A) \[\frac{1}{4}\]
B) \[4\sqrt{2}\]
C) \[\frac{1}{8}\]
D) \[2\sqrt{2}\]
Correct Answer: A
Solution :
| In the expansion of \[{{\left( 1+{{x}^{{{\log }_{2}}}}^{x} \right)}^{5}}\] |
| Third term say \[{{T}_{3}}={}^{5}{{C}_{2}}{{\left( {{x}^{1o{{g}_{2}}x}} \right)}^{2}}=2560\] |
| \[\Rightarrow \] \[{{\left( {{x}^{{{\log }_{2}}x}} \right)}^{2}}=256\] |
| Taking logarithm to the base 2 on both sides |
| \[\Rightarrow \] \[2{{\left( {{\log }_{2}}x \right)}^{2}}=8\] |
| \[\Rightarrow \] \[({{\log }_{2}}x)=\pm 2\] |
| \[\Rightarrow \] \[x=4,\frac{1}{4}\] |
| Here, \[x=\frac{1}{4}.\] |
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