If A, B and C are mutually exclusive and exhaustive events of a random experiment such that P(B) = $\frac{3}{2}$P(A) and P(C) = $\frac{1}{2}$P(B), then P(A ∪ C) equals to

• $\frac{10}{13}$

• $\frac{3}{13}$

• $\frac{6}{13}$

• $\frac{7}{13}$

If ${\mathrm{x}}_1, {\mathrm{x}}_2, ... , {\mathrm{x}}_{\mathrm{n}}$ are n observations such that $\sum _{\mathrm{i} = 1}^{\mathrm{n}}{{\mathrm{x}}_1}^2 = 400 \mathrm{and} \sum _{\mathrm{i} = 1}^{\mathrm{n}}{\mathrm{x}}_1 = 80$, then the least value of n is 

• 18

• 12

• 15

• 16

The mean of four observations is 3. If the sum of the squares of these observations is 48, then their standard deviation is

• $\sqrt{7}$

• $\sqrt{2}$

• $\sqrt{3}$

• $\sqrt{5}$

54.

$$\begin{gathered} \mathrm{The} \mathrm{shortest} \mathrm{distance} \mathrm{between} \mathrm{the} \mathrm{skew} \mathrm{lines} \\ \hat{\mathrm{r}} = \left(\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}\right) + \mathrm{t}\left(\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right) \mathrm{and} \hat{\mathrm{r}} = \left(4\hat{\mathrm{i}} + 5\hat{\mathrm{j}} + 6\hat{\mathrm{k}}\right) + \mathrm{t}\left(2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + \hat{\mathrm{k}}\right) \mathrm{is} \end{gathered}$$

• $\sqrt{6}$

• 3

• $2\sqrt{3}$

• $\sqrt{3}$

$\underset{\mathrm{x}\to 0}{\lim }\frac{\sqrt{1 + {\mathrm{x}}^2} - \sqrt{1 - \mathrm{x} + {\mathrm{x}}^2}}{3^{\mathrm{x}} - 1} = ?$

• $\frac{1}{{\log }_{\mathrm{e}}\left(3\right)}$

• ${\log }_{\mathrm{e}}\left(9\right)$

• $\frac{1}{{\log }_{\mathrm{e}}\left(9\right)}$

• ${\log }_{\mathrm{e}}\left(3\right)$

$\mathrm{If} \mathrm{y} = {\tan }^{-1}\left(\frac{\sqrt{1 +\hspace{0.17em}{\mathrm{a}}^2{\mathrm{x}}^2} - 1}{\mathrm{ax}}\right), \mathrm{then} \left(1 + {\mathrm{a}}^2{\mathrm{x}}^2\right)\mathrm{y}\text{'}\text{'} +\hspace{0.17em}2{\mathrm{a}}^2\mathrm{y}\text{'} = ?$

• - 2a²

• a²

• 2a²

• 0

The condition that f(x) = a + bx² + cx + d has no extreme value, is

• ${\mathrm{b}}^2 > 3\mathrm{ac}$

• ${\mathrm{b}}^2 = 4\mathrm{ac}$

• ${\mathrm{b}}^2 = 3\mathrm{ac}$

• ${\mathrm{b}}^2 < 3\mathrm{ac}$

If there is an error of ± 0.04cm in the measurement of the diameter of a sphere, then the approximate percentage error in its volume, when the radius is 10cm, is

• $\pm 1.2$

• $\pm 0 . 06$

• $\pm 0 . 006$

• $\pm 0 . 6$

If a, b, c and d are real numbers such that ${\mathrm{a}}^2 + {\mathrm{b}}^2 + {\mathrm{c}}^2 + {\mathrm{d}}^2 = 1 \mathrm{and} \mathrm{A} = \left[\begin{array}{cc}\mathrm{a} + \mathrm{id}& \mathrm{c} + \mathrm{id}\\ - \mathrm{c} + \mathrm{id}& \mathrm{a} - \mathrm{ib}\end{array}\right], \mathrm{then} {\mathrm{A}}^{ - 1} = ?$

• $\left[\begin{array}{cc}\mathrm{a} +ib& - \mathrm{c} - \mathrm{id}\\ - \mathrm{c} - \mathrm{id}& \mathrm{a} - \mathrm{ib}\end{array}\right]$

• $\left[\begin{array}{cc}\mathrm{a} - \mathrm{ib}& \mathrm{c} - \mathrm{id}\\ - \mathrm{c} + \mathrm{id}& \mathrm{a} + \mathrm{ib}\end{array}\right]$

• $\left[\begin{array}{cc}\mathrm{a} - \mathrm{ib}& - \mathrm{c} - \mathrm{id}\\ \mathrm{c} - \mathrm{id}& \mathrm{a} + \mathrm{ib}\end{array}\right]$

• $\left[\begin{array}{cc}\mathrm{a} +\mathrm{id}& \mathrm{c} + \mathrm{id}\\ \mathrm{c} - \mathrm{id}& \mathrm{a} - \mathrm{ib}\end{array}\right]$

$\mathrm{If} \mathrm{the} \mathrm{matrix} \mathrm{A} = \left[\begin{array}{cccc}1& 2& 3& 0\\ 2& 4& 3& 2\\ 3& 2& 1& 3\\ 6& 8& 7& \mathrm{\alpha }\end{array}\right] \mathrm{is} \mathrm{of} \mathrm{rank} 3, \mathrm{then} \mathrm{\alpha } = ?$

• - 5

• 5

• 4

• 1