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Recent questions tagged tifr2014
+31
votes
3
answers
1
TIFR2014B20
Consider the following game. There is a list of distinct numbers. At any round, a player arbitrarily chooses two numbers $a, b$ from the list and generates a new number $c$ by subtracting the smaller number from the larger one. The numbers $a$ and $b$ are put back in the list. If ... $273$. What is the score of the best player for this game? $40$ $16$ $33$ $91$ $123$
asked
Nov 20, 2015
in
Algorithms
by
makhdoom ghaya
Boss
(
30.7k
points)

919
views
tifr2014
algorithms
identifyfunction
+28
votes
4
answers
2
TIFR2014B19
Consider the following tree with $13$ nodes. Suppose the nodes of the tree are randomly assigned distinct labels from $\left\{1, 2,\ldots,13\right\}$, each permutation being equally likely. What is the probability that the labels form a minheap (i.e., every node receives the minimum label in ... $\frac{2}{13}$ $\frac{1}{2^{13}}$
asked
Nov 20, 2015
in
DS
by
makhdoom ghaya
Boss
(
30.7k
points)

1.4k
views
tifr2014
heap
+15
votes
3
answers
3
TIFR2014B18
Let $k$ be an integer at least $4$ and let $\left[k\right]= \left\{1, 2,...,k\right\}$. Let $f:\left[k\right]^{4}\rightarrow \left\{0, 1\right\}$ be defined as follows: $f(y_{1}, y_{2}, y_{3}, y_{4}) = 1$ if an only if the $y_{i} 's$ are all distinct. For each ... $2^{\left(\frac{k}{3}\right)}$ $\left(\frac{k}{3}\right)$ $\left(\frac{k}{3}\right)+1$ $4 \left(\frac{k}{3}\right)$
asked
Nov 20, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
30.7k
points)

727
views
tifr2014
settheory&algebra
functions
+8
votes
3
answers
4
TIFR2014B17
Let $f: \left\{0, 1\right\}^{n} \rightarrow \left\{0, 1\right\}$ ... $f$ is the MAJORITY function. $f$ is the PARITY function. $f$ outputs $1$ at exactly one assignment of the input bits.
asked
Nov 20, 2015
in
Digital Logic
by
makhdoom ghaya
Boss
(
30.7k
points)

677
views
tifr2014
booleanalgebra
+13
votes
4
answers
5
TIFR2014B16
Consider the ordering relation $x\mid y \subseteq N \times N$ over natural numbers $N$ such that $x \mid y$ if there exists $z \in N$ such that $x ∙ z = y$. A set is called lattice if every finite subset has a least upper bound and greatest lower bound. It ... $\mid$ is a total order. $(N, \mid)$ is a complete lattice. $(N, \mid)$ is a lattice but not a complete lattice.
asked
Nov 20, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
30.7k
points)

986
views
tifr2014
settheory&algebra
partialorder
+13
votes
3
answers
6
TIFR2014B15
Consider the set $N^{*}$ of finite sequences of natural numbers with $x \leq_{p}y$ denoting that sequence $x$ is a prefix of sequence $y$. Then, which of the following is true? $N^{*}$ is uncountable. $\leq_{p}$ is a total order. Every nonempty subset of ... Every nonempty subset of $N^{*}$ has a greatest lower bound. Every nonempty finite subset of $N^{*}$ has a least upper bound.
asked
Nov 20, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
30.7k
points)

759
views
tifr2014
settheory&algebra
partialorder
+19
votes
1
answer
7
TIFR2014B14
Which the following is FALSE? Complement of a recursive language is recursive. A language recognized by a nondeterministic Turing machine can also be recognized by a deterministic Turing machine. Complement of a context free language can be recognized ... enumerable then it is recursive. Complement of a nonrecursive language can never be recognized by any Turing machine.
asked
Nov 20, 2015
in
Theory of Computation
by
makhdoom ghaya
Boss
(
30.7k
points)

1.6k
views
tifr2014
theoryofcomputation
closureproperty
+25
votes
2
answers
8
TIFR2014B13
Let $L$ be a given contextfree language over the alphabet $\left\{a, b\right\}$. Construct $L_{1},L_{2}$ as follows. Let $L_{1}= L  \left\{xyx\mid x, y \in \left\{a, b\right\}^*\right\}$, and $L_{2} = L \cdot L$. Then, Both $L_{1}$ and $L_{2}$ ... and $L_{2}$ is context free. $L_{1}$ and $L_{2}$ both may not be context free. $L_{1}$ is regular but $L_{2}$ may not be context free.
asked
Nov 20, 2015
in
Theory of Computation
by
makhdoom ghaya
Boss
(
30.7k
points)

929
views
tifr2014
theoryofcomputation
identifyclasslanguage
+17
votes
3
answers
9
TIFR2014B12
Consider the following three statements: Intersection of infinitely many regular languages must be regular. Every subset of a regular language is regular. If $L$ is regular and $M$ is not regular then $L ∙ M$ is necessarily not regular. Which of the following gives the ... the above? true, false, true. false, false, true. true, false, true. false, false, false. true, true, true.
asked
Nov 20, 2015
in
Theory of Computation
by
makhdoom ghaya
Boss
(
30.7k
points)

1.4k
views
tifr2014
theoryofcomputation
regularlanguages
+21
votes
3
answers
10
TIFR2014B11
Consider the following recurrence relation: $T\left(n\right)= \begin{cases} T\left(\frac{n}{k}\right)+ T\left(\frac{3n}{4}\right)+ n & \text{if } n \geq 2 \\ 1& \text{if } n=1 \end{cases}$ Which of the following statements is FALSE? $T(n)$ is $O(n^{3/2})$ when $k=3$. $T(n)$ is ... $O(n \log n)$ when $k=4$. $T(n)$ is $O(n \log n)$ when $k=5$. $T(n)$ is $O(n)$ when $k=5$.
asked
Nov 20, 2015
in
Algorithms
by
makhdoom ghaya
Boss
(
30.7k
points)

1.8k
views
tifr2014
algorithms
recurrence
+10
votes
2
answers
11
TIFR2014B10
Given a set of $n$ distinct numbers, we would like to determine both the smallest and the largest number. Which of the following statements is TRUE? These two elements can be determined using $O\left(\log^{100}n\right)$ ... $2(n  1)$ comparisons. None of the above.
asked
Nov 19, 2015
in
Algorithms
by
makhdoom ghaya
Boss
(
30.7k
points)

1.4k
views
tifr2014
algorithms
minimummaximum
+31
votes
6
answers
12
TIFR2014B9
Given a set of $n$ distinct numbers, we would like to determine the smallest three numbers in this set using comparisons. Which of the following statements is TRUE? These three elements can be determined using $O\left(\log^{2}n\right)$ ... $O(n)$ comparisons. None of the above.
asked
Nov 19, 2015
in
Algorithms
by
makhdoom ghaya
Boss
(
30.7k
points)

2.4k
views
tifr2014
algorithms
minimummaximum
+20
votes
3
answers
13
TIFR2014B8
Which of these functions grows fastest with $n$? $e^{n}/n$. $e^{n0.9 \log n}$. $2^{n}$. $\left(\log n\right)^{n1}$. None of the above.
asked
Nov 19, 2015
in
Algorithms
by
makhdoom ghaya
Boss
(
30.7k
points)

1.1k
views
tifr2014
algorithms
asymptoticnotations
+10
votes
5
answers
14
TIFR2014B7
Which of the following statements is TRUE for all sufficiently large $n$? $\displaystyle \left(\log n\right)^{\log\log n} < 2^{\sqrt{\log n}} < n^{1/4}$ $\displaystyle 2^{\sqrt{\log n}} < n^{1/4} < \left(\log n\right)^{\log\log n}$ ... $\displaystyle 2^{\sqrt{\log n}} < \left(\log n\right)^{\log\log n} < n^{1/4}$
asked
Nov 19, 2015
in
Algorithms
by
makhdoom ghaya
Boss
(
30.7k
points)

865
views
tifr2014
algorithms
timecomplexity
+22
votes
3
answers
15
TIFR2014B6
Consider the problem of computing the minimum of a set of $n$ distinct numbers. We choose a permutation uniformly at random (i.e., each of the n! permutations of $\left \langle 1,....,n \right \rangle$ is chosen with probability $(1/n!)$ and we inspect the numbers in the order given by this ... number of times MIN is updated? $O (1)$ $H_{n}=\sum ^{n}_{i=1} 1/i$ $\sqrt{n}$ $n/2$ $n$
asked
Nov 19, 2015
in
Algorithms
by
makhdoom ghaya
Boss
(
30.7k
points)

650
views
tifr2014
algorithms
minimummaximum
+22
votes
1
answer
16
TIFR2014B5
Let $G = (V,E)$ be an undirected connected simple (i.e., no parallel edges or selfloops) graph with the weight function $w: E \rightarrow \mathbb{R}$ on its edge set. Let $w(e_{1}) < w(e_{2}) < · · · < w(e_{m})$ ... each edge weight $w_{i} = w(e_{i})$ by its square $w^{2}_{i}$ , then $T$ must still be a minimum spanning tree of this new instance.
asked
Nov 19, 2015
in
Algorithms
by
makhdoom ghaya
Boss
(
30.7k
points)

1.1k
views
tifr2014
algorithms
spanningtree
+24
votes
4
answers
17
TIFR2014B4
Consider the following undirected graph with some edge costs missing. Suppose the wavy edges form a Minimum Cost Spanning Tree for $G$. Then, which of the following inequalities NEED NOT hold? cost$(a, b) \geq 6$. cost$(b, e) \geq 5$. cost$(e, f) \geq 5$. cost$(a, d) \geq 4$. cost$(b, c) \geq 4$.
asked
Nov 19, 2015
in
Algorithms
by
makhdoom ghaya
Boss
(
30.7k
points)

1.6k
views
tifr2014
algorithms
graphalgorithms
spanningtree
+20
votes
1
answer
18
TIFR2014B3
Consider the following directed graph. Suppose a depthfirst traversal of this graph is performed, assuming that whenever there is a choice, the vertex earlier in the alphabetical order is to be chosen. Suppose the number of tree edges is $T$, the number of back edges is $B$ and the number of cross edges is $C$. ... $B = 1$, $C = 2$, and $T = 3$. $B = 2$, $C = 2$, and $T = 1$.
asked
Nov 19, 2015
in
Algorithms
by
makhdoom ghaya
Boss
(
30.7k
points)

1.2k
views
tifr2014
algorithms
graphalgorithms
+6
votes
3
answers
19
TIFR2014B2
Consider the following code. def brian(n): count = 0 while ( n ! = 0 ) n = n & ( n1 ) count = count + 1 return count Here $n$ is meant to be an unsigned integer. The operator & considers its arguments in binary and computes their bit wise ... of $n$. The code might go into an infinite loop for some $n$. The result depends on the number of bits used to store unsigned integers.
asked
Nov 19, 2015
in
Algorithms
by
makhdoom ghaya
Boss
(
30.7k
points)

447
views
tifr2014
algorithms
identifyfunction
+7
votes
2
answers
20
TIFR2014B1
Let $T$ be a rooted binary tree whose vertices are labelled with symbols $a, b, c, d, e, f, g, h, i, j, k$. Suppose the inorder (visit left subtree, visit root, visit right subtree) and postorder (visit left subtree, visit right subtree, visit root) ... How many leaves does the tree have? THREE. FOUR. FIVE. SIX. Cannot be determined uniquely from the given information.
asked
Nov 19, 2015
in
DS
by
makhdoom ghaya
Boss
(
30.7k
points)

410
views
tifr2014
binarytree
datastructures
easy
+2
votes
1
answer
21
TIFR2014A20
Consider the equation $x^{2}+y^{2}3z^{2}3t^{2}=0$. The total number of integral solutions of this equation in the range of the first $10000$ numbers, i.e., $1 \leq x, y, z, t \leq 10000$, is $200$ $55$ $100$ $1$ None of the above
asked
Nov 19, 2015
in
Numerical Ability
by
makhdoom ghaya
Boss
(
30.7k
points)

211
views
tifr2014
numbertheory
numericalability
+4
votes
2
answers
22
TIFR2014A19
Consider the following random function of $x$ $F(x) = 1 + Ux + Vx^{2} \bmod 5$, where $U$ and $V$ are independent random variables uniformly distributed over $\left\{0, 1, 2, 3, 4\right\}$. Which of the following is FALSE? $F(1)$ is uniformly ... $F(1), F(2), F(3)$ are independent and identically distributed random variables. All of the above. None of the above.
asked
Nov 19, 2015
in
Probability
by
makhdoom ghaya
Boss
(
30.7k
points)

328
views
tifr2014
probability
randomvariable
+1
vote
2
answers
23
TIFR2014A18
We are given a collection of real numbers where a real number $a_{i}\neq 0$ occurs $n_{i}$ times. Let the collection be enumerated as $\left\{x_{1}, x_{2},...x_{n}\right\}$ so that $x_{1}=x_{2}=...=x_{n_{1}}=a_{1}$ and so on, and $n=\sum _{i}n_{i}$ is finite. What is ... $\min_{i} a_{i}$ $\min_{i} \left(n_{i}a_{i}\right)$ $\max_{i} a_{i}$ None of the above.
asked
Nov 19, 2015
in
Calculus
by
makhdoom ghaya
Boss
(
30.7k
points)

209
views
tifr2014
limits
+9
votes
2
answers
24
TIFR2014A17
A fair dice (with faces numbered $1, . . . , 6$) is independently rolled repeatedly. Let $X$ denote the number of rolls till an even number is seen and let $Y$ denote the number of rolls till $3$ is seen. Evaluate $E(Y X = 2)$. $6\frac{5}{6}$ $6$ $5\frac{1}{2}$ $6\frac{1}{3}$ $5\frac{2}{3}$
asked
Nov 19, 2015
in
Probability
by
makhdoom ghaya
Boss
(
30.7k
points)

1.1k
views
tifr2014
expectation
+2
votes
2
answers
25
TIFR2014A16
Let $x_{0}=1$ and $x_{n+1}= \frac{3+2x_{n}}{3+x_{n}}, n\geq 0$. $x_{\infty}=\lim_{n\rightarrow \infty}x_{n}$ is $\left(\sqrt{5}1\right) / 2$ $\left(\sqrt{5}+1\right) / 2$ $\left(\sqrt{13}1\right) / 2$ $\left(\sqrt{13}1\right) / 2$ None of the above.
asked
Nov 19, 2015
in
Calculus
by
makhdoom ghaya
Boss
(
30.7k
points)

335
views
tifr2014
limits
+2
votes
1
answer
26
TIFR2014A15
Consider the following statements: $b_{1}= \sqrt{2}$, series with each $b_{i}= \sqrt{b_{i1}+ \sqrt{2}}, i \geq 2$, converges. $\sum ^{\infty} _{i=1} \frac{\cos (i)}{i^{2}}$ converges. $\sum ^{\infty} _{i=0} b_{i}$ ... $(2)$ and $(3)$ but not $(1)$. Statements $(1)$ and $(3)$ but not $(2)$. All the three statements. None of the three statements.
asked
Nov 14, 2015
in
Calculus
by
makhdoom ghaya
Boss
(
30.7k
points)

188
views
tifr2014
convergence
nongate
+3
votes
1
answer
27
TIFR2014A14
Let $m$ and $n$ be any two positive integers. Then, which of the following is FALSE? $m + 1$ divides $m^{2n} − 1$. For any prime $p$, $m^{p} \equiv m (\mod p)$. If one of $m$, $n$ is prime, then there are integers $x, y$ such that $mx + ny = 1$. If $m < n$, then $m!$ divides $n(n − 1)(n − 2) \ldots (n − m + 1)$. If $2^{n} − 1$ is prime, then $n$ is prime.
asked
Nov 14, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
30.7k
points)

195
views
tifr2014
numbertheory
settheory&algebra
+3
votes
1
answer
28
TIFR2014A13
Let $L$ be a line on the two dimensional plane. $L'$s intercepts with the $X$ and $Y$ axes are respectively $a$ and $b$. After rotating the coordinate system (and leaving $L$ untouched), the new intercepts are $a'$ and $b'$ ... $\frac{b}{a}+\frac{a}{b}=\frac{b'}{a'}+\frac{a'}{b'}$. None of the above.
asked
Nov 14, 2015
in
Numerical Ability
by
makhdoom ghaya
Boss
(
30.7k
points)

271
views
tifr2014
geometry
cartesiancoordinates
+2
votes
0
answers
29
TIFR2014A12
Let $f(x)= 2^{x}$. Consider the following inequality for real numbers $a, b$ and $0 < \lambda < 1$: $f(\lambda a + b) \leq \lambda f(a) + (1  \lambda) f (\frac{b}{1  \lambda})$. Consider the following 3 conditions: ... $(3)$ but not under condition $(2)$. The above inequality holds under all the three conditions. The above inequality holds under none of the three conditions.
asked
Nov 14, 2015
in
Numerical Ability
by
makhdoom ghaya
Boss
(
30.7k
points)

122
views
tifr2014
numericalability
convexsetsfunctions
nongate
+9
votes
1
answer
30
TIFR2014A11
A large community practices birth control in the following peculiar fashion. Each set of parents continues having children until a son is born; then they stop. What is the ratio of boys to girls in the community if, in the absence of birth control, $51\%$ of the babies are born male? $51:49$ $1:1$ $49:51$ $51:98$ $98:51$
asked
Nov 13, 2015
in
Numerical Ability
by
makhdoom ghaya
Boss
(
30.7k
points)

441
views
tifr2014
numericalability
fractions
tricky
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