Q. If $T_{0},\ T_{1},T_{2},\ldots,T_{n}$ represents the terms in the expansion of $(x + a)^{n}$, then $\left( T_{0} - \ T_{2} + T_{4} - ... \right)^{2} + \left( T_{1} - \ T_{3} + T_{5} - ... \right)^{2}$ is equal to
A) $(x^{2} + a^{2})$
B) $\left( x^{2} + a^{2} \right)^{n}$
C) ${(x^{2} + a^{2})}^{\frac{1}{n}}$
D) ${(x^{2} + a^{2})}^{\frac{- 1}{n}}$
Explanation
(b)
Given expansion is${(x + a)}^{n}$
On replacing $a$ by $ai$ and $–ai$ respectively, we get
${(x + ai)}^{n} = \left( T_{0} - T_{2} + T_{4} - ... \right) + i(T_{1} - T_{3} + T_{5} - ...)$ …(i)
and ${(x - ai)}^{n} = \left( T_{0} - T_{2} + T_{4} - ... \right) + i(T_{1} - T_{3} + T_{5} - ...)$ …(ii)
On multiplying Eqs. (ii) and (i), we get required result
$\left( x^{2} + a^{2} \right)^{n} = \left( T_{0} - T_{2} + T_{4} - ... \right)^{2} + \left( T_{1} - T_{3} + T_{5} - ... \right)^{2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.