JEE Main 2025 (Online) 4th April Evening Shift | Complex Numbers Question 5 | Mathematics

If $\alpha$ is a root of the equation $x^2+x+1=0$ and $\sum_\limits{\mathrm{k}=1}^{\mathrm{n}}\left(\alpha^{\mathrm{k}}+\frac{1}{\alpha^{\mathrm{k}}}\right)^2=20$, then n is equal to _________.

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JEE Main 2025 (Online) 4th April Morning Shift

Numerical

-1

Let $\mathrm{A}=\{z \in \mathrm{C}:|z-2-i|=3\}, \mathrm{B}=\{z \in \mathrm{C}: \operatorname{Re}(z-i z)=2\}$ and $\mathrm{S}=\mathrm{A} \cap \mathrm{B}$. Then $\sum_{z \in S}|z|^2$ is equal to _________.

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JEE Main 2025 (Online) 29th January Evening Shift

Numerical

-1

Let integers $\mathrm{a}, \mathrm{b} \in[-3,3]$ be such that $\mathrm{a}+\mathrm{b} \neq 0$. Then the number of all possible ordered pairs (a, b), for which $\left|\frac{z-\mathrm{a}}{z+\mathrm{b}}\right|=1$ and $\left|\begin{array}{ccc}z+1 & \omega & \omega^2 \\ \omega & z+\omega^2 & 1 \\ \omega^2 & 1 & z+\omega\end{array}\right|=1, z \in \mathrm{C}$, where $\omega$ and $\omega^2$ are the roots of $x^2+x+1=0$, is equal to _____________ .

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JEE Main 2025 (Online) 23rd January Evening Shift

Numerical

-1

Let $\alpha, \beta$ be the roots of the equation $x^2-\mathrm{ax}-\mathrm{b}=0$ with $\operatorname{Im}(\alpha)<\operatorname{Im}(\beta)$. Let $\mathrm{P}_{\mathrm{n}}=\alpha^{\mathrm{n}}-\beta^{\mathrm{n}}$. If $\mathrm{P}_3=-5 \sqrt{7} i, \mathrm{P}_4=-3 \sqrt{7} i, \mathrm{P}_5=11 \sqrt{7} i$ and $\mathrm{P}_6=45 \sqrt{7} i$, then $\left|\alpha^4+\beta^4\right|$ is equal to __________.

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