21.

Ram is visiting a friend. Ram knows that his friend has 2 children and 1 of them is a boy. Assuming that a child is equally likely to be a boy or a girl, then the probability that the other child is a girl, is

• 1/2

• 1/3

• 2/3

• 7/10

The value of the sum ${\left({}^{\mathrm{n}}\mathrm{C}_1\right)}^2 + {\left({}^{\mathrm{n}}\mathrm{C}_2\right)}^2 + {\left({}^{\mathrm{n}}\mathrm{C}_3\right)}^2 + ... + {\left({}^{\mathrm{n}}\mathrm{C}_{\mathrm{n}}\right)}^2$ is

• ${\left({}^{2\mathrm{n}}\mathrm{C}_{\mathrm{n}}\right)}^2$

• ${}^{2\mathrm{n}}\mathrm{C}_{\mathrm{n}}$

• ${}^{2\mathrm{n}}\mathrm{C}_{\mathrm{n}} + 1$

• ${}^{2\mathrm{n}}\mathrm{C}_{\mathrm{n}} - 1$

The remainder obtained when 1! + 2! + 3! + ... + 11! is divided by 12 is

• 9

• 8

• 7

• 6

Out of 7 consonants and 4 vowels, the number of words (not necessarily meaningful) that can be made, each consisting of 3 consonants and 2 vowels is,

• 24800

• 25100

• 25200

• 25400

Let S = $\frac{2}{1}{}^{\mathrm{n}}\mathrm{C}_0 + \frac{2^2}{2}{}^{\mathrm{n}}\mathrm{C}_1 + \frac{2^3}{3}{}^{\mathrm{n}}\mathrm{C}_2 + ... + \frac{2^{\mathrm{n} + 1}}{\mathrm{n} + 1}{}^{\mathrm{n}}\mathrm{C}_{\mathrm{n}}$. Then, S equals

• $\frac{2^{\mathrm{n} + 1} - 1}{\mathrm{n} + 1}$

• $\frac{3^{\mathrm{n} + 1} - 1}{\mathrm{n} + 1}$

• $\frac{3^{\mathrm{n}} - 1}{\mathrm{n}}$

• $\frac{2^{\mathrm{n}} - 1}{\mathrm{n}}$

If a, b and c are positive numbers in a GP, then the roots of the quadratic equation

$\left({\log }_{\mathrm{e}}\left(\mathrm{a}\right)\right){\mathrm{x}}^2 - \left(2{\log }_{\mathrm{e}}\left(\mathrm{b}\right)\right)\mathrm{x} + {\log }_{\mathrm{e}}\left(\mathrm{c}\right) = 0$ are

• $- 1 \mathrm{and} \frac{{\log }_{\mathrm{e}}\left(\mathrm{c}\right)}{{\log }_{\mathrm{e}}\left(\mathrm{a}\right)}$

• $1 \mathrm{and} - \frac{{\log }_{\mathrm{e}}\left(\mathrm{c}\right)}{{\log }_{\mathrm{e}}\left(\mathrm{a}\right)}$

• $1 \mathrm{and} {\log }_{\mathrm{a}}\left(\mathrm{c}\right)$

• $- 1 \mathrm{and} {\log }_{\mathrm{c}}\left(\mathrm{a}\right)$

The range of the function y = $3\sin \left(\sqrt{\frac{{\mathrm{\pi }}^2}{16} - {\mathrm{x}}^2}\right)$ is

• $\left[0, \sqrt{3/2}\right]$

• [0, 1]

• $\left[0, 3/\sqrt{2}\right]$

• $\left[0, \infty \right]$

Let f(x) be a differentiable function and f'(4) = 5. Then

$\underset{\mathrm{x}\to 2}{\lim }\frac{\mathrm{f}\left(4\right) - \mathrm{f}\left({\mathrm{x}}^2\right)}{\mathrm{x} - 2}$ equals

• 0

• 5

• 20

• - 20

The sum of the series $\sum _{\mathrm{n} = 1}^{\infty }\sin \left(\frac{\mathrm{n}! \mathrm{\pi }}{720}\right)$ is

• $\sin \left(\frac{\mathrm{\pi }}{180}\right) + \sin \left(\frac{\mathrm{\pi }}{360}\right) + \sin \left(\frac{\mathrm{\pi }}{540}\right)$

• $\sin \left(\frac{\mathrm{\pi }}{6}\right) + \sin \left(\frac{\mathrm{\pi }}{30}\right) + \sin \left(\frac{\mathrm{\pi }}{120}\right) + \sin \left(\frac{\mathrm{\pi }}{360}\right)$

• $\sin \left(\frac{\mathrm{\pi }}{6}\right) + \sin \left(\frac{\mathrm{\pi }}{30}\right) + \sin \left(\frac{\mathrm{\pi }}{120}\right) + \sin \left(\frac{\mathrm{\pi }}{360}\right) + \sin \left(\frac{\mathrm{\pi }}{720}\right)$

• $\sin \left(\frac{\mathrm{\pi }}{180}\right) + \sin \left(\frac{\mathrm{\pi }}{360}\right)$

The coefficient of x³ in the infinite series  expansion of 

$\frac{2}{\left(1 - \mathrm{x}\right)\left(2 - \mathrm{x}\right)}, \mathrm{for} \left|\mathrm{x}\right| < 1$, is

• $- \frac{1}{16}$

• $\frac{15}{8}$

• $- \frac{1}{8}$

• $\frac{15}{16}$