# Recent questions tagged tifr2019 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
2
How many proper divisors (that is, divisors other than $1$ or $7200$) does $7200$ have ? $18$ $20$ $52$ $54$ $60$
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 $(iii)$ $(i),(ii) \text{ and }(iii)$
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 $\alpha + \beta \leq 1$? $\frac{1}{\pi}$ $\frac{3}{4} + \frac{1}{4} \cdot \frac{1}{\pi}$ $\frac{3}{4}+ \frac{1}{4} \cdot \frac{2}{\pi}$ $1$ $\frac{2}{\pi}$
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 number of ... ask within which she can always find out the number Lata has thought of? $10$ $32$ $100$ $999$ $\text{None of the above}$
1 vote
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}→ ... $p,q$ and $r$ must be convex? Only $p$ Only $q$ Only $r$ Only $p$ and $r$ Only $q$ and $r$
7
What are the last two digits of $1! + 2! + \dots +100!$? $00$ $13$ $30$ $33$ $73$
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 for the ... . Which of the following graphs most accurately captures the relationship between the speed $v$ and the density $\rho$ in this model ?
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 container $B$ is ... $V_{AY} > V_{BX}$ $V_{AY} = V_{BX}$ $V_{AY} + V_{BX}=30$ $V_{AY} + V_{BX}=20$
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 the ... has 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
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 hands with ... $2 \mid \text{Even} \mid - \mid \text{Odd} \mid$ $2 \mid \text{Odd} \mid - \mid \text{Even} \mid$
1 vote
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: There exists ... statements must be TRUE for ALL such functions $f$ and $g$ ? Only $(i)$ Only $(i)$ and $(ii)$ Only $(iii)$ None of them All of them
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? $0.00$ $0.02$ $0.10$ $0.33$ $1.00$
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 ? $7/12$ $1/6$ $1/12$ $1/81$ $\text{None of the above}$
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 $\lim_{n→\infty}$A^n$?$\begin{bmatrix} \ 0 & 0 & 0\\ 0& 0 & 0\\ 0 & 0 & 0 \end{bmatrix}\begin{bmatrix} ... $\text{The limit exists, but it is none of the above}$
1 vote
16
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
17
How many distinct minimum weight spanning trees does the following undirected, weighted graph have ? $8$ $16$ $32$ $64$ None of the above
18
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
1 vote
19
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 $\mu$ ... is a Craig interpolant for $\varphi$ and $\psi$ ? $q$ $\varphi$ itself $q \vee s$ $q \vee r$ $\neg q \wedge s$
20
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 $n!$ $?$ $n! = \Omega(n^n) \text{ and } n! = \mathcal{O}(n^{n+\frac{1}{2}})$ ... $n! =\Theta((\frac{n}{e})^{n+\frac{1}{2}})$ $n! =\Theta(n^{n+\frac{1}{2}}2^{-n})$
21
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); } } $625$ $256$ $120$ $24$ $5$
22
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. Define the ... $\text{3-DF-SAT}$ is NP-complete Neither $\text{3-CF-SAT}$ nor $\text{3-DF-SAT}$ are in P
23
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
24
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
25
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 following must ... $D$ 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
26
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 accepted by ... $(a+b)^*ba^*$ $(a+b)^*ba(aa)^*$ $(a+b)^*$ $(a+b)^*baa^*$
27
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 $G$ ... $H^*$ is acyclic $w^*$ is less than the weight of the minimum weight directed Hamiltonian cycle in $G$, when $G$ has a directed Hamiltonian cycle
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? $199$ $683$ $1365$ $3^{10}-2^{10}$ $3^{10}$
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
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 ? $\sum_{k=1}^{n-1} k!(n-k)!$ $\frac{n!}{2}\bigg[\sum_{k=1}^{n-1} \frac{1}{k(n-k)}\bigg]$ $n!\bigg[\sum_{k=1}^{n-1} \frac{1}{k}\bigg]$ $\frac{n!(n-1)}{2}$ None of the above