If ${\log }_{\mathrm{x}}\left(\mathrm{a}\right), {\mathrm{a}}^{\raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} \mathrm{and} {\log }_{\mathrm{b}}\left(\mathrm{x}\right)$ are in GP, then x is equal to

• ${\log }_{\mathrm{b}}\left({\log }_{\mathrm{a}}\left(\mathrm{b}\right)\right)$

• $\log \left(\log \left(\mathrm{a}\right)\right)$

• $\log \left(\log \left(\mathrm{a}\right)\right) + \log \left(\log \left(\mathrm{b}\right)\right)$

• ${\log }_{\mathrm{a}}\left({\log }_{\mathrm{b}}\left(\mathrm{a}\right)\right)$

At the foot of a mountain the elevation of its summit is 45°. After ascending 2 km towards the mountain up an incline of 30°, the elevation changes to 60°. The height ofmountain will be

• $\sqrt{3} - 1$

• $1 - \sqrt{3}$

• $1 + \sqrt{3}$

• $\sqrt{3}$

If $\frac{\mathrm{a} + \mathrm{b}}{1 - \mathrm{ab}}, \frac{\mathrm{b} + \mathrm{c}}{1 - \mathrm{bc}}$ are in AP, then $\mathrm{a}, \frac{1}{\mathrm{b}}$ and c are in

• AP

• GP

• HP

• None of these

The number of terms in the series 105 + 103 + 101 + ... + 49 + 47 is

• 28

• 30

• 25

• 22

The geometric mean of roots of the equation x² - 18x + 81 = 0 is

• 18

• 6

• 9

• 3

The sum of n terms of the series 1 + 5 + 12 + 22 + 35 + ... is

• $\frac{{\mathrm{n}}^2\left(\mathrm{n} + 1\right)}{8}$

• $\frac{{\mathrm{n}}^2\left(\mathrm{n} + 1\right)}{6}$

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

• None of these

287.

If geometric mean and harmonic mean of two numbers a and b are 16 and 64/5 respectively, then the value of a : b is

• 4 : 1

• 3 : 2

• 2 : 3

• 1 : 4

If the sum offour numbers in GP is 60 and the arithmetic mean of the first and last numbers is 18, then the numbers are

• 3, 9, 27, 81

• 4, 8, 16, 32

• 2, 6, 18, 54

• None of these

iF a,b and c are in HP, then for any n $\in$ N, which one of the following is true ?

• aⁿ + cⁿ < 2bⁿ

• aⁿ + cⁿ > 2bⁿ

• aⁿ + cⁿ = 2bⁿ

• None of the above

The sum of the series $\frac{1}{2 . 3} + \frac{1}{4 . 5} + \frac{1}{6 . 7} + ...$ is

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

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

• $\frac{\mathrm{e}}{2}$

• $\frac{2}{\mathrm{e}}$