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
Hide Solution
Verified By Collegedunia
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 $
Was this answer helpful?
0
0
Top Questions on Binomial theorem
- If the number of terms in the binomial expansion of \((2x + 3)^n\) is 22, then the value of \(n\) is:
- KCET - 2025
- Mathematics
- Binomial theorem
- A radar system can detect an enemy plane in one out of ten consecutive scans. The probability that it can detect an enemy plane at least twice in four consecutive scans is:
- AP EAPCET - 2025
- Mathematics
- Binomial theorem
- A string of letters is to be formed by using 4 letters from all the letters of the word “MATHEMATICS”. The number of ways this can be done such that two letters are of same kind and the other two are of different kind is
- AP EAPCET - 2025
- Mathematics
- Binomial theorem
- The number of trials conducted in a binomial distribution is 6. If the difference between the mean and variance of this variate is \(\frac{27}{8}\), then the probability of getting at most 2 successes is:
- AP EAPCET - 2025
- Mathematics
- Binomial theorem
- The number of ways of dividing 15 persons into 3 groups containing 3, 5 and 7 persons so that two particular persons are not included into the 5 persons group is
- AP EAPCET - 2025
- Mathematics
- Binomial theorem
View More Questions
Questions Asked in KEAM exam
- Benzene when treated with Br\(_2\) in the presence of FeBr\(_3\), gives 1,4-dibromobenzene and 1,2-dibromobenzene. Which type of reaction is this?
- KEAM - 2025
- Haloalkanes and Haloarenes
- 10 g of (90 percent pure) \( \text{CaCO}_3 \), treated with excess of HCl, gives what mass of \( \text{CO}_2 \)?
- KEAM - 2025
- Stoichiometry and Stoichiometric Calculations
- Evaluate the following limit: $ \lim_{x \to 0} \frac{1 + \cos(4x)}{\tan(x)} $
- KEAM - 2025
- Limits
- Given that \( \mathbf{a} \times (2\hat{i} + 3\hat{j} + 4\hat{k}) = (2\hat{i} + 3\hat{j} + 4\hat{k}) \times \mathbf{b} \), \( |\mathbf{a} + \mathbf{b}| = \sqrt{29} \), \( \mathbf{a} \cdot \mathbf{b} = ? \)
- KEAM - 2025
- Vector Algebra
- In an equilateral triangle with each side having resistance \( R \), what is the effective resistance between two sides?
- KEAM - 2025
- Combination of Resistors - Series and Parallel
View More Questions
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.