If the constant term in the binomial expansion of (x^(5/2)/2-4/xˡ)⁹ is -84 and the coefficient of x⁻³ˡ is 2^α β, where β<0 is an odd number, then |α l-β| is equal to ________.

Solution

Maths sir · 4 steps

Write one general term with all the powers of 2 combined, and both conditions become easy to read off.

1. General term. Take (x^(5/2)/2)⁹⁻ʳ(-4/xˡ)ʳ and gather the powers of 2, since 4ʳ = 2²ʳ and 2⁻⁽⁹⁻ʳ⁾ = 2ʳ⁻⁹.

   Tᵣ₊₁ = C(9, r)(-1)ʳ 2³ʳ⁻⁹ x^(5(9-r)/2 - lr)

2. Constant term. The coefficient must be -84 = -C(9, 3). With r = 3 the sign is negative and 2³ʳ⁻⁹ = 2⁰ = 1, so this works. For the constant term the power is zero: (5 × 6)/2 = 3l, which gives l = 5.
3. Second condition. Now x⁻³ˡ = x⁻¹⁵. Set 5(9-r)/2 - 5r = -15, so 45 - 15r = -30 and r = 5. The coefficient is C(9, 5)(-1)⁵ 2⁶ = -126 × 64 = -2⁷ × 63.
4. So. α = 7 and β = -63, which is negative and odd. Then |α l - β| = |35 + 63| = 98.

The common slip: 28. Dropping the (-1)ʳ from (-4)ʳ makes β = 63, and then |35 - 63| gives 28.

The idea to keep. In binomial coefficient problems, first combine every number into one sign and one power of a base. Then the condition that is fully known fixes r, and the other gives the unknown.

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