Question 7

The number of real numbers $$x$$ which satisfy $$\frac{8^x+27^x}{12^x+18^x}=\frac{7}{6}$$ is

Solution

Put $$a=2^x$$ and $$b=3^x$$. The equation becomes $$\frac{a^3+b^3}{a^2b+ab^2}=\frac{7}{6}$$, which simplifies to $$6a^2-13ab+6b^2=0$$. Hence $$(3a-2b)(2a-3b)=0$$, giving $$\left(\frac{2}{3}\right)^x=\frac{2}{3}$$ or $$\left(\frac{2}{3}\right)^x=\frac{3}{2}$$. Therefore $$x=1$$ or $$x=-1$$, so there are $$2$$ real solutions.

Get AI Help

Video Solution

Other Cracku’s Useful Resources:

JEE Questions JEE Question Bank JEE Mains Previous Papers JEE Advanced Previous Papers JEE Study Material PDF JEE Mains Mock Test JEE Advanced Mock Test JEE Daily Target

Free JEE Questions

JEE Physics Questions JEE Maths Questions JEE Chemistry Questions