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Recent questions tagged isi2020-pcb-mathematics

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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.

admin asked in Others Aug 25, 2022

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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. $

admin asked in Others Aug 8, 2022

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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.

admin asked in Others Aug 8, 2022

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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 $A$.

admin asked in Linear Algebra Aug 8, 2022

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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 $c=\alpha^{2}-\alpha$.

admin asked in Others Aug 8, 2022

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6

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 $x y=y x, \forall x \in N, \forall y \in K$.

admin asked in Others Aug 8, 2022

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7

ISI 2020 | PCB Mathematics | Question: 6
Find all possible integers $n$ for which $n^{2}+20 n+15$ is a perfect square.

admin asked in Others Aug 8, 2022

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8

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 $125?$

admin asked in Others Aug 8, 2022

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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 $x$.

admin asked in Others Aug 8, 2022

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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 $n \rightarrow \infty.$

admin asked in Others Aug 8, 2022

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11

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 $T_{1}, T_{2}, \ldots, T_{k}$.

admin asked in Others Aug 8, 2022

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