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${\left( {\frac{x}{2} - \frac{3}{{{x^2}}}} \right)^{10}}$ ના વિસ્તરણમાં ${x^4}$ નો સહગુણક મેળવો.
$\frac{{405}}{{256}}$
$\frac{{504}}{{259}}$
$\frac{{450}}{{263}}$
એકપણ નહીં.
Solution
(a) In the expansion of ${\left( {\frac{x}{2} – \frac{3}{{{x^2}}}} \right)^{10}}$,
the general term is ${T_{r + 1}} = {\,^{10}}{C_r}{\left( {\frac{x}{2}} \right)^{10 – r}}.\,\,{\left( { – \frac{3}{{{x^2}}}} \right)^r}$
${ = ^{10}}{C_r}{( – 1)^r}.\frac{{{3^r}}}{{{2^{10 – r}}}}{x^{10 – r – 2r}}$
Here, the exponent of $x$ is $10 – 3r = 4 \Rightarrow r = 2$
$\therefore \,\,\,\,{T_{2 + 1}}{ = ^{10}}{C_2}{\left( {\frac{x}{{\rm{2}}}} \right)^8}{\left( { – \frac{3}{{{x^2}}}} \right)^2}$
= $\frac{{10.9}}{{1.2}}.\frac{1}{{{2^8}}}{.3^2}.{x^4}$
= $\frac{{405}}{{256}}{x^4}$
$\therefore $ The required coefficient $ = \frac{{405}}{{256}}$.
