# Recent questions tagged tifr2010

1
Which of the following statement is FALSE? All recursive sets are recursively enumerable. The complement of every recursively enumerable sets is recursively enumerable. Every Non-empty recursively enumerable set is the range of some totally recursive function. All finite sets are recursive. The complement of every recursive set is recursive.
2
Suppose a language $L$ is $\mathbf{NP}$ complete. Then which of the following is FALSE? $L \in \mathbf{NP}$ Every problem in $\mathbf{P}$ is polynomial time reducible to $L$. Every problem in $\mathbf{NP}$ is polynomial time reducible to $L$. The Hamilton cycle problem is polynomial time reducible to $L$. $\mathbf{P} \neq \mathbf{NP}$ and $L \in \mathbf{P}$.
3
Suppose three coins are lying on a table, two of them with heads facing up and one with tails facing up. One coin is chosen at random and flipped. What is the probability that after the flip the majority of the coins(i.e., at least two of them) will have heads facing up? ... $\left(\frac{1}{4}\right)$ $\left(\frac{1}{4}+\frac{1}{8}\right)$ $\left(\frac{2}{3}\right)$
4
Consider the program where $a, b$ are integers with $b > 0$. x:=a; y:=b; z:=0; while y > 0 do if odd (x) then z:= z + x; y:= y - 1; else y:= y % 2; x:= 2 * x; fi Invariant of the loop is a condition which is true before and after every ... will not terminate for some values of $a, b$ but when it does terminate, the condition $z = a * b$ will hold. The program will terminate with $z=a^{b}$
5
In a directed graph, every vertex has exactly seven edges coming in. What can one always say about the number of edges going out of its vertices? Exactly seven edges leave every vertex. Exactly seven edges leave some vertex. Some vertex has at least seven edges leaving it. The number of edges coming out of vertex is odd. None of the above.
6
Consider the following languages over the alphabet $\{0, 1\}$. $L1=\left \{ x.x^{R}\mid x\in \left \{ 0, 1 \right \}^* \right \}$ $L2=\left \{ x.x\mid x\in \left \{ 0, 1 \right \}^* \right \}$ Where $x^{R}$ ... . Both $L1$ and $L2$ are context free and neither is regular. $L1$ is context free but $L2$ is not context free. Both $L1$ and $L2$ are not context free.
7
Let $r, s, t$ be regular expressions. Which of the following identities is correct? $(r + s)^* = r^*s^*$ $r(s + t) = rs + t$ $(r + s)^* = r^* + s^*$ $(rs + r)^* r = r (sr + r)^*$ $(r^*s)^* = (rs)^*$
8
In a relational database there are three relations: Customers = C (C Name) Shops = S (S Name) Buys = B (C Name, S Name) Then the Relational Algebra expression ( $\Pi$ is the projection operator). $C-\Pi _{C Name}((C \times S)-B)$ returns the names of ... . Customers who buy from at least two shops. Customers who buy from all shops. Customers who do not buy buy anything at all. None of the above.
9
Consider the following solution (expressed in Dijkstra's guarded command notation) to the mutual exclusion problem. process P1 is begin loop Non_critical_section; while not (Turn=1) do skip od; Critical_section_1; Turn:=2; end loop end ∥ process P2 is begin loop Non_critical_section; while ... and (3), but does not satisfies the requirement (2). Satisfies all the requirement (1), (2), and (3).
10
Consider the following computation rules. Parallel-outermost rule: Replace all the outermost occurrences of F (i.e., all occurrences of F which do not occur as arguments of other F's) simultaneously. Parallel - innermost rule: Replace all the innermost occurrences of F (i.e.,all ... and $0$ respectively $0$ and $0$ respectively $w$ and $w$ respectively $w$ and $1$ respectively none of the above
11
Consider the following program for summing the entries of the array $b$: array $[0 .. N-1]$ of integers, where $N$ is a positive integer. (The symbol '$<>$' denotes 'not equal to'). var i, s: integer; Program i:= 0; s:= 0; [*] while i <> N do s := s + b[i]; i := i + 1; od Which of the ... $s = \sum\limits^{i-1}_{j=0}b[j] \;\&\; 0 \leq i \leq N$
12
Suppose you are given an array $A$ with $2n$ numbers. The numbers in odd positions are sorted in ascending order, that is, $A[1] \leq A[3] \leq \ldots \leq A[2n - 1]$. The numbers in even positions are sorted in descending order, that ... binary search on the entire array. Perform separate binary searches on the odd positions and the even positions. Search sequentially from the end of the array.
13
Consider the concurrent program: x: 1; cobegin x:= x + 3 || x := x + x + 2 coend Reading and writing of variables is atomic, but the evaluation of an expression is not atomic. The set of possible values of variable $x$ at the end of the execution of the program is: $\{4\}$ $\{8\}$ $\{4, 7, 10\}$ $\{4, 7, 8, 10\}$ $\{4, 7, 8\}$
14
Consider the Insertion Sort procedure given below, which sorts an array $L$ of size $n\left ( \geq 2 \right )$ in ascending order: begin for xindex:= 2 to n do x := L [xindex]; j:= xindex - 1; while j > 0 and L[j] > x do L[j + 1]:= L[j]; j:= j - ... insertion sort makes $n (n - 1) / 2$ comparisons. Insertion Sort makes $n (n - 1) / 2$ comparisons whenever all the elements of $L$ are not distinct.
15
Suppose there is a balanced binary search tree with $n$ nodes, where at each node, in addition to the key, we store the number of elements in the sub tree rooted at that node. Now, given two elements $a$ and $b$, such that $a < b$, we want to find the ... additions. $O(\log n)$ comparisons but a constant number of additions. $O(n)$ comparisons and $O(n)$ additions, using depth-first- search.
16
Which of the following problems is decidable? (Here, CFG means context free grammar and CFL means context free language.) Given a CFG $G$, find whether $L(G) = R$, where $R$ is regular set. Given a CFG $G$, find whether $L(G) = \{\}$. Find whether the intersection of two ... CFL. Find whether CFG $G_1$ and CFG $G_2$ generate the same language, i.e, $L\left (G_1 \right ) = L\left (G_2 \right)$.
17
Consider the following program operating on four variables $u, v, x, y$, and two constants $X$ and $Y$. x, y, u, v:= X, Y, Y, X; While (x ≠ y) do if (x > y) then x, v := x - y, v + u; else if (y > x) then y, u:= y - x, u + v; od; print ((x + ... $\frac1 2 \times \text{gcd}(X, Y)$ followed by $\frac1 2 \times \text{lcm}(X, Y)$. The program does none of the above.
18
Suppose you are given $n$ numbers and you sort them in descending order as follows: First find the maximum. Remove this element from the list and find the maximum of the remaining elements, remove this element, and so on, until all elements are exhausted. How many comparisons does this method require in ... $O\left (n \log n \right )$ Same as heap sort. $O\left ( n^{1.5} \right )$ but not better.
19
Let $L$ consist of all binary strings beginning with a $1$ such that its value when converted to decimal is divisible by $5$. Which of the following is true? $L$ can be recognized by a deterministic finite state automaton. $L$ can be recognized by a ... by a non-deterministic push-down automaton but not by a deterministic push-down automaton. $L$ cannot be recognized by any push-down automaton.
20
For $x \in \{0,1\}$, let $\lnot x$ denote the negation of $x$, that is $\lnot \, x = \begin{cases}1 & \mbox{iff } x = 0\\ 0 & \mbox{iff } x = 1\end{cases}$. If $x \in \{0,1\}^n$, then $\lnot \, x$ denotes the component wise negation of $x$ ... $g(x) = f(x) \land f(\lnot x)$ $g(x) = f(x) \lor f(\lnot x)$ $g(x) = \lnot f(\lnot x)$ None of the above.
21
How many integers from $1$ to $1000$ are divisible by $30$ but not by $16$? $29$ $31$ $32$ $33$ $25$
22
Karan tells truth with probability $\dfrac{1}{3}$ and lies with probability $\dfrac{2}{3}.$ Independently, Arjun tells truth with probability $\dfrac{3}{4}$ and lies with probability $\dfrac{1}{4}.$ Both watch a cricket match. Arjun tells you that India won, Karan tells you that India lost. What ... $\left(\dfrac{3}{4}\right)$ $\left(\dfrac{5}{6}\right)$ $\left(\dfrac{6}{7}\right)$
23
Let $X$ be a set of size $n$. How many pairs of sets (A, B) are there that satisfy the condition $A\subseteq B \subseteq X$ ? $2^{n+1}$ $2^{2n}$ $3^{n}$ $2^{n} + 1$ $3^{n + 1}$
24
Suppose there is a sphere with diameter at least $6$ inches. Through this sphere we drill a hole along a diameter. The part of the sphere lost in the process of drilling the hole looks like two caps joined to a cylinder, where the cylindrical part has length $6$ inches. It turns out that the ... part. $24\pi$ cu. inches $36\pi$ cu. inches $27\pi$ cu. inches $32\pi$ cu. inches $35\pi$ cu. inches
25
Let the characteristic equation of matrix $M$ be $\lambda ^{2} - \lambda - 1 = 0$. Then. $M^{-1}$ does not exist. $M^{-1}$ exists but cannot be determined from the data. $M^{-1} = M + I$ $M^{-1} = M - I$ $M^{-1}$ exists and can be determined from the data but the choices (c) and (d) are incorrect.
26
Let $A, B$ be sets. Let $\bar{A}$ denote the compliment of set $A$ (with respect to some fixed universe), and $( A - B)$ denote the set of elements in $A$ which are not in $B$. Set $(A - (A - B))$ is equal to: $B$ $A\cap \bar{B}$ $A - B$ $A\cap B$ $\bar{B}$
27
A marine biologist wanted to estimate the number of fish in a large lake. He threw a net and found $30$ fish in the net. He marked all these fish and released them into the lake. The next morning he again threw the net and this time caught $40$ fish, of which two were found to be marked. The (approximate) number of fish in the lake is: $600$ $1200$ $68$ $800$ $120$
A cube whose faces are colored is split into $1000$ small cubes of equal size. The cubes thus obtained are mixed thoroughly. The probability that a cube drawn at random will have exactly two colored faces is: $0.096$ $0.12$ $0.104$ $0.24$ None of the above
The coefficient of $x^{3}$ in the expansion of $(1 + x)^{3} (2 + x^{2})^{10}$ is. $2^{14}$ $31$ $\left ( \frac{3}{3} \right ) + \left ( \frac{10}{1} \right )$ $\left ( \frac{3}{3} \right ) + 2\left ( \frac{10}{1} \right )$ $\left ( \frac{3}{3} \right ) \left ( \frac{10}{1} \right ) 2^{9}$
The length of a vector $x = (x_{1},\ldots,x_{n})$ is defined as $\left \| x\right \| = \sqrt{\sum ^{n}_{i=1}x^{2}_{i}}$. Given two vectors $x=(x_{1},\ldots, x_{n})$ and $y=(y_{1},\ldots, y_{n})$, which of the following measures of discrepancy between $x$ and $y$ is ... $\left \| \frac{X}{\left \| X \right \|}-\frac{Y}{\left \| Y \right \|} \right \|$ None of the above.