Binomial Theorem And Its Simple Applications Question 4
Question: What are the values of k if the term independent of x in the expansion of $ {{( \sqrt{x}+\frac{k}{x^{2}} )}^{10}} $ is 405-
Options:
A) $ \pm 3 $
B) $ \pm 6 $
C) $ \pm 5 $
D) $ \pm 4 $
Show Answer
Answer:
Correct Answer: A
Solution:
- [a] Given expansion is $ {{( \sqrt{x}+\frac{k}{x^{2}} )}^{10}} $
$ {{(r+1)} _{th}}term,{T _{r+1}}={{,}^{10}}C _{r}{{(\sqrt{x})}^{10-r}}{{( \frac{k}{x^{2}} )}^{r}} $
$ \Rightarrow {T _{r+1}}={{,}^{10}}C _{r}{x^{5-r/2}}.{{(k)}^{r}}.{x^{-2r}} $
$ \therefore {T _{r+1}}={{,}^{10}}C _{r}{x^{(10-5r)/2}}{{(k)}^{r}} $
Since, $ {T _{r+1}} $ is independent of x
$ \therefore ,\frac{10-5r}{2}=0\Rightarrow r=2\therefore ,405={{,}^{10}}C_2{{(k)}^{2}} $
$ 405=45\times k^{2}\Rightarrow k^{2}=9\Rightarrow k=\pm 3 $
BETA
