# Recent questions tagged tifr2014

1
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 the number ... $273$. What is the score of the best player for this game? $40$ $16$ $33$ $91$ $123$
2
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 minimum label in its subtree)? ... $\frac{2}{13}$ $\frac{1}{2^{13}}$
3
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 $y_{i} 's$ ... $2^{\left(\frac{k}{3}\right)}$ $\left(\frac{k}{3}\right)$ $\left(\frac{k}{3}\right)+1$ $4 \left(\frac{k}{3}\right)$
4
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.
5
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 is called a complete ... $|$. $\mid$ is a total order. $(N, \mid)$ is a complete lattice. $(N, \mid)$ is a lattice but not a complete lattice.
6
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. $\leq_{p}$ ... bound. Every non-empty subset of $N^{*}$ has a greatest lower bound. Every non-empty finite subset of $N^{*}$ has a least upper bound.
7
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 be recognized by a ... both recursively enumerable then it is recursive. Complement of a non-recursive language can never be recognized by any Turing machine.
8
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 $L_{1}$ and $L_{2}$ are regular. ... 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.
9
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 correct true/ ... of the above? true, false, true. false, false, true. true, false, true. false, false, false. true, true, true.
10
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 $O(n \log n)$ ... $T(n)$ is $O(n \log n)$ when $k=4$. $T(n)$ is $O(n \log n)$ when $k=5$. $T(n)$ is $O(n)$ when $k=5$.
11
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. $O\left(\log^{100}n\right)$ ... $2(n - 1)$ comparisons. None of the above.
12
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.
13
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.
14
Which of the following statements is TRUE for all sufficiently large $n$? $\displaystyle \left(\log n\right)^{\log\log n} < 2^{\sqrt{\log n}} < n^{1/4}$ $\displaystyle 2^{\sqrt{\log n}} < n^{1/4} < \left(\log n\right)^{\log\log n}$ ... $\displaystyle 2^{\sqrt{\log n}} < \left(\log n\right)^{\log\log n} < n^{1/4}$
15
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 permutation. ... number of times MIN is updated? $O (1)$ $H_{n}=\sum ^{n}_{i=1} 1/i$ $\sqrt{n}$ $n/2$ $n$
16
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 $w(e_{1}) < w(e_{2}) < · · · < w(e_{m})$ ... we replace 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.
17
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$(b, c) \geq 4$.
18
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 cross edges is $C$ ... $T = 3$. $B = 1$, $C = 2$, and $T = 3$. $B = 2$, $C = 2$, and $T = 1$.
19
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 computes their bit wise $AND$. ... 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.
20
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 subtree, visit root) traversals ... How many leaves does the tree have? THREE. FOUR. FIVE. SIX. Cannot be determined uniquely from the given information.
21
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
22
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.
23
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.
24
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 $E(Y |X = 2)$. $6\frac{5}{6}$ $6$ $5\frac{1}{2}$ $6\frac{1}{3}$ $5\frac{2}{3}$
25
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.
26
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. $\sum ^{\infty} _{i=0} b_{i}$ ... Statements $(2)$ and $(3)$ but not $(1)$. Statements $(1)$ and $(3)$ but not $(2)$. All the three statements. None of the three statements.
27
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.
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.
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: $\lambda= 0.5$ ... $(3)$ but not under condition $(2)$. The above inequality holds under all the three conditions. The above inequality holds under none of the three conditions.
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? $51:49$ $1:1$ $49:51$ $51:98$ $98:51$