Aspire Faculty ID #16675 · Topic: CUET 2023 · Just now
CUET 2023

Let $a=\cos \dfrac{2\pi}{7}+i\sin \dfrac{2\pi}{7}$, $\alpha=a+a^2+a^4$ and $\beta=a^3+a^5+a^6$. Then the equation whose roots are $\alpha,\beta$ is

Solution

Here $a^7=1$ and $1+a+a^2+\cdots+a^6=0$. $\alpha+\beta=a+a^2+a^3+a^4+a^5+a^6=-1$ Also, $\alpha\beta=(a+a^2+a^4)(a^3+a^5+a^6)=2$ Required equation: $x^2-(\alpha+\beta)x+\alpha\beta=0$ $\Rightarrow x^2+x+2=0$
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