The coefficient of the middle term in the binomial expansion in powers

1.	The coefficient of the middle term in the binomial expansion in powers of x of (1 + αx)4 and of (1 – αx)6 is the same if α equals(a) – \(\frac{3}{10}\)(b) − \(\frac{5}{3}\)(c) \(\frac{3}{5}\)(d) \(\frac{10}{3}\)
Answer» Answer : (a) - \(\frac{3}{10}\) The Middle term in the expansion of (1 + αx)⁴ is the \((\frac{4}{2} +1)\)th term, i.e., 3rd term. ∴ t₃ = t_{2 + 1} = ⁴C₂ (αx)² The Middle term in the expansion of (1 – αx)⁶ is the \((\frac{6}{2} +1)\)th term, i.e., 4th term. ∴ T₄= T_{3 + 1} = ⁶C₃ (– 1)³ (αx)³ ∴ Coefficient of t₃ = Coefficient of T₄ ⇒ ⁴C₂ α² = ⁶C₃ (-1)³ α³ ⇒ \(\frac{4\times 3}{2} \alpha^2 = (-1) \times \frac{6\times 5\times 4}{3\times 2} \alpha^3\) ⇒ 6α² = -20α³ ⇒ α = - \(\frac{6}{20}\) = - \(\frac{3}{10}\)

The coefficient of the middle term in the binomial expansion in powers of x of (1 + αx)4 and of (1 – αx)6 is the same if α equals(a) – \(\frac{3}{10}\)(b) − \(\frac{5}{3}\)(c) \(\frac{3}{5}\)(d) \(\frac{10}{3}\)

Answer»

Answer : (a) - \(\frac{3}{10}\)

The Middle term in the expansion of (1 + αx)⁴ is the \((\frac{4}{2} +1)\)th term, i.e., 3rd term.

∴ t₃ = t_{2 + 1} = ⁴C₂ (αx)²

The Middle term in the expansion of (1 – αx)⁶ is the \((\frac{6}{2} +1)\)th term, i.e., 4th term.

∴ T₄= T_{3 + 1} = ⁶C₃ (– 1)³ (αx)³

∴ Coefficient of t₃ = Coefficient of T₄

⇒ ⁴C₂ α² = ⁶C₃ (-1)³ α³

⇒ \(\frac{4\times 3}{2} \alpha^2 = (-1) \times \frac{6\times 5\times 4}{3\times 2} \alpha^3\)

⇒ 6α² = -20α³

⇒ α = - \(\frac{6}{20}\)

= - \(\frac{3}{10}\)