21.

• $\left[\frac{1}{2}, \frac{5}{8}\right]$

• $\left[\frac{1}{4}, 1\right]$

• $\left[\frac{1}{4}, \frac{1}{2}\right]$

The most general value of 0 which satisfies both the equations tan 8 = - 1 and $\cos \left(\mathrm{\theta }\right) = \frac{1}{\sqrt{2}}$ is

• $\mathrm{n\pi } + \frac{7\mathrm{\pi }}{4}$

• $2\mathrm{n\pi } + \frac{7\mathrm{\pi }}{4}$

• $\mathrm{n\pi } + {\left(- 1\right)}^{\mathrm{n}}\frac{7\mathrm{\pi }}{4}$

• $\frac{7\mathrm{n\pi }}{4}$

$\mathrm{In} \mathrm{a} ∆\mathrm{ABC}, \mathrm{if} \mathrm{a} {\cos }^2\frac{\mathrm{C}}{2} + {\mathrm{ccos}}^2\frac{\mathrm{A}}{2} = \frac{3\mathrm{B}}{2}, \mathrm{then} \mathrm{the} \mathrm{sides} \mathrm{of} \mathrm{the} \mathrm{triangle} \mathrm{are} \mathrm{in}$

• an arithmetic progression

• a geometric progression

• a harmonic progression

• an arithmetico-geometric progression

$\mathrm{In} \mathrm{a} ∆\mathrm{ABC}, \mathrm{if} \frac{\cos \left(\mathrm{A}\right)}{\mathrm{a}} = \frac{\cos \left(\mathrm{B}\right)}{\mathrm{b}} = \frac{\cos \left(\mathrm{C}\right)}{\mathrm{c}}, \mathrm{then} ∆\mathrm{ABC} \mathrm{is}$

• right angled

• isosceles right angled

• equilateral

• Scalene

The angle of elevation of a stationary cloud from a point 2500m above a lake is 15° and from the same point the angle of depression of its reflection in the lake is 45°. The height(in metres) of the cloud above the lake, given that cot(15°) = 2 + $\sqrt{3}$, is

• 2500

• $2500\sqrt{2}$

• $2500\sqrt{3}$

• 5000

Seven white balls and three black balls are randomly arranged in a row. The probability that no two black balls are placed adjacently is

• $\frac{1}{2}$

• $\frac{7}{15}$

• $\frac{2}{15}$

• $\frac{1}{3}$

The probability distribution of a random variable X is given below

X = x	0	1	2	3
P(X) = x	$\frac{1}{10}$	$\frac{2}{10}$	$\frac{3}{10}$	$\frac{4}{10}$

Then the variance of X is

• 1

• 2

• 3

• 4

The probability that an individual suffers a bad reaction from an injection is 0.001. The probability that out of 2000 individuals exactly three will suffer bad reaction is

• $\frac{1}{{\mathrm{e}}^2}$

• $\frac{2}{3{\mathrm{e}}^2}$

• $\frac{8}{3{\mathrm{e}}^2}$

• $\frac{4}{3{\mathrm{e}}^2}$

The locus of a point such that the sum of its distances from the points (0, 2) and (0, - 2) is 6, is

• 9x² - 5y² = 45

• 5x² + 9y² = 45

• 9x² + 5y² = 45

• 5x² - 9y² = 45

The number of points P(x, y) with natural numbers as coordinates that lie inside the quadrilateral formed by the lines 2x + y = 2, x = 0, y = 0 and x + y = 5 is

• 12

• 10

• 6

• 4