61.

Suppose A and B are two events such that  is equal to

• $\frac{3}{11}$

• $\frac{1}{11}$

• $\frac{9}{11}$

If three unit vectors $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ satisfy $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}$, then the angle between a and b is

• $\frac{2\mathrm{\pi }}{3}$

• $\frac{5\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{6}$

Suppose that a random variable X follows Poisson distribution. If P(X = 1) = P(X = 2) then P(X = 5) is equal to

• $\frac{2}{3}{\mathrm{e}}^{- 2}$

• $\frac{3}{4}{\mathrm{e}}^{- 2}$

• $\frac{4}{15}{\mathrm{e}}^{- 2}$

• $\frac{7}{8}{\mathrm{e}}^{- 2}$

If the mean and variance of a binomial variable X are 2 and 1 respectively, then P(X $\ge$ 1) is equal to

• $\frac{2}{3}$

• $\frac{15}{16}$

• $\frac{7}{8}$

• $\frac{4}{5}$

$\left(\overrightarrow{\mathrm{a}} + \overrightarrow{2\mathrm{b}} - \overrightarrow{\mathrm{c}}\right) . \left(\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}}\right) \times \left(\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}} - \overrightarrow{\mathrm{c}}\right) = ?$

• $- \left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

• $2\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

• $3\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

$\mathrm{If} \overrightarrow{\mathrm{u}} = \overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{v}} = \overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}, \left|\overrightarrow{\mathrm{a}}\right| = \left|\overrightarrow{\mathrm{b}}\right| = 2, \mathrm{then} \left|\overrightarrow{\mathrm{u}} \times \overrightarrow{\mathrm{v}}\right| = ?$

• $2\sqrt{16 - {\left(\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}\right)}^2}$

• $2\sqrt{16 + {\left(\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}}\right)}^2 }$

• $2\sqrt{4 - {\left(\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}}\right)}^2 }$

• $2\sqrt{4 + {\left(\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}}\right)}^2 }$

$$\begin{gathered} \mathrm{If} \mathrm{the} \mathrm{angle} \mathrm{\theta } \mathrm{between} \mathrm{the} \mathrm{vectors} \overrightarrow{\mathrm{a}} = 2{\mathrm{x}}^2\hat{\mathrm{i}} + 4\mathrm{x}\hat{\mathrm{j}} + \hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{b}} = 7\hat{\mathrm{i}} - 2\hat{\mathrm{j}} +\mathrm{x}\hat{\mathrm{k}} \\ \mathrm{is} \mathrm{such} \mathrm{that} 90° <\mathrm{\theta } < 180°, \mathrm{then} \mathrm{x} \mathrm{lies} \mathrm{in} \mathrm{the} \mathrm{interval} \end{gathered}$$

• $\left(0, \frac{1}{2}\right)$

• $\left(\frac{1}{2}, 1\right)$

• $\left(1, \frac{3}{2}\right)$

• $\left(\frac{1}{2}, \frac{3}{2}\right)$

Let OA, OB, QC be the co-terminal edges of a rectangular parallelopiped of volume V and let P be the vertex opposite to O. Then, $\left[\overrightarrow{\mathrm{AP}} \overrightarrow{\mathrm{BP}} \overrightarrow{\mathrm{CP}}\right]$ is equal to

• 2V

• 12V

• $3\sqrt{3}\mathrm{V}$

• 0

$\mathrm{If} \int \frac{7{\mathrm{x}}^8 + 8{\mathrm{x}}^7}{{\left(1 + \mathrm{x} + {\mathrm{x}}^8\right)}^2}\mathrm{dx} = \mathrm{f}\left(\mathrm{x}\right) + \mathrm{c}, \mathrm{then} \mathrm{f}\left(\mathrm{x}\right) = ?$

• $\frac{{\mathrm{x}}^8}{1 + \mathrm{x} + {\mathrm{x}}^8}$

• $28\log \left(1 + \mathrm{x} + {\mathrm{x}}^8\right)$

• $\frac{1}{1 + \mathrm{x} + {\mathrm{x}}^8}$

• $\frac{- 1}{1 + \mathrm{x} + {\mathrm{x}}^8}$

$$\begin{gathered} \mathrm{If} {\mathrm{f}}_{\mathrm{n}}\left(\mathrm{x}\right) = \log \log \log . . .\log \left(\mathrm{x}\right) \left(\log \mathrm{is} \mathrm{repeated} \mathrm{n}-\mathrm{times}\right), \mathrm{then} \\ \int {\left({\mathrm{xf}}_1\left(\mathrm{x}\right){\mathrm{f}}_2\left(\mathrm{x}\right) . . . {\mathrm{f}}_{\mathrm{n}}\left(\mathrm{x}\right)\right)}^{ - 1}\mathrm{dx} \mathrm{is} \mathrm{equal} \mathrm{to} \end{gathered}$$

• ${\mathrm{f}}_{\mathrm{n} + 1} + \mathrm{c}$

• $\frac{{\mathrm{f}}_{\mathrm{n} + 1}\left(\mathrm{x}\right)}{\mathrm{n} + 1} + \mathrm{c}$

• ${\mathrm{nf}}_{\mathrm{n}\left(\mathrm{x}\right) }+ \mathrm{c}$

• $\frac{{\mathrm{f}}_{\mathrm{n}}\left(\mathrm{x}\right)}{\mathrm{n}} + \mathrm{c}$