If the co-efficient of x⁹ in ( α x³ + 1/(β x) )¹¹ and the co-efficient of x⁻⁹ in ( α x - 1/(β x³) )¹¹ are equal, then (αβ)² is equal to ___________.

Solution

Maths sir · 4 steps

Write the general term of each expansion, match the power of x to find r, then equate the two coefficients.

1. First expansion. Tᵣ₊₁ = C(11, r)(α x³)¹¹⁻ʳ(1/(β x))ʳ = C(11, r)α¹¹⁻ʳ/βʳ x³³⁻⁴ʳ. Setting 33-4r = 9 gives r = 6, so the coefficient of x⁹ is C(11, 6)α⁵/β⁶.
2. Second expansion. Tᵣ₊₁ = C(11, r)(α x)¹¹⁻ʳ(-1/(β x³))ʳ = (-1)ʳC(11, r)α¹¹⁻ʳ/βʳ x¹¹⁻⁴ʳ. Setting 11-4r = -9 gives r = 5, so the coefficient of x⁻⁹ is -C(11, 5)α⁶/β⁵.
3. Equate them. Since C(11, 6) = C(11, 5) = 462, the binomial coefficients cancel. Divide both sides by α⁵/β⁵, which is not zero because a coefficient containing it is being compared.

   α⁵/β⁶ = -α⁶/β⁵ ⇒ 1/β = -α ⇒ αβ = -1

4. So. (αβ)² = (-1)² = 1.

The idea to keep. For a specific power of x, write the general term, collect all powers of x into one exponent and solve for r. Remember C(n, r) = C(n, n-r), which often makes the numbers cancel.

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