Question

If a^x = b^y = c^z, where x, y, z are unequal positive numbers and a, b, c are in G.P., then x³ + z³

• A. ³ 2y³
• B. £ 2y³
• C. > 2y³
• D. None of these

Answer 1

Answer: (C)

> 2y³

Answer 2

Given a^x = b^y = c^z = k

\ a = $k^{\frac{1}{x}}$, b = $k^{\frac{1}{y}}$, c = $k^{\frac{1}{z}}$

Q a, b, c are in G.P. Ž $\frac{b}{a}$= $\frac{c}{b}$

Ž $k^{\frac{1}{y} - \frac{1}{x}}$ = $k^{\frac{1}{z} - \frac{1}{y}}$Ž $\frac{1}{y}$– $\frac{1}{x}$= $\frac{1}{z}$–$\frac{1}{x}$Ž x, y, z are in H.P.

Ž y = H.M. of x and z Ž $\sqrt{xy}$> y

and $\frac{x^{3} + z^{3}}{2}$> ($\sqrt{xy}$)³ > y³ Ž x³ + z³ > 2y³

\ [C] is correct.