51.

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

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 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

• $- \frac{1}{16}$

• $\frac{15}{8}$

• $- \frac{1}{8}$

• $\frac{15}{16}$

For every real number x, 

let f(x) = $\frac{\mathrm{x}}{1!} + \frac{3}{2!}{\mathrm{x}}^2 + \frac{7}{3!}{\mathrm{x}}^3 + \frac{15}{4!}{\mathrm{x}}^4 + ...$ Then, the equation f(x) = 0 has

• no real solution

• exactly one real solution

• exactly two real solutions

• infinite number of real solutions

Let S denote the sum of the infinite series $1 + \frac{8}{2!} + \frac{21}{3!} + \frac{40}{4!} + \frac{65}{5!} + ...$

• $\mathrm{S} < 8$

• $\mathrm{S} > 12$

• 8 < S < 12

• S = 8

Let t_n denotes the nth term of the infinite series $\frac{1}{1!} + \frac{10}{2!} + \frac{21}{3!} + \frac{34}{4!} + \frac{49}{5!} +\hspace{0.17em}...$ . Then, $\underset{\mathrm{n}\to \infty }{\lim }{\mathrm{t}}_{\mathrm{n}}$ is

• e

• 0

• e²

• 1

Five numbers are in HP. The middle term is 1 and the ratio of the second and the fourth terms is 2 : 1. Then, the sum of the first three terms is

• $\frac{11}{2}$

• 5

• 2

• $\frac{14}{3}$