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Let $\alpha$ and $\beta$ be the sum and the product of all the non-zero solutions of the equation $(\bar{z})^{2}+|z|=0, z \in C$. Then $4\left(\alpha^{2}+\beta^{2}\right)$ is equal to
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in: Jee Mains
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- 2 weeks, 2 days ago
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If $\mathrm{z}=\frac{1}{2}-2 \mathrm{i}$ is such that $|\mathrm{z}+1|=\alpha \mathrm{z}+\beta(1+\mathrm{i})$ and $\alpha, \beta \in \mathrm{R}$, then $\alpha+\beta$ is equal to
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- 1
- 2 weeks, 2 days ago
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The number of complex numbers z, satisfying $|z|=1$ and $\left|\frac{z}{\bar{z}}+\frac{\bar{z}}{z}\right|=1$, is
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- 1
- 2 weeks, 2 days ago
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Let $\alpha, \beta$ be the roots of the equation $x^{2}-a x-b=0$ with $\operatorname{Im}(\alpha)<\operatorname{Im}(\beta)$. Let $P_{n}=\alpha^{n}-\beta^{n}$. If $P_{3}=-5 \sqrt{7} i, P_{4}=-3 \sqrt{7} i$,
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- 1
- 2 weeks, 2 days ago
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If $\alpha$ and $\beta$ are theroots of theequation $2 z^{2}-3 z-2 i=0$, where $i=\sqrt{-1}$,
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- 1
- 2 weeks, 2 days ago
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If $\alpha+\mathrm{i} \beta$ and $\gamma+\mathrm{i} \delta$ aretherootsof $\mathrm{x}^{2}-(3-2 \mathrm{i}) \mathrm{x}-(2 \mathrm{i}-2)=0, \mathrm{i}=\sqrt{-1}$,then $\alpha \gamma+\beta \delta$ is equal to
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- 1
- 2 weeks, 2 days ago
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Let $\mathrm{A}=\{\mathrm{z} \in \mathrm{C}:|\mathrm{z}-2-\mathrm{i}|=3\}, \mathrm{B}=\{\mathrm{z} \in \mathrm{C}: \operatorname{Re}(\mathrm{z}-\mathrm{iz})=2\}$ and $\mathrm{S}=\mathrm{A} \cap \mathrm{B}$.
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- 1
- 2 weeks, 2 days ago
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Let the product of $\omega_{1}=(8+\mathrm{i}) \sin \theta+(7+4 \mathrm{i}) \cos \theta$ and $\omega_{2}=(1+8 \mathrm{i}) \sin \theta+(4+7 \mathrm{i}) \cos \theta$ be $\alpha+\mathrm{i} \beta, \mathrm{i}=\sqrt{-1}$.
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- 1
- 2 weeks, 2 days ago
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Among the statements(S1) : The set $\left\{\mathrm{z} \in \mathrm{C}-\{-\mathrm{i}\}:|\mathrm{z}|=1\right.$ and $\frac{\mathrm{z}-\mathrm{i}}{\mathrm{z}+\mathrm{i}}$ is purely real\} contains exactly two elements,
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- 1
- 2 weeks, 2 days ago
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Let $\mathrm{A}=\left\{\theta \in[0,2 \pi]: 1+10 \operatorname{Re}\left(\frac{2 \cos \theta+\mathrm{i} \sin \theta}{\cos \theta-3 \mathrm{i} \sin \theta}\right)=0\right\}$.
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- 1
- 2 weeks, 2 days ago
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Let $\mathrm{z}_{1}, \mathrm{z}_{2}$ and $\mathrm{z}_{3}$ be three complex numbers on the circle $|\mathrm{z}|=1$ with $\arg \left(\mathrm{z}_{1}\right)=-\frac{\pi}{4}, \arg \left(\mathrm{z}_{2}\right)=0$ and $\arg \left(\mathrm{z}_{3}\right)=\frac{\pi}{4}$.
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- 1
- 2 weeks, 2 days ago
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Let $\mathrm{z}=(1+\mathrm{i})(1+2 \mathrm{i})(1+3 \mathrm{i}) \ldots(1+\mathrm{ni})$, where $\mathrm{i}=\sqrt{-1}$. If $|\mathrm{z}|^{2}=44200$, then n is equal to
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- 1
- 2 weeks, 2 days ago
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Let $\mathrm{z} \in \mathrm{C}$ be such that $\frac{\mathrm{z}^{2}+3 \mathrm{i}}{\mathrm{z}-2+\mathrm{i}}=2+3 \mathrm{i}$. Then the sum of all possible values of $\mathrm{z}^{2}$ is
Started by:
- 1
- 2 weeks, 2 days ago
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Let $\alpha$ and $\beta$ be the sum and the product of all the non-zero solutions of the equation $(\bar{z})^{2}+|z|=0, z \in C$. Then $4\left(\alpha^{2}+\beta^{2}\right)$ is equal to
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