Binomial Theorem 1 Question 8
9.
The total number of irrational terms in the binomial expansion of $\left(7^{1 / 5}-3^{1 / 10}\right)^{60}$ is
(2019 Main, 12 Jan II)
(a) 49
(b) 48
(c) 54
(d) 55
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Answer:
Correct Answer: 9. (c)
Solution:
- The general term in the binomial expansion of $(a+b)^{n}$ is $T_{r+1}={ }^{n} C_{r} a^{n-r} b^{r}$.
So, the general term in the binomial expansion of $\left(7^{1 / 5}-3^{1 / 10}\right)^{60}$ is
$ \begin{aligned} T_{r+1} & ={ }^{60} C_{r}\left(7^{1 / 5}\right)^{60-r}\left(-3^{1 / 10}\right)^{r} \\ & ={ }^{60} C_{r} 7^{\frac{60-r}{5}}(-1)^{r} 3^{\frac{r}{10}}=(-1)^{r}{ }^{60} C_{r} 7^{12-\frac{r}{5}} 3^{\frac{r}{10}} \end{aligned} $
The possible non-negative integral values of ’ $r$ ’ for which $\frac{r}{5}$ and $\frac{r}{10}$ are integer, where $r \leq 60$, are $r=0,10,20,30,40,50,60$.
$\therefore$ There are 7 rational terms in the binomial expansion and remaining $61-7=54$ terms are irrational terms.
