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Let z be a complex number such that $|z-6|=5$ and $|z+2-6 i|=5$. Then the value of $z^{3}+3 z^{2}-15 z+141$ is equal to
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in: Jee Mains
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- 1 week, 2 days ago
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Let z be the complex number satisfying $|z-5| \leq 3$ and having maximum positive principal argument. Then $34\left|\frac{5 z-12}{5 i z+16}\right|^{2}$ is equal to
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- 1
- 1 week, 2 days ago
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Let $\mathrm{S}=\{\mathrm{z}: 3 \leq|2 \mathrm{z}-3(1+\mathrm{i})| \leq 7\}$ be a set of complex numbers. Then $\operatorname{Min}_{\mathrm{z} \in \mathrm{S}}\left|\left(\mathrm{z}+\frac{1}{2}(5+3 \mathrm{i})\right)\right|$ is equal to
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- 1
- 1 week, 2 days ago
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Let $\mathrm{S}=\left\{\mathrm{z} \in \mathrm{C}:\left|\frac{\mathrm{z}-6 \mathrm{i}}{\mathrm{z}-2 \mathrm{i}}\right|=1\right.$ and $\left.\left|\frac{\mathrm{z}-8+2 \mathrm{i}}{\mathrm{z}+2 \mathrm{i}}\right|=\frac{3}{5}\right\}$. Then $\sum_{\mathrm{z} \in \mathrm{s}}|\mathrm{z}|^{2}$ is equal to
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- 1
- 1 week, 2 days ago
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Let the circle $\mathrm{C}_{1}:|\mathrm{z}|=\mathrm{r}$ and $\mathrm{C}_{2}:|\mathrm{z}-3-4 \mathrm{i}|=5, \mathrm{z} \in \mathrm{C}$, be such that $\mathrm{C}_{2}$ lies within $\mathrm{C}_{1}$. If $\mathrm{z}_{1}$ moves on $\mathrm{C}_{1},
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- 1
- 1 week, 2 days ago
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The number of values of $\mathrm{z} \in \mathrm{C}$, satisfying the equations $|\mathrm{z}-(4+8 \mathrm{i})|=\sqrt{10}$ and $|\mathrm{z}-(3+5 \mathrm{i})|+|\mathrm{z}-(5+11 \mathrm{i})|=4 \sqrt{5}$, is
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- 1
- 1 week, 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
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- 1
- 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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- 1
- 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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- 1
- 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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- 1
- 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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- 1
- 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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- 1
- 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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- 1
- 2 weeks, 2 days ago
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Let z be a complex number such that $|z-6|=5$ and $|z+2-6 i|=5$. Then the value of $z^{3}+3 z^{2}-15 z+141$ is equal to
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