21.

$$\begin{gathered} \mathrm{If} \frac{1}{2 \times 4} + \frac{1}{4 \times 6} + \frac{1}{6 \times 8} + . . . \left(\mathrm{n} \mathrm{terms}\right) = \frac{\mathrm{kn}}{\mathrm{n} \times 1}, \\ \mathrm{then} \mathrm{k} =? \end{gathered}$$

• $\frac{1}{4}$

• $\frac{1}{2}$

• 1

• $\frac{1}{8}$

A regular polygon of n sides has 170 diagonals,then n is equal to

• 12

• 17

• 20

• 25

A committee of 12 members is to be formed from 9 women and 8 men. The number of committees in which the women are in majority is

• 2720

• 2702

• 2270

• 2278

A student has to answer 10 out of 13 questions in an examination choosing atleast 5 questions from the first 6 questions. The number of choice available to the student is

• 63

• 91

• 161

• 196

$\sum _{\mathrm{k} = 1}^{\infty }\sum _{\mathrm{r} = 0}^{\mathrm{k}}\frac{1}{3^{\mathrm{k}}}\left({}^{\mathrm{k}}\mathrm{C}_{\mathrm{r}}\right) = ?$

• $\frac{1}{3}$

• $\frac{2}{3}$

• 1

• 2

$$\begin{gathered} \mathrm{If} \mathrm{ab} \ne 0 \mathrm{and} \mathrm{the} \mathrm{sum} \mathrm{of} \mathrm{the} \mathrm{coefficients} \mathrm{of} {\mathrm{x}}^7 \\ \mathrm{and} {\mathrm{x}}^4 \mathrm{in} \mathrm{the} \mathrm{expansion} \mathrm{of} {\left(\frac{{\mathrm{x}}^2}{\mathrm{a}} - \frac{\mathrm{b}}{\mathrm{x}}\right)}^{11} \mathrm{is} 0, \mathrm{then} \end{gathered}$$

• a = b

• a + b = 0

• ab = - 1

• ab = 1

$\frac{1}{\mathrm{x}\left(\mathrm{x} + 1\right)\left(\mathrm{x} + 2\right) . . . \left(\mathrm{x} + \mathrm{n}\right)} = \frac{{\mathrm{A}}_0}{\mathrm{x}} + \frac{{\mathrm{A}}_1}{\mathrm{x} + 1} + \frac{{{\displaystyle \mathrm{A}}}_{\mathrm{n}}}{\mathrm{x} + \mathrm{n}}, 0 \le \mathrm{i} \le \mathrm{r} \Rightarrow {\mathrm{A}}_{\mathrm{r}} = ?$

• $\frac{{\left(- 1\right)}^{\mathrm{r}}\mathrm{r}!}{\left(\mathrm{n} - \mathrm{r}\right)!}$

• $\frac{{\left(- 1\right)}^{\mathrm{r}}}{\mathrm{r}!\left(\mathrm{n} - \mathrm{r}\right)!}$

• $\frac{1}{\mathrm{r}!\left(\mathrm{n} - \mathrm{r}\right)!}$

• $\frac{\mathrm{r}!}{\left(\mathrm{n} - \mathrm{r}\right)!}$

$1 + \frac{1}{3 . 2^2} + \frac{1}{5 . 2^4} + \frac{1}{7 . 2^6} + ... =?$

• ${\log }_{\mathrm{e}}\left(2\right)$

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

• ${\log }_{\mathrm{e}}\left(4\right)$

• ${\log }_{\mathrm{e}}\left(5\right)$

$\mathrm{In} ∆\mathrm{PQR}, \angle \mathrm{R} = \frac{\mathrm{\pi }}{4}, \tan \left(\frac{\mathrm{P}}{3}\right), \tan \left(\frac{\mathrm{Q}}{3}\right) \mathrm{are} \mathrm{the} \mathrm{roots} \mathrm{of} \mathrm{the} \mathrm{equation} {\mathrm{ax}}^2 + \mathrm{bx} + \mathrm{c} = 0, \mathrm{then}$

• a +b = c

• b + c = 0

• a + c = 0

• b = c

$\mathrm{The} \mathrm{product} \mathrm{of} \mathrm{real} \mathrm{of} \mathrm{the} \mathrm{equation} {\left|\mathrm{x}\right|}^{\raisebox{1ex}{$6$}\!\left/ \!\raisebox{-1ex}{$5$}\right.} - 26{\left|\mathrm{x}\right|}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.} - 27 = 0$

• - 3¹⁰

• $- 3^{12}$

• - 3^12/5

• - 3^12/5