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Questions by Lakshman Patel RJIT
0
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1
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1
TIFR CSE 2022 | Part A | Question: 1
A snail crawls up a vertical pole $75$ feet high, starting from the ground. Each day it crawls up $5$ feet, and each night it slides down $4$ feet. When will it first reach the top of the pole? $75^{\text {th}}$ day $74^{\text {th}}$ day $73^{ \text{rd}}$ day $72^{\text {nd }}$ day $71^{\text {st }}$ day
asked
in
Combinatory
Sep 1
164
views
tifr2022
combinatory
counting
0
votes
2
answers
2
TIFR CSE 2022 | Part A | Question: 2
We would like to invite a minimum number $n$ of people (their birthdays are independent of each other) to a party such that the expected number of pairs of people that share the same birthday is at least $1.$ What should $n$ be? (Ignore leap years, so ... birthdays fall with equal probability on each of the $365$ days of the year.) $23$ $28$ $92$ $183$ $366$
asked
in
Probability
Sep 1
183
views
tifr2022
probability
expectation
0
votes
1
answer
3
TIFR CSE 2022 | Part A | Question: 3
A binary string is a sequence of $0 \text{'s}$ and $1\text{'s}.$ A binary string is finite if the sequence is finite, otherwise it is infinite. Examples of finite binary strings include $00010100$, and $1111101010.$ Which of ... set of all finite binary strings is countable while whether the set of all infinite binary strings is countable or not is not known
asked
in
Theory of Computation
Sep 1
92
views
tifr2022
theory-of-computation
countable-uncountable-set
1
vote
1
answer
4
TIFR CSE 2022 | Part A | Question: 4
Consider the polynomial $p(x)=x^3-x^2+x-1$. How many symmetric matrices with integer entries are there whose characteristic polynomial is $p$? (Recall that the characteristic polynomial of a square matrix $A$ in the variable $x$ is defined to be the determinant of the matrix $(A-x I)$ where $I$ is the identity matrix.) $0$ $1$ $2$ $4$ Infinitely many
asked
in
Linear Algebra
Sep 1
109
views
tifr2022
linear-algebra
matrix
determinant
2
votes
1
answer
5
TIFR CSE 2022 | Part A | Question: 5
Let $\mathcal{F}$ be the set of all functions mapping $\{1, \ldots, n\}$ to $\{1, \ldots, m\}$. Let $f$ be a function that is chosen uniformly at random from $\mathcal{F}$. Let $x, y$ be distinct elements from the set $\{1, \ldots, n\}$. Let $p$ denote the probability ... Then, $p=0$ $p=\frac{1}{n^m}$ $0<p \leq \frac{1}{m^n}$ $p=\frac{1}{m}$ $p=\frac{1}{n}$
asked
in
Probability
Sep 1
85
views
tifr2022
probability
uniform-distribution
functions
0
votes
0
answers
6
TIFR CSE 2022 | Part A | Question: 6
Let $f$ be a polynomial of degree $n \geq 3$ all of whose roots are non-positive real numbers. Suppose that $f(1)=1$. What is the maximum possible value of $f^{\prime}(1)?$ $1$ $n$ $n+1$ $\frac{n(n+1)}{2}$ $f^{\prime}(1)$ can be arbitrarily large given only the constraints in the question
asked
in
Calculus
Sep 1
67
views
tifr2022
calculus
maxima-minima
1
vote
1
answer
7
TIFR CSE 2022 | Part A | Question: 7
Initially, $N$ white beads are arranged in a circle. A number $k$ is chosen uniformly at random from $\{1, \ldots, N-1\}$. Then a set of $k$ beads is chosen uniformly from the white beads, and these $k$ beads are coloured black. The position of the beads remains ... None of the above
asked
in
Probability
Sep 1
110
views
tifr2022
probability
uniform-distribution
13
votes
1
answer
8
TIFR CSE 2022 | Part A | Question: 8
Let $A$ be the $(n+1) \times(n+1)$ matrix given below, where $n \geq 1$. For $i \leq n$, the $i$-th row of $A$ has every entry equal to $2i-1$ and the last row, i.e., the $(n+1)$-th row of $A$ has every entry equal to $-n^2$ ... $A$ has rank $n$ $A^2$ has rank $1$ All the eigenvalues of $A$ are distinct All the eigenvalues of $A$ are $0$ None of the above
asked
in
Linear Algebra
Sep 1
243
views
tifr2022
linear-algebra
rank-of-matrix
eigen-value
0
votes
1
answer
9
TIFR CSE 2022 | Part A | Question: 9
You are given the following properties of sets $A, B, X$, and $Y$. For notation, $|A|$ denotes the cardinality of set $A$ (i.e., the number of elements in $A$ ), and $A \backslash B$ denotes the set of elements that are in $A$ but not in $B$. $A \cup B=X \cup Y$ ... $|X|=5$ $|Y|=5$ $|A \cup X|=|B \cup Y|$ $|A \cap X|=|B \cap Y|$ $|A|=|B|$
asked
in
Set Theory & Algebra
Sep 1
90
views
tifr2022
set-theory&algebra
set-theory
0
votes
1
answer
10
TIFR CSE 2022 | Part A | Question: 10
Consider a bag containing colored marbles. There are $n$ marbles in the bag such that there is exactly one pair of marbles of color $i$ for each $i \in\{1, \ldots, m\}$ and the rest of the marbles are of distinct colors (different from colors $\{1, \ldots, m\}$ ). You draw ... $\frac{2m}{n}$ $\frac{2m}{n(n-1)}$ $\frac{2m}{n^2}$ $\frac{m}{n(n-1)}$
asked
in
Probability
Sep 1
70
views
tifr2022
probability
conditional-probability
0
votes
0
answers
11
TIFR CSE 2022 | Part A | Question: 11
Let $X$ be a finite set. A family $\mathcal{F}$ of subsets of $X$ is said to be upward closed if the following holds for all sets $A, B \subseteq X$ ... $\mathcal{F} \sqcup \mathcal{G}=\mathcal{G} \backslash \mathcal{F}$ None of the above
asked
in
Set Theory & Algebra
Sep 1
61
views
tifr2022
set-theory&algebra
set-theory
0
votes
1
answer
12
TIFR CSE 2022 | Part A | Question: 12
Alice plays the following game on a math show. There are $7$ boxes and identical prizes are hidden inside $3$ of the boxes. Alice is asked to choose a box where a prize might be. She chooses a box uniformly at random. From the unchosen boxes which do not have a prize, ... ). Her probability of winning the prize is $3 / 7$ $1 / 2$ $17 / 30$ $18 / 35$ $9 / 19$
asked
in
Probability
Sep 1
94
views
tifr2022
probability
conditional-probability
0
votes
0
answers
13
TIFR CSE 2022 | Part A | Question: 13
Consider the transition system shown in the figure below with the initial state $s_1$. A token is initially placed at $s_1$, and it moves to $s_2$ with probability $\frac{2}{3}$, and to $s_3$ with probability $\frac{1}{3}$. From $s_2$ and $s_3$, the token always ... appear in the run? $\frac{1}{7}$ $\frac{2}{7}$ $\frac{3}{7}$ $\frac{5}{7}$ None of the above
asked
in
Theory of Computation
Sep 1
58
views
tifr2022
theory-of-computation
finite-automata
probability
0
votes
0
answers
14
TIFR CSE 2022 | Part A | Question: 14
Suppose $w(t)=4 e^{i t}, x(t)=3 e^{i(t+\pi / 3)}, y(t)=3 e^{i(t-\pi / 3)}$ and $z(t)=3 e^{i(t+\pi)}$ are points that move in the complex plane as the time $t$ varies in $(-\infty, \infty)$. Let $c(t)$ ... $\frac{1}{2 \pi}$ $2 \pi$ $\sqrt{3} \pi$ $\frac{1}{\sqrt{3} \pi}$ $1$
asked
in
Calculus
Sep 1
53
views
tifr2022
calculus
differentiation
0
votes
1
answer
15
TIFR CSE 2022 | Part A | Question: 15
Fix $n \geq 4$. Suppose there is a particle that moves randomly on the number line, but never leaves the set $\{1,2, \ldots, n\}$. The initial probability distribution of the particle is $\pi$ i.e., the probability that particle is in location $i$ is given by $\pi(i)$. In the ... $\pi(1)=1$ and $\pi(i)=0$ for $i \neq 1$ $\pi(n)=1$ and $\pi(i)=0$ for $i \neq n$
asked
in
Probability
Sep 1
101
views
tifr2022
probability
uniform-distribution
0
votes
0
answers
16
ISI 2020 | PCB Mathematics | Question: 5.2
Deduce that if $N, H, K$ are normal subgroups of a group $G$ such that $ N \bigcap H=N \bigcap K=H \bigcap K=\left\{e_{G}\right\} $ and $G=H K$, then $N$ is an Abelian group.
asked
in
Others
Aug 25
26
views
isi2020-pcb-mathematics
descriptive
0
votes
0
answers
17
ISI2020-PCB-CS: 8.2
Consider the following state diagram of a sequential circuit, where each of a, b, c, d, e, f and g represents a state. Represent the state diagram with minimum number of states without altering the input-output relationships. Justify your answer.
asked
in
Others
Aug 25
31
views
isi2020-pcb-cs
descriptive
0
votes
1
answer
18
ISI 2021 | PCB CS | Question: 7.b
Consider a $4$-way set associative cache mapping, in which the cache blocks are grouped into sets and each set has $4$ blocks. There are $16$ cache blocks in total. The following memory block requests arrive in order when ... . Show the cache configuration (along with intermediate configurations) on meeting the above memory requirements. What is the hit ratio?
asked
in
CO and Architecture
Aug 24
98
views
isi2021-pcb-cs
descriptive
co-and-architecture
cache-memory
1
vote
3
answers
19
ISI2020-PCB-CS: 1.3
What does the following function compute for $x \neq 0?$ float isi1(float x, int y){ if (y==0){ return 1 ;} else if (y>0) {return isi1(x,-y);} else { return isi1(x, y+1)/x;} }
asked
in
Programming
Aug 18
216
views
isi2020-pcb-cs
identify-function
descriptive
0
votes
1
answer
20
ISI2020-PCB-CS: 1.2
What will be the output of the following C program? Justify your answer. Negative numbers are represented in $2$'s complement, #include<stdio.h> int main() { if (-~-1) printf("COVID"); if ((~7 & Ox000f ) == 8) printf (“19”); printf ("*"); }
asked
in
Programming
Aug 18
181
views
isi2020-pcb-cs
programming
programming-in-c
number-representation
descriptive
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