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Recent questions tagged tifr2018
+21
votes
2
answers
1
TIFR2018A9
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? $r^{4}$ $r^{4}  4r^{3}$ $r^{4}5r^{3}+8r^{2}4r$ $r^{4}4r^{3}+9r^{2}3r$ $r^{4}5r^{3}+10r^{2}15r$
asked
Dec 10, 2017
in
Graph Theory
by
Rohit Gupta 8
Active
(
2k
points)

961
views
tifr2018
graphtheory
graphcoloring
+4
votes
2
answers
2
TIFR2018A14
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 $A^{1}$ ... $(1)$ and $(2)$ are correct but not necessarily statement $(3)$ all the three statements $(1), (2),$ and $(3)$ are correct
asked
Dec 10, 2017
in
Linear Algebra
by
Rohit Gupta 8
Active
(
2k
points)

506
views
tifr2018
matrices
linearalgebra
+11
votes
2
answers
3
TIFR2018A13
A hacker knows that the password to the TIFR server is 10letter string consisting of lowercase letters from the English alphabet. He guesses a set of $5$ distinct 10letter strings (with lowercase letters) uniformly at random. What is the probability that one of the guesses of the hacker is ... $ \frac{1}{(26)^{10}}$ None of the above
asked
Dec 10, 2017
in
Probability
by
Rohit Gupta 8
Active
(
2k
points)

619
views
tifr2018
probability
+10
votes
1
answer
4
TIFR2018A15
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 ... ball with number $``2"$ on it is present in the box? $9/20$ $9/19$ $1/2$ $10/19$ None of the above
asked
Dec 10, 2017
in
Probability
by
Rohit Gupta 8
Active
(
2k
points)

655
views
tifr2018
probability
+7
votes
2
answers
5
TIFR2018B11
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 ... $L,$ defined as $\overline{L} = \left \{ a,b,c \right \}^{*}\backslash L,$ is regular. $L$ is regular, contextfree and decidable
asked
Dec 10, 2017
in
Theory of Computation
by
Arjun
Veteran
(
424k
points)

497
views
tifr2018
identifyclasslanguage
theoryofcomputation
+3
votes
1
answer
6
TIFR2018B15
$G$ 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 manytoone reduction from $SCYCLE$ to $LCYCLE$).
asked
Dec 10, 2017
in
Algorithms
by
Arjun
Veteran
(
424k
points)

297
views
tifr2018
algorithms
pnpnpcnph
nongate
+4
votes
3
answers
7
TIFR2018B14
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 of the following ... regular. It is Turing decidable (recursive). It is Turing recognizable but not decidable. Its complement is Turing recognizable but it is not decidable.
asked
Dec 10, 2017
in
Theory of Computation
by
Arjun
Veteran
(
424k
points)

489
views
tifr2018
identifyclasslanguage
+4
votes
2
answers
8
TIFR2018B13
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 ... weight edge of $C$ is not in $T.$ $T$ can be found in $O(n)$ time from the adjacency list representation of $G.$
asked
Dec 10, 2017
in
Algorithms
by
Arjun
Veteran
(
424k
points)

386
views
tifr2018
graphalgorithms
minimumspanningtrees
+6
votes
1
answer
9
TIFR2018B12
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 correct. Only statement (ii) is correct. Only statement (iii) is correct. Only statement (iv) is correct. None of the statements are correct.
asked
Dec 10, 2017
in
Theory of Computation
by
Arjun
Veteran
(
424k
points)

388
views
tifr2018
regularlanguages
+7
votes
1
answer
10
TIFR2018B10
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 $z_{i}=x_{i}+y_{i} \bmod 2$ ... The number of such linear functions for $n \geq 2$ is: $2^{n}$ $2^{n^{2}}$ $\large2^{\frac{n}{2}}$ $2^{4n}$ $2^{n^{2}+n}$
asked
Dec 10, 2017
in
Set Theory & Algebra
by
Arjun
Veteran
(
424k
points)

282
views
tifr2018
functions
+5
votes
1
answer
11
TIFR2018B9
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 $\large s$ ... If $P$ is NOT a shortest path in $G,$ then $\omega'(P)<2\times \omega(P).$ All of the above options are necessarily TRUE.
asked
Dec 10, 2017
in
Algorithms
by
Arjun
Veteran
(
424k
points)

328
views
tifr2018
graphalgorithms
shortestpath
+14
votes
2
answers
12
TIFR2018B8
In an undirected graph $G$ with $n$ vertices, vertex $1$ has degree $1$, while each vertex $2,\ldots,n1$ 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 $1$ to ... $n$ has degree $1$. The diameter of the graph is at most $\frac{n}{10}$ All of the above choices must be TRUE
asked
Dec 10, 2017
in
Graph Theory
by
Arjun
Veteran
(
424k
points)

899
views
tifr2018
graphtheory
degreeofgraph
+9
votes
1
answer
13
TIFR2018B7
Consider the recursive quicksort algorithm with "random pivoting". That is, in each recursive call, a pivot is chosen uniformly at random from the subarray being sorted.When this randomized algorithm is applied to an array of size $n$ all whose elements are distinct, what is the probability ... ${O} \left(\dfrac{1}{n^{2}}\right)$ $\Theta\left(\dfrac{1}{n \log^{2} n}\right)$
asked
Dec 10, 2017
in
Algorithms
by
Arjun
Veteran
(
424k
points)

493
views
tifr2018
algorithms
sorting
quicksort
+3
votes
2
answers
14
TIFR2018B6
Consider the following implementation of a binary tree data strucrure. The operator $+$ denotes listconcatenation. 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 ... $[12,6,7,9,10,15,2,1,4,8,11,14,3,13,5]$ $\text{Cannot be uniquely determined from given information.}$
asked
Dec 10, 2017
in
DS
by
Arjun
Veteran
(
424k
points)

211
views
tifr2018
datastructure
binarytree
+12
votes
1
answer
15
TIFR2018B5
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
asked
Dec 10, 2017
in
Algorithms
by
Arjun
Veteran
(
424k
points)

594
views
tifr2018
asymptoticnotations
recurrence
+1
vote
3
answers
16
TIFR2018B4
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 $Y$ ... $Y$ is not satisfiable. $X$ is not tautology and $Y$ is satisfiable. $X$ is a tautology and $Y$ is satisfiable,
asked
Dec 10, 2017
in
Mathematical Logic
by
Arjun
Veteran
(
424k
points)

336
views
tifr2018
mathematicallogic
propositionallogic
+5
votes
2
answers
17
TIFR2018B3
How many distinct minimum weight spanning trees does the following undirected, weighted graph have ? $1$ $2$ $4$ $6$ $8$
asked
Dec 10, 2017
in
Graph Theory
by
Arjun
Veteran
(
424k
points)

394
views
tifr2018
graphtheory
minimumspanningtrees
+5
votes
1
answer
18
TIFR2018B2
Consider the following nondeterministic 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 ... $aba(a+b)^{*}aba$ $(a+b)aba(b+a)^{*}$ $aba(a+b)^{*}$ $(ab)^{*}aba$
asked
Dec 10, 2017
in
Theory of Computation
by
Arjun
Veteran
(
424k
points)

187
views
tifr2018
regularexpressions
finiteautomata
+5
votes
7
answers
19
TIFR2018B1
What is the remainder when $4444^{4444}$ is divided by $9?$ $1$ $2$ $5$ $7$ $8$
asked
Dec 10, 2017
in
Combinatory
by
Arjun
Veteran
(
424k
points)

473
views
tifr2018
modulararithmetic
permutationandcombination
+1
vote
1
answer
20
TIFR2018A12
An $n \times n$ matrix $M$ with real entries is said to be positive definite if for every nonzero $n$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 ... $(3)$ Only $(1)$ and $(3)$ None of the above matrices are positive definite All of the above matrices are positive definite
asked
Dec 10, 2017
in
Linear Algebra
by
Arjun
Veteran
(
424k
points)

263
views
tifr2018
matrices
linearalgebra
+1
vote
3
answers
21
TIFR2018A11
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 ... $(i) \text{ and } (iii)$ only None of the statements must be TRUE All of the statements must be TRUE
asked
Dec 10, 2017
in
Numerical Ability
by
Arjun
Veteran
(
424k
points)

327
views
tifr2018
numericalability
logicalreasoning
+4
votes
2
answers
22
TIFR2018A10
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 $n \in \{0,1,2,\ldots \},$ ... $p_{n}=1 \text{ if } n \text{ is odd and } 0 \text{ otherwise}.$
asked
Dec 10, 2017
in
Probability
by
Arjun
Veteran
(
424k
points)

344
views
tifr2018
probability
+7
votes
2
answers
23
TIFR2018A8
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 ... $\text{Chacko}$ $\text{Desmond}$ $\text{Either Anuj or Binky; the information is insufficient to pinpoint the criminal}$
asked
Dec 10, 2017
in
Numerical Ability
by
Arjun
Veteran
(
424k
points)

300
views
tifr2018
logicalreasoning
+6
votes
3
answers
24
TIFR2018A7
Consider the following function definition. void greet(int n) { if(n>0) { printf("hello"); greet(n1); } printf("world"); } If you run greet(n) for some nonnegative integer n, what would it print? n times ... by n times "world" n times "helloworld" n+1 times "helloworld" n times "helloworld", followed by "world"
asked
Dec 10, 2017
in
Programming
by
Arjun
Veteran
(
424k
points)

373
views
tifr2018
programminginc
+8
votes
5
answers
25
TIFR2018A6
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 ? $6$ $23$ $61$ $72$ $91$
asked
Dec 10, 2017
in
Combinatory
by
Arjun
Veteran
(
424k
points)

427
views
tifr2018
pigeonholeprinciple
permutationandcombination
+1
vote
1
answer
26
TIFR2018A5
Which of the following id the derivative of $f(x)=x^{x}$ when $x>0$ ? $x^{x}$ $x^{x} \ln \;x$ $x^{x}+x^{x}\ln\;x$ $(x^{x}) (x^{x}\ln\;x)$ $\text{None of the above;function is not differentiable for }x>0$
asked
Dec 10, 2017
in
Calculus
by
Arjun
Veteran
(
424k
points)

212
views
tifr2018
calculus
differentiability
+1
vote
2
answers
27
TIFR2018A4
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 $5$ ... and $T_{1}>T_{3}$ $T_{1}=T_{2}$ and $T_{1}<T_{3}$ $T_{1}<T_{2}$ and $T_{1}=T_{3}$
asked
Dec 10, 2017
in
Others
by
Arjun
Veteran
(
424k
points)

191
views
tifr2018
numericalability
worktime
+9
votes
2
answers
28
TIFR2018A3
Which of the following statements is TRUE for all sufficiently large integers n ? $2^{2^{\sqrt{\log \log n}}} <2^{\sqrt{\log n}} < n$ $2^{\sqrt{\log n}}<n<2^{2^{\sqrt{\log \log n}}}$ $n<2^{\sqrt{\log n}}<2^{2^{\sqrt{\log \log n}}}$ $n<2^{2^{\sqrt{\log \log n}}} <2^{\sqrt{\log n}}$ $2^{\sqrt{\log n}}<2^{2^{\sqrt{\log \log n}}} < n$
asked
Dec 10, 2017
in
Algorithms
by
Arjun
Veteran
(
424k
points)

409
views
tifr2018
asymptoticnotations
+1
vote
1
answer
29
TIFR2018A2
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
asked
Dec 10, 2017
in
Numerical Ability
by
Arjun
Veteran
(
424k
points)

225
views
tifr2018
numericalability
geometry
threedimensionalgeometry
nongate
+1
vote
1
answer
30
TIFR2018A1
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 $A?$ $2$ $6$ $7$ $12$ $14$
asked
Dec 10, 2017
in
Numerical Ability
by
Arjun
Veteran
(
424k
points)

507
views
tifr2018
numericalability
geometry
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