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

• 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

• $\frac{1}{8}$

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

• $\frac{1}{3}$

• $\frac{2}{3}$

• 1

• 2

186.

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

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

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

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

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

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

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

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

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

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

Given that, ${\mathrm{a\alpha }}^2 + 2\mathrm{b\alpha } + \mathrm{c} \ne 0 \mathrm{and} \mathrm{that} \mathrm{the} \mathrm{system} \mathrm{of} \mathrm{equations}$

$$\begin{gathered} \left(\mathrm{a\alpha } + \mathrm{b}\right)\mathrm{x} + \mathrm{ay} + \mathrm{bz} = 0; \\ \left(\mathrm{b\alpha } + \mathrm{c}\right)\mathrm{x} +\hspace{0.17em}\mathrm{by} +\hspace{0.17em}\mathrm{cz} = 0; \\ \left(\mathrm{a\alpha } + \mathrm{b}\right)\mathrm{y} + \left(\mathrm{b\alpha } +\hspace{0.17em}\mathrm{c}\right)\mathrm{z} = 0 \\ \mathrm{has} \mathrm{a} \mathrm{non}-\mathrm{trival} \mathrm{solution}, \mathrm{then} \mathrm{a}, \mathrm{b} \mathrm{and} \mathrm{c} \mathrm{lie} \mathrm{in} \end{gathered}$$

• Arithmetic Progression

• Geometric Progression

   

• Harmonic Progression

• Arithmetico- geometric Progression

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

The point (3, 2) undergoes the following three transformations in the order given

(i) Reflection about the line y = x.

(ii) Translation by the distance 1 unit in the positive direction of x-axis.

(iii) Rotation by an angle $\frac{\mathrm{\pi }}{4}$ about the origin in  anti-clockwise direction.

Then, the final position of the point is

• $\left(- \sqrt{18}, \sqrt{18}\right)$

• (- 2, 3)

• $\left(0, \sqrt{18}\right)$

• (0, 3)