The coefficient of the middle term in the binomial expansion in powers of x of (1+α x)4 and of (1-α x)6 is the same if α equals

Question

The coefficient of the middle term in the binomial expansion in powers of x of (1+α x)4 and of (1-α x)6 is the same if α equals (A) -(5/3) (B) (10/3) (C) -(3/10) (D) (3/5)

Tardigrade Admin · Accepted Answer

T 3 in (1+α x )4 = 4 C 2 α2 x 2 ; T 4 in (1-α x )6= 6 C 3(-α x )3=- 6 C 3 α3 x 3 hence 4 C 2 α2=- 6 C 3 α3 α=-( 4 C 2/ 6 C 3) ; α=-(6/20) ; α=-(3/10)

Q. 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

$- \frac{5}{3}$

$\frac{10}{3}$

$- \frac{3}{10}$

$\frac{3}{5}$

$T_{3}$ in $(1 + αx)^{4} =^{4} C_{2} α^{2} x^{2}; T_{4}$ in $(1 - αx)^{6} =^{6} C_{3} (- αx)^{3} = -^{6} C_{3} α^{3} x^{3}$
hence $^{4} C_{2} α^{2} = -^{6} C_{3} α^{3}$
$α = - \frac{^{4} C _{2}}{^{6} C _{3}}; α = - \frac{6}{20}; α = - \frac{3}{10}$

Q. 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

−35​

310​

−103​

53​