bounds that can be placed on $n!$ $?$ ... $n! =\Theta((\frac{n}{e})^{n+\frac{1}{2}})$ $n! =\Theta(n^{n+\frac{1}{2}}2^{-n})$

Arjun asked in Algorithms Dec 18, 2018

by Arjun

2.6k views

12 votes

3 answers

21

TIFR CSE 2019 | Part B | Question: 6
Given the following pseudocode for function $\text{printx()}$ below, how many times is $x$ printed if we execute $\text{printx(5)}?$ void printx(int n) { if(n==0){ printf(“x”); } for(int i=0;i<=n-1;++i){ printx(n-1); } } $625$ $256$ $120$ $24$ $5$

Arjun asked in Programming Dec 18, 2018

by Arjun

2.1k views

3 votes

1 answer

22

TIFR CSE 2019 | Part B | Question: 7
A formula is said to be a $3$-CF-formula if it is a conjunction (i.e., an AND) of clauses, and each clause has at most $3$ literals. Analogously, a formula is said to be a $3$ ... $\text{3-DF-SAT}$ is NP-complete Neither $\text{3-CF-SAT}$ nor $\text{3-DF-SAT}$ are in P

Arjun asked in Theory of Computation Dec 18, 2018

by Arjun

881 views

5 votes

2 answers

23

TIFR CSE 2019 | Part B | Question: 8
Consider the following program fragment: var a,b : integer; procedure G(c,d: integer); begin c:=c-d; d:=c+d; c:=d-c end; a:=2; b:=3; G(a,b); If both parameters to $G$ are passed by reference, what are the values of $a$ and $b$ at the end of the above program fragment ? $a=0$ and $b=2$ $a=3$ and $b=2$ $a=2$ and $b=3$ $a=1$ and $b=5$ None of the above

Arjun asked in Programming Dec 18, 2018

by Arjun

1.6k views

11 votes

2 answers

24

TIFR CSE 2019 | Part B | Question: 9
Consider the following program fragment: var x, y: integer; x := 1; y := 0; while y < x do begin x := 2*x; y := y+1 end; For the above fragment , which of the following is a loop invariant ? $x=y+1$ $x=(y+1)^2$ $x=(y+1)2^y$ $x=2^y$ None of the above, since the loop does not terminate

Arjun asked in Programming Dec 18, 2018

by Arjun

2.7k views

9 votes

2 answers

25

TIFR CSE 2019 | Part B | Question: 10
Let the language $D$ be defined in the binary alphabet $\{0,1\}$ as follows: $D:= \{ w \in \{0,1\}^* \mid \text{ substrings 01 and 10 occur an equal number of times in w} \}$ For example , $101 \in D$ while $1010 \notin D$ ... $D$ is context-free but not regular $D$ is decidable but not context-free $D$ is decidable but not in NP $D$ is undecidable

Arjun asked in Theory of Computation Dec 18, 2018

by Arjun

1.6k views

10 votes

3 answers

26

TIFR CSE 2019 | Part B | Question: 11
Consider the following non-deterministic automaton,where $s_1$ is the start state and $s_4$ is the final (accepting) state. The alphabet is $\{a,b\}$. A transition with label $\epsilon$ can be taken without consuming any symbol from the input. Which of the following regular expressions correspond ... $(a+b)^*ba^*$ $(a+b)^*ba(aa)^*$ $(a+b)^*$ $(a+b)^*baa^*$

Arjun asked in Theory of Computation Dec 18, 2018

by Arjun

1.7k views

5 votes

1 answer

27

TIFR CSE 2019 | Part B | Question: 12
Let $G=(V,E)$ be a directed graph with $n(\geq 2)$ vertices, including a special vertex $r$. Each edge $e \in E$ has a strictly positive edge weight $w(e)$. An arborescence in $G$ rooted at $r$ is a subgraph $H$ of $G$ ... is acyclic $w^*$ is less than the weight of the minimum weight directed Hamiltonian cycle in $G$, when $G$ has a directed Hamiltonian cycle

Arjun asked in Graph Theory Dec 18, 2018

by Arjun

1.8k views

20 votes

7 answers

28

TIFR CSE 2019 | Part B | Question: 13
A row of $10$ houses has to be painted using the colours red, blue, and green so that each house is a single colour, and any house that is immediately to the right of a red or a blue house must be green. How many ways are there to paint the houses? $199$ $683$ $1365$ $3^{10}-2^{10}$ $3^{10}$

Arjun asked in Combinatory Dec 18, 2018

by Arjun

3.7k views

3 votes

2 answers

29

TIFR CSE 2019 | Part B | Question: 14
Let $m$ and $n$ be two positive integers. Which of the following is NOT always true? If $m$ and $n$ are co-prime, there exist integers $a$ and $b$ such that $am + bn=1$ $m^{n-1} \equiv 1 (\text{ mod } n)$ ... $m+1$ is a factor of $m^{n(n+1)}-1$ If $2^n -1$ is prime, then $n$ is prime

Arjun asked in Quantitative Aptitude Dec 18, 2018

by Arjun

1.0k views

6 votes

2 answers

30

TIFR CSE 2019 | Part B | Question: 15
Consider directed graphs on $n$ labelled vertices $\{1,2, \dots ,n\}$, where each vertex has exactly one edge coming in and exactly one edge going out. We allow self-loops. How many graphs have exactly two cycles ? $\displaystyle \sum_{k=1}^{n-1} k!(n-k)!$ ... $n!\bigg[\displaystyle \sum_{k=1}^{n-1} \frac{1}{k}\bigg]$ $\frac{n!(n-1)}{2}$ None of the above

Arjun asked in Graph Theory Dec 18, 2018

by Arjun

1.7k views

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