A) \[10,\,\,{{10}^{-3/2}}\]
B) \[100\,\,or\,\,{{10}^{-3/2}}\]
C) \[10\,\,or\,\,{{10}^{-5/2}}\]
D) None of these
Correct Answer: C
Solution :
\[\log x\]is defined only when x > 0 Now, the\[{{3}^{rd}}\]term in the expansion \[{{T}_{2+1}}{{=}^{5}}{{C}_{2}}\cdot {{x}^{5-2}}{{({{x}^{{{\log }_{10}}x}})}^{2}}=1,000,000\](given) \[\Rightarrow \]\[{{x}^{3+2{{\log }_{10}}x}}={{10}^{5}}\] Taking logarithm of both sides, we get \[(3+2{{\log }_{10}}x)\cdot {{\log }_{10}}x=5\] \[\Rightarrow \] \[2{{y}^{2}}+3y-5=0\] Where \[{{\log }_{10}}x=y\] \[\Rightarrow \] \[(y-1)(2y+5)=0\] \[\Rightarrow \] \[y=1\]or\[-5/2\] \[\Rightarrow \] \[{{\log }_{10}}x=1\]or\[-5/2\] \[\Rightarrow \] \[x={{10}^{1}}=10\]or\[{{10}^{-5/2}}\]
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