81.

A random variable X takes the values 0, 1 and 2. If P(X = 1) = P(X = 2) and P(X = 0) = 0.4, then the mean of the random variable X is

• 0.2

• 0.7

• 0.5

• 0.9

The acute angle between the two lines whose direction ratios are given by l + m - n = 0 and  ${\mathrm{l}}^2 + {\mathrm{m}}^2 + {\mathrm{n}}^2 = 0, \mathrm{is}$

• 0

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

The direction ratios of normal to the plane passing through (0, 0, 1), (0, 1, 2) and (1, 0, 3) are

• (2, 1, - 1)

• (1, 0, 1)

• (0, 0, - 1)

• (1, 0, 0)

If P = (0, 1, 0), Q = (0, 1, 0), then the projection of PQ on the plane x + y + z = 3 is

• 2

• $\sqrt{2}$

• 3

• $\sqrt{3}$

In the space the equation by + cz + d = 0 represents a plane perpendicular to the

• YOZ-plane

• ZOX-plane

• XOY-plane

• None of these

A plane x passes through the point (1, 1, 1). If b, c, a are the direction ratios of a normal to the plane, where a, b, c (a < b < c) are the factors of 2001, then the equation of plane is 

• 29x + 31y + 3z = 63

• 23x + 29y - 29z = 23

• 23x + 29y + 3z = 55

• 31x + 37y + 3z = 71

If the plane 7x + 11y + 13z = 3003 meets the co-ordinate axes in A, B, C, then the centroid of the $∆\mathrm{ABC}$ is

• (143, 91, 77)

• (143, 77, 91)

• (91, 143, 77)

• (143, 66, 91)

$\int \frac{\mathrm{dx}}{1 - \cos \left(\mathrm{x}\right) - \sin \left(\mathrm{x}\right)} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\log \left(\left|\begin{array}{c}1 + \cot \left(\frac{\mathrm{x}}{2}\right)\end{array}\right|\right) + \mathrm{c}$

• $\log \left(\left|\begin{array}{c}1 - \tan \left(\frac{\mathrm{x}}{2}\right)\end{array}\right|\right) + \mathrm{c}$

• $\log \left(\left|\begin{array}{c}1 - \cot \left(\frac{\mathrm{x}}{2}\right)\end{array}\right|\right) + \mathrm{c}$

• $\log \left(\left|\begin{array}{c}1 + \tan \left(\frac{\mathrm{x}}{2}\right)\end{array}\right|\right) + \mathrm{c}$

$\int \frac{\mathrm{dx}}{7 + 5\cos \left(\mathrm{x}\right)} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\frac{1}{\sqrt{3}}{\tan }^{-1}\left(\frac{{\displaystyle 1}}{{\displaystyle \sqrt{3}}}\tan \frac{\mathrm{x}}{2}\right) + \mathrm{c}$

• $\frac{1}{\sqrt{6}}{\tan }^{-1}\left(\frac{{\displaystyle 1}}{{\displaystyle \sqrt{6}}}\tan \frac{\mathrm{x}}{2}\right) + \mathrm{c}$

• $\frac{1}{7}{\tan }^{-1}\left(\tan \frac{\mathrm{x}}{2}\right) + \mathrm{c}$

• $\frac{1}{4}\tan { }^{-1}\left(\tan \frac{\mathrm{x}}{2}\right) + \mathrm{c}$

$\int \frac{3^{\mathrm{x}}\mathrm{dx}}{\sqrt{9^{\mathrm{x}} - 1}} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\frac{1}{\log \left(3\right)}\log \left(\left|\begin{array}{c}{3}^{\mathrm{x}} + \sqrt{9^{\mathrm{x}} - 1}\end{array}\right|\right) + \mathrm{c}$

• $\frac{1}{\log \left(3\right)}\log \left(\left|\begin{array}{c}{3}^{\mathrm{x}} - \sqrt{9^{\mathrm{x}} - 1}\end{array}\right|\right) + \mathrm{c}$

• $\frac{1}{\log \left(9\right)}\log \left(\left|\begin{array}{c}{3}^{\mathrm{x}} + \sqrt{9^{\mathrm{x}} - 1}\end{array}\right|\right) + \mathrm{c}$

• $\frac{1}{\log \left(3\right)}\log \left(\left|\begin{array}{c}{9}^{\mathrm{x}} + \sqrt{9^{\mathrm{x}} - 1}\end{array}\right|\right) + \mathrm{c}$