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

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

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

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

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

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Solution

The correct option is B $(x^{2} + a^{2})^{n}$
$(x + a)^{n} = x^{n} + C_{1} x^{n - 1} a + C_{2} x^{n - 2} a^{2} + C_{3} x^{n - 3} a^{3} + \dots \dots$ (1)
$= T_{0} + T_{1} + T_{2} + T_{3} + \dots$
$= (T_{0} + T_{2} + T_{4} + \dots) + (T_{1} + T_{3} + T_{5} \dots)$
The first bracket involves even powers of a and 2nd bracket involves odd powers of a. Also $i^{2} = - 1, i^{4} = 1, i^{3} = - i, i^{5} = i$ Replacing a by ai and -ai respectively in (1), we get
$(x + a i)^{n} = (T_{0} - T_{2} + T_{4} \dots) + i (T_{1} - T_{3} + T_{5} - \dots)$ .... (i)
$(x - a i)^{n} = (T_{0} - T_{2} + T_{4} \dots) - i (T_{1} - T_{3} + T_{5} - \dots)$ ....(ii)
Multiply (i) & (ii)
$(x^{2} + a^{2})^{n} = (T_{0} - T_{2} + T_{4} - \dots)^{2} + (T_{1} - T_{3} + T_{5} - \dots)^{2}$

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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