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Let $S=\left\{z \in C: \bar{z}=i\left(z^{2}+\operatorname{Re}(\bar{z})\right)\right\}$. Then $\sum_{z \in S}|z|^{2}$ is equal to
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in: Jee Mains
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- 2 weeks, 2 days ago
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Let $\alpha, \beta$ be the roots of the quadratic equation $x^{2}+\sqrt{6} x+3=0$. Then $\frac{\alpha^{23}+\beta^{23}+\alpha^{14}+\beta^{14}}{\alpha^{15}+\beta^{15}+\alpha^{10}+\beta^{10}}$ is equal to
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- 1
- 2 weeks, 2 days ago
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For two non-zero complex numbers $\mathrm{z}_{1}$ and $\mathrm{z}_{2}$, if $\operatorname{Re}\left(\mathrm{z}_{1} \mathrm{z}_{2}\right)=0$ and $\operatorname{Re}\left(\mathrm{z}_{1}+\mathrm{z}_{2}\right)=0$,
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- 1
- 2 weeks, 2 days ago
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Let $\mathrm{p}, \mathrm{q} \in \mathrm{R}$ and $(1-\sqrt{3} \mathrm{i})^{200}=2^{199}(\mathrm{p}+\mathrm{iq}), \mathrm{i}=\sqrt{-1}$.
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- 1
- 2 weeks, 2 days ago
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The value of $\left(\frac{1+\sin \frac{2 \pi}{9}+\mathrm{i} \cos \frac{2 \pi}{9}}{1+\sin \frac{2 \pi}{9}-\mathrm{i} \cos \frac{2 \pi}{9}}\right)^{3}$ is
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- 1
- 2 weeks, 2 days ago
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Let $z=1+i$ and $z_{1}=\frac{1+i \bar{z}}{\bar{z}(1-z)+1 / z}$. Then $\frac{12}{\pi} \arg \left(z_{1}\right)$ is equal to \_\_\_\_.
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- 1
- 2 weeks, 2 days ago
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Let $\alpha, \beta$ be the roots of the equation $\mathrm{x}^{2}-\mathrm{x}+2=0$ with $\operatorname{Im}(\alpha)>\operatorname{Im}(\beta)$. Then $\alpha^{6}+\alpha^{4}+\beta^{4}-5 \alpha^{2}$ is equal to
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- 1
- 2 weeks, 2 days ago
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Let z be a complex number such that $|z+2|=1$ and $\operatorname{Im}\left(\frac{z+1}{z+2}\right)=\frac{1}{5}$. Then the value of $|\operatorname{Re}(\overline{z+2})|$ is
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- 1
- 2 weeks, 2 days ago
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If the set $\mathrm{R}=\{(\mathrm{a}, \mathrm{b}): \mathrm{a}+5 \mathrm{~b}=42, \mathrm{a}, \mathrm{b} \in \mathrm{N}\}$ has m elements and $\sum_{\mathrm{n}=1}^{\mathrm{m}}\left(1-\mathrm{i}^{\mathrm{n}!}\right)=\mathrm{x}+\mathrm{i}$ y
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- 1
- 2 weeks, 2 days ago
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The sum of all possible values of $\theta \in[-\pi, 2 \pi]$, for which $\frac{1+\mathrm{i} \cos \theta}{1-2 \mathrm{i} \cos \theta}$ is purely imaginary, is equal to
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- 1
- 2 weeks, 2 days ago
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Let $\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}, \mathrm{x}_{4}$ be the solution of the equation $4 \mathrm{x}^{4}+8 \mathrm{x}^{3}-17 \mathrm{x}^{2}-12 \mathrm{x}+9=0$
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- 1
- 2 weeks, 2 days ago
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If $\mathrm{z}=\mathrm{x}+\mathrm{iy}, \mathrm{xy} \neq 0$, satisfies the equation $\mathrm{z}^{2}+\mathrm{i} \overline{\mathrm{z}}=0$, then $\left|\mathrm{z}^{2}\right|$ is equal to
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- 1
- 2 weeks, 2 days ago
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Let $\mathrm{z}_{1}$ and $\mathrm{z}_{2}$ be two complex numbers such that $\mathrm{z}_{1}+\mathrm{z}_{2}=5$ and $\mathrm{z}_{1}^{3}+\mathrm{z}_{2}^{3}=20+15 \mathrm{i}$. Then,
Started by:
- 1
- 2 weeks, 2 days ago
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Let $S=\left\{z \in C: \bar{z}=i\left(z^{2}+\operatorname{Re}(\bar{z})\right)\right\}$. Then $\sum_{z \in S}|z|^{2}$ is equal to
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