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The sum of the square of the modulus of the elements in the set $\{\mathrm{z}=\mathrm{a}+\mathrm{ib}: \mathrm{a}, \mathrm{b} \in \mathrm{Z}, \mathrm{z} \in \mathrm{C},|\mathrm{z}-1| \leq 1,|\mathrm{z}-5| \leq|\mathrm{z}-5 \mathrm{i}|\}$ is \_\_\_\_.
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in: Jee Mains
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- 1 week, 2 days ago
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Let $\mathrm{P}=\{\mathrm{z} \in \mathrm{C}:|\mathrm{z}+2-3 \mathrm{i}| \leq 1\}$ and $\mathrm{Q}=\{\mathrm{z} \in \mathrm{C}: \mathrm{z}(1+\mathrm{i})+\overline{\mathrm{z}}(1-\mathrm{i}) \leq-8\}$. Let in $\mathrm{P} \cap \mathrm{Q},|\mathrm{z}-3+2 \mathrm{i}|$ be maximum and minimum at $z_{1}$ and $z_{2}$ respectively. If $\left|z_{1}\right|^{2}+2\left|z_{2}\right|^{2}=\alpha+\beta \sqrt{2}$, where $\alpha, \beta$ are integers, then $\alpha+\beta$ equals
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- 1
- 1 week, 2 days ago
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The area (in sq. units) of the region $\mathrm{S}=\{\mathrm{z} \in \mathrm{C} ;|\mathrm{z}-1| \leq 2 ;(\mathrm{z}+\overline{\mathrm{z}})+\mathrm{i}(\mathrm{z}-\overline{\mathrm{z}}) \leq 2, \operatorname{Im}(\mathrm{z}) \geq 0\}$ is
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- 1
- 1 week, 2 days ago
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Let the complex numbers $\alpha$ and $\frac{1}{\bar{\alpha}}$ lie on the circles $\left|\mathrm{z}-\mathrm{z}_{0}\right|^{2}=4$ and $\left|\mathrm{z}-\mathrm{z}_{0}\right|^{2}=16$ respectively, where $\mathrm{z}_{0}=1+\mathrm{i}$. Then, the value of $100|\alpha|^{2}$ is
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- 1
- 1 week, 2 days ago
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If $\alpha$ denotes the number of solutions of $|1-\mathrm{i}|^{\mathrm{x}}=2^{\mathrm{x}}$ and $\beta=\left(\frac{|\mathrm{z}|}{\arg (\mathrm{z})}\right)$, where $\mathrm{z}=\frac{\pi}{4}(1+\mathrm{i})^{4}\left[\frac{1-\sqrt{\pi \mathrm{i}}}{\sqrt{\pi}+\mathrm{i}}+\frac{\sqrt{\pi}-\mathrm{i}}{1+\sqrt{\pi \mathrm{i}}}\right], \mathrm{i}=\sqrt{-1}$ then the distance of the point $(\alpha, \beta)$ from the line $4 \mathrm{x}-3 \mathrm{y}=7$ is
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- 1
- 1 week, 2 days ago
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If z is a complex number such that $|\mathrm{z}| \geq 1$, then the minimum value of $\left|\mathrm{z}+\frac{1}{2}(3+4 \mathrm{i})\right|$ is
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- 1
- 1 week, 2 days ago
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If $\mathrm{S}=\{\mathrm{z} \in \mathrm{C}:|\mathrm{z}-\mathrm{i}|=|\mathrm{z}+\mathrm{i}|=|\mathrm{z}-1|\}$, then, $\mathrm{n}(\mathrm{S})$ is :
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- 1
- 1 week, 2 days ago
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Let the curve $\mathrm{z}(1+\mathrm{i})+\overline{\mathrm{z}}(1-\mathrm{i})=4, \mathrm{z} \in \mathrm{C}$, divide the region $|\mathrm{z}-3| \leq 1$ into two parts of areas $\alpha$ and $\beta$. Then $|\alpha-\beta|$ equals
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- 1
- 1 week, 2 days ago
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Let $\left|\frac{\bar{z}-\mathrm{i}}{2 \bar{z}+\mathrm{i}}\right|=\frac{1}{3}, \mathrm{z} \in \mathrm{C}$, be the equation of a circle with center at C. If the area of the triangle, whose vertices are at the points (0, 0), C and $(\alpha, 0)$ is 11 square units, then $\alpha^{2}$ equals
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- 1
- 1 week, 2 days ago
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Let O be the origin, the point A be $\mathrm{z}_{1}=\sqrt{3}+2 \sqrt{2} \mathrm{i}$, the point $\mathrm{B}\left(\mathrm{z}_{2}\right)$ be such that $\sqrt{3}\left|\mathrm{z}_{2}\right|=\left|\mathrm{z}_{1}\right|$ and $\arg \left(\mathrm{z}_{2}\right)=\arg \left(\mathrm{z}_{1}\right)+\frac{\pi}{6}$. Then
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- 1 week, 2 days ago
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Let $\left|\mathrm{z}_{1}-8-2 \mathrm{i}\right| \leq 1$ and $\left|\mathrm{z}_{2}-2+6 \mathrm{i}\right| \leq 2, \mathrm{z}_{1}, \mathrm{z}_{2} \in \mathrm{C}$. Then the minimum value of $\left|\mathrm{z}_{1}-\mathrm{z}_{2}\right|$ is
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- 1
- 1 week, 2 days ago
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If the locus of $\mathrm{z} \in \mathrm{C}$, such that $\operatorname{Re}\left(\frac{\mathrm{z}-1}{2 \mathrm{z}+\mathrm{i}}\right)+\operatorname{Re}\left(\frac{\overline{\mathrm{z}}-1}{2 \overline{\mathrm{z}}-\mathrm{i}}\right)=2$, is a circle of radius r and center (a,b), then $\frac{15 \mathrm{ab}}{\mathrm{r}^{2}}$ is equal to
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- 1
- 1 week, 2 days ago
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Let $\mathrm{A}=\{\mathrm{z} \in \mathrm{C}:|\mathrm{z}-2| \leq 4\}$ and $\mathrm{B}=\{\mathrm{z} \in \mathrm{C}:|\mathrm{z}-2|+|\mathrm{z}+2|=5\}$. Then the max $\left\{\left|\mathrm{z}_{1}-\mathrm{z}_{2}\right|: \mathrm{z}_{1} \in \mathrm{~A}\right.$ and $\left.\mathrm{z}_{2} \in \mathrm{~B}\right\}$ is
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- 1
- 1 week, 2 days ago
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Let z be a complex number such that $|\mathrm{z}|=1$. If $\frac{2+\mathrm{k}^{2} \mathrm{z}}{\mathrm{k}+\overline{\mathrm{z}}}=\mathrm{kz}, \mathrm{k} \in \mathrm{R}$, then the maximum distance of $\mathrm{k}+\mathrm{ik}^{2}$ from the circle $|\mathrm{z}-(1+2 \mathrm{i})|=1$ is
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- 1
- 1 week, 2 days ago
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The sum of the square of the modulus of the elements in the set $\{\mathrm{z}=\mathrm{a}+\mathrm{ib}: \mathrm{a}, \mathrm{b} \in \mathrm{Z}, \mathrm{z} \in \mathrm{C},|\mathrm{z}-1| \leq 1,|\mathrm{z}-5| \leq|\mathrm{z}-5 \mathrm{i}|\}$ is \_\_\_\_.
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