The coefficient of x^4 in ( x 2 -3 x^ 2 ) is: - Vidyadip

MCQ

The coefficient of $x^4$ in $\Big(\frac{\text{x}}{2}-\frac{3}{\text{x}^{2}}\Big)$ is:

✓
$\frac{405}{256}$
B
$\frac{504}{259}$
C
$\frac{450}{263}$
D
None of these.

✓

Answer

Correct option: A.

$\frac{405}{256}$

Suppose $x^4$ occurs at the $(r + 1)^{th}$ term in the given expansion.
Then, we have
$\text{T}_{\text{r}+1}={^\text{10}}\text{C}_{\text{r}}\Big(\frac{\text{x}}{2}\Big)^{10-\text{r}}\Big(\frac{-3}{2\text{x}^{2}}\Big)$
$=(-1)^{\text{r}}\ {^\text{10}}\text{C}_{\text{r}}\frac{3^{\text{r}}}{2^{10-\text{r}}}\text{x}^{10-\text{r}-2\text{r}}$
For this term to contain $x^4$, we must have
$10-3\text{r}=4$
$\Rightarrow \text{r}=2$
$\therefore$ Required coefficient $={^\text{10}}\text{C}_{\text{2}}\frac{3^{2}}{2^{8}}=\frac{10\times9\times9}{2\times2^{8}}=\frac{405}{256}$

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