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1

IITB Practice Set: 1
Given four distinct points $A, B, C,$ and $D$ on the directed ray denoting the positive $x$-axis, the cross ratio of $A,B$ with respect to $C,D$ denoted by $(A,B;C,D)$ is defined to be $\frac{AC}{BC}/\frac{AD}{BD}$. Here, $AC$, for example, ... A colleague pretends that this ray of slope $1$ is the $x$-axis and computes the cross-ratio as $p$. The value of of $r/p$ is ______.

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2

IITB Practice Set: 2
Consider the following in $2D:$ Two lines $AB$ and $CD$ intersect at a point $K$, such that the angle between them is $t$ degrees. A triangle $\Delta PQR$ is first reflected about line $AB$, then about line $CD$. After the reflections, the ... $2\times 2$ rotation transformation $T_R$. $T_R$ rotates the triangle by an angle _____ degrees, about the point _____.

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3

IITB Practice Set: 3
Consider the real-valued function $f(x, y):= x + y$ defined on points $(x, y)$ in a $2-$dimensional Euclidean plane. Suppose you want to find all possible $(x^{\ast}, y^{\ast})$ that maximize the value of the function $f\left(\cdot,\cdot\right)$ ... $f(\cdot,\cdot)$ at all possible solutions $(x', y'):$______

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4

IITB Practice Set: 4
Kamala tosses $11$ fair coins and Rajani tosses $10$ fair coins. The probability that the number of heads obtained by Kamala is more than the number of tails obtained by Rajani is _____.

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5

IITB Practice Set: 5
Let matrix $X = \begin{bmatrix}x_1 & x_2 \end{bmatrix}, \; \Sigma = \begin{bmatrix}a_{1,1} & a_{1,2} a_{2,1} & a_{2,2}\end{bmatrix}$. Assume none of the elements are zero in $X,\Sigma$. Let $X^T$ denote the transpose of vector $X.$ Select all ... $X^T \Sigma X$. It is non-singular. None of the above are true.

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6

IITB Practice Set: 6
Consider a binary classification problem $(y \in \{0,1\})$ and two dimensional data where both dimensions $x_1$ and $x_2$ are binary. Let $n^y_{00},n^y_{10},n^y_{01},n^y_{11}$ denote the number of training instances in class $y$ where $[x_1 \; x_2]$ ... $\dfrac{n^1_{00}+n^1_{01}}{n^1_{00}+n^1_{10}+n^1_{01}+n^1_{11}}$

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7

IITB Practice Set: 7
This question is about the behaviour of lexical analysers generated by Lex. Consider the following Lex-like specification of two tokens: %% abc printf("Token 1"); (abc)*d printf("Token 2"); %% The time taken for the complete tokenization (extraction of all tokens) of the input string ... alternatives: $(a) O(1), (b) O(m), (c) O(m^2) \text{ and } (d) O(m^3).$

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8

IITB Practice Set: 8
Consider the grammar: $S \rightarrow aSB$ $S \rightarrow d$ $B \rightarrow b$ During a bottom-up parsing of the sentence $aaadbbb$, consider all the states of the parser in which the contents of the stack, read as a string from the bottom of the stack to the ... for $\textit{shift symbol x,} \text{ or } \textit{r n}$, standing for $\textit{reduce using production number n}$.

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9

IITB Practice Set: 9
Consider expression trees made up of variables and the operator $+$ ... $4$ registers to evaluate is _______.

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10

IITB Practice Set: 10
Let $\Sigma = \{0,1\}$ and $L$ be the language given by the regular expression $(01)^\ast$. We define the following equivalence relation $\sim_L$ on $\Sigma ^{\ast}$: given two words $x, y,$ we say $x \sim_L$ y iff $\text{for all } z \in \Sigma^{\ast}: xz \in L \text{ iff } yz \in L.$ The number of equivalence classes for $\sim_L$ is ______.

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11

IITB Practice Set: 11
Let $F$ be a propositional formula. Let $G(x)$ be a propositional formula containing propositional variable $x.$ For some propositional formula $H,$ let $G(H)$ be the formula obtained by replacing all occurrences of $x$ by $H$ ... $F \Rightarrow G(F) \equiv F \Rightarrow G\text{(False)}$

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12

IITB Practice Set: 12
Which of the following identities are true for arbitrary regular languages $R$ and $S$. Choose ALL correct answers. $(R^\ast S^\ast)^{\ast}= (R+S)^{\ast}$ $(RS)^{\ast}R = R(SR)^{\ast}$ $(R+RS)^{\ast} = \left(R^{\ast} +RS^{\ast}\right)^{\ast}$ $(R+S)^{\ast} S = (R^{\ast} S)^{\ast}$

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13

IITB Practice Set: 13
A $\textbf{CFG G}$ satisfies the following properties: The number of non-terminals (variables) in $G$ is $3$. The maximum number of symbols in the right hand side of any production rule is $2$. There is a word $w \in L(G)$ ... Turing machine. $L(G)$ can be accepted (by empty-stack acceptance condition) by a Nondeterministic pushdown automaton which has only one state.

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14

IITB Practice Set: 14
Suppose the ground is the $(x, y)$ plane and the sun is located at infinity in the direction of the vector $(1,2,1)$. When a flying bird is at the point $(1,1,1)$, its shadow on the ground will fall on the coordinates $(x, y)=$ ______.

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15

IITB Practice Set: 15
Suppose G is a $\textit{group}$ formed by all $2\times 2$ matrices with the group operation being matrix addition. Which all of the following are subgroups of $G$? The set of all $2\times 2$ invertible matrices. The set of all $2\times 2$ symmetric ... $0$. The set of all $2\times2$ matrices with the top right entry being $1$.

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16

IITB Practice Set: 16
A $\textit{proper edge coloring}$ of a graph is an assignment of colors to the edges so that any two edges sharing a common vertex should get different colors. Let $G$ be a graph with $20$ vertices that has a proper edge coloring with $6$ colors. The maximum number of edges in $G$ can be ______.

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17

IITB Practice Set: 17
Let $v$ be a column vector with $n > 1$ elements of which at least one is non-zero. Consider the matrix $B = vv^t$ where $v^t$ refers to the transpose of $v$. Which of the following is true about matrix $B$? $B$ has two ... one eigenvalue which is zero and another which is strictly negative. $B$ has one eigenvalue which is zero and another which may be positive or negative.

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18

IITB Practice Set: 18
In a certain town, there exist $100$ taxis out of which $1$ is red and $99$ are blue. A person $ABC$ observes a serious accident caused by a taxi at night and remembers that the taxi was red in color. It turns out that $ABC$ sees red objects ... The probability (correct to three decimal places) that the taxi was really a red one, when $ABC$ observed it to be red is _________.

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19

IITB Practice Set: 19
Consider a $4\times 4$ matrix $A$ with eigenvectors $u,v,w,x$ corresponding to distinct eigenvalues $\lambda{_1},\lambda{_2},\lambda{_3},\lambda{_4}$ where $1 = \lambda{_1} > \lambda{_2} > \lambda{_3} > \lambda{_4} > 0$. Consider a vector $z = 2u + v - w + 3x$. Then, $\lim_{k\rightarrow \infty} A^kz = $_____.

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20

IITB Practice Set: 20
Suppose the attenuation of a $10\; MHz$ signal on a $\text{CAT5}$ twisted pair cable is $20\; dB$ per $300$ metres. Suppose a $10 \;MHz$ signal of amplitude $1$ volt is input to one end of a $\text{CAT5}$ twisted pair cable of length $6,000$ ... amplitude of the signal at the other end of the cable? Express your answer as $10^X$. Specify what $X$ is in the given blank. ______

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21

IITB Practice Set: 21
Consider a $TCP$ connection which starts with initial congestion window equal to $2\ast \text{MSS}$ where the maximum segment size $\text{(MSS)}$ is $1460$ bytes. Suppose that initially two packets are sent out, each of size $\text{MSS}$ ... . Assuming that $\text{TCP}$ is still in slow start, the congestion window after receiving the second $\text{ACK}$ is: _____

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22

IITB Practice Set: 22
Suppose the stop and wait protocol is employed over an asymmetric link $A$ to $B$. The $A$ to $B$ link bandwidth is $8\text{ Mbps}$ with a propagation delay of $20\;ms$, however the $B$ to $A$ link bandwidth is $800\;\text{Kbps}$ ... other delays. Express answer in kbps. ($1\text{ Mbps} = 10^6 \text {bps and } 1\text{ Kbps} = 10^3 \text{bps})$

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23

IITB Practice Set: 23
A router can reach four different organizations via the same interface. The IP prefixes used by the different organizations are $108.25.224.0/21, 108.25.232.0/21, 108.25.240.0/21$ and $108.25.248.0/21 $. What $\textbf{IP}$ prefix can the router aggregate the ... $224$ is $11100000, 232 \text{ is } 11101000, 240 \text{ is } 11110000 \text{ and } 248 \text{ is } 11111000.$

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24

IITB Practice Set: 24
The test-and-set hardware instruction atomically sets a value and returns the old value stored at a memory location. The $C$-style pseudo-code of this atomic instruction is as follows: int test-and-set(int* ptr, int newvalue) { int old = *ptr; *ptr = newvalue; return old; } The ... ; *lk = 1; while(test-and-set(&lk, 1)); *lk = 0; test-and-set(&lk, 0); *lk = 1;

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25

IITB Practice Set: 25
Assume the following setup on a machine — latency of single memory access is $45\;ns$, a two-level page table is used for memory translations, and $\textbf{TLB}$ lookup latency is $5\;ns$. Assuming no page faults, what should be the $\textbf{TLB}$ hit rate to achieve an average memory access latency of $95\;ns$ ?

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26

IITB Practice Set: 26
Consider a filesystem in which an inode of a file contains pointers to $12$ direct blocks, one single indirect block, and one double indirect block. The block size in the system is $64$ ... numbers (pointers) and no other metadata. What is the maximum size of a file (in blocks) that can be stored in this filesystem?

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27

IITB Practice Set: 27
Consider a program using the fork and exec system calls shown below. Assume that the code shown below executes correctly, and that the fork and exec system calls succeed. The exec system call (invoked via the execv function) correctly runs the sleep command to sleep for $1$ second. #include < ... $\textsf{a = 1}\\ \textsf{a = 0}$ $\textsf{a = 1}$ $\textsf{a = 0}$

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28

IITB Practice Set: 28
Consider the following algorithm. What does the above algorithm compute?

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29

IITB Practice Set: 29
The input is an array $A$ of size $n\geq 5$. As output we require indices $i < j < k$ such that either $A[i] \leq A[ j] \leq A[k]\text{ or }A[i] \geq A[ j] \geq A[k]$. The worst case complexity of the best algorithm for this problem is $\theta$ ______ .

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30

IITB Practice Set: 30
How many ordered pairs of positive integers $(a,b)$ are there such that $\text{lcm}(a,b) = n \text{ and gcd}(a,b) = 1?$ Express your answer in terms of the prime factorization of $n$, given as $n = p_1^{d_1} \dots p_t^{d_t}.$

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Recent questions tagged iitb-practice-set