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Recent questions tagged tifr2014

39 votes

4 answers

1

TIFR CSE 2014 | Part B | Question: 20
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 ... $273$. What is the score of the best player for this game? $40$ $16$ $33$ $91$ $123$

makhdoom ghaya asked in Algorithms Nov 20, 2015

by makhdoom ghaya

2.6k views

40 votes

4 answers

2

TIFR CSE 2014 | Part B | Question: 19
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 min-heap (i.e., every node receives the ... $\frac{2}{13}$ $\frac{1}{2^{13}}$

makhdoom ghaya asked in DS Nov 20, 2015

by makhdoom ghaya

4.8k views

19 votes

3 answers

3

TIFR CSE 2014 | Part B | Question: 18
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 ... $2^{\left(\frac{k}{3}\right)}$ $\left(\frac{k}{3}\right)$ $\left(\frac{k}{3}\right)+1$ $4 \left(\frac{k}{3}\right)$

makhdoom ghaya asked in Set Theory & Algebra Nov 20, 2015

by makhdoom ghaya

1.8k views

12 votes

3 answers

4

TIFR CSE 2014 | Part B | Question: 17
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.

makhdoom ghaya asked in Digital Logic Nov 20, 2015

by makhdoom ghaya

2.4k views

17 votes

5 answers

5

TIFR CSE 2014 | Part B | Question: 16
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 ... $(N, \mid)$ is a complete lattice. $(N, \mid)$ is a lattice but not a complete lattice.

makhdoom ghaya asked in Set Theory & Algebra Nov 20, 2015

by makhdoom ghaya

4.0k views

17 votes

4 answers

6

TIFR CSE 2014 | Part B | Question: 15
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 non ... -empty subset of $N^{*}$ has a greatest lower bound. Every non-empty finite subset of $N^{*}$ has a least upper bound.

makhdoom ghaya asked in Set Theory & Algebra Nov 20, 2015

by makhdoom ghaya

2.9k views

24 votes

2 answers

7

TIFR CSE 2014 | Part B | Question: 14
Which the following is FALSE? Complement of a recursive language is recursive. A language recognized by a non-deterministic Turing machine can also be recognized by a deterministic Turing machine. Complement of a context free language can ... enumerable then it is recursive. Complement of a non-recursive language can never be recognized by any Turing machine.

makhdoom ghaya asked in Theory of Computation Nov 20, 2015

by makhdoom ghaya

6.6k views

27 votes

2 answers

8

TIFR CSE 2014 | Part B | Question: 13
Let $L$ be a given context-free 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 $L_{1}$ ... $L_{1}$ and $L_{2}$ both may not be context free. $L_{1}$ is regular but $L_{2}$ may not be context free.

makhdoom ghaya asked in Theory of Computation Nov 20, 2015

by makhdoom ghaya

3.0k views

21 votes

3 answers

9

TIFR CSE 2014 | Part B | Question: 12
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 ... above? true, false, true. false, false, true. true, false, true. false, false, false. true, true, true.

makhdoom ghaya asked in Theory of Computation Nov 20, 2015

by makhdoom ghaya

4.1k views

30 votes

4 answers

10

TIFR CSE 2014 | Part B | Question: 11
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 $O(n^{3/2})$ ... $k=4$. $T(n)$ is $O(n \log n)$ when $k=5$. $T(n)$ is $O(n)$ when $k=5$.

makhdoom ghaya asked in Algorithms Nov 20, 2015

by makhdoom ghaya

4.9k views

19 votes

2 answers

11

TIFR CSE 2014 | Part B | Question: 10
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 $O\left(\log^{100}n\right)$ ... comparisons do not suffice, however these two elements can be determined using $2(n - 1)$ comparisons. None of the above.

makhdoom ghaya asked in Algorithms Nov 19, 2015

by makhdoom ghaya

4.3k views

47 votes

7 answers

12

TIFR CSE 2014 | Part B | Question: 9
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.

makhdoom ghaya asked in Algorithms Nov 19, 2015

by makhdoom ghaya

7.6k views

28 votes

3 answers

13

TIFR CSE 2014 | Part B | Question: 8
Which of these functions grows fastest with $n$? $e^{n}/n$. $e^{n-0.9 \log n}$. $2^{n}$. $\left(\log n\right)^{n-1}$. None of the above.

makhdoom ghaya asked in Algorithms Nov 19, 2015

by makhdoom ghaya

3.9k views

19 votes

5 answers

14

TIFR CSE 2014 | Part B | Question: 7
Which of the following statements is TRUE for all sufficiently large $n$? $\displaystyle \left(\log n\right)^{\log\log n} < 2^{\sqrt{\log n}} < n^{1/4}$ $\displaystyle 2^{\sqrt{\log n}} < n^{1/4} < \left(\log n\right)^{\log\log n}$ ... $\displaystyle 2^{\sqrt{\log n}} < \left(\log n\right)^{\log\log n} < n^{1/4}$

makhdoom ghaya asked in Algorithms Nov 19, 2015

by makhdoom ghaya

3.4k views

30 votes

3 answers

15

TIFR CSE 2014 | Part B | Question: 6
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 ... of times MIN is updated? $O (1)$ $H_{n}=\sum ^{n}_{i=1} 1/i$ $\sqrt{n}$ $n/2$ $n$

makhdoom ghaya asked in Algorithms Nov 19, 2015

by makhdoom ghaya

2.1k views

31 votes

1 answer

16

TIFR CSE 2014 | Part B | Question: 5
Let $G = (V,E)$ be an undirected connected simple (i.e., no parallel edges or self-loops) graph with the weight function $w: E \rightarrow \mathbb{R}$ on its edge set. Let $w(e_{1}) < w(e_{2}) < · · · < w(e_{m})$ ... $w_{i} = w(e_{i})$ by its square $w^{2}_{i}$ , then $T$ must still be a minimum spanning tree of this new instance.

makhdoom ghaya asked in Algorithms Nov 19, 2015

by makhdoom ghaya

2.6k views

31 votes

4 answers

17

TIFR CSE 2014 | Part B | Question: 4
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$(b, c) \geq 4$.

makhdoom ghaya asked in Algorithms Nov 19, 2015

by makhdoom ghaya

4.2k views

28 votes

1 answer

18

TIFR CSE 2014 | Part B | Question: 3
Consider the following directed graph. Suppose a depth-first 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 ... $B = 1$, $C = 2$, and $T = 3$. $B = 2$, $C = 2$, and $T = 1$.

makhdoom ghaya asked in Algorithms Nov 19, 2015

by makhdoom ghaya

4.3k views

7 votes

3 answers

19

TIFR CSE 2014 | Part B | Question: 2
Consider the following code. def brian(n): count = 0 while ( n ! = 0 ) n = n & ( n-1 ) count = count + 1 return count Here $n$ is meant to be an unsigned integer. The operator & considers its arguments in binary and ... $n$. The result depends on the number of bits used to store unsigned integers.

makhdoom ghaya asked in Algorithms Nov 19, 2015

by makhdoom ghaya

1.8k views

9 votes

2 answers

20

TIFR CSE 2014 | Part B | Question: 1
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 in-order (visit left subtree, visit root, visit right subtree) and post-order (visit left subtree, visit right ... How many leaves does the tree have? THREE. FOUR. FIVE. SIX. Cannot be determined uniquely from the given information.

makhdoom ghaya asked in DS Nov 19, 2015

by makhdoom ghaya

1.9k views

3 votes

1 answer

21

TIFR CSE 2014 | Part A | Question: 20
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

makhdoom ghaya asked in Quantitative Aptitude Nov 19, 2015

by makhdoom ghaya

775 views

7 votes

2 answers

22

TIFR CSE 2014 | Part A | Question: 19
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? $F(1)$ ... $F(1), F(2), F(3)$ are independent and identically distributed random variables. All of the above. None of the above.

makhdoom ghaya asked in Probability Nov 19, 2015

by makhdoom ghaya

1.2k views

2 votes

2 answers

23

TIFR CSE 2014 | Part A | Question: 18
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 $n=\sum _{i}n_{i}$ ... $\min_{i} |a_{i}|$ $\min_{i} \left(n_{i}|a_{i}|\right)$ $\max_{i} |a_{i}|$ None of the above

makhdoom ghaya asked in Calculus Nov 19, 2015

by makhdoom ghaya

813 views

15 votes

2 answers

24

TIFR CSE 2014 | Part A | Question: 17
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 $E(Y |X = 2)$. $6\frac{5}{6}$ $6$ $5\frac{1}{2}$ $6\frac{1}{3}$ $5\frac{2}{3}$

makhdoom ghaya asked in Probability Nov 19, 2015

by makhdoom ghaya

3.4k views

6 votes

2 answers

25

TIFR CSE 2014 | Part A | Question: 16
Let $x_{0}=1$ and $x_{n+1}= \frac{3+2x_{n}}{3+x_{n}}, n\geq 0$. $x_{\infty}=\displaystyle \lim_{n\rightarrow \infty}x_{n}$ 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

makhdoom ghaya asked in Calculus Nov 19, 2015

by makhdoom ghaya

1.3k views

3 votes

1 answer

26

TIFR CSE 2014 | Part A | Question: 15
Consider the following statements: $b_{1}= \sqrt{2}$, series with each $b_{i}= \sqrt{b_{i-1}+ \sqrt{2}}, i \geq 2$, converges. $\sum ^{\infty} _{i=1} \frac{\cos (i)}{i^{2}}$ converges. $\sum ^{\infty} _{i=0} b_{i}$ ... $(3)$ but not $(1)$. Statements $(1)$ and $(3)$ but not $(2)$. All the three statements. None of the three statements.

makhdoom ghaya asked in Calculus Nov 14, 2015

by makhdoom ghaya

509 views

5 votes

1 answer

27

TIFR CSE 2014 | Part A | Question: 14
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.

makhdoom ghaya asked in Combinatory Nov 14, 2015

by makhdoom ghaya

901 views

4 votes

1 answer

28

TIFR CSE 2014 | Part A | Question: 13
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 co-ordinate 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.

makhdoom ghaya asked in Quantitative Aptitude Nov 14, 2015

by makhdoom ghaya

1.0k views

2 votes

0 answers

29

TIFR CSE 2014 | Part A | Question: 12
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 ... $(2)$. The above inequality holds under all the three conditions. The above inequality holds under none of the three conditions.

makhdoom ghaya asked in Quantitative Aptitude Nov 14, 2015

by makhdoom ghaya

374 views

11 votes

3 answers

30

TIFR CSE 2014 | Part A | Question: 11
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? $51:49$ $1:1$ $49:51$ $51:98$ $98:51$

makhdoom ghaya asked in Quantitative Aptitude Nov 13, 2015

by makhdoom ghaya

1.2k views

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