If the term without x in the expansion of (x^(2/3)+α/x³)²² is 7315 , then |α| is equal to ___________.

Solution

Maths sir · 4 steps

Write the general term, set the power of x to zero, and the binomial coefficient does the rest.

1. General term. The (r+1)th term is Tᵣ₊₁ = C(22, r)(x^(2/3))²²⁻ʳ(α/x³)ʳ = C(22, r) αʳ x^(2(22-r)/3 - 3r).
2. Kill the power. For the term without x, the exponent must be zero: 2(22-r)/3 = 3r. So 44 - 2r = 9r, which gives r = 4.

   T₅ = C(22, 4)α⁴ = (22 × 21 × 20 × 19)/(4 × 3 × 2 × 1) α⁴ = 7315 α⁴

3. Match the value. Setting 7315 α⁴ = 7315 gives α⁴ = 1, so α = ± 1 for real α.
4. So. |α| = 1.

The idea to keep. For any term-independent-of-x question, collect the full power of x in Tᵣ₊₁ as one expression in r, set it to zero, and solve for r first.

Answer from the NTA final answer key. Solution by PrepWiser.