Statement-1: The sum of rational terms in ( 2 + 3 3 + 5 6 ) 10 is 1262 Statement-2: The number of irrational terms in the expansion of ( 5 1 / 6 + 2 1 / 8 ) 100 is 97 The correct statements among the following are

Question

Statement-1: The sum of rational terms in ${(\sqrt{2} + \sqrt[3]{3} + \sqrt[6]{5})}^{10}$ is 1262

Statement-2: The number of irrational terms in the expansion of ${(5^{1 / 6} + 2^{1 / 8})}^{100}$ is 97

The correct statements among the following are

Smart Achievers · Accepted Answer

(3)

$Statement-1 General term = \frac{10!}{α! β! γ!} 2^{α / 2} 3^{β / 3} 5^{γ / 6} for rational terms$

$α = 0, 2, 4, 6, 8, 10, β = 0, 3, 6, γ = 0, 6$

$Hence possible sets = (4, 6, 0), (4, 0, 6), (10, 0, 0)$

$Hence, there are 5 rational terms.$

$∴$ $required = \frac{10!}{4! 6!} 2^{24} \frac{10!}{10!} 2^{5} = 12632$

$Statement-3 t_{r + 1}, the (r + 1) th in the expansion of$ ${(5^{1 / 6} + 2^{1 / 8})}^{10} is$ $t_{r + 1} = {}^{10}C_{r} {(5)}^{\frac{100 - r}{6}} {(2^{1 / 8})}^{r}$

$As 5 and 2 are relatively prime, t_{r + 1} will be rational if \frac{100 - r}{6} and \frac{r}{8} are both integers.$

$i.e. If 100 - r is a multiple of 6 and r is a multiple of 8.$

$As 0 \leq r \leq 100, multiple of 8 upto 100 and corresponding value of 100 - r$

$r = 0, 8, 16, 24, \dots, 88, 96$

$100 - r = 100, 92, 84, 76, \dots, 12, 4$

$Out of 100 - r, multiple of 6 are 84, 60, 36, 12 .$

$∴$ $There are just four rational terms$

$\Rightarrow Number of irrational terms = 101 - 4 = 97$

Solution

Q.
Statement-1: The sum of rational terms in ${(\sqrt{2} + \sqrt[3]{3} + \sqrt[6]{5})}^{10}$ is 1262

Statement-2: The number of irrational terms in the expansion of ${(5^{1 / 6} + 2^{1 / 8})}^{100}$ is 97

The correct statements among the following are

1 Statement-1 is true, statement-2 is true

2 Statement-1 is true, statement-2 is false

3 Statement-1 is false, statement-2 is true

4 Statement-1 is false, statement-2 is false

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