JEE Main 2021 · 16 March · morning · Single correct · +4 / −1
If n is the number of irrational terms in the
expansion of ( 3^(1/4) + 5^(1/8) )⁶⁰, then (n - 1) is divisible by :
- (a) 30
- (b) 8
- (c) 7
- (d) 26
Answer and solution
Answer (d) 26
Solution
Maths sir · 4 steps
Count the rational terms first. They are few, and the rest of the 61 terms are irrational.
Tᵣ₊₁ = C(60, r) 3^((60-r)/4) 5^(r/8), r = 0, 1, …, 60
- Condition for rational. A term is rational only when both powers are whole numbers. So r/8 must be an integer, and (60-r)/4 must be an integer.
- Find such r. If r is a multiple of 8, it is also a multiple of 4. Since 60 is a multiple of 4, 60 - r is then divisible by 4 as well. So r = 0, 8, 16, 24, 32, 40, 48, 56, which gives 8 rational terms.
- Count irrational terms. The expansion has 60 + 1 = 61 terms. So n = 61 - 8 = 53, and n - 1 = 52 = 2 × 26.
- So. 52 is divisible by 26 but not by 30, 8 or 7. The answer is (d).
The idea to keep. For irrational terms, count the rational terms, where every exponent becomes an integer, and subtract that count from the total of n+1 terms.
Answer from the NTA final answer key. Solution by PrepWiser.