Constant term in the expansion of (x-1 x )^ 10 is: - Vidyadip

MCQ

Constant term in the expansion of $\Big(\text{x}-\frac{1}{\text{x}}\Big)^{10}$ is:

A
$152$
B
$-152$
✓
$-252$
D
$252$

✓

Answer

Correct option: C.

$-252$

Suppose $(r + 1)^{th}$ term is the constant term in the given expansion.
Then, we have
$\text{T}_{\text{r}+1}={^\text{10}}\text{C}_{\text{r}}\ \text{x}^{10-\text{r}}\ \Big(\frac{\text{1}}{\text{x}}\Big)^{\text{r}}$
$={^\text{10}}\text{C}_{\text{r}} (-1)^{\text{r}}\ \text{x}^{10-\text{r}-\text{r}}$
For this term to be constant, we must have
$10-2\text{r}=0$
$\Rightarrow \text{r}=5$
Required term $={^\text{10}}\text{C}_{\text{5}}=-252$

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