Question

The number of the solutions of the equation $ {{5}^{2x-1}}+{{5}^{x+1}}=250, $ is/are

• A. $ 0 $
• B. $ 1 $
• C. $ 2 $
• D. $infinitely \,many$

Answer 1

Answer: (B)

Given equation is $ {{5}^{2x-1}}+{{5}^{x+1}}=250 $
$ \Rightarrow $ $ {{5}^{2x}}{{.5}^{-1}}+{{5}^{x}}.5=250 $
$ \Rightarrow $ $ \frac{{{({{5}^{x}})}^{2}}}{5}+{{5}^{x}}.5=250 $ ?.(i) Let $ {{5}^{x}}=t $ Then Eq. (i) becomes $ \frac{{{t}^{2}}}{5}+t.5=250 $
$ \Rightarrow $ $ {{t}^{2}}+25t=250\times 5 $
$ \Rightarrow $ $ {{t}^{2}}+25t-1250=0 $
$ \therefore $ $ t=\frac{-25\pm \sqrt{{{(25)}^{2}}-4\times 1\times (-1250)}}{2\times 1} $
$ \Rightarrow $ $ t=\frac{-25\pm \sqrt{625+5000}}{2} $
$ \Rightarrow $ $ t=\frac{-25\pm \sqrt{5625}}{2} $
$ \Rightarrow $ $ t=\frac{-25\pm 75}{2} $ On taking $ +ve $ sign, we get $ t={{5}^{x}}=\frac{-25+75}{2}=\frac{50}{2} $
$ \Rightarrow $ $ {{5}^{x}}=25 $
$ \Rightarrow $ $ {{5}^{x}}={{5}^{2}} $
$ \Rightarrow $ $ x=2 $ On taking $ -ve $ sign, we get $ t={{5}^{x}}=\frac{-25-75}{2} $ $ {{5}^{x}}=\frac{-100}{2} $ $ {{5}^{x}}=-50 $ Cannot express in terms of power of 5. Hence number of solution of given equation is 1.