The position vectors of P and Q are $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ respectively. If R is a point on PQ such that $\overrightarrow{\mathrm{PR}} = 5\overrightarrow{\mathrm{PQ}}$, then the position vector of R is

• $5\overrightarrow{\mathrm{b}} - 4\overrightarrow{\mathrm{a}}$

• $5\overrightarrow{\mathrm{b}} + 4\overrightarrow{\mathrm{a}}$

• $4\overrightarrow{\mathrm{b}} - 5\overrightarrow{\mathrm{a}}$

• $4\overrightarrow{\mathrm{b}} + 5\overrightarrow{\mathrm{a}}$

If the points with position vectors $60\hat{\mathrm{i}} + 3\hat{\mathrm{j}}, 40\hat{\mathrm{i}} - 8\hat{\mathrm{j}} \mathrm{and} \mathrm{a}\hat{\mathrm{i}} - 52\hat{\mathrm{j}} \mathrm{are} \mathrm{collinear}, \mathrm{then} \mathrm{a} \mathrm{is} \mathrm{equal} \mathrm{to}$  are collinear,then a is equal to

• - 40

• - 20

• 20

• 40

If the position vectors of A, B and C are respectively $2\hat{\mathrm{I}} - \hat{\mathrm{J}} + \hat{\mathrm{K}}, \hat{\mathrm{I}} - 3\hat{\mathrm{J}} - 5\hat{\mathrm{K}} \mathrm{and} 3\hat{\mathrm{i}} - 4\hat{\mathrm{j}} - 4\hat{\mathrm{k}}, \mathrm{then} {\cos }^2\left(\mathrm{A}\right) = ?$ 

• 0

• $\frac{6}{41}$

• $\frac{35}{41}$

• 1

$$\begin{gathered} \mathrm{Let} \overrightarrow{\mathrm{a}} \mathrm{be} \mathrm{a} \mathrm{unit} \mathrm{vector}, \overrightarrow{\mathrm{b}} = 2\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{c}} =\hat{\mathrm{i}} + 3\hat{\mathrm{k}}. \\ \mathrm{Then}, \mathrm{maximum} \mathrm{value} \mathrm{of}[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b} }\overrightarrow{\mathrm{c}}] \mathrm{is} \end{gathered}$$

• - 1

• $\sqrt{10} + \sqrt{6}$

• $\sqrt{10} - \sqrt{6}$

• $\sqrt{59}$

$$\begin{gathered} \mathrm{If} \mathrm{A} \mathrm{and} \mathrm{B} \mathrm{are} \mathrm{independent} \mathrm{events} \mathrm{of} \mathrm{arandom} \mathrm{experiment} \mathrm{such} \mathrm{that} \\ \mathrm{P}(\mathrm{A} \cap \mathrm{B}) = \frac{1}{6} \mathrm{and} \mathrm{P}\left(\overline{\mathrm{A}} \cap \overline{\mathrm{B}}\right) = \frac{1}{3}, \mathrm{then} \mathrm{P}\left(\mathrm{A}\right) = ? \end{gathered}$$

• $\frac{1}{4}$

• $\frac{1}{3}$

• $\frac{1}{2}$

• $\frac{2}{3}$

66.

Let S be the sample space of the random experiment of throwing simultaneously two unbiased dice with six faces (numbered1 to 6) and let E_k = {(a, b) ∈ S : ab = k} for k $\ge$ 1. If p_k + P(E_k) for k $\ge$ 1, then the correct among the following, is

• ${\mathrm{p}}_1 < {\mathrm{P}}_{30} < {\mathrm{P}}_4 < {\mathrm{P}}_6$

• ${\mathrm{p}}_{36} < {\mathrm{P}}_6 < {\mathrm{P}}_2 < {\mathrm{P}}_4$

• ${\mathrm{p}}_1 < {\mathrm{P}}_{11} < {\mathrm{P}}_4 < {\mathrm{P}}_6$

• ${\mathrm{p}}_{36} < {\mathrm{P}}_{11} < {\mathrm{P}}_6 < {\mathrm{P}}_4$

For k = 1, 2, 3 the box B_k contains k red balls and        (k + 1) white balls. Let P(B₁) = $\frac{1}{2}$, P(B₂) = $\frac{1}{3}$ and P(B₃) = $\frac{1}{6}$. A box is selected at 36 random and a ball is drawn from it. If a redball is drawn, then the probability that it has come from box B, is

• $\frac{35}{78}$

• $\frac{14}{39}$

• $\frac{10}{13}$

• $\frac{12}{13}$

$\mathrm{The} \mathrm{distribution} \mathrm{of} \mathrm{a} \mathrm{random} \mathrm{variable} \mathrm{X} \mathrm{is} \mathrm{given} \mathrm{below}$

• $\frac{1}{10}$

• $\frac{2}{10}$

• $\frac{3}{10}$

• $\frac{7}{10}$

If X is a Poisson variate such that P(X = 1) = P(X = 2), then P(X = 4) is equal to

• $\frac{1}{2{\mathrm{e}}^2}$

• $\frac{1}{3{\mathrm{e}}^2}$

• $\frac{2}{3{\mathrm{e}}^2}$

• $\frac{1}{{\mathrm{e}}^2}$

The angle between the lines whose direction 

cosine are $\left(\frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{\sqrt{3}}{2}\right) \mathrm{and} \left(\frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{- \sqrt{3}}{2}\right), \mathrm{is}$

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$