$$\begin{gathered} \mathrm{If} \frac{3\mathrm{x} + 2}{\left(\mathrm{x} + 1\right)\left(2{\mathrm{x}}^2 + 3\right)} = \frac{\mathrm{A}}{\mathrm{x} + 1} + \frac{\mathrm{Bx} + \mathrm{C}\hspace{0.17em}}{2{\mathrm{x}}^2 +\hspace{0.17em}3}, \mathrm{then} \\ \mathrm{A} + \mathrm{C} - \mathrm{B} \mathrm{is} \mathrm{equal} \mathrm{to} : \end{gathered}$$

• 0

• 2

• 3

• 5

$\mathrm{The} \mathrm{coefficient} \mathrm{of} {\mathrm{x}}^{\mathrm{n}} \mathrm{in} \frac{1 - 2\mathrm{x}}{{\mathrm{e}}^{\mathrm{x}}} \mathrm{is} :$

• $\frac{\left(1 + 2\mathrm{n}\right)}{\mathrm{n}!}$

• ${\left(- 1\right)}^{\mathrm{n}}\frac{\left(1 + 2\mathrm{n}\right)}{\mathrm{n}!}$

• ${\left(- 1\right)}^{\mathrm{n}}\frac{\left(1 - 2\mathrm{n}\right)}{\mathrm{n}!}$

• ${\left(- 1\right)}^{\mathrm{n}}\frac{\left(1 + 4\mathrm{n}\right)}{\mathrm{n}!}$

$\mathrm{If} \left|\mathrm{x}\right| < 1 \mathrm{and} \mathrm{y} = \mathrm{x} - \frac{{\mathrm{x}}^2}{2} + \frac{{\mathrm{x}}^3}{3} - \frac{{\mathrm{x}}^4}{4} + . . . , \mathrm{then} \mathrm{x} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\mathrm{y} + \frac{{\mathrm{y}}^2}{2} + \frac{{\mathrm{y}}^3}{3} + . . .$

• $\mathrm{y} - \frac{{\mathrm{y}}^2}{2} + \frac{{\mathrm{y}}^3}{3} - \frac{{\mathrm{y}}^4}{4} + . . .$

• $\mathrm{y} + \frac{{\mathrm{y}}^2}{2!} + \frac{{\mathrm{y}}^3}{3!} + . . .$

• $\mathrm{y} - \frac{{\mathrm{y}}^2}{2!} + \frac{{\mathrm{y}}^3}{3!} - \frac{{\mathrm{y}}^4}{4!} + . . .$

• $\mathrm{x} \in \left(- \frac{3}{2}, \frac{1}{4}\right)$

• $\mathrm{x} \in \left(- \frac{3}{4}, \frac{1}{4}\right)$

• $\mathrm{x} \in \left(- \frac{3}{4}, \frac{1}{2}\right)$

• $\mathrm{x} < \frac{1}{4}$

The difference between two roots of the equation x³ - 13x² + 15x + 189 = 0 is 2. Then  the roots of the equation are :

• - 3, 5, 7

• - 3, - 7, - 9

• 3, - 5, 7

• - 3, - 7, 9

16.

$$\begin{gathered} \mathrm{If} \mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma } \mathrm{are} \mathrm{the} \mathrm{roots} \mathrm{of} \mathrm{the} \mathrm{equation} {\mathrm{x}}^3 - 6{\mathrm{x}}^2 + 11\mathrm{x} + 6 = 0, \\ \mathrm{then} \sum _{}{\mathrm{\alpha }}^2\mathrm{\beta } + \sum _{}{\mathrm{\alpha \beta }}^2 \mathrm{is} \mathrm{equal} \mathrm{to}\hspace{0.17em}: \end{gathered}$$

• 80

• 84

• 90

• - 84

The locus of the point z = x + iy satisfying the equation

$\left|\frac{\mathrm{z} - 1}{\mathrm{z} + 1}\right| = 1 \mathrm{is} \mathrm{given} \mathrm{by}$ :

• x = 0

• y = 0

• x = y

• x + y = 0

The equation of the locus of z such that $\left|\frac{\mathrm{z} + \mathrm{i}}{\mathrm{z} - \mathrm{i}}\right| = 2$, where z= x + iy is a complex number, is

• $3{\mathrm{x}}^2 + 3{\mathrm{y}}^2 + 10\mathrm{y} - 3 = 0$

• $3{\mathrm{x}}^2 + 3{\mathrm{y}}^2 + 10\mathrm{y} + 3 = 0$

• $3{\mathrm{x}}^2 - 3{\mathrm{y}}^2 - 10\mathrm{y} - 3 = 0$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 - 5\mathrm{y} + 3 = 0$

The product of the distinct (2n)th roots of 1 + i$\sqrt{3}$ is equal to :

• 0

• $- 1 - \mathrm{i}\sqrt{3}$

• $1 + \mathrm{i}\sqrt{3}$

• $- 1 + \mathrm{i}\sqrt{3}$

$\sin \left(120°\right)\cos \left(150°\right) - \cos \left(240°\right)\sin \left(330°\right) \mathrm{is} \mathrm{equal} \mathrm{to} :$

• 1

• - 1

• $\frac{2}{3}$

• $- \left(\frac{\sqrt{3} + 1}{4}\right)$