The coefficient of the middle term in the binomial expansion in powers of $x$ of $(1+\alpha x)^{4}$ and of $(1-\alpha x)^{6}$ is the same if $\alpha$ equals

JEE Main · 2004 · 24 Apr 2004 · Offline · Binomial Theorem

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

x

(1 + α x)^{4}

and of

(1 - α x)^{6}

is the same if

α

equals

Answer: (C)

The middle term of

(1 + α x)^{4}

is the term in

x^{2}

, whose coefficient is

(2 4) α^{2} = 6 α^{2}

. The middle term of

(1 - α x)^{6}

is the term in

x^{3}

, whose coefficient is

(3 6) (- α)^{3} = - 20 α^{3}

. Setting the two coefficients equal,

6 α^{2} = - 20 α^{3}

, and dividing by

2 α^{2}

gives

3 = - 10 α

, so

α = - \frac{3}{10}

JEE Main · 2004 · 24 Apr 2004 · Offline · Single correct option