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 

विकल्प

• (a) 1 

• (b) 2 

• (c) $$\frac{1}{2}$$ 

• (d) none of these

   

Answer 1

(b) 2 

$$\text{ Let the two numbers be a and b }.$$
$$\text{ a, x and b are in A . P }.$$
$$\therefore 2x=a+b(i)$$
$$\text{ Also, a, y, z and b are in G . P }.$$
$$\therefore \frac{y}{a}=\frac{z}{y}=\frac{b}{z}$$
$$\Rightarrow y^2=az,yz=ab,z^2=by(ii)$$
$$\text{ Now },\frac{y^3+z^3}{xyz}$$
$$=\frac{y^2}{xz}+\frac{z^2}{xy}$$
$$=\frac{1}{x}(\frac{y^2}{z}+\frac{z^2}{y})$$
$$=\frac{1}{x}(\frac{az}{z}+\frac{by}{y})[\text{ Using }(ii)]$$
$$=\frac{1}{x}(a+b)$$
$$=\frac{2}{(a+b)}(a+b)[\text{ Using }(i)]$$
$$=2$$
$$$$