Statement – I: If = cos ( 2 7 ) + i sin ( 2 7 ) , p = + 2 + 4 , q = 3 + 5 + 6 , then the equation whose roots are p and q is x 2 + x + 2 . Statement – II: If is a root of z 7 = 1 , then 1 + + 2 + … + 6 = 0

Question

Statement – I: If $α = \cos (\frac{2 π}{7}) + i \sin (\frac{2 π}{7}), p = α + α^{2} + α^{4}, q = α^{3} + α^{5} + α^{6}$ , then the equation whose roots are $p$ and $q$ is $x^{2} + x + 2$ .

Statement – II: If $α$ is a root of $z^{7} = 1$ , then $1 + α + α^{2} + \dots + α^{6} = 0$

Smart Achievers · Accepted Answer

(1)

$α is 7^{th} root of unity \Rightarrow 1 + α + α^{2} + \dots + α^{6} = 0, p + q = - 1$

$p q = α^{4} + α^{6} + α^{5} + α^{7} + α^{8} + α^{7} + α^{9} + α^{10}$

$= 3 + (α + α^{2} + α^{3} + \dots + α^{6}) = 3 + (- 1) = 2$

$\Rightarrow x^{2} + x + 2 = 0$

$Both I and II are true and II is the correct explanation.$

Solution

Q.
Statement – I: If $α = \cos (\frac{2 π}{7}) + i \sin (\frac{2 π}{7}), p = α + α^{2} + α^{4}, q = α^{3} + α^{5} + α^{6}$ , then the equation whose roots are $p$ and $q$ is $x^{2} + x + 2$ .

Statement – II: If $α$ is a root of $z^{7} = 1$ , then $1 + α + α^{2} + \dots + α^{6} = 0$

1 Statement 1 is true, Statement – 2 is true

2 Statement 1 is false, Statement – 2 is false

3 Statement 1 is true, Statement – 2 is false

4 Statement 1 is false, Statement – 2 is true

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