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7.Binomial Theorem
normal
The term independent of $' x '$ in the expansion of ${\left( {9\,x\,\, - \,\,\frac{1}{{3\,\sqrt x }}} \right)^{18}}, x > 0$ , is $\alpha$ times the corresponding binomial co-efficient . Then $' \alpha '$ is :
A
$3$
B
$\frac{1}{3}$
C
$-\frac{1}{3}$
D
$1$
Solution
$T_{r+1}={ }^{18} C_{r}\left(q_{x}\right)^{18-r} \cdot\left(\frac{-1}{3 \sqrt{x}}\right)^{r}$
Power of $x$ in $T_{r}+1$ should $6 p O$ $\frac{x^{18-r}}{x^{1 / 2 r}} \Rightarrow \begin{array}{l}x^{18-r-\frac{1}{2} r} \\ \Rightarrow 18-r-\frac{1}{2} r=0\end{array}$
$18-\frac{3}{2} r=0$
$36-3 r=0$
$r=12$
$16 / 129^{6}\left(\frac{-1}{3}\right)^{12}$
$\Rightarrow{ }^{18} C _{12} \frac{9^{1}}{9^{6}} \Rightarrow{ }^{18} C _{12}= T _{13}$
${ }^{18} C_{12}=\alpha \cdot{ }^{18} C_{12}$
$\alpha=1$
Standard 11
Mathematics
