Question:
If \( a = \cos \frac{2\pi}{7} + i \sin \frac{2\pi}{7} \), then the quadratic equation whose roots are \( \alpha = a + a^2 + a^4 \) and \( \beta = a^3 + a^5 + a^6 \) is
If \( a = \cos \frac{2\pi}{7} + i \sin \frac{2\pi}{7} \), then the quadratic equation whose roots are \( \alpha = a + a^2 + a^4 \) and \( \beta = a^3 + a^5 + a^6 \) is
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Sum of all $n$th roots of unity = 0.
Updated On: Apr 23, 2026
- $x^2 - x + 2 = 0$
- $x^2 + 2x + 2 = 0$
- $x^2 + x + 2 = 0$
- $x^2 + x - 2 = 0$
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The Correct Option is C
Solution and Explanation
Concept:
Roots of unity properties
Step 1: $a$ is 7th root of unity $\Rightarrow a^7 = 1$.
Step 2: Sum of all roots: \[ 1 + a + a^2 + \cdots + a^6 = 0 \]
Step 3: $\alpha + \beta$: \[ (a + a^2 + a^4) + (a^3 + a^5 + a^6) = -1 \]
Step 4: $\alpha\beta$ (using symmetry of roots) = $2$.
Step 5: Equation: \[ x^2 - (\alpha+\beta)x + \alpha\beta = 0 \] \[ x^2 + x + 2 = 0 \] Conclusion:
Required equation = $x^2 + x + 2 = 0$
Step 1: $a$ is 7th root of unity $\Rightarrow a^7 = 1$.
Step 2: Sum of all roots: \[ 1 + a + a^2 + \cdots + a^6 = 0 \]
Step 3: $\alpha + \beta$: \[ (a + a^2 + a^4) + (a^3 + a^5 + a^6) = -1 \]
Step 4: $\alpha\beta$ (using symmetry of roots) = $2$.
Step 5: Equation: \[ x^2 - (\alpha+\beta)x + \alpha\beta = 0 \] \[ x^2 + x + 2 = 0 \] Conclusion:
Required equation = $x^2 + x + 2 = 0$
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