81.

a, b, c, d are coplanar vectors, then (a x b) x (c x d) is equal to

• 0

• 1

• a

• b

Find the binomial probability distribution whose mean is 3 and variance is 2.

• ${\left(\frac{2}{3} + \frac{1}{3}\right)}^9$

• ${\left(\frac{5}{3} + \frac{2}{3}\right)}^9$

• ${\left(\frac{3}{3} + \frac{1}{2}\right)}^9$

• None of these

For a binominal variate X, if n = 4 and P(X = 4) = 6P(X = 2), then the value of p is

• $\frac{3}{7}$

• $\frac{4}{7}$

• $\frac{6}{7}$

• $\frac{5}{7}$

If the foot of the perpendicular from (0, 0, 0) to the plane is (1, 2, 2), then the equation ofthe plane is

• - x + 2y + 8z - 9 = 0

• x + 2y + 2z - 9 = 0

• x + y + z - 5 = 0

• x + 2y - 3z + 1 = 0

If P = (0, 1, 2), Q =(4, - 2, 1), O =(0, 0, 0), then $\angle$POQ is equal to

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{3}$

A variable plane is at a constant distance h from the origin and meets the coordinate axes in A, B, C. Locus of centroid of ABC is

• x² + y² + z² = h^{- 2}

• x² + y² + z² = 4h^{- 2}

• x² + y² + z² = 16h²

• $\frac{1}{{\mathrm{x}}^2} + \frac{1}{{\mathrm{y}}^2} + \frac{1}{{\mathrm{z}}^2} = \frac{9}{{\mathrm{h}}^2}$

Evaluate ${\int }_{- 2}^1\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$, where f(x) = $$\begin{gathered} \left\{1 - 2\mathrm{x}, \mathrm{x} \le 0 \\ 1 + 2\mathrm{x}, \mathrm{x} \ge 0\right\} \end{gathered}$$

• 0

• 2

• 4

• 6

The area (in square units) of the region bounded by x² = 8y, x = 4 and X-axis, is

• $\frac{2}{3}$

• $\frac{4}{3}$

• $\frac{8}{3}$

• $\frac{10}{3}$

$\int \frac{\mathrm{dx}}{\sqrt{\mathrm{x}}\left(\mathrm{x} +\hspace{0.17em}9\right)}$ is equal to

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

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

• ${\tan }^{-1}\left(\sqrt{\mathrm{x}}\right) + \mathrm{C}$

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

$\int {\left(\mathrm{x} + 1\right)}^2{\mathrm{e}}^{\mathrm{x}}\mathrm{dx}$ is equal to

• xe^x + C

• x²x^x + C

• (x + 1)e^x + C

• (x² + 1)e^x + C