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121
ISI 2021 | PCB CS | Question: 3
Let $G$ be a simple undirected graph having $n$ vertices with the property that the average of the degrees of the vertices in $G$ is at least $4.$ ... $\textsf{deg(x)}$ denotes the degree of the vertex $\textsf{x}$.
Let $G$ be a simple undirected graph having $n$ vertices with the property that the average of the degrees of the vertices in $G$ is at least $4.$ Consider the following ...
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126
ISI 2021 | PCB CS | Question: 8
Consider the following language $L$ over the alphabet $\Sigma=\{a, b, c\}$. $ L=\left\{a^{i} b^{j} c^{k} \mid i, j, k \geq 0 \text {, and if } i=1 \text { then } j=k\right\} $ Show that $L$ is not regular. Show that $L$ is a context-free language.
Consider the following language $L$ over the alphabet $\Sigma=\{a, b, c\}$. $$ L=\left\{a^{i} b^{j} c^{k} \mid i, j, k \geq 0 \text {, and if } i=1 \text { then } j=k\rig...
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127
ISI 2021 | PCB CS | Question: 9
In relational algebra, for any pair of relations $R_{1}$ and $R_{2}$, the standard division operation is denoted by $\div$ ...
In relational algebra, for any pair of relations $R_{1}$ and $R_{2}$, the standard division operation is denoted by $\div$ and defined as follows: $$ R_{1} \div R_{2}=\pi...
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128
ISI 2021 | PCB CS | Question: 10
Consider sending a message of $10,000$ bits from the source node $S$ to the destination node $D$ passing through the two routers $R 1$ and $R 2$ ... -end latency of the message when it is broken into $10$ packets each of size $1000$ bits, and then transmitted to the destination.
Consider sending a message of $10,000$ bits from the source node $S$ to the destination node $D$ passing through the two routers $R 1$ and $R 2$ as shown in the figure. E...
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130
ISI 2021 | PCB Mathematics | Question: 2
Consider three real numbers $a \geq b \geq c>0$. If $\left(a^{x}-b^{x}-c^{x}\right)(x-2)>0$ for any rational number $x \neq 2$, show that $a, b$ and $c$ can be the lengths of the three sides of a triangle $A B C;$ $A B C$ is a right-angled triangle.
Consider three real numbers $a \geq b \geq c>0$. If $\left(a^{x}-b^{x}-c^{x}\right)(x-2)>0$ for any rational number $x \neq 2$, show that$a, b$ and $c$ can be the lengths...
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133
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 $m 2^{m}+1$.
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 $m 2^{m}+1$.
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1 answer
134
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.
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,...
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137
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 $\sqrt{a}.$
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 seque...
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138
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. $
Let $\epsilon>0$. Prove that there exists $n_{0} \in \mathbb{N}$ such that $$ n \geq n_{0} \Rightarrow 2-\epsilon<\frac{2 n+1}{n+2}<2+\epsilon. $$
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139
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.
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.
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3 votes
1 answer
141
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 $c=\alpha^{2}-\alpha$.
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 $c=\alpha^{2}-\alpha$...
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142
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 $x y=y x, \forall x \in N, \forall y \in K$.
Prove that if $N$ and $K$ are normal subgroups of a group $G$ such that $N \cap K=\left\{e_{G}\right\}$, then $x y=y x, \forall x \in N, \forall y \in K$.
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143
ISI 2020 | PCB Mathematics | Question: 6
Find all possible integers $n$ for which $n^{2}+20 n+15$ is a perfect square.
Find all possible integers $n$ for which $n^{2}+20 n+15$ is a perfect square.
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144
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 $125?$
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 $125?$
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145
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 $x$.
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 $x$.
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147
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 $T_{1}, T_{2}, \ldots, T_{k}$.
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 $T_{1}, T_{2}, \ldots, T_{k}$.
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