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

3^{10}

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2^{10}

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2^{9}

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None of these

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Solution

The correct option is C $2^{10}$
$(1 + x)^{10} = a_{0} + a_{1} x + a_{2} x^{2} + . . . a_{10} x^{10}$
$(1 - x)^{10} = a_{0} - a_{1} x + a_{2} x^{2} - a_{3} x^{3} . . . a_{10} x^{10}$
Adding both we get
$(1 + x)^{10} + (1 - x)^{10} = 2 (a_{0} + a_{2} x^{2} + a_{4} x^{4} + . . . a_{10} x^{10})$
Substituting, $x = \sqrt{- 1} = i$
we get
$2^{5} [e^{\frac{i π}{4} \times 2} + e^{\frac{- i π}{4} \times 2}] = 2 (a_{0} - a_{2} + a_{4} + . . . - a_{10})$
$2^{4} [e^{\frac{i π}{2}} + e^{i \frac{- π}{2}}] = a_{0} - a_{2} + a_{4} + . . . - a_{10}$
$2^{4} [i - i] = a_{0} - a_{2} + a_{4} + . . . - a_{10}$
$a_{0} - a_{2} + a_{4} + . . . - a_{10} = 0$
$(a_{0} - a_{2} + a_{4} + . . . - a_{10})^{2} = 0$ ...(i)
Similarly
$(1 + x)^{10} - (1 - x)^{10} = 2 (a_{1} x + a_{2} x^{3} + a_{5} x^{5} + . . . a_{9} x^{9})$
Substituting $x = i$ we get
$2^{5} [e^{\frac{i π}{4} \times 2} - e^{\frac{- i π}{4} \times 2}] = 2 i [a_{1} - a_{3} + a_{5} - a_{7} + a_{9}]$
$2^{4} [e^{\frac{i π}{2}} - e^{i \frac{- π}{2}}] = i [a_{1} - a_{3} + a_{5} - a_{7} + a_{9}]$
$2^{4} [i - (- i)] = i [a_{1} - a_{3} + a_{5} - a_{7} + a_{9}]$
$2^{5} (i) = i (a_{1} - a_{3} + a_{5} - a_{7} + a_{9})$
$2^{5} = (a_{1} - a_{3} + a_{5} - a_{7} + a_{9})$
$(a_{1} - a_{3} + a_{5} - a_{7} + a_{9})^{2} = (2^{5})^{2} = 2^{10}$ ...(ii)
Adding i and ii, we get $2^{10}$ .

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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^{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}

(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),

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