The value of $\left(\frac{1}{{\log }_3\left(12\right)} + \frac{1}{{\log }_4\left(12\right)}\right)$ is

• 0

• $\frac{1}{2}$

• 1

• 2

If $\mathrm{x} = {\log }_{\mathrm{a}}\left(\mathrm{bc}\right), \mathrm{y} = {\log }_{\mathrm{b}}\left(\mathrm{ca}\right), \mathrm{z} = {\log }_{\mathrm{c}}\left(\mathrm{ab}\right)$,  then the value of $\frac{1}{1 + \mathrm{x}} + \frac{1}{1 + \mathrm{y}} + \frac{1}{1 + \mathrm{z}}$ will be

• x + y + z

• 1

• ab + bc + ca

• abc

Find the sum of the first n terms of the series 0.2 + 0.22 + 0.222 + ...

If the mth term and the nth term of an AP are respectively $\frac{1}{\mathrm{n}}$ and $\frac{1}{\mathrm{m}}$, then the mnth term of the AP is

• $\frac{1}{\mathrm{mn}}$

• $\frac{\mathrm{m}}{\mathrm{n}}$

• 1

• $\frac{{\displaystyle \mathrm{n}}}{{\displaystyle \mathrm{m}}}$

${\int }_{- \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\sin }^9\left(\mathrm{x}\right){\cos }^5\left(\mathrm{x}\right)d\mathrm{x}$ equals

• $\frac{1}{20}$

• 20

• 0

• $\frac{1}{330}$

The sum of the series (1 + 2) + (1 + 2 + 2²) + (1 + 2 + 2² + 2³) + ...upto n terms is

• 2^{n + 2} - n - 4

• 2(2ⁿ - 1) - n

• 2^{n + 1} - n

• 2^{n + 1} - 1

The 5th term of the series $\frac{10}{9}, \frac{1}{3}\sqrt{\frac{20}{3}}, \frac{2}{3}, ...$ is

• $\frac{1}{3}$

• 1

• $\frac{2}{5}$

• $\sqrt{\frac{2}{3}}$

The sum of the infinite series

${\left(\frac{1}{3}\right)}^2 + \frac{1}{3}{\left(\frac{1}{3}\right)}^4 + \frac{1}{5}{\left(\frac{1}{3}\right)}^6 + ...$ is

• $\frac{1}{4}{\log }_{\mathrm{e}}\left(2\right)$

• $\frac{1}{2}{\log }_{\mathrm{e}}\left(2\right)$

• $\frac{1}{6}{\log }_{\mathrm{e}}\left(2\right)$

• $\frac{1}{4}{\log }_{\mathrm{e}}\left(\frac{3}{2}\right)$

109.

If a, b, c be in arithmetic progression, then the value of (a + 2b - c)(2b + c - a)(a + 2b + c) is

• 16abc

• 4abc

• 8abc

• 3abc

If three real numbers a, b, c are in harmonic progression, then which of the following is true ?

• $\frac{1}{\mathrm{a}}, \mathrm{b}, \frac{1}{\mathrm{c}}$ are in AP

• $\frac{1}{\mathrm{bc}}, \frac{1}{\mathrm{ca}}, \frac{1}{\mathrm{ab}}$ are in HP

• ab, bc, ca are in HP

• $\frac{\mathrm{a}}{\mathrm{b}}, \frac{\mathrm{b}}{\mathrm{c}}, \frac{\mathrm{c}}{\mathrm{a}}$ are in HP