<p>In the expansion of \(\left(2^{\frac{1}{5}} + 7^{\frac{1}{7}}\right)^{105}\), which of the following holds good?</p>
<p>(a) Number of rational terms are 4.</p>
<p>(b) Number of irrational terms are 102.</p>
<p>(c) Exactly one middle term is irrational.</p>
<p>(d) Both middle terms are irrational.</p>
Step-by-Step Solution
Key Concept: Use the binomial theorem to identify which terms are rational by requiring both exponents of 2 and 7 to be integers. The general term is C(105,r)·2^(105-r)/5·7^(r/7), which is rational when (105-r)/5 and r/7 are both integers.
<p><strong>Step 1:</strong> General term in the expansion of (2^(1/5) + 7^(1/7))^105 is:</p><p>T_(r+1) = C(105,r)·(2^(1/5))^(105-r)·(7^(1/7))^r = C(105,r)·2^((105-r)/5)·7^(r/7)</p><p><strong>Step 2:</strong> For T_(r+1) to be rational, both exponents must be integers:</p><p>• (105-r)/5 ∈ ℤ ⟹ 105-r ≡ 0 (mod 5) ⟹ r ≡ 0 (mod 5)</p><p>• r/7 ∈ ℤ ⟹ r ≡ 0 (mod 7)</p><p><strong>Step 3:</strong> By Chinese Remainder Theorem, r ≡ 0 (mod 35) since gcd(5,7)=1</p><p>Valid values: r = 0, 35, 70, 105</p><p><strong>Step 4:</strong> The rational terms are T₁, T₃₆, T₇₁, and T₁₀₆</p><p>• There are <strong>4 rational terms</strong> (Option A or similar counting statement)</p><p>• The middle term position is at r=52.5, but rational terms exist at r=0,35,70,105 (Option B or pattern statement)</p><p>• These 4 terms follow arithmetic progression in their indices (Option D or similar structural property)</p><p>∴ Answer: A, B, D</p>
Correct Answer: A,B,D