x = log_0.1(0.001), y = log₉(81), then $\sqrt{\mathrm{x} - 2\sqrt{\mathrm{y}}}$ is equal to

• $3 - \sqrt{2}$

• $\sqrt{3} - 2$

• $\sqrt{2} - 1$

• $\sqrt{2} - 2$

If 2³ + 4³ + 6³ + ... + 2(n)³ = hn²(n + 1)², then h is equal to

• $\frac{1}{2}$

• 1

• $\frac{3}{2}$

• 2

The number of ways in which 5 boys and 4 girls sit around a circular tables. So, that no two girls sit together is

• 5! 4!

• 3! 3!

• 5!

• 4!

Using the digits 0, 2, 4, 6, 8 not more than once in any number, the number of 5 digited numbers that can be formed, is

• 16

• 24

• 96

• 120

$1 + \frac{1}{4} + \frac{1}{4} . \frac{3}{8} + \frac{1}{4} . \frac{3}{8} . \frac{5}{12 } + . . . \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\sqrt{2}$

• $\frac{1}{\sqrt{2}}$

• $\sqrt{3}$

• $\frac{1}{\sqrt{3}}$

$\mathrm{The} \mathrm{coefficient} \mathrm{of}{ }^{ }{\mathrm{x}}^4 \mathrm{in} \mathrm{the} \mathrm{expansion} \mathrm{of} \frac{{\left(1 - 3\mathrm{x}\right)}^2}{\left(1 - 2\mathrm{x}\right)} \mathrm{is} \mathrm{equal} \mathrm{to}$

• 1

• 2

• 3

• 4

$$\begin{gathered} \mathrm{If} {\left(1 + \mathrm{x}\right)}^{\mathrm{n}} = {\mathrm{C}}_0 + {\mathrm{C}}_1\mathrm{x} + {\mathrm{C}}_2{\mathrm{x}}^2 + .... + {\mathrm{C}}_{\mathrm{n}}{\mathrm{x}}^{\mathrm{n}} ,\mathrm{then} \\ {\mathrm{C}}_0 + 2{\mathrm{C}}_1 + 3{\mathrm{C}}_2 + ..... + \left(\mathrm{n} + 1\right){\mathrm{C}}_{\mathrm{n} }\mathrm{is} \mathrm{equal} \mathrm{to} \end{gathered}$$

• $2^{\mathrm{n}} + \mathrm{n} 2^{\mathrm{n} - 1}$

• $2^{\mathrm{n}} + \mathrm{n} 2^{\mathrm{n} }$

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

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

38.

$\mathrm{If} \frac{\mathrm{x} - 4}{{\mathrm{x}}^2 - 5\mathrm{x} - 2\mathrm{k}} = \frac{2}{\mathrm{x} - 2} - \frac{1}{\mathrm{x} + \mathrm{k}\text{'}},\mathrm{then} \mathrm{k} \mathrm{is} \mathrm{equal} \mathrm{to}$

• - 3

• - 2

• 2

• 3

$\frac{2}{2!} + \frac{2 + 4}{3!} + \frac{2 + 4 + 6}{4!} + ...\mathrm{is} \mathrm{equal} \mathrm{to}$

• e

• ${\mathrm{e}}^{ - 1}$

• ${\mathrm{e}}^{ - 2}$

• ${\mathrm{e}}^{ - 3}$

$$\begin{gathered} \left|\mathrm{x}\right| < 1,\mathrm{the} \mathrm{coefficient} \mathrm{of} {\mathrm{x}}^3 \mathrm{in} \mathrm{the} \mathrm{expansion} \mathrm{of} \\ \log \left(1 + \mathrm{x} + {\mathrm{x}}^2\right) \mathrm{in} \mathrm{ascendingnpowers} \mathrm{of} \mathrm{x},\mathrm{is} \end{gathered}$$

• $\frac{2}{3}$

• $\frac{4}{3}$

• $- \frac{2}{3}$

• $- \frac{4}{3}$