Binomial Theorem 1 Question 23
25.
The coefficient of $x^{4}$ in $(\frac{x}{2}-{\frac{3}{x^{2}}})^{10}$ is
(1983, 1M)
(a) $\frac{405}{256}$
(b) $\frac{504}{259}$
(c) $\frac{450}{263}$
(d) None of these
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Answer:
Correct Answer: 25. (a)
Solution:
- The general term in $(\frac{x}{2}-{\frac{3}{x^{2}}})^{10}$ is
$t_{r+1}=(-1)^{r} \quad { }^{10} C_{r} (\frac{x}{2})^{10-r} \quad (\frac{3}{x^{2}}){ }^{r}=(-1)^{r}{ }^{10} C_{r} \cdot \frac{3^{r}}{2^{10-r}} \cdot x^{10-3 r}$
For coefficient of $x^{4}$, we put $10-3 r=4$
$ \begin{aligned} \Rightarrow & 3 r =6 \\ \Rightarrow & r =2 \end{aligned} $
$\therefore$ Coefficient of $x^{4}$ in $(\frac{x}{2}-\frac{3}{x^{2}} ){ }^{10}=(-1)^{2} \cdot{ }^{10} C_{2} \cdot \frac{3^{2}}{2^{8}}$
$ =\frac{45 \times 9}{256}=\frac{405}{256} $
