Topic 1 Complex Number JEE MAIN 2023
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If the set $\left\{\operatorname{Re}\left(\frac{z-\bar{z}+z \bar{z}}{2-3 z+5 \bar{z}}\right): z \in C, \operatorname{Re}(z)=3\right\}$ is equal to the interval $(\alpha, \beta]$, then $24(\beta-\alpha)$ is equal to
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- 2 weeks, 2 days ago
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Let $\mathrm{A}=\left\{\theta \in(0,2 \pi): \frac{1+2 \mathrm{i} \sin \theta}{1-\mathrm{i} \sin \theta}\right.$ is purely imaginary $\}$. Then the sum of the elements in A is
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- 2 weeks, 2 days ago
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If for $\mathrm{z}=\alpha+\mathrm{i} \beta,|\mathrm{z}+2|=\mathrm{z}+4(1+\mathrm{i})$, then $\alpha+\beta$ and $\alpha \beta$ are the roots of the equation
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- 2 weeks, 2 days ago
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Let $\mathrm{S}=\left\{\mathrm{z}=\mathrm{x}+\mathrm{iy}: \frac{2 \mathrm{z}-3 \mathrm{i}}{4 \mathrm{z}+2 \mathrm{i}}\right.$ is a real number $\}$. Then which of the following is NOT correct?
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- 2 weeks, 2 days ago
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Let $\alpha, \beta$ be the roots of the equation $x^{2}-\sqrt{2} x+2=0$. Then $\alpha^{14}+\beta^{14}$ is equal to
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- 2 weeks, 2 days ago
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The complex number $\mathrm{z}=\frac{\mathrm{i}-1}{\cos \frac{\pi}{3}+\mathrm{i} \sin \frac{\pi}{3}}$ is equal to
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- 2 weeks, 2 days ago
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Let $\mathrm{a} \neq \mathrm{b}$ be two non-zero real numbers.
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- 2 weeks, 2 days ago
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Let the complex number $\mathrm{z}=\mathrm{x}+\mathrm{iy}$ be such that $\frac{2 \mathrm{z}-3 \mathrm{i}}{2 \mathrm{z}+\mathrm{i}}$ is purely imaginary. If $\mathrm{x}+\mathrm{y}^{2}=0$,
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- 2 weeks, 2 days ago
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For $\mathrm{a} \in \mathrm{C}$, let $\mathrm{A}=\{\mathrm{z} \in \mathrm{C}: \operatorname{Re}(\mathrm{a}+\overline{\mathrm{z}})>\operatorname{lm}(\overline{\mathrm{a}}+\mathrm{z})\}$
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- 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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- 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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- 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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- 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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- 2 weeks, 2 days ago
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If the set $\left\{\operatorname{Re}\left(\frac{z-\bar{z}+z \bar{z}}{2-3 z+5 \bar{z}}\right): z \in C, \operatorname{Re}(z)=3\right\}$ is equal to the interval $(\alpha, \beta]$, then $24(\beta-\alpha)$ is equal to
