A) \[0\]
B) \[1\]
C) \[2\]
D) infinitely many
Correct Answer: B
Solution :
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.
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