Let a = $\frac{{10}^{\mathrm{n}}}{\mathrm{n}!}$ for n = 1, 2, 3 . . . then the greatest  value of n for which a_n  is the greatest is

• 11

• 20

• 10

• 8

$\mathrm{If} (1 + 2\mathrm{x} + 3{\mathrm{x}}^2{)}^{10} = {\mathrm{a}}_0 + {\mathrm{a}}_1\mathrm{x} + {\mathrm{a}}_2{\mathrm{x}}^2 + . . . + {\mathrm{a}}_{20}{\mathrm{x}}^{20}, \mathrm{then} \frac{{\mathrm{a}}_2}{{\mathrm{a}}_1} = ?$

• 10.5

• 21

• 10

• 5.5

The condition that the x³ - bx² + cx - d = 0 are in progression is

• ${\mathrm{c}}^3 = {\mathrm{b}}^3\mathrm{d}$

• ${\mathrm{c}}^2 = {\mathrm{b}}^2\mathrm{d}$

• $\mathrm{c} = {\mathrm{bd}}^3$

• $\mathrm{c} = {\mathrm{bd}}^2$

$\sum _{\mathrm{n} = 1}^{\infty } \frac{2\mathrm{n}}{\left(2\mathrm{n} + 1\right){\displaystyle !}} = ?$

• $\frac{1}{\mathrm{e}}$

• $\frac{\mathrm{e}}{2}$

• e

• 2e

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

• $\frac{1}{4}$

• $\frac{1}{2}$

• 1

• $\frac{1}{8}$

176.

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

• $\frac{1}{3}$

• $\frac{2}{3}$

• 1

• 2

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

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

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

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

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

If a, b and c form a geometric progression with common ratio r, then the sum of the ordinates of the points of intersection of the line ax + by + c = 0 and the curve x + 2y² = 0 is

• $- \frac{{\mathrm{r}}^2}{2}$

• $- \frac{\mathrm{r}}{2}$

• $\frac{\mathrm{r}}{2}$

• r

If X is a poisson variate such that a = P(X = 1) = P(X = 2), then P(X = 4) is equal to

• $2\mathrm{\alpha }$

• $\frac{\mathrm{\alpha }}{3}$

• ${\mathrm{\alpha e}}^{ - 2}$

• ${\mathrm{\alpha e}}^2$

If pth, qth, rth terms of a geometric progression are the positive numbers a, b and c respectively,then the angle· between the vectors (log(a))²i + (log(b))²j + (log(c))²k and (q - r)i + (r - p)j + (p - q)k is

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

• ${\sin }^{-1}\left(\frac{1}{\sqrt{{\mathrm{a}}^2 + {\mathrm{b}}^2 + {\mathrm{c}}^2}}\right)$

• $\frac{\mathrm{\pi }}{4}$