If 1- 2-n are the nth roots of unity then nC1- nC2- - nC3-2 -nCn-n is equal to

The correct option is D 1/α[ (1+α)^n 1] ^nC_1 + ^nC_2.α + ^nC_3.α^2 + ......... + ^nC_n.α^n1 ⇒1/α [^n C_1. α + ^n C_2 . α^2 + ....... + ^n C_n ...

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Byju's Answer

Standard XII

Mathematics

Integration Using Substitution

If 1,α,α 2....

Question

If $1, α, α^{2} . . . . . . α^{n}$ are the $n^{t h}$ roots of unity then $^{n} C_{1} +^{n} C_{2} . α +^{n} C_{3} . α^{2} . . . . . . . . +^{n} C_{n} . α^{n}$ is equal to

\frac{1}{α}

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\frac{1}{α} (2^{n} - 1)

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α

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\frac{1}{α} [(1 + α)^{n} - 1]

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Solution

The correct option is D $\frac{1}{α} [(1 + α)^{n} - 1]$
$^{n} C_{1} +^{n} C_{2} . α +^{n} C_{3} . α^{2} + . . . . . . . . . +^{n} C_{n} . α^{n 1}$
$\Rightarrow \frac{1}{α} [^{n} C_{1} . α +^{n} C_{2} . α^{2} + . . . . . . . +^{n} C_{n} . α^{n}]$
$\Rightarrow \frac{1}{α} [^{n} C_{0} .1 +^{n} C_{1} . α +^{n} C_{2} . α^{2} + . . . . . . . +^{n} C_{n} . α^{n} - 1] \dots (^{n} C_{0} = 1)$
$\Rightarrow \frac{1}{α} [(1 + α)^{n} - 1]$

$^{n} C_{1} +^{n} C_{2} . α +^{n} C_{3} . α^{2} + . . . . . . . . . +^{n} C_{n} . α^{n 1} = \frac{1}{α} [(1 + α)^{n} - 1]$

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If 1,α,α2......αn are the nth roots of unity then nC1+nC2.α+nC3.α2........+nCn.αn is equal to

The correct option is D 1α[(1+α)n−1]nC1+nC2.α+nC3.α2+.........+nCn.αn1⇒1α[nC1.α+nC2.α2+.......+nCn.αn]⇒1α[nC0.1+nC1.α+nC2.α2+.......+nCn.αn−1]…(nC0=1)⇒1α[(1+α)n−1]nC1+nC2.α+nC3.α2+.........+nCn.αn1=1α[(1+α)n−1]

If $1, α, α^{2} . . . . . . α^{n}$ are the $n^{t h}$ roots of unity then $^{n} C_{1} +^{n} C_{2} . α +^{n} C_{3} . α^{2} . . . . . . . . +^{n} C_{n} . α^{n}$ is equal to