If P(A)=7 13 ,P(B)=9 13 and P(A )=4 13 , find P ( A B ).

Evaluate the following:

$\int \cos (\sqrt[3]{x}) d x$

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Form the differential equation from the following primitives where constants are arbitrart:$\text{y}=\text{ax}^2+\text{bx}+\text{c}$

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If $y=\log \left[\cos \left(x^5\right)\right]$ then find $\frac{ d y}{ d x}$

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Show that the sum of the length of projections of $\hat{p} \hat{i}+q \hat{j}+r \hat{k}$ on the coordinate axes,where p = 2, q = 3 and r = 4, is 9.

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Evaluate the following:

$\int \frac{\log (\log x)}{x} d x$

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Compute the indicated products:
$\begin{bmatrix}2&3&4\\3&4&5\\4&5&6\end{bmatrix}\begin{bmatrix}1&-3&5\\0&2&4\\3&0&5\end{bmatrix}$

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If the position vector of a point (-4, -3) be $\vec{\text{a}}$, find $\big|\vec{\text{a}}\big|$.

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if $A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]$ and $X$ is a $2 \times 2$ matrix such that $A X=1$, then find $X$.

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Evaluate the following definite integrals:$\int_{-2}^\limits{3}\frac{1}{\text{x}+7} \text{ dx}$

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Find the approximate values of:

$f(x)=x^3-3 x+5$ at $x=1.99$

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Answer