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Recent questions tagged descriptive
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61
ISI 2021 | PCB Mathematics | Question: 6
Let $p$ be an odd prime and let $n=(p-1)(p+1)$. Show that $p$ divides $n 2^{n}+1$. Show that there are infinitely many integers $m$ such that $p$ divides $m 2^{m}+1$.
Lakshman Patel RJIT
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Aug 8
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Lakshman Patel RJIT
15
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isi2021-pcb-mathematics
descriptive
1
vote
1
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62
ISI 2021 | PCB Mathematics | Question: 7
Let $G$ be a cubic graph, that is, every vertex has degree exactly $3$. Prove that the number of vertices of $G$ cannot be $101$. Prove that if $G$ contains $100$ vertices, then it contains a bipartite subgraph that has at least $75$ edges.
Lakshman Patel RJIT
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Others
Aug 8
by
Lakshman Patel RJIT
43
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isi2021-pcb-mathematics
descriptive
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63
ISI 2021 | PCB Mathematics | Question: 8
Calculate the number of different ways you can divide $2 n$ elements of the set $S=\{1,2, \ldots, 2 n\}$ to form $n$ disjoint subsets, each containing a pair of elements. Calculate the number of different ways in which the above division can be done if each subset is required to contain an even number and an odd number.
Lakshman Patel RJIT
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Others
Aug 8
by
Lakshman Patel RJIT
22
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isi2021-pcb-mathematics
descriptive
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64
ISI 2021 | PCB Mathematics | Question: 9
Consider a $4 \times 4$ positive semi-definite matrix $A$ with all diagonal elements equal to $1$ and all off-diagonal elements equal to $\rho$. If $\rho<0$, show that the largest eigenvalue of $A$ cannot exceed $4 / 3$ Give an eigenvector of $A$ other than $(1,1,1,1)^{\top}$.
Lakshman Patel RJIT
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Others
Aug 8
by
Lakshman Patel RJIT
13
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isi2021-pcb-mathematics
descriptive
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65
ISI 2021 | PCB Mathematics | Question: 10
Let $a>0$ and $x_{1}>0$. Define $x_{n+1}=\frac{1}{2}\left(x_{n}+\frac{a}{x_{n}}\right)$ for all $n \in \mathbb{N}$. Show that $x_{n}>\sqrt{a}$ for all $n \geq 2;$ the sequence $\left\{x_{n}: n \geq 1\right\}$ converges to $\sqrt{a}.$
Lakshman Patel RJIT
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Others
Aug 8
by
Lakshman Patel RJIT
20
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isi2021-pcb-mathematics
descriptive
0
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66
ISI 2020 | PCB Mathematics | Question: 1
Let $\epsilon>0$. Prove that there exists $n_{0} \in \mathbb{N}$ such that $ n \geq n_{0} \Rightarrow 2-\epsilon<\frac{2 n+1}{n+2}<2+\epsilon. $
Lakshman Patel RJIT
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in
Others
Aug 8
by
Lakshman Patel RJIT
20
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isi2020-pcb-mathematics
descriptive
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67
ISI 2020 | PCB Mathematics | Question: 2
Show that the sequence $\left\{x_{n}\right\}, n>0$, defined by $ x_{n}=\int_{1}^{n} \frac{\cos (t)}{t^{2}} d t $ is convergent.
Lakshman Patel RJIT
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in
Others
Aug 8
by
Lakshman Patel RJIT
20
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isi2020-pcb-mathematics
descriptive
0
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1
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68
ISI 2020 | PCB Mathematics | Question: 3
Suppose $A$ is an $(n \times n)$ matrix over $\mathbb{R}$ such that $A^{p}=0$ for some positive integer $p$. Prove that $I+A$ is an invertible matrix, where $I$ is the $(n \times n)$ identity matrix. Find the characteristic polynomial of $A$.
Lakshman Patel RJIT
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in
Linear Algebra
Aug 8
by
Lakshman Patel RJIT
55
views
isi2020-pcb-mathematics
descriptive
linear-algebra
matrix
0
votes
1
answer
69
ISI 2020 | PCB Mathematics | Question: 4
Let $c$ be a positive real number for which the equation $ x^{4}-x^{3}+x^{2}-(c+1) x-\left(c^{2}+c\right)=0 $ has a real root $\alpha$. Prove that $c=\alpha^{2}-\alpha$.
Lakshman Patel RJIT
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Others
Aug 8
by
Lakshman Patel RJIT
28
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isi2020-pcb-mathematics
descriptive
0
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70
ISI 2020 | PCB Mathematics | Question: 5.1
Prove that if $N$ and $K$ are normal subgroups of a group $G$ such that $N \cap K=\left\{e_{G}\right\}$, then $x y=y x, \forall x \in N, \forall y \in K$.
Lakshman Patel RJIT
asked
in
Others
Aug 8
by
Lakshman Patel RJIT
33
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isi2020-pcb-mathematics
descriptive
0
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71
ISI 2020 | PCB Mathematics | Question: 6
Find all possible integers $n$ for which $n^{2}+20 n+15$ is a perfect square.
Lakshman Patel RJIT
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in
Others
Aug 8
by
Lakshman Patel RJIT
82
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isi2020-pcb-mathematics
descriptive
0
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72
ISI 2020 | PCB Mathematics | Question: 7
Let $A=\{1,2,3, \cdots, 50\}$. In how many ways can three distinct numbers $x<y<z$ be chosen from $A$ such that the product $x y z$ is divisible by $125?$
Lakshman Patel RJIT
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in
Others
Aug 8
by
Lakshman Patel RJIT
82
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isi2020-pcb-mathematics
descriptive
0
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0
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73
ISI 2020 | PCB Mathematics | Question: 8
Prove that the function defined by $ f(x)=\sum_{n=0}^{\infty}\left(\frac{x^{n}}{n !}\right)^{2} $ is continuous on $\mathbb{R}$, for any real number $x$.
Lakshman Patel RJIT
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in
Others
Aug 8
by
Lakshman Patel RJIT
19
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isi2020-pcb-mathematics
descriptive
0
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74
ISI 2020 | PCB Mathematics | Question: 9
Let $X_{i} \sim\left(i . i . d\right.$.) Bernoulli $\left(\frac{\lambda}{n}\right), n \geq \lambda \geq 0$. $Y_{i} \sim\left(\right. i. i. d.)$ Poisson $\left(\frac{\lambda}{n}\right),\left\{X_{i}\right\}$ and $\left\{Y_{i}\right\}$ ... $\dfrac{T_{n}}{S_{n}}$ as $n \rightarrow \infty.$
Lakshman Patel RJIT
asked
in
Others
Aug 8
by
Lakshman Patel RJIT
13
views
isi2020-pcb-mathematics
descriptive
0
votes
0
answers
75
ISI 2020 | PCB Mathematics | Question: 10
Prove that if $T_{1}, T_{2}, \ldots, T_{k}$ are pairwise-intersecting subtrees of a tree $T$, then $T$ has a vertex that belongs to all of $T_{1}, T_{2}, \ldots, T_{k}$.
Lakshman Patel RJIT
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in
Others
Aug 8
by
Lakshman Patel RJIT
17
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isi2020-pcb-mathematics
descriptive
0
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76
ISI2020-PCB-CS: 2
An $n \times n$ binary matrix $M$ is called a NICE matrix, if each row of $M$ has exactly one non-zero element and each column also has exactly one non-zero element. Suggest a method of storing a NICE matrix in an $O(n)$ size array. Design an $O(n)$ time algorithm ( ... computing $R=P Q$, where $P$ and $Q$ are both NICE matrices each stored in an array of size $O(n)$ as in (i).
Lakshman Patel RJIT
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in
Others
Aug 8
by
Lakshman Patel RJIT
19
views
isi2020-pcb-cs
descriptive
0
votes
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77
ISI2020-PCB-CS: 3
You are given two sorted arrays $X[\;]$ and $Y[\;]$ of positive integers. The array sizes are not given. Accessing any index beyond the last element of the arrays returns $-1$. The elements in each array are distinct but the two arrays may have common ... marks will be awarded if the time complexity of your algorithm is linear (or higher) in the maximum size of $X$ and $Y.$
Lakshman Patel RJIT
asked
in
Others
Aug 8
by
Lakshman Patel RJIT
51
views
isi2020-pcb-cs
descriptive
0
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78
ISI2020-PCB-CS: 4
Given a graph $G$ and a vertex $u$ in it, let $N(u)$ denote the set of neighbours of $u$ in $G$. A graph $G$ having $n$ vertices is said to be $k$ degenerate if there is a linear ordering $v_{1}, v_{2}, \ldots, v_{n}$ of the vertices, in which each ... removing a vertex of degree at most $k$ from $H$. Prove that $H$ is $k$-degenerate if and only if $H^{\prime}$ is $k$-degenerate.
Lakshman Patel RJIT
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in
Others
Aug 8
by
Lakshman Patel RJIT
15
views
isi2020-pcb-cs
descriptive
0
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0
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79
ISI2020-PCB-CS: 5
In a conference, the relation, $\textsf{registered (participant, topic)}$ stores the names of participants and the topics registered by them. The primary key for this relation is $\textsf{(participant, topic)}.$ ... . Which strategy is faster for $x=3000?$ Justify your answer. Which strategy has less disk access time? Justify your answer.
Lakshman Patel RJIT
asked
in
Databases
Aug 8
by
Lakshman Patel RJIT
47
views
isi2020-pcb-cs
descriptive
databases
sql
0
votes
0
answers
80
ISI2020-PCB-CS: 6
Let $L=\left\{s \overline{s_{R}} \mid s \in\{0,1\}^{*}\right\}$ be a language over alphabet $\{0,1\}$, where $\overline{s_{R}}$ describes the reverse complement of $s$. For an $s, \overline{s_{R}}$ is obtained by reversing the ... $0100.$ Give a context-free grammar which generates $L$. Draw a Pushdown Automata that recognizes $L$. Is $L$ a regular language? Justify your answer.
Lakshman Patel RJIT
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in
Others
Aug 8
by
Lakshman Patel RJIT
21
views
isi2020-pcb-cs
descriptive
0
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0
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81
ISI2020-PCB-CS: 7
Consider two hosts, A and B, which may belong to the same network or two different networks connected through a router. Recall the Internet protocol where a sender host sends packets directly to a destination if the sender finds the network ID of the destination host same as that of its ...
Lakshman Patel RJIT
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in
Others
Aug 8
by
Lakshman Patel RJIT
17
views
isi2020-pcb-cs
descriptive
1
vote
1
answer
82
ISI2020-PCB-CS: 8.1
Simplify the following Boolean function in product-of-sums form: $ F(A, B, C, D)=\sum(0,1,2,5,8,9,10) . $
Lakshman Patel RJIT
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in
Digital Logic
Aug 8
by
Lakshman Patel RJIT
158
views
isi2020-pcb-cs
descriptive
digital-logic
boolean-algebra
0
votes
0
answers
83
ISI2020-PCB-CS: 9
Consider a $\textsf{RISC}$ ... in the above $\textsf{RISC}$ machine? Explain your answer (no marks will be awarded if justification is not given).
Lakshman Patel RJIT
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Others
Aug 8
by
Lakshman Patel RJIT
17
views
isi2020-pcb-cs
descriptive
0
votes
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answer
84
ISI2020-PCB-CS: 10
Suppose instead of a decoder with $n$ input bits ( $n$ is even) to access a memory of size $2^{n}$, one uses two decoders of input sizes $k$ bits and $(n-k)$ bits. Explain how these two decoders can be used to access the ... address decoding time. Justify your answer. Assume that the time complexity of the decoder is measured by the number of output lines of that decoder.
Lakshman Patel RJIT
asked
in
Digital Logic
Aug 8
by
Lakshman Patel RJIT
69
views
isi2020-pcb-cs
digital-logic
combinational-circuit
decoder
descriptive
0
votes
0
answers
85
CMI-2021-DataScience-B: 5
Show that every selection of $503$ numbers from $\{1,2,3, \dots,987\}$ has two numbers with g.c.d. $1$. Recall that the g.c.d. of two positive integers $x, y$ is the largest positive integer smaller than $x, y$ which divides both $x$ and $y$.
Lakshman Patel RJIT
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Others
Jul 23
by
Lakshman Patel RJIT
27
views
cmi2021-datascience
descriptive
0
votes
0
answers
86
CMI-2021-DataScience-B: 10
Let $A$ be the $3 \times 3$ real matrix $\left(\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & I \end{array}\right)$. Suppose $x^{T} Ax \geq 0$ for every $x \in \mathbb{R}^{3}$. Then show that all of $a, e$ and $i$ are non-negative.
Lakshman Patel RJIT
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Others
Jul 23
by
Lakshman Patel RJIT
25
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cmi2021-datascience
descriptive
1
vote
0
answers
87
CMI2022-B: 1
A Muller automaton is defined as a tuple $\text{M} = (\text{Q}, \text{I}, \Sigma, \rightarrow, \text{T})$ where: $\text{Q}$ is a finite set of states; $\text{I} \subseteq \text{Q}$ is the set of initial states; $\Sigma$ ... $a^{\ast}?$
Lakshman Patel RJIT
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in
Others
Jul 22
by
Lakshman Patel RJIT
72
views
cmi2022
descriptive
0
votes
0
answers
88
CMI2022-B: 2
Consider the language $\text{L}$ over the alphabet $\left \{ a, b \right \}$ given below. $\text{L}= \{ w \mid w \;\text{has equal number of $a$'s and $b$'s, and there are no adjacent $a$'s.}\}$ For instance, the words $abba, abab$ are in language ... $baab$. Prove that $\text{L}$ does not contain any word that starts and ends with $a$ $b$. Give a context-free grammar for $\text{L}$.
Lakshman Patel RJIT
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Others
Jul 22
by
Lakshman Patel RJIT
50
views
cmi2022
descriptive
0
votes
0
answers
89
CMI2022-B: 3
We say that an integer $a$ is co-prime to another integer $b$ if $\gcd(a, b) = 1$. For any integer $n, \varphi (n)$ is the number of integers from $1$ up to $|n|$ that are co-prime to $n$. Calculate $\varphi (5), \varphi (10)$ and $\varphi (20)$. Show that ... for any prime $p$. Prove that if $a$ is co-prime to $b$ then the remainder of $a$ when divided by $b$ is also co-prime to $b$.
Lakshman Patel RJIT
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Others
Jul 22
by
Lakshman Patel RJIT
44
views
cmi2022
descriptive
0
votes
0
answers
90
CMI2022-B: 5
For any set $\text{S}$ of natural numbers, we say that a relation $\text{R} \subseteq \text{S} \times \text{S}$ is a $2$-spanner of $\text{S}$ if it satisfies the following conditions: $\left ( i, j \right ) \in R \Rightarrow i < j$ ... any set $\text{S}$ of size $2^{k} - 1$ (for $k > 2)$ has a $2$-spanner of size $(k - 2) 2^{k} + 2$.
Lakshman Patel RJIT
asked
in
Others
Jul 22
by
Lakshman Patel RJIT
28
views
cmi2022
descriptive
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