JEE MAIN - Mathematics (2021 - 16th March Morning Shift - No. 7)
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 :
expansion of $${\left( {{3^{1/4}} + {5^{1/8}}} \right)^{60}}$$, then (n $$-$$ 1) is divisible by :
30
8
7
26
Explanation
$${T_{r + 1}} = {}^{60}{C_r}{\left( {{3^{1/4}}} \right)^{60 - r}}{\left( {{5^{1/8}}} \right)^r}$$
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
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
Comments (0)
