SATHEE: Binomial Theorem 1 Question 8

Binomial Theorem 1 Question 8

9.

The total number of irrational terms in the binomial expansion of ${(7^{1 / 5} - 3^{1 / 10})}^{60}$ is

(2019 Main, 12 Jan II)

(a) 49

(b) 48

(d) 55

Show Answer

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 ${(7^{1 / 5} - 3^{1 / 10})}^{60}$ is

$\begin{aligned} T_{r + 1} & =^{60} C_{r} {(7^{1 / 5})}^{60 - r} {(- 3^{1 / 10})}^{r} \\ =^{60} C_{r} 7^{\frac{60 - r}{5}} (- 1)^{r} 3^{\frac{r}{10}} = (- 1)^{r}^{60} C_{r} 7^{12 - \frac{r}{5}} 3^{\frac{r}{10}} \end{aligned}$

The possible non-negative integral values of ’ $r$ ’ for which $\frac{r}{5}$ and $\frac{r}{10}$ are integer, where $r \leq 60$ , are $r = 0, 10, 20, 30, 40, 50, 60$ .

$∴$ There are 7 rational terms in the binomial expansion and remaining $61 - 7 = 54$ terms are irrational terms.

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Binomial Theorem 1 Question 8

9.

Answer:

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