Two numbers are chosen at random from{1, 2, 3, 4, 5, 6, 7, 8} at a time. The probability that smaller of the two numbers is less than 4 is

• $\frac{7}{14}$

• $\frac{8}{14}$

• $\frac{9}{14}$

• $\frac{10}{14}$

Two fair dice are rolled. The probability of the sum of digits on their faces to be greater than or equal to 10 is

• $\frac{1}{5}$

• $\frac{1}{4}$

• $\frac{1}{8}$

• $\frac{1}{6}$

A bag contains 2n + 1 corns. It is known that n of these coins have a head on both sides, whereas the remaining n + 1 coins are fair. A coin is picked up at random from the bag and tossed. If the probability that the toss results in a head is $\frac{31}{42}$, then n is equal to

• 10

• 11

• 12

• 13

The random variable takes the values 1, 2, 3, 1 ..., m. If P(X = n) = $\frac{1}{\mathrm{m}}$ to each n, then the variance of X is

• $\frac{\left(\mathrm{m} + 1\right)\left(2\mathrm{m} + 1\right)}{6}$

• $\frac{{\mathrm{m}}^2 - 1}{12}$

• $\frac{\mathrm{m} + 1}{2}$

• $\frac{{\mathrm{m}}^2 + 1}{12}$

$\mathrm{If} \mathrm{X} \mathrm{is} \mathrm{a} \mathrm{poisson} \mathrm{variate} \mathrm{P}\left(\mathrm{X} = 1\right) = 2\mathrm{P}\left(\mathrm{X} = 2\right), \mathrm{then} \mathrm{P}\left(\mathrm{X} = 3\right) = ?$

• $\frac{{\mathrm{e}}^{ - 1}}{6}$

• $\frac{\mathrm{e}{ }^{ - 2}}{2}$

• $\frac{{\mathrm{e}}^{ - 1}}{2}$

• $\frac{{\mathrm{e}}^{ - 1}}{3}$

The direction ratios of the two lines AB and AC are 1, - 1, - 1 and 2, - 1, 1. The direction ratios of the normal to the plane ABC are

• 2, 3, -1

• 2, 2, 1

• 3, 2, - 1

• - 1, 2, 3

67.

A plane passing through(- 1, 2, 3) and whose normal makes equal angles with the coordinate axes is

• x + y + z + 4 = 0

• x - y + z + 4 = 0

• x + y + z - 4 = 0

• x + y + z = 0

A variable plane passes through a fixed point (1, 2, 3). Then, the foot of the perpendicular from the origin to the plane lies on

• a circle

• a sphere

• an ellipse

• a parabola

$$\begin{gathered} \mathrm{If} {\cos }^{-1}\left(\frac{\mathrm{y}}{\mathrm{b}}\right) = 2\log \left(\frac{\mathrm{x}}{2}\right), \mathrm{where} \mathrm{x} > 0, \mathrm{then} \\ {\mathrm{x}}^2\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + \mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}} = ? \end{gathered}$$

• 4y

• - 4y

• 0

• - 8y

If the curves x² + py² = l and qx² + y² = l are orthogonal to each other, then

• p - q = 2

• $\frac{1}{\mathrm{p}} - \frac{1}{\mathrm{q}} = 2$

• $\frac{1}{\mathrm{p}} + \frac{1}{\mathrm{q}} = - 2$

• $\frac{1}{\mathrm{p}} + \frac{1}{\mathrm{q}} = 2$