If $\frac{3\mathrm{x}}{\left(\mathrm{x} - \mathrm{a}\right)\left(\mathrm{x} - \mathrm{b}\right)} = \frac{2}{\mathrm{x} - \mathrm{a}} + \frac{1}{\mathrm{x} - \mathrm{b}}$, then a : b is equal to

• 1 : 2

• - 2 : 1

• 1 : 3

• 3 : 1

The coefficient of x^k in the expansion of $\frac{1 - 2\mathrm{x} - {\mathrm{x}}^2}{{\mathrm{e}}^{- \mathrm{x}}}$ is

• $\frac{1 - \mathrm{k} - {\mathrm{k}}^2}{\mathrm{k}!}$

• $\frac{{\mathrm{k}}^2 + 1}{\mathrm{k}!}$

• $\frac{1 - \mathrm{k}}{\mathrm{k}!}$

• $\frac{1}{\mathrm{k}!}$

$\frac{1}{2} - \frac{{\displaystyle 1}}{{\displaystyle 2 . 2^2}} + \frac{1}{3 . 2^3} - \frac{1}{4 . 2^4} + ...$ is equal to

• $\frac{1}{4}$

• ${\log }_{\mathrm{e}}\left(\frac{{\displaystyle 3}}{{\displaystyle 4}}\right)$

• ${\log }_{\mathrm{e}}\left(\frac{{\displaystyle 3}}{{\displaystyle 2}}\right)$

• ${\log }_{\mathrm{e}}\left(\frac{{\displaystyle 2}}{{\displaystyle 3}}\right)$

If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ are the roots of the equation ax² + bx + c = 0 and, if px² + qx + r = 0 has roots $\frac{1 - \mathrm{\alpha }}{\mathrm{\alpha }} \mathrm{and} \frac{1 - \mathrm{\beta }}{\mathrm{\beta }}$, then r is equal to

• a + 2b

• a + b + c

• ab + bc + ca

• abc

The set of values of x for which the inequalities x² - 3x - 10 < 0, 10x - x² - 16 > 0 hold simultaneously, is

• (- 2, 5)

• (2, 8)

• (- 2, 8)

• (2, 5)

If 1, 2, 3 and 4 are the roots of the equation x⁴ + ax³ + bx² + cx + d = 0, then a + 2b + c is equal to

• - 25

• 0

• 10

• 24

If $\mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma }$ are the roots of x³ - 2x² + 3x - 4 = 0, then the value of ${\mathrm{\alpha }}^2{\mathrm{\beta }}^2 + {\mathrm{\beta }}^2{\mathrm{\gamma }}^2 + {\mathrm{\gamma }}^2{\mathrm{\alpha }}^2$ is

• - 7

• - 5

• - 3

• 0

18.

$\mathrm{The} \mathrm{locus} \mathrm{of} \mathrm{the} \mathrm{point} \mathrm{z} = \mathrm{x} + \mathrm{iy} \mathrm{satisfying} \left|\begin{array}{c}\frac{\mathrm{z} - 2\mathrm{i}}{\mathrm{z} + 2\mathrm{i}}\end{array}\right| = 1 \mathrm{is}$

• x-axis

• y-axis

• y = 0

• x = 2

$\mathrm{A} \mathrm{value} \mathrm{of} \mathrm{n} \mathrm{such} \mathrm{that} {\left(\frac{\sqrt{3}}{2} +\hspace{0.17em}\frac{\mathrm{i}}{2}\right)}^{\mathrm{n}} = 1 \mathrm{is}$

• 12

• 3

• 2

• 1

$$\begin{gathered} \mathrm{If} \mathrm{\theta } \mathrm{Lies} \mathrm{in} \mathrm{the} \mathrm{first} \mathrm{quadrant} \mathrm{and} 5\mathrm{tan\theta } = 4, \\ \mathrm{then} \frac{5\sin \left(\mathrm{\theta }\right) - 3\cos \left(\mathrm{\theta }\right)}{5\sin \left(\mathrm{\theta }\right)- 3\cos \left(\mathrm{\theta }\right)} = ? \end{gathered}$$

• $\frac{5}{14}$

• $\frac{3}{14}$

• $\frac{1}{14}$

• 0