The harmonic mean of two numbers is 4. Their arithmetic mean A and geometric mean G satisfy the relation 2A +G² = 27. Find the numbers.

82.

The sum of the series $\frac{1}{1 . 2} - \frac{1}{2 . 3} + \frac{1}{3 . 4} - ... \infty$

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

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

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

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

If ${}^{\mathrm{n}}\mathrm{C}_4, {}^{\mathrm{n}}\mathrm{C}_5 \mathrm{and} {}^{\mathrm{n}}\mathrm{C}_6$ are in AP, then n is

• 7 or 14

• 7

• 14

• 14 or 21

If w $\ne$ 1 is a cube root of unity, then the sum of the series S = 1 + 2w + 3w² + ... + 3nw^{3n -1} is

• $\frac{3\mathrm{n}}{\mathrm{w} - 1}$

• 3n(w - 1)

• $\frac{\mathrm{w} - 1}{3\mathrm{n}}$

• 0

If ${\log }_3\left(\mathrm{x}\right) + {\log }_3\left(\mathrm{y}\right) = 2 + {\log }_3\left(2\right) \mathrm{and} {\log }_3\left(\mathrm{x} + \mathrm{y}\right) = 2$, then

• x = 1, y = 8

• x = 8, y = 1

• x = 3, y = 6

• x = 9, y = 3

If ${\log }_7\left(2\right) = \mathrm{\lambda }$, then the value of ${\log }_{49}\left(28\right)$ is

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

• $\left(2\mathrm{\lambda } + 3\right)$

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

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

The sequence $\log \left(\mathrm{a}\right), \log \left(\frac{{\mathrm{a}}^2}{\mathrm{b}}\right), \log \left(\frac{{\mathrm{a}}^3}{{\mathrm{b}}^2}\right), ...$ is

• a GP

• an AP

• a HP

• both a GP and a HP

If in a $∆\mathrm{ABC}$. ABC, sin(A), sin(B), sin(C) are in AP, then

• the altitudes are in AP

• the altitudes are in HP

• the angles are in AP

• the angles are in HP

The value of $\frac{2}{3!} +\hspace{0.17em}\frac{4}{5!} + \frac{6}{7!} + ...$

• ${\mathrm{e}}^{\frac{1}{2}}$

• e^{- 1}

• e

• ${\mathrm{e}}^{- \frac{1}{3}}$

If sum of an infinite geometric series is 4/3 and its Ist term is 3/4, then its common ratio is

• 7/16

• 9/16

• 1/9

• 7/9