# Answers by Sukanya Das

1
Consider the following grammar: $S \rightarrow S$ $S \rightarrow SS \mid a \mid \epsilon$ Construct the collection of sets of LR (0) items for this grammar and draw its goto graph.
2
Which of the following functions given by their recurrences grows the fastest asymptotically? $T(n) = 8T(n/4) + 100n^2$ $T(n) = 81T(n/9) + 10n^2$ $T(n) = 16T(n/4)+ 100(n \log n)^{1.99}$ $T(n) = 100T(n/100)+ n \log^2 n$
3
Consider a well functioning clock where the hour, minute and the seconds needles are exactly at zero. How much time later will the minutes needle be exactly one minute ahead ($1/60$ th of the circumference) of the hours needle and the seconds needle again exactly at zero? ... an integer multiple of $1/60$ th of the circumference. $144$ minutes $66$ minutes $96$ minutes $72$ minutes $132$ minutes
1 vote
4
Toss three fair coins. What is the probability of exactly one Heads$(H)$ ?
5
If $a,b,c$ and $d$ satisfy the equations $a+7b+3c+5d =16$ $8a+4b+6c+2d = -16$ $2a+6b+4c+8d = 16$ $5a+3b+7c+d= -16$ Then $(a+d)(b+c)$ equals $-4$ $0$ $16$ $-16$
1 vote
6
How many complex numbers $z$ are there such that $\mid z+1 \mid = \mid z+i \mid$ and $\mid z \mid = 5$ ? $0$ $1$ $2$ $3$
1 vote
7
5. Consider the graph given below : Use Kruskal’s algorithm to find a minimal spanning tree for the graph. The List of the edges of the tree in the order in which they are choosen is ? (1) AD, AE, AG, GC, GB, BF (2) GC, GB, BF, GA, AD, AE (3) GC, AD, GB, GA, BF, AE (4) AD, AG, GC, AE, GB, BF
1 vote
8
Two trains start at the same time from Mumbai and Pune and proceed towards each other at the rate of $60\hspace{0.1cm} km$ and $40\hspace{0.1cm}km$ $\text{per hour}$ respectively. When they meet, it is found that one train has travelled $20\hspace{0.1cm} km$ more than the other. Find the distance between Mumbai and Pune.
9
how many minimum number of state is required to construct a DFA of regular expression (a*+a*b*).
10
Suppose the rank of the matrix $\begin{pmatrix}1&1&2&2\\1&1&1&3\\a&b&b&1\end{pmatrix}$ is $2$ for some real numbers $a$ and $b$. Then $b$ equals $1$ $3$ $1/2$ $1/3$
11
If $A$ is a $2 \times 2$ matrix such that trace $A = det \ A = 3,$ then what is the trace of $A^{-1}$? $1$ $\left(\dfrac{1}{3}\right)$ $\left(\dfrac{1}{6}\right)$ $\left(\dfrac{1}{2}\right)$
12
In a class of $80$ students, $40$ are girls and $40$ are boys. Also, exactly $50$ students wear glasses. Then the set of all possible number of boys without glasses is $\{0,.....,30\}$ $\{10,....,30\}$ $\{0,.....,40\}$ $\text{none of these}$
13
The number of polynomial functions $f$ of degree $\geq1$ satisfying $f(x^2) = (f(x))^2 = f(f(x))$ for all real $x$, is $0$ $1$ $2$ $\text{infinitely many}$
14
There are four machines and it is known that exactly two of them are faulty. They are tested one by one in a random order till both the faulty machines are identified. The probability that only two tests are required is $\left(\dfrac{1}{2}\right)$ $\left(\dfrac{1}{3}\right)$ $\left(\dfrac{1}{4}\right)$ $\left(\dfrac{1}{6}\right)$
15
Which of the following statements is true? There are three consecutive integers with sum $2015$ There are four consecutive integers with sum $2015$ There are five consecutive integers with sum $2015$ There are three consecutive integers with product $2015$
16
A Shopkeeper cheats to the extends of $20\%$ while buying as well as selling.By using false weight.His gain or loss is? Loss $50\%$ Gain $44\%$ Loss $44\%$ Gain $40\%$
17
If $A$ and $B$ run at $6km/hr$ and $12 km/hr$ on a circular track $6 km$ long.When will they meet for the first time if they are running in opposite direction?
18
Two trains separated by $480 \hspace{0.1cm}km$ are approaching each other with speed $70\hspace{0.1cm}kmph$ and $50\hspace{0.1cm}kmph$ respectively. A bird with speed $100\hspace{0.1cm} km/hr$ started from the front of first train goes to the front of second train and comes ... distance covered by the bird!? $500\hspace{0.1cm}km$ $400\hspace{0.1cm}km$ $480\hspace{0.1cm}km$ $600\hspace{0.1cm}km$
1 vote
19
In a $200m$ race, $A$ beats $B$ by $20m$.$B$ beates $C$ by $10m$ in a $250m$ race.By how many meters will $A$ beat $C$ in a $1 km$ race?
20
The maximum value of $\theta$ until which the approximation $\sin\theta \approx \theta$ holds to within $10\%$ error is $10^{\circ}$ $18^{\circ}$ $50^{\circ}$ $90^{\circ}$
21
Encode / Decode data for physical transmission is done by which layer of OSI reference model?
22
Consider the following regular expression: $a^*b^*b(a+(ab)^*)^*b^*$ $a^*(ab+ba)^*b^*$ What is the length of shortest string which is in both (i) and (ii) $2$ $3$ $4$ $none$
23
If events $B$ and $C$ are dependent on event $A$ and $P(A \hspace{0.1cm}and\hspace{0.1cm} B) = 0.30$, $P(A\hspace{0.1cm} and\hspace{0.1cm} C) = 0.20$ and the dependent events $B$ and $C$ are mutually exclusive and collectively exhaustive, then $P(C/A)$ is equal to ________ ?
24
$\text{If Direct Broadcast Address of subnet is}$ $201.15.16.31$. $\text{Which of the following is subnet mask?}$ $255.255.255.240$ $255.255.255.192$ $255.255.255.198$ $\text{None Of the Above}$
1 vote
25
A club with $x$ members is organized into four committees such that, each member is in exactly two committees, any two committees have exactly one member in common. Then $x$ has exactly two values both between $4$ and $8$ exactly one value and this lies between $4$ and $8$ exactly two values both between $8$ and $16$ exactly one value and this lies between $8$ and $16$
How many strings with seven or more characters can be formed from the letters of the word $\text{EVERGREEN}$ ?
One Hundred tickets, numbered $1,2,3,...,100$, are sold $100$ different people for a drawing. Four different prizes are awarded, including a grand prize(a trip to Tahiti).How many ways are there to award the prizes if the people holding tickets $19$ and $47$ both win prizes? the people holding tickets $19,47,$ and $73$ all win prizes?
A project requires $40$ hours from each of three employees to complete. Each employee is paid $Rs.25$ per hour. If the company hires a contractor for $Rs.10$ per hour, and the four divide the total amount of work equally, how much would is cost to complete the audit?