# Recent questions and answers in Others

1
What are the last two digits of $7^{2021}$? $67$ $07$ $27$ $01$ $77$
2
A schema describes : data elements records and files record relationship all of the above
1 vote
3
Consider the following greedy algorithm for colouring an $n$-vertex undirected graph $G$ with colours $c_{1}, c_{2}, \dots:$ consider the vertices of $G$ in any sequence and assign the chosen vertex the first colour that has not already been assigned to any of its neighbours. Let $m(n, r)$ be the minimum ... $m\left ( n, r \right ) = nr$ $m\left ( n, r \right ) = n\binom{r}{2}$
4
Let $A$ be a $3 \times 6$ matrix with real-valued entries. Matrix $A$ has rank $3$. We construct a graph with $6$ vertices where each vertex represents distinct column in $A$, and there is an edge between two vertices if the two columns represented by the vertices are ... The graph is connected. There is a clique of size $3$. The graph has a cycle of length $4$. The graph is $3$-colourable.
1 vote
5
Let $G$ be an undirected graph. For any two vertices $u, v$ in $G$, let $\textrm{cut} (u, v)$ be the minimum number of edges that should be deleted from $G$ so that there is no path between $u$ and $v$ in the resulting graph. Let $a, b, c, d$ be $4$ vertices in $G$. Which of the following ... $\textrm{cut} (c,d) = 2$ $\textrm{cut} (b,d) = 2$, $\textrm{cut} (b,c) = 2$ and $\textrm{cut} (c,d) = 1$
6
Let $G$ be a connected bipartite simple graph (i.e., no parallel edges) with distinct edge weights. Which of the following statements on $\text{MST}$ (minimum spanning tree) need $\text{NOT}$ be true? $G$ hasa unique $\text{MST}$. Every $\text{MST}$ in $G$ ... second lightest edge. Every $\text{MST}$ in $G$ contains the third lightest edge. No $\text{MST}$ in $G$ contains the heaviest edge.
7
Consider the following statements about propositional formulas. $\left ( p\wedge q \right )\rightarrow r$ and $\left ( p \rightarrow r \right )\wedge \left ( q\rightarrow r \right )$ are $\textit{not }$ ... Depending on the values $p$ and $q$, $\text{(i)}$ can be either true or false, while $\text{(ii)}$ is always false.
1 vote
8
Five married couples attended a party. In the party, each person shook hands with those they did not know. Everyone knows his or her spouse. At the end of the party, Shyamal, one of the attendees, listed the number of hands that other attendees including his spouse shook. He ... $8$ once in the list. How many persons shook hands with Shyamal at the party? $2$ $4$ $6$ $8$ Insufficient information
9
Fix $n\geq 6$. Consider the set $\mathcal{C}$ of binary strings $x_{1}, x_{2} \dots x_{n}$ of length $n$ such that the bits satisfy the following set of equalities, all modulo $2$: $x_{i}+x_{i+1}+x_{i+2}=0$ for all $1\leq i\leq n-2, x_{n-1}+x_{n}+x_{1}=0,$ ... $3$ $\left | \mathcal{C} \right |=4$. If $n\geq 6$ is not divisible by $3$ then $\left | \mathcal{C} \right |=1$.
10
A matching in a graph is a set of edges such that no two edges in the set share a common vertex. Let $G$ be a graph on $n$ $\textit{vertices}$ in which there is a subset $M$ of $m$ $\textit{edges}$ which is a matching. Consider a random process where each vertex in the graph is independently selected ... $p^{2m}$ $\left ( 1-p^{2} \right )^{m}$ $1-\left ( 1-p\left ( 1-p \right ) \right )^{m}$
11
What is the area of a rectangle with the largest perimeter that can be inscribed in the unit circle (i.e., all the vertices of the rectangle are on the circle with radius $1$)? $1$ $2$ $3$ $4$ $5$
12
Consider the following regular expressions over alphabet$\{a,b\}$, where the notation $(a+b)^+$ means $(a+b)(a+b)^*$: $r_1=(a+b)^+a(a+b)^*$ $r_2=(a+b)^*b(a+b)^+$ Let $L_1$ and $L_2$ be the languages defined by $r_1$ and $r_2$, respectively. Which of the following regular expressions define $L_1\cap L_2$ ... $(a+b)^*a\;b(a+b)^*$ $(a+b)^*b(a+b)^*a(a+b)^*$ $(a+b)^*a(a+b)^*b(a+b)^*$
13
We have a procedure $P(n)$ that makes multiple calls to a procedure $Q(m)$, and runs in polynomial time in $n$. Unfortunately, a significant flaw was discovered in $Q(m)$, and it had to be replaced by $R(m)$, which runs in exponential time in $m$. Thankfully, $P$ is still correct ... $log\;n.$ $P(n)$ runs in polynomial time in $n$ if, for each call $Q(m),m \underline<log \;n.$
14
consider the following directed graph,find the number of paths from A to I
15
Which of the following regular expressions defines a language that is different from the other choices? $b^{\ast }\left ( a+b \right )^\ast a\left ( a+b \right )^ \ast ab^\ast \left ( a+b \right )^{\ast }$ ... $\left ( a+b \right )^{\ast }b^{\ast }a \left ( a+b\right )^{\ast }b^{\ast }\left ( a+b \right )^{\ast }$
16
Let $L$ be a context-free language generated by the context-free grammar $G = (V, \Sigma, R, S)$ where $V$ is the finite set of variables, $\Sigma$ the finite set of terminals (disjoints from $V$), $R$ the finite set of rules and $S \in V$ the start variable. Consider the context-free grammar ${G}'$ ... ${L}'=LL$ ${L}'=L$ ${L}'=L^{\ast }$ ${L}'=\left \{ xx \mid x \in L \right \}$ None of the above
17
For a language $L$ over the alphabet $\{a, b\}$, let $\overline{L}$ denote the complement of $L$ and let $L^{\ast}$ denote the Kleene-closure of $L$. Consider the following sentences. $\overline{L}$ and $L^{\ast}$ are both context-free. $\overline{L}$ ... ? Both (i) and (iii) Only (i) Only (iii) Only (ii) None of the above
18
Consider the following pseudocode: procedure HowManyDash(n) if n=0 then print '-' else if n=1 then print '-' else HowManyDash(n-1) HowManyDash(n-2) end if end procedure How many ‘-’ does HowManyDash$(10)$ print? $9$ $10$ $55$ $89$ $1024$
19
What is the prefix expression corresponding to the expression: $\left ( \left ( 9+8 \right ) \ast 7+\left ( 6\ast \left ( 5+4 \right ) \right )\ast 3\right )+2?$ You may assume that $\ast$ has precedence over $+$? $\ast + +\: 987 \ast \ast \: 6 + + \:5432$ ... $+ + \ast +\: 987 \ast \ast \: 6 + \:5432$ $+ \ast + \ast \: 987+ + \: 6 \ast \:5432$
20
A box contains $5$ red marbles, $8$ green marbles, $11$ blue marbles, and $15$ yellow marbles. We draw marbles uniformly at random without replacement from the box. What is the minimum number of marbles to be drawn to ensure that out of the marbles drawn, at least $7$ are of the same colour? $7$ $8$ $23$ $24$ $39$
21
Let $L$ be a singly-linked list $X$ and $Y$ be additional pointer variables such that $X$ points to the first element of $L$ and $Y$ points to the last element of $L$. Which of the following operations cannot be done in time that is bound above by a constant? ... Add an element after the last element of $L$. Add an element before the first element of $L$. Interchange the first two elements of $L$.
22
How many numbers in the range ${0, 1, \dots , 1365}$ have exactly four $1$'s in their binary representation? (Hint: $1365_{10}$ is $10101010101_{2}$, that is, $1365=2^{10} + 2^{8}+2^{6}+2^{4}+2^{2}+2^{0}.)$ In the following, the binomial coefficient $\binom{n}{k}$ counts the number of ... $\binom{11}{4}+\binom{9}{3}+\binom{7}{2}+\binom{5}{1}$ $1024$
1 vote
23
Find the following sum. $\frac{1}{2^{2}-1}+\frac{1}{4^{2}-1}+\frac{1}{6^{2}-1}+\cdots+\frac{1}{40^{2}-1}$ $\frac{20}{41}$ $\frac{10}{41}$ $\frac{10}{21}$ $\frac{20}{21}$ $1$
24
Lavanya and Ketak each flip a fair coin (i.e., both heads and tails have equal probability of appearing) $n$ times. What is the probability that Lavanya sees more heads than ketak? In the following, the binomial coefficient $\binom{n}{k}$ counts the number of $k$-element subsets of an $n$ ... $\sum_{i=0}^{n}\frac{\binom{n}{i}}{2^{n}}$
25
Let $d$ be the positive square integers (that is, it is a square of some integer) that are factors of $20^{5} \times 21^{5}$. Which of the following is true about $d$? $50\leq d< 100$ $100\leq d< 150$ $150\leq d< 200$ $200\leq d< 300$ $300\leq d$
26
Let $n, m$ and $k$ be three positive integers such that $n \geq m \geq k$. Let $S$ be a subset of $\left \{ 1, 2,\dots, n \right \}$ of size $k$. Consider sampling a function $f$ uniformly at random from the set of all functions mapping $\left \{ 1,\dots, n \right \}$ ... $1-\frac{k!\binom{m}{k}}{m^{k}}$ $1-\frac{k!\binom{n}{k}}{n^{k}}$ $1-\frac{k!\binom{n}{k}}{m^{k}}$
27
What is the probability that at least two out of four people have their birthdays in the same month, assuming their birthdays are uniformly distributed over the twelve months? $\frac{25}{48}$ $\frac{5}{8}$ $\frac{5}{12}$ $\frac{41}{96}$ $\frac{55}{96}$
1 vote
28
Let $M$ be an $n \times m$ real matrix. Consider the following: Let $k_{1}$ be the smallest number such that $M$ can be factorized as $A \cdot B$, where $A$ is an $n \times k_{1}$ and $B$ is a $k_{1} \times m$ matrix. Let $k_{2}$ ... $k_{2}= k_{3}< k_{1}$ $k_{1}= k_{2}= k_{3}$ No general relationship exists among $k_{1}, k_{2}$ and $k_{3}$
29
Consider the sequence $y_{n}=\frac{1}{\int_{1}^{n}\frac{1}{\left ( 1+x/n \right )^{3}}dx}$ for $\text{n} = 2,3,4, \dots$ Which of the following is $\text{TRUE}$? The sequence $\{y_{n}\}$ does not have a limit as $n\rightarrow \infty$. $y_{n}\leq 1$ for all ... $0$. The sequence $\{y_{n}\}$ first increases and then decreases as $\text{n}$ takes values $2, 3, 4, \dots$
30
Let $P$ be a convex polygon with sides $5, 4, 4, 3$. For example, the following: Consider the shape in the plane that consists of all points within distance $1$ from some point in $P$. If $\ell$ is the perimeter of the shape, which of the following is always correct? $\ell$ cannot be determined from the given information. $20\leq \ell < 21$ $21\leq \ell< 22$ $22\leq \ell< 23$ $23\leq \ell< 24$
31
Consider the following two languages. $\begin{array}{rcl} \text{PRIME} & = & \{ 1^{n} \mid n \text{ is a prime number} \}, \\ \text{FACTOR} & = & \{ 1^{n}0 1^{a} 01^{b} \mid n \text{ has a factor in the range }[a,b] \} \end{array}$. What can you say ... $\text{P}$. None of the above since we can answer this question only if we resolve the status of the $\text{NP}$ vs. $\text{P}$ question.
32
Let $A$ and $B$ be two matrices of size $n \times n$ and with real-valued entries. Consider the following statements. If $AB = B$, the $A$ must be the identity matrix. If $A$ is an idempotent (i.e. $A^{2} = A$) nonsingular matrix, then $A$ must be the identity matrix. If $A^{-1} = A$, the ... $\text{MUST}$ be true of $A$? $1, 2$ and $3$ Only $2$ and $3$ Only $1$ and $2$ Only $1$ and $3$ Only $2$
33
Suppose we toss a fair coin (i.e., both beads and tails have equal probability of appearing) repeatedly until the first time by which at least $\textit{two}$ heads and at least $\textit{two}$ tails have appeared in the sequence of tosses made. What is the expected number of coin tosses that we would have to make? $8$ $4$ $5.5$ $7.5$ $4.5$
34
Let $A\left [ i \right ] : i=0, 1, 2, \dots , n-1$ be an array of $n$ distinct integers. We wish to sort $A$ in ascending order. We are given that each element in the array is at a position that is at most $k$ ... $t\left ( n \right ) = \Theta \left ( nk \right )$
35
A $n\times n$ matrix $A$ is said to be $symmetric$ if $A^T=A$. Suppose $A$ is an arbitrary $2\times 2$ matrix. Then which of the following matrices are symmetric (here $0$ denotes the $2\times 2$ matrix consisting of zeros): $A^TA$ $\begin{bmatrix} 0&A^T \\ A & 0 \end{bmatrix}$ $AA^T$ $\begin{bmatrix} A & 0 \\ 0 & A^T \end{bmatrix}$
36
In an entrance examination with multiple choice questions, with each question having four options and a single correct answer, suppose that only $20\%$ candidates think they know the answer to one difficult question and only half of them know it correctly and the ... tick the same. If a candidate has correctly answered the question, what is the (conditional) probability that she knew the answer?
37
Consider the following set of $3n$ linear equations in $3n$ variables: $\begin{matrix} x_1-x_2=0 & x_4-x_5 =0 & \dots & x_{3n-2}-x_{3n-1}=0 \\ x_2-x_3=0 & x_5-x_6=0 & & x_{3n-1}-x_{3n}=0 \\ x_1-x_3=0 & x_4-x_6 =0 & & x_{3n-2}=x_{3n}=0 \end{matrix}$ ... $S$ is a subspace of $\mathbb{R}^{3n}$ of dimension n $S$ is a subspace of $\mathbb{R}^{3n}$ of dimension $n-1$ $S$ has exactly $n$ elements
Let $x=\begin{bmatrix} 3& 1 & 2 \end{bmatrix}$. Which of the following statements are true? $x^Tx$ is a $3\times 3$ matrix $xx^T$ is a $3\times 3$ matrix $xx^T$ is a $1\times 1$ matrix $xx^T=x^Tx$
Let $A=\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}, B=\begin{bmatrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{bmatrix}, C=\begin{bmatrix} 4 & 5 & 6 \\ 1 & 2 & 3 \\ 7 & 8 & 9 \end{bmatrix}$ and $D=\begin{bmatrix} -1 & 2 & 3 \\ 4 & -5 & 6 \\ 7 & 8 & -9 \end{bmatrix}$. Which of the following statements are true? $|A|=|B|$ $|C|=|D|$ $|B|=-|C|$ $|A|=-|D|$
Let $A=\begin{bmatrix} 1& 1& 1\\0&2&2\\0&0&3 \end{bmatrix}, B=\begin{bmatrix} 5&5&5\\0&10&10\\0&0&15\end{bmatrix}, C=\begin{bmatrix} 3&0&0\\3&6&0\\3&6&9 \end{bmatrix}$. Which of the following statements are true? $|A|=|B|$ $|B|=125|A|$ $|C|=27|A|$ $|C|=\frac{|A|}{3}$