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Recent questions tagged tifr2018
30
votes
6
answers
1
TIFR CSE 2018 | Part A | Question: 9
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? $r^{4}$ $r^{4} - 4r^{3}$ $r^{4}-5r^{3}+8r^{2}-4r$ $r^{4}-4r^{3}+9r^{2}-3r$ $r^{4}-5r^{3}+10r^{2}-15r$
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?...
Rohit Gupta 8
4.5k
views
Rohit Gupta 8
asked
Dec 10, 2017
Graph Theory
tifr2018
graph-theory
graph-coloring
+
–
8
votes
5
answers
2
TIFR CSE 2018 | Part A | Question: 14
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 ... and $(2)$ are correct but not necessarily statement $(3)$ all the three statements $(1), (2),$ and $(3)$ are correct
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 ...
Rohit Gupta 8
3.4k
views
Rohit Gupta 8
asked
Dec 10, 2017
Linear Algebra
tifr2018
matrix
linear-algebra
+
–
17
votes
2
answers
3
TIFR CSE 2018 | Part A | Question: 13
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 ... $ \frac{1}{(26)^{10}}$ None of the above
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...
Rohit Gupta 8
1.9k
views
Rohit Gupta 8
asked
Dec 10, 2017
Probability
tifr2018
probability
conditional-probability
+
–
17
votes
3
answers
4
TIFR CSE 2018 | Part A | Question: 15
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 $\text{ 1"}$ ... $\text{ 2"}$ on it is present in the box? $9/20$ $9/19$ $1/2$ $10/19$ None of the above
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 ra...
Rohit Gupta 8
2.6k
views
Rohit Gupta 8
asked
Dec 10, 2017
Probability
tifr2018
probability
conditional-probability
+
–
11
votes
3
answers
5
TIFR CSE 2018 | Part B | Question: 13
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 ... edge of $C$ is not in $T.$ $T$ can be found in $O(n)$ time from the adjacency list representation of $G.$
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 associate...
Arjun
2.7k
views
Arjun
asked
Dec 10, 2017
Algorithms
tifr2018
graph-algorithm
minimum-spanning-tree
+
–
4
votes
1
answer
6
TIFR CSE 2018 | Part B | Question: 15
$G$ respresents an undirected graph and a cycle refers to a simple cycle (no repeated edges or vertices). Define the following two languages. $\text{SCYCLE}=\{(G,k)\mid G \text{ contains a cycle of length at most k}\}$ ... $\text{SCYCLE}$ to $\text{LCYCLE}$).
$G$ respresents an undirected graph and a cycle refers to a simple cycle (no repeated edges or vertices). Define the following two languages.$\text{SCYCLE}=\{(G,k)\mid G ...
Arjun
957
views
Arjun
asked
Dec 10, 2017
Theory of Computation
tifr2018
theory-of-computation
reduction
p-np-npc-nph
non-gate
+
–
13
votes
3
answers
7
TIFR CSE 2018 | Part B | Question: 14
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 ... It is Turing decidable (recursive). It is Turing recognizable but not decidable. Its complement is Turing recognizable but it is not decidable.
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 det...
Arjun
3.5k
views
Arjun
asked
Dec 10, 2017
Theory of Computation
tifr2018
theory-of-computation
identify-class-language
+
–
9
votes
1
answer
8
TIFR CSE 2018 | Part B | Question: 12
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 a universal constant ... . Only statement (ii) is correct. Only statement (iii) is correct. Only statement (iv) is correct. None of the statements are correct.
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 eve...
Arjun
2.0k
views
Arjun
asked
Dec 10, 2017
Theory of Computation
tifr2018
theory-of-computation
regular-language
+
–
12
votes
2
answers
9
TIFR CSE 2018 | Part B | Question: 11
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 ... defined as $\overline{L} = \left \{ a,b,c \right \}^{*}\backslash L,$ is regular. $L$ is regular, context-free and decidable
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 f...
Arjun
2.4k
views
Arjun
asked
Dec 10, 2017
Theory of Computation
tifr2018
identify-class-language
theory-of-computation
+
–
12
votes
1
answer
10
TIFR CSE 2018 | Part B | Question: 9
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 ... is NOT a shortest path in $G,$ then $\omega'(P)<2\times \omega(P).$ All of the above options are necessarily TRUE.
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 ve...
Arjun
2.4k
views
Arjun
asked
Dec 10, 2017
Algorithms
tifr2018
graph-algorithm
shortest-path
+
–
14
votes
1
answer
11
TIFR CSE 2018 | Part B | Question: 5
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
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} \...
Arjun
3.5k
views
Arjun
asked
Dec 10, 2017
Algorithms
tifr2018
asymptotic-notation
recurrence-relation
+
–
11
votes
1
answer
12
TIFR CSE 2018 | Part B | Question: 10
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$ ... such linear functions for $n \geq 2$ is: $2^{n}$ $2^{n^{2}}$ $\large2^{\frac{n}{2}}$ $2^{4n}$ $2^{n^{2}+n}$
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 $...
Arjun
1.5k
views
Arjun
asked
Dec 10, 2017
Set Theory & Algebra
tifr2018
set-theory&algebra
functions
+
–
32
votes
3
answers
13
TIFR CSE 2018 | Part B | Question: 8
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 ... Vertex $n$ has degree $1$. The diameter of the graph is at most $\frac{n}{10}$ All of the above choices must be TRUE
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 o...
Arjun
5.4k
views
Arjun
asked
Dec 10, 2017
Graph Theory
tifr2018
graph-theory
degree-of-graph
+
–
13
votes
1
answer
14
TIFR CSE 2018 | Part B | Question: 7
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, ... $\Theta\left(\dfrac{1}{n \log^{2} n}\right)$
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....
Arjun
8.2k
views
Arjun
asked
Dec 10, 2017
Algorithms
tifr2018
algorithms
sorting
quick-sort
+
–
4
votes
2
answers
15
TIFR CSE 2018 | Part B | Question: 6
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: ... $\text{Cannot be uniquely determined from given information.}$
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: i...
Arjun
1.4k
views
Arjun
asked
Dec 10, 2017
DS
tifr2018
data-structures
binary-tree
+
–
4
votes
3
answers
16
TIFR CSE 2018 | Part B | Question: 4
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: ... is not satisfiable. $X$ is not tautology and $Y$ is satisfiable. $X$ is a tautology and $Y$ is satisfiable,
The notation "$\Rightarrow$" denotes "implies" and "$\wedge$" denotes "and" in the following formulae.Let $X$ denote the formula: $(b \Rightarrow a ) \Rightarrow ( a \Ri...
Arjun
1.9k
views
Arjun
asked
Dec 10, 2017
Mathematical Logic
tifr2018
mathematical-logic
propositional-logic
+
–
9
votes
2
answers
17
TIFR CSE 2018 | Part B | Question: 3
How many distinct minimum weight spanning trees does the following undirected, weighted graph have ? $1$ $2$ $4$ $6$ $8$
How many distinct minimum weight spanning trees does the following undirected, weighted graph have ?$1$$2$$4$$6$$8$
Arjun
2.6k
views
Arjun
asked
Dec 10, 2017
Algorithms
tifr2018
algorithms
minimum-spanning-tree
+
–
8
votes
1
answer
18
TIFR CSE 2018 | Part B | Question: 2
Consider the following non-deterministic automation, where $\large s_{1}$ is the start state and $\large s_{4}$ is the final (accepting) state. The alphabet is $\{a,b\}.$ A transition with label $\large\epsilon$ can be taken without consuming any symbol from the input. Which of the ... $aba(a+b)^{*}aba$ $(a+b)aba(b+a)^{*}$ $aba(a+b)^{*}$ $(ab)^{*}aba$
Consider the following non-deterministic automation, where $\large s_{1}$ is the start state and $\large s_{4}$ is the final (accepting) state. The alphabet is $\{a,b\}.$...
Arjun
1.4k
views
Arjun
asked
Dec 10, 2017
Theory of Computation
tifr2018
regular-expression
finite-automata
+
–
17
votes
12
answers
19
TIFR CSE 2018 | Part B | Question: 1
What is the remainder when $4444^{4444}$ is divided by $9?$ $1$ $2$ $5$ $7$ $8$
What is the remainder when $4444^{4444}$ is divided by $9?$$1$$2$$5$$7$$8$
Arjun
3.2k
views
Arjun
asked
Dec 10, 2017
Quantitative Aptitude
tifr2018
quantitative-aptitude
modular-arithmetic
+
–
8
votes
1
answer
20
TIFR CSE 2018 | Part A | Question: 12
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 ... $(3)$ Only $(1)$ and $(3)$ None of the above matrices are positive definite All of the above matrices are positive definite
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 ...
Arjun
1.6k
views
Arjun
asked
Dec 10, 2017
Linear Algebra
tifr2018
matrix
linear-algebra
+
–
3
votes
3
answers
21
TIFR CSE 2018 | Part A | Question: 11
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 ... $(i) \text{ and } (iii)$ only None of the statements must be TRUE All of the statements must be TRUE
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 ...
Arjun
1.6k
views
Arjun
asked
Dec 10, 2017
Analytical Aptitude
tifr2018
analytical-aptitude
logical-reasoning
+
–
7
votes
2
answers
22
TIFR CSE 2018 | Part A | Question: 10
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 $n \in \{0,1,2,\ldots \},$ ... $p_{n}=1 \text{ if } n \text{ is odd and } 0 \text{ otherwise}.$
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 $...
Arjun
2.0k
views
Arjun
asked
Dec 10, 2017
Probability
tifr2018
probability
binomial-distribution
+
–
13
votes
2
answers
23
TIFR CSE 2018 | Part A | Question: 8
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. ... $\text{Desmond}$ $\text{Either Anuj or Binky; the information is insufficient to pinpoint the criminal}$
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...
Arjun
1.5k
views
Arjun
asked
Dec 10, 2017
Analytical Aptitude
tifr2018
analytical-aptitude
logical-reasoning
+
–
16
votes
5
answers
24
TIFR CSE 2018 | Part A | Question: 7
Consider the following function definition. void greet(int n) { if(n>0) { printf("hello"); greet(n-1); } printf("world"); } If you run $\textsf{greet(n)}$ ... "helloworld" $\textsf{n+1}$ times "helloworld" $\textsf{n}$ times "helloworld", followed by "world"
Consider the following function definition. void greet(int n) { if(n>0) { printf("hello"); greet(n-1); } printf("world"); }If you run $\textsf{greet(n)}$ for some non-neg...
Arjun
2.8k
views
Arjun
asked
Dec 10, 2017
Programming in C
tifr2018
programming
programming-in-c
recursion
+
–
16
votes
6
answers
25
TIFR CSE 2018 | Part A | Question: 6
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 ? $6$ $23$ $61$ $72$ $91$
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$
Arjun
3.3k
views
Arjun
asked
Dec 10, 2017
Combinatory
tifr2018
pigeonhole-principle
combinatory
+
–
3
votes
1
answer
26
TIFR CSE 2018 | Part A | Question: 5
Which of the following is the derivative of $f(x)=x^{x}$ when $x>0$ ? $x^{x}$ $x^{x} \ln \;x$ $x^{x}+x^{x}\ln\;x$ $(x^{x}) (x^{x}\ln\;x)$ $\text{None of the above; function is not differentiable for }x>0$
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...
Arjun
1.3k
views
Arjun
asked
Dec 10, 2017
Calculus
tifr2018
calculus
differentiation
+
–
22
votes
3
answers
27
TIFR CSE 2018 | Part A | Question: 3
Which of the following statements is TRUE for all sufficiently large integers n ? $2^{2^{\sqrt{\log \log n}}} <2^{\sqrt{\log n}} < n$ $2^{\sqrt{\log n}}<n<2^{2^{\sqrt{\log \log n}}}$ $n<2^{\sqrt{\log n}}<2^{2^{\sqrt{\log \log n}}}$ $n<2^{2^{\sqrt{\log \log n}}} <2^{\sqrt{\log n}}$ $2^{\sqrt{\log n}}<2^{2^{\sqrt{\log \log n}}} < n$
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 ...
Arjun
3.0k
views
Arjun
asked
Dec 10, 2017
Algorithms
tifr2018
algorithms
asymptotic-notation
+
–
2
votes
2
answers
28
TIFR CSE 2018 | Part A | Question: 4
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 ... $T_{1}=T_{2}$ and $T_{1}<T_{3}$ $T_{1}<T_{2}$ and $T_{1}=T_{3}$
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 ...
Arjun
627
views
Arjun
asked
Dec 10, 2017
Quantitative Aptitude
tifr2018
quantitative-aptitude
work-time
+
–
2
votes
1
answer
29
TIFR CSE 2018 | Part A | Question: 2
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
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 \};...
Arjun
833
views
Arjun
asked
Dec 10, 2017
Quantitative Aptitude
tifr2018
quantitative-aptitude
geometry
three-dimensional-geometry
non-gate
+
–
4
votes
1
answer
30
TIFR CSE 2018 | Part A | Question: 1
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 $A?$ $2$ $6$ $7$ $12$ $14$
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 $A$ wi...
Arjun
1.4k
views
Arjun
asked
Dec 10, 2017
Quantitative Aptitude
tifr2018
quantitative-aptitude
geometry
circle
+
–
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