Recent questions tagged tifr2019

6 votes

6 answers

1

TIFR2019-A-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

1.7k views

5 votes

2 answers

2

TIFR2019-A-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

833 views

7 votes

2 answers

3

TIFR2019-A-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 TRUE? 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

1.7k views

5 votes

1 answer

4

TIFR2019-A-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

953 views

7 votes

5 answers

5

TIFR2019-A-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 the least ... within which 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

2.2k views

1 vote

1 answer

6

TIFR2019-A-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 $f:$\mathbb{R}$ $→$ $\mathbb ... $p,q$ 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

566 views

11 votes

2 answers

7

TIFR2019-A-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

680 views

0 votes

1 answer

8

TIFR2019-A-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 $v$ that satisfies the following constraints: The speed is no more than the speed limit $v_{max}$ mandated ... Which of the following graphs most accurately captures the relationship between the speed $v$ and the density $\rho$ in this model ?

Arjun asked in Verbal Aptitude Dec 18, 2018

by Arjun

535 views

2 votes

1 answer

9

TIFR2019-A-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 $B$. The mixture in ... $V_{AY} > V_{BX}$ $V_{AY} = V_{BX}$ $V_{AY} + V_{BX}=30$ $V_{AY} + V_{BX}=20$

Arjun asked in Quantitative Aptitude Dec 18, 2018

by Arjun

504 views

3 votes

1 answer

10

TIFR2019-A-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 if she chooses ... a winning 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

600 views

5 votes

5 answers

11

TIFR2019-A-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, they don't shake ... $2 \mid \text{Even} \mid - \mid \text{Odd} \mid$ $2 \mid \text{Odd} \mid - \mid \text{Even} \mid$

Arjun asked in Analytical Aptitude Dec 18, 2018

by Arjun

817 views

2 votes

1 answer

12

TIFR2019-A-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 $x \geq y$ ... $f$ 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

1.1k views

6 votes

2 answers

13

TIFR2019-A-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

1.2k views

5 votes

1 answer

14

TIFR2019-A-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 the probability 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

1k views

6 votes

5 answers

15

TIFR2019-A-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 & 0 & 0\\ ... $\text{The limit exists, but it is none of the above}$

Arjun asked in Calculus Dec 18, 2018

by Arjun

1.3k views

2 votes

2 answers

16

TIFR2019-B-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

1.9k views

9 votes

5 answers

17

TIFR2019-B-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

2.1k views

3 votes

1 answer

18

TIFR2019-B-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

897 views

2 votes

1 answer

19

TIFR2019-B-4
Let $\varphi$ be a propositional formula on a set of variables $A$ and $\varphi$ 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 $\mu$ on ... Craig 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

715 views

7 votes

2 answers

20

TIFR2019-B-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

1.7k views

12 votes

3 answers

21

TIFR2019-B-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

1.5k views

2 votes

1 answer

22

TIFR2019-B-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 $3$-DF-formula if it is a disjunction (i.e., an OR) of clauses of at most $3$ literals each. ... $\text{3-DF-SAT}$ is NP-complete Neither $\text{3-CF-SAT}$ nor $\text{3-DF-SAT}$ are in P

Arjun asked in Algorithms Dec 18, 2018

by Arjun

514 views

5 votes

2 answers

23

TIFR2019-B-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

1k views

10 votes

2 answers

24

TIFR2019-B-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

1.6k views

9 votes

2 answers

25

TIFR2019-B-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 $1010 \notin D$. Which of the ... ? $D$ is regular $D$ 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

1.1k views

10 votes

3 answers

26

TIFR2019-B-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 to the language ... $(a+b)^*ba^*$ $(a+b)^*ba(aa)^*$ $(a+b)^*$ $(a+b)^*baa^*$

Arjun asked in Theory of Computation Dec 18, 2018

by Arjun

1.2k views

5 votes

1 answer

27

TIFR2019-B-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 $G$ in which every ... $w^*$ 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

1.2k views

18 votes

7 answers

28

TIFR2019-B-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

2.2k views

3 votes

2 answers

29

TIFR2019-B-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

646 views

5 votes

2 answers

30

TIFR2019-B-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

1.1k views

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