Discuss the applicability of the Rolle's theorem for the following function on the indicated interval f(x)= -4x+5,&0 1 2x-3,&1<x 2

Discuss the applicability of the Rolle's theorem for the followin…

If $|\vec{\text{a}}|=\text{a}$ and $\big|\vec{\text{b}}\big|=\text{b},$ prove that $\Big(\frac{\vec{\text{a}}}{\text{a}^2}-\frac{\vec{\text{b}}}{\text{b}^2}\Big)^2=\Big(\frac{\vec{\text{a}}-\vec{\text{b}}}{\text{ab}}\Big)^2.$

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Show that $\text{f}(\text{x})=\log\sin\text{x}$ is increasing on $\Big(0,\frac{\pi}{2}\Big)$ and decreasing on $\Big(\frac{\pi}{2},\pi\Big).$

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Evaluate the following integrals:
$\int\sqrt{\text{x}^2-2\text{x}}\text{dx}$

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Evaluate the following integrals:$\int\text{e}^{\text{x}}\big(\cot\text{x}-\text{cosec}^2\text{x}\big)\text{dx}$

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It the lines $\frac{\text{x}-1}{-3}=\frac{\text{y}-2}{2\lambda}=\frac{\text{z}-3}{2}$ and $\frac{\text{x}-1}{3\lambda}=\frac{\text{y}-2}{1}=\frac{\text{z}-6}{-5}$ are perpendicular, find the value of $\lambda.$

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Evaluate the following integrals:
$\int_{1}^\limits{3}\frac{\cos(\log\text{x})}{\text{x}}\text{ dx}$

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Write the direction cosines of the line $\frac{\text{x}-2}{2}=\frac{2\text{y}-5}{-3},\text{z}=2.$

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Let A and B be two independent events such that $P(A) = p_1$ and $P(B) = p_2$. Describe in words the events whose probabilities are:
$p_1 + p_2 = 2p_1p_2.$

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If $\int\limits^{\text{a}}_0\frac{1}{4+\text{x}^2}\text{ dx}=\frac{\pi}{8},$ find the value of a.

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If $|\vec{\text{a}}|=2,\big|\vec{\text{b}}\big|=7$ and $\vec{\text{a}}\times\vec{\text{b}}=3\hat{\text{i}}+2\hat{\text{j}}+6\hat{\text{k}},$ find the angle between $\vec{\text{a}}$ and $\vec{\text{b}}.$

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