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ISI 2019 | PCB Mathematics | Question: 11

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ISI 2019 | PCB Mathematics | Question: 1
Let $\left\{a_{n}\right\}$ be a decreasing sequence such that $\displaystyle{}\sum_{n=1}^{\infty} a_{n}$ is convergent. Prove that the sequence $\left\{n a_{n}\right\}$ g...
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ISI 2019 | PCB Mathematics | Question: 2
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ISI 2019 | PCB Mathematics | Question: 3
Let $A$ and $B$ be two invertible real matrices of order $n$. Show that $\det(x A+(1-x) B)=0$ has finitely many solutions for $x.$
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ISI 2019 | PCB Mathematics | Question: 4
Show that for every $\theta \in\left(0, \frac{\pi}{2}\right),$ there exists a unique real number $x_{\theta}$ such that $$ (\sin \theta)^{x_{\theta}}+(\cos \theta)^{x_{\t...