Question:
The term independent of $x$ in the expansion of $\left(x+\frac{1}{x^{2}}\right)^{6}$ is
The term independent of $x$ in the expansion of $\left(x+\frac{1}{x^{2}}\right)^{6}$ is
Updated On: May 18, 2024
- 20
- $15$
- 6
- 1
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The Correct Option is B
Solution and Explanation
The general term of $\left(x+\frac{1}{x^{2}}\right)^{6}$ is
$T_{r+1}={ }^{6} C_{r} x^{r}\left(\frac{1}{x^{2}}\right)^{6-r}$
$={ }^{6} C_{r} x^{r-12+2 r}$
For independent of $x$,
$r-12+2 r=0 $
$\Rightarrow 3 r=12 $
$\Rightarrow r=4$
$\therefore$ Required term $={ }^{6} C_{4}=\frac{6 \times 5}{2 \times 1}=15 $
$T_{r+1}={ }^{6} C_{r} x^{r}\left(\frac{1}{x^{2}}\right)^{6-r}$
$={ }^{6} C_{r} x^{r-12+2 r}$
For independent of $x$,
$r-12+2 r=0 $
$\Rightarrow 3 r=12 $
$\Rightarrow r=4$
$\therefore$ Required term $={ }^{6} C_{4}=\frac{6 \times 5}{2 \times 1}=15 $
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.