# Recent questions tagged tifr2018

1
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? $r^{4}$ $r^{4} - 4r^{3}$ $r^{4}-5r^{3}+8r^{2}-4r$ $r^{4}-4r^{3}+9r^{2}-3r$ $r^{4}-5r^{3}+10r^{2}-15r$
2
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 the matrix $A^{-1}$ sums ... statement $(1)$ and $(2)$ are correct but not necessarily statement $(3)$ all the three statements $(1), (2),$ and $(3)$ are correct
3
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 guesses of the hacker is correct ... $\frac{1}{(26)^{10}}$ None of the above
4
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 $1"$ on it is present in the box. If it is ... that the ball with number $2"$ on it is present in the box? $9/20$ $9/19$ $1/2$ $10/19$ None of the above
5
$G$ respresents an undirected graph and a cycle refers to a simple cycle (no repeated edges or vertices). Define the following two languages. $SCYCLE=\{(G,k)\mid G \text{ contains a cycle of length at most k}\}$ ... $SCYCLE \leq_{p} LCYCLE$ (i.e, there is a polynomial time many-to-one reduction from $SCYCLE$ to $LCYCLE$).
6
Define the language $\text{INFINITE}_{DFA}\equiv \{(A)\mid A \text{ is a DFA and } L(A) \text{ is an infinite language}\},$ where $(A)$ ... not regular. It is Turing decidable (recursive). It is Turing recognizable but not decidable. Its complement is Turing recognizable but it is not decidable.
7
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 following statements is NOT ... maximum weight edge of $C$ is not in $T.$ $T$ can be found in $O(n)$ time from the adjacency list representation of $G.$
8
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 $p.$ $\large\pi$ ... (i) is correct. Only statement (ii) is correct. Only statement (iii) is correct. Only statement (iv) is correct. None of the statements are correct.
9
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? $L$ is regular but not ... of $L,$ defined as $\overline{L} = \left \{ a,b,c \right \}^{*}\backslash L,$ is regular. $L$ is regular, context-free and decidable
10
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
11
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 $z_{i}=x_{i}+y_{i} \bmod 2$). A function $h:\{0,1\}^{n} \to \{0,1\}^{n}$ ... $n \geq 2$ is: $2^{n}$ $2^{n^{2}}$ $\large2^{\frac{n}{2}}$ $2^{4n}$ $2^{n^{2}+n}$
12
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 path from $\large s$ ... $P$ is NOT a shortest path in $G,$ then $\omega'(P)<2\times \omega(P).$ All of the above options are necessarily TRUE.
13
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 from vertex $1$ ... $n$ has degree $1$. The diameter of the graph is at most $\frac{n}{10}$ All of the above choices must be TRUE
14
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, what is the probability that the smallest ... ${O} \left(\dfrac{1}{n^{2}}\right)$ $\Theta\left(\dfrac{1}{n \log^{2} n}\right)$
15
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: return [] else: return [T.value ... $[12,6,7,9,10,15,2,1,4,8,11,14,3,13,5]$ $\text{Cannot be uniquely determined from given information.}$
16
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: $(a \Rightarrow b) \wedge b$ Which of the following is ... not tautology and $Y$ is not satisfiable. $X$ is not tautology and $Y$ is satisfiable. $X$ is a tautology and $Y$ is satisfiable,
17
How many distinct minimum weight spanning trees does the following undirected, weighted graph have ? $1$ $2$ $4$ $6$ $8$
18
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 following regular expressions ... $aba(a+b)^{*}aba$ $(a+b)aba(b+a)^{*}$ $aba(a+b)^{*}$ $(ab)^{*}aba$
19
What is the remainder when $4444^{4444}$ is divided by $9?$ $1$ $2$ $5$ $7$ $8$
1 vote
20
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 object is also circular. Not all ... $(i) \text{ and } (iii)$ only None of the statements must be TRUE All of the statements must be TRUE
21
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 $n\times n$ with real entries. ... $(2)$ Only $(3)$ Only $(1)$ and $(3)$ None of the above matrices are positive definite All of the above matrices are positive definite
22
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 $n \in \{0,1,2,\ldots \},$ ... $p_{n}=1 \text{ if } n \text{ is odd and } 0 \text{ otherwise}.$
23
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. Anuj says that Binky did it. ... $\text{Chacko}$ $\text{Desmond}$ $\text{Either Anuj or Binky; the information is insufficient to pinpoint the criminal}$
24
Consider the following function definition. void greet(int n) { if(n>0) { printf("hello"); greet(n-1); } printf("world"); } If you run greet(n) for some non-negative integer n, what would it print? n times "hello", followed by n+1 times "world" n times "hello", followed by n times "world" n times "helloworld" n+1 times "helloworld" n times "helloworld", followed by "world"
25
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 ? $6$ $23$ $61$ $72$ $91$
26
Which of the following is the derivative of $f(x)=x^{x}$ when $x>0$ ? $x^{x}$ $x^{x} \ln \;x$ $x^{x}+x^{x}\ln\;x$ $(x^{x}) (x^{x}\ln\;x)$ $\text{None of the above; function is not differentiable for }x>0$
1 vote
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 $A?$ $2$ $6$ $7$ $12$ $14$
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 a speed of $5$ Km/hr for the ... $T_{1}>T_{3}$ $T_{1}=T_{2}$ and $T_{1}<T_{3}$ $T_{1}<T_{2}$ and $T_{1}=T_{3}$
Consider the following subset of $\mathbb{R} ^{3}$ (the first two are cylinder, the third is a plane): $C_{1}=\left \{ \left ( x,y,z \right ): y^{2}+z^{2}\leq 1 \right \};$ ... Let $A = C_{1}\cap C_{2}\cap H.$ Which of the following best describe the shape of set $A?$ Circle Ellipse Triangle Square An octagonal convex figure with curved sides