Verify Rolle's theorem for the following function on the indicated intervalsf(x)= 2 (x- 4 ) on [0, 2 ]

Verify Rolle's theorem for the following function on the indicate…

Evaluate the following integrals as limit of sum:$\int\limits^{3}_{0}\big(2\text{x}^2+3\text{x}+5\big)\text{dx}$

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

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Find the inverse of $A=\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 2 & 3 \\ 1 & 2 & 1\end{array}\right]$ by elementary column transformations.

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Let $\vec{\text{u}},\vec{\text{v}}$ and $\vec{\text{w}}$ be vectors such $\vec{\text{u}}+\vec{\text{v}}+\vec{\text{w}}=\vec{0}.$ If $|\vec{\text{u}}|=3,|\vec{\text{v}}|=4$ and $|\vec{\text{w}}|=5,$ then find $\vec{\text{u}}.\vec{\text{v}}+\vec{\text{v}}.\vec{\text{w}}+\vec{\text{w}}.\vec{\text{u}}.$

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Find the distance between the lines $l_1$ and$ l_2$ given by$\vec{\text{r}}=\hat{\text{i}}+2\hat{\text{j}}-4\hat{\text{k}}+\lambda\big(2\hat{\text{i}}+3\hat{\text{j}}+6\hat{\text{k}}\big)$ and, $\vec{\text{r}}=3\hat{\text{i}}+3\hat{\text{j}}-5\hat{\text{k}}+\mu\big(2\hat{\text{i}}+3\hat{\text{j}}+6\hat{\text{k}}\big)$

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Evaluate the following integrals:$\int\limits^{2\pi}_0\log(\sec\text{x}+\tan\text{x})\text{dx}$

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Find the inverse of the following matrices:$\begin{bmatrix}0 & 1 & -1 \\ 4 & -3 & 4 \\ 3 & -3 & 4 \end{bmatrix}$

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Find the intervals in which the following functions are increasing or decreasing.
$f(x) = 10 - 6x - 2x^2$

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If $\text{x}=\text{a}(1-\cos^3\theta),\text{y}=\text{a}\sin^3\theta,$ Prove that $\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{32}{27\text{a}}\text{ at}\ \theta=\frac{\pi}{6}$

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Evaluate the following integrals:
$\int\frac{1}{\sin\text{x}+\sin2\text{x}}\ \text{dx}$

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