If $T_{0}, T_{1}, T_{2}, . . ., T_{n}$ represent in 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 equal to

{(x^{2} - a^{2})}^{2}

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{(x^{2} + a^{2})}^{n}

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(a^{2} - x^{2})^{n}

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n o n e

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Solution

The correct option is B ${(x^{2} + a^{2})}^{n}$
$(x + a)^{n} = x^{n} +^{n} C_{0} (x)^{n - 1} (a) + . . . +^{n} C_{n} (a)^{n}$
Hence,
$T_{0} = x^{n}$
$T_{1} =^{n} C_{0} (x)^{n - 1} (a)$
.
.
.
$T_{n} =^{n} C_{n} (a)^{n}$
Replace $a$ by $i a$ .
$(x + i a)^{n} = x^{n} + i \times^{n} C_{1} \times (x)^{n - 1} (a) -^{n} C_{2} \times (x)^{n - 2} (a)^{2} + . . . + i^{n} \times^{n} C_{n} a^{n}$
$\Rightarrow (x + i a)^{n} = (x^{n} - n C_{2} (x)^{n - 2} + . . .) + i (^{n} C_{1} (x)^{n - 1} (a) -^{n} C_{3} (x)^{n - 3} (a)^{3} + . . .)$
$\Rightarrow (x + i a)^{n} = (T_{0} - T_{2} + T_{4} - . . .) + i (T_{1} - T_{3} + T_{5} - . . .)$
Consider the modulus of $(x + i a)^{n}$
$\Rightarrow {| (x + i a)^{n} |}^{2} = (T_{0} - T_{2} + T_{4} - . . .)^{2} + (T_{1} - T_{3} + T_{5} - . . .)^{2}$
$\Rightarrow {| (x + i a) |}^{2 n} = (T_{0} - T_{2} + T_{4} - . . .)^{2} + (T_{1} - T_{3} + T_{5} - . . .)^{2}$
$\Rightarrow (x^{2} + a^{2})^{n} = (T_{0} - T_{2} + T_{4} - . . .)^{2} + (T_{1} - T_{3} + T_{5} - . . .)^{2}$
Hence, option B is the correct answer.

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Similar questions

Q. 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}

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