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ISI 2020 | PCB Mathematics | Question: 1

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admin asked Aug 8, 2022
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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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admin asked Aug 25, 2022
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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.
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 ...
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admin asked 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.
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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admin asked 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 $c=\alpha^{2}-\alpha$.
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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