Question

Let x be the A.M. and y, z be two G.M.s between two positive numbers. Then, $\frac{y^3+z^3}{xyz}$ is equal to

• A. 1
• B. 2
• C. $\frac{1}{2}$
• D. none of these

Answer 1

The correct option is B

2

(b) 2

Let the two numbers be a and b

a, x and b are in A.P.

$\therefore 2x=a+b$ ...... (i)

Also, a, y, z and b are in G.P.

$\therefore \frac{y}{a}=\frac{z}{y}=\frac{b}{z}$

$\Rightarrow y^2=az,yz=ab,z^2=by$ ...... (ii)

Now, $\frac{y^3+z^3}{xyz}$

$=\frac{y^2}{xz}+\frac{z^2}{xy}$

$=\frac{1}{x}(\frac{y^2}{z}+\frac{z^2}{y})$

$=\frac{1}{x}(\frac{az}{z}+\frac{by}{y})$ [Using (ii)]

$=\frac{1}{x}(a+b)$

$=\frac{2}{(a+b)}(a+b)$ [Using (i)]

$=2$