QJEEM21Ma6854: JEE Main 2021 Mathematics - Class 11 JEEM21Ma6854 Solution: If n is the number of irrational terms in the expansion of ,…

Question

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If n is the number of irrational terms in the
expansion of

{\left( {{3^{1/4}} + {5^{1/8}}} \right)^{60}}

, then (n

-

1) is divisible by :

Accepted Answer

{T_{r + 1}} = {}^{60}{C_r}{\left( {{3^{1/4}}} \right)^{60 - r}}{\left( {{5^{1/8}}} \right)^r}

rational if

{{60 - r} \over 4},{r \over 8}

, both are whole numbers,

r \in \{ 0,1,2,......60\}

{{60 - r} \over 4} \in W \Rightarrow r \in \{ 0,4,8,....60\}

and

{r \over 8} \in W \Rightarrow r \in \{ 0,8,16,.....56\}

\therefore

Common terms

r \in \{ 0,8,16,.....56\}

So, 8 terms are rational

Then Irrational terms =

61 - 8 = 53 = n

\therefore

n - 1 = 52 = 13 \times {2^2}

Factors 1, 2, 4, 13, 26, 52

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