If ${(1 + x)}^{10} = a_{0} + a_{1} x + a_{2} x^{2} + . . . . . . a_{10} x^{10}$ , then value of
${(a_{0} - a_{2} + a_{4} - a_{6} + a_{8} - a_{10})}^{2} + {(a_{1} - a_{3} + a_{5} - a_{7} + a_{9})}^{2}$ is

2^{10}

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2

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2^{20}

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None of these

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Solution

The correct option is B None of these
$x = i$
$(1 + i)^{10} = a_{0} + a_{1} i - a_{2} - a_{3} i + a_{4} . . . . . . a_{10}$ ………. $(1)$
$x = - i$
$(1 - i)^{10} = a_{0} - a_{1} i - a_{2} + a_{3} i + a_{4} . . . . . a_{10}$ …….. $(2)$
$(1)$ $+$ $(2)$
$\Rightarrow (1 + i)^{10} + (1 - i)^{10} = 2 (a_{0} - a_{2} + a_{4} . . . . . . a_{10})$
$(\sqrt{2})^{10} ((e^{i π / 4})^{10} + (e^{- i π / 4})^{10}) = 2 (a_{0} + a_{2} + a_{4} . . . .)$
$2^{5} = 2 (a_{0} - a_{1} . . . \dots)$
$(a_{0} + a_{2} + a_{4} . . . . . . a_{10})^{2} = 0$
$(1)$ - $(2)$
$(1 + i)^{10} - (1 - i)^{10} = (a_{1} - a_{3} + a_{5} . . . \dots . . a_{n}) i$
$(\sqrt{2})^{10}$ $(2 sin \frac{10 π}{4}) i = (a_{1} - a_{3} + a_{5} . . . . . a_{9}) i$
$(2^{6})^{2} = (a_{1} - a_{3} + a_{5} . . . . . a_{9})^{2} = 2^{12}$
$(a_{0} - a_{2} + a_{4} . . . . . a_{10})^{2} + (a_{1} - a_{3} + a_{5} . . . \dots + a_{9})^{2} = 2^{12}$ .

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Similar questions

(1 + x)^{10} = a_{0} + a_{1} x + a_{2} x^{2} + . . . . . + a_{10} x^{10}

(a_{0} - a_{2} + a_{4} - a_{6} + a_{8} - a_{10})^{2} + (a_{1} - a_{3} + a_{5} - a_{7} + a_{9})^{2}

i f (1 + x)^{10} = a_{0} + a_{1} x + . . . . a_{10} x^{10}, t h e n (a_{0} - a_{2} + a_{4} - a_{6} + a_{8} - a_{10})^{2} + (a_{1} - a_{3} + a_{5} - a_{7} + a_{9})^{2}

I f (1 + x)^{10} = a_{0} + a_{1} x + . . . . a_{10} x^{10}, t h e n (a_{0} - a_{2} + a_{4} - a_{6} + a_{8} - a_{10})^{2} + (a_{1} - a_{3} + a_{5} - a_{7} + a_{9})^{2}

(1 + x + x^{2}) n = a 0 + a^{1} x + a^{2} x + a^{3} x^{3} + . . . . . . . a (2^{n}) x (2^{n})

(a^{0} + a^{3} + a^{6} + . . . . . . . . .) = (a^{1} + a^{4} + a^{7}) = (a^{2} + a^{5} + a^{8}) = 3^{(} n - 1),

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

a_{0} + a_{1} x + a_{2} x^{2} + . . . . . + a_{r} x^{r} + . . . . . + a_{2 n} x^{2 n}

a_{0} + a_{3} + a_{6} + . . . . . = a_{1} + a_{4} + a_{7} + . . . . . = a_{2} + a_{5} + a_{8} + . . . . = 3^{n - 1}

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