If $T_{0}, T_{1}, T_{2}, . . . . T_{n}$ represent the terms in the expansion of ${(x + a)}^{n}$ , then the value of
${(T_{0} - T_{2} + T_{4} - T_{6} + . . . .)}_{0}^{2}$ + ${(T_{1} - T_{3} + T_{5} - . . . .)}_{1}^{2}$ is

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Solution

The correct option is B
$T_{r + 1} =^{n} C_{r} x^{n - r} . a^{r} {(x + i a)}^{n} =^{n} C_{0} x^{n} +^{n} C_{1} x^{n - 1} . a . i +^{n} C_{2} x^{n - 2} a^{2} i^{2} + . . . . = (T_{0} - T_{2} + T_{4} . . . .) + i (T_{0} - T_{2} + T_{4} . . . .) ∴ {(T_{0} - T_{2} + . . . .)}_{0}^{2} + {(T_{1} - T_{3} + . . . .)}_{1}^{2} = {(x^{2} + a^{2})}^{n}$

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If $T_{0}, T_{1}, T_{2}, . . . . . . . . . . . . . . . . . . T_{n}$ represent the terms in the expansion of $(x + a)^{n}$ , then the value of $(T_{0} - T_{2} + T_{4} - T_{6} + . . . . . . . . . . . . . . . . .)^{2}$ + $(T_{1} - T_{3} + T_{5} - . . . . . . . . . . . . . . . . .)^{2}$ is