Find k if the function f(x) is defined by : f(x)=k x(1-x), for 0<x<1 =0, otherwise. is the probability density function (p.d.f.) of a random variable (r.v.) X. Also find P (X<1 2 )

Find k if the function f(x) is defined by : f(x)=k x(1-x), for 0<…

$\int \frac{ d x}{x^3-1}$

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Solve the following equations by using inversion method.$x + y + z = −1, x − y + z = 2$ and $x + y − z = 3$

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$\int x \sin 2 x \cos 5 x d x$

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Find the vector equation of the plane passing through the point A(-2, 7, 5) and parallel to

vectors $\hat{4}-\hat{j}+3 \hat{k}$ and $\hat{i}+\hat{j}+\hat{k}$.

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A computer installation has 10 terminals. Independently, the probability that anyone terminal will require attention during a week is 0.1. Find the probabilities that (i) 0 (ii) 1 (iii) 2 (iv) 3 or more, terminals will require attention during the next week.

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If $\tan ^{-1}\left(\frac{x-1}{x-2}\right)+\cot ^{-1}\left(\frac{x+2}{x+1}\right)=\frac{\pi}{4}$; find $x$.

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Show the equation $2 x^2+x y-y^2+x+4 y-3=0$ represents a pair of lines. Also find the acute angle between them.

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$\int \frac{3 e ^{2 x}+5}{4 e ^{2 x}-5} d x$

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Differentiate the following w. r. t. x.$y=\log \left[\sqrt{\frac{1-\cos \left(\frac{3 x}{2}\right)}{1+\cos \left(\frac{3 x}{2}\right)}}\right]$

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Determine whether $\bar{a}$ and $\bar{b}$ are orthogonal, parallel or neither.

1.$\bar{a}=-9 \hat{i}+6 \hat{j}+15 \hat{k} \bar{b}=6 \hat{i}-4 \hat{j}-10 \hat{k}$

2.$\bar{a}=2 \hat{i}+3 \hat{j}-\hat{k}, \bar{b}=5 \hat{i}-2 \hat{j}+4 \hat{k}$

3.$\bar{a}=-\frac{3}{5} \hat{i}+\frac{1}{2} \hat{j}+\frac{1}{3} \hat{k}, \bar{b}=5 \hat{i}+4 \hat{j}+3 \hat{k}$

4.$\bar{a}=4 \hat{i}-\hat{j}+6 \hat{k}, \bar{a}=5 \hat{i}-2 \hat{j}+4 \hat{k}$

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