Question

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 1

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 t ₃ = 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 = - \(\frac{6}{20}\)  

= - \(\frac{3}{10}\)