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ISI 2019 | PCB Mathematics | Question: 8
Lakshman Patel RJIT
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Let $f$ be a real valued function on $\mathbb{R}$. If for all real $x$, $$ f(x)+3 f(1-x)=5 $$ holds, then show that $f$ is a constant function.
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Lakshman Patel RJIT
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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 $\left\{n a_{n}\right\}$ goes to zero as $n \rightarrow \infty$.
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ISI 2019 | PCB Mathematics | Question: 2
Consider an $n \times n$ matrix $A=I_{n}-\alpha \alpha^{T}$, where $I_{n}$ is the identity matrix of order $n$ and $\alpha$ is an $n \times 1$ column vector such that $\alpha^{T} \alpha=1$. Prove that $A^{2}=A.$
Lakshman Patel RJIT
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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 $x.$
Lakshman Patel RJIT
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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_{\theta}}=\frac{3}{2} . $
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