Binomial Theorem And Its Simple Applications Question 52
Question: If the third term in the expansion of $ {{[x+{x^{{\log_{,10}},x}}]}^{5}} $ is $ 10^{6} $ , then x may be
Options:
A) 1
B) $ \sqrt{10} $
C) $\mathrm{x}=10^1$ or $10^{-\frac{5}{2}}$
D) $ {10^{-2/5}} $
Show Answer
Answer:
Correct Answer: C
Solution:
- [c] Put $ {\log_{10}}x=y $ , the given expansion becomes $ {{(x+x^{y})}^{5}} $ . $ T_3={{,}^{5}}C_2,.x^{3}{{(x^{y})}^{2}}=10{x^{3+2y}}=10^{6}(given) $
$ \Rightarrow (3+2y)log_{10}x=5,{\log_{10}}10=5 $
$ \Rightarrow (3+2y)y=5 $
$ \Rightarrow y=1,-\frac{5}{2}\Rightarrow {\log_{10}}x=1 $ or $ {\log_{10}}x=-\frac{5}{2} $
$ \therefore ,x=10 $ or $ x={{(10)}^{-5/2}} $
BETA
