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- For $\alpha, \beta, z \in C$ and $\lambda>1$, if $\sqrt{\lambda-1}$ is the radius of the circle $|z-\alpha|^{2}+|z-\beta|^{2}=2 \lambda$, then $|\alpha-\beta|$ is equal to \_\_\_\_.
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- 1 week, 2 days ago
- Let C be the circle in the complex plane with centre $\mathrm{z}_{0}=\frac{1}{2}(1+3 \mathrm{i})$ and radius r = 1. Let $\mathrm{z}_{1}=1+\mathrm{i}$ and the complex number $\mathrm{z}_{2}$ be outside the circle C such that $\left|\mathrm{z}_{1}-\mathrm{z}_{0}\right|\left|\mathrm{z}_{2}-\mathrm{z}_{0}\right|=1$.
- 1
- 1 week, 2 days ago
- Let $\mathrm{w}=\mathrm{z} \overline{\mathrm{z}}+\mathrm{k}_{1} \mathrm{z}+\mathrm{k}_{2} \mathrm{iz}+\lambda(1+\mathrm{i}), \mathrm{k}_{1}, \mathrm{k}_{2} \in \mathrm{R}$. Let $\mathrm{Re}(\mathrm{w})=0$ be the circle C of radius 1 in the first quadrant touching the line $\mathrm{y}=1$ and the $y$-axis. If the curve $\operatorname{Im}(w)=0$ intersects C at A and B, then $30(A B)^{2}$ is equal to \_\_\_\_.
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- 1 week, 2 days ago
- Let $\alpha=8-14 i, A=\left\{z \in C: \frac{a z-\bar{\alpha} \bar{z}}{z^{2}-(\bar{z})^{2}-112 i}=1\right\}$ and $B=\{z \in C:|z+3 i|=4\}$. The $\sum_{z \in A \cap B}(\operatorname{Rez}-\operatorname{Imz})$ is equal to \_\_\_\_.
- 1
- 1 week, 2 days ago
- For all $\mathrm{z} \in \mathrm{C}$ on the curve $\mathrm{C}_{1}:|\mathrm{z}|=4$, let the locus of the point $\mathrm{z}+\frac{1}{\mathrm{z}}$ be the curve $\mathrm{C}_{2}$. Then
- 1
- 1 week, 2 days ago
- Let $\mathrm{w}_{1}$ be the point obtained by the rotation of $\mathrm{z}_{1}=5+4 \mathrm{i}$ about the origin through a right angle in the anticlockwise direction,
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- 1 week, 2 days ago
- Let $\mathrm{z}_{1}=2+3 \mathrm{i}$ and $\mathrm{z}_{2}=3+4 \mathrm{i}$. The set $\mathrm{S}=\left\{\mathrm{z} \in \mathrm{C}:\left|\mathrm{z}-\mathrm{z}_{1}\right|^{2}-\left|\mathrm{z}-\mathrm{z}_{2}\right|^{2}=\left|\mathrm{z}_{1}-\mathrm{z}_{2}\right|^{2}\right\}$ represents a
- 1
- 1 week, 2 days ago
- Let z be a complex number such that $\left|\frac{z-2 i}{z+i}\right|=2, z \neq-i$. Then z lies on the circle of radius 2 and centre
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- 1 week, 2 days ago
- Let $\mathrm{S}=\{\mathrm{z} \in \mathrm{C},|\mathrm{z}-1|=1$ and $(\sqrt{2}-1)(\mathrm{z}+\overline{\mathrm{z}})-\mathrm{i}(\mathrm{z}-\overline{\mathrm{z}})=2 \sqrt{2}\}$. Let $\mathrm{z}_{1}, \mathrm{z}_{2} \in \mathrm{~S}$ be such that $\left|\mathrm{z}_{1}\right|=\max _{\mathrm{Z} \in \mathrm{S}}|\mathrm{z}|$ and $\left|\mathrm{z}_{2}\right|=\min _{\mathrm{Z} \in \mathrm{S}}|\mathrm{z}|$. Then $\left|\sqrt{2} \mathrm{z}_{1}-\mathrm{z}_{2}\right|^{2}$ equals
- 1
- 1 week, 2 days ago
- If the center and radius of the circle $\left|\frac{\mathrm{z}-2}{\mathrm{z}-3}\right|=2$ are respectively $(\alpha, \beta)$ and $\gamma$, then $3(\alpha, \beta, \gamma)$ is equal to
- 1
- 1 week, 2 days ago
- Let a, b be two real numbers such that $\mathrm{ab}<0$. If the complex number $\frac{1+\mathrm{ai}}{\mathrm{b}+\mathrm{i}}$ is of unit modulus and a + ib lies on the circle $|z-1|=|2 z|$, then a possible value of $\frac{1+[a]}{4 b}$, where [t] is greatest integer function, is
- 1
- 1 week, 2 days ago
- Let z be a complex number such that the real part of $\frac{\mathrm{z}-2 \mathrm{i}}{\mathrm{z}+2 \mathrm{i}}$ is zero. Then, the maximum value of $|\mathrm{z}-(6+8 \mathrm{i})|$ is equal to
- 1
- 1 week, 2 days ago
- Consider the following two statements : Statement I : For any two non-zero complex numbers $\mathrm{z}_{1}, \mathrm{z}_{2},\left(\left|\mathrm{z}_{1}\right|+\left|\mathrm{z}_{2}\right|\right)\left|\frac{\mathrm{z}_{1}}{\left|\mathrm{z}_{1}\right|}+\frac{\mathrm{z}_{2}}{\left|\mathrm{z}_{2}\right|}\right| \leq 2\left(\left|\mathrm{z}_{1}\right|+\left|\mathrm{z}_{2}\right|\right)$,
- 1
- 1 week, 2 days ago
- Let $\mathrm{S}_{1}=\{\mathrm{z} \in \mathrm{C}:|\mathrm{z}| \leq 5\}, \mathrm{S}_{2}=\left\{\mathrm{z} \in \mathrm{C}: \operatorname{Im}\left(\frac{\mathrm{z}+1-\sqrt{3} \mathrm{i}}{1-\sqrt{3} \mathrm{i}}\right) \geq 0\right\}$ and $\mathrm{S}_{3}=\{\mathrm{z} \in \mathrm{C}: \operatorname{Re}(\mathrm{z}) \geq 0\}$. Then the area of the region $\mathrm{S}_{1} \cap \mathrm{~S}_{2} \cap \mathrm{~S}_{3}$ is
- 1
- 1 week, 2 days ago
- If $\mathrm{z}_{1}, \mathrm{z}_{2}$ are two distinct complex number such that $\left|\frac{\mathrm{z}_{1}-2 \mathrm{z}_{2}}{\frac{1}{2}-\mathrm{z}_{1} \overline{\mathrm{z}}_{2}}\right|=2$, then
- 1
- 1 week, 2 days ago
- 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 \_\_\_\_.
- 1
- 1 week, 2 days ago
- 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
- 1
- 1 week, 2 days ago
- 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
- 1
- 1 week, 2 days ago
- 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
- 1
- 1 week, 2 days ago
- 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
- 1
- 1 week, 2 days ago
- 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
- 1
- 1 week, 2 days ago
- 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 :
- 1
- 1 week, 2 days ago
- 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
- 1
- 1 week, 2 days ago
- 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
- 1
- 1 week, 2 days ago
- 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
- 1
- 1 week, 2 days ago
- 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
- 1
- 1 week, 2 days ago
- 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
- 1
- 1 week, 2 days ago
- 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
- 1
- 1 week, 2 days ago
- If $\mathrm{z}_{1}, \mathrm{z}_{2}, \mathrm{z}_{3} \in \mathrm{C}$ are the vertices of an equilateral triangle, whosecentroids $\mathrm{z}_{0}$, then $\sum_{\mathrm{k}=1}^{3}\left(\mathrm{z}_{\mathrm{k}}-\mathrm{z}_{0}\right)^{2}$ is equal to
- 1
- 1 week, 2 days ago
- 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
- 1
- 1 week, 2 days ago
- 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
- 1
- 1 week, 2 days ago
- 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
- 1
- 1 week, 2 days ago
- 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
- 1
- 1 week, 2 days ago
- 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
- 1
- 1 week, 2 days ago
- 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},
- 1
- 1 week, 2 days ago
- 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
- 1
- 1 week, 2 days ago
- 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
- 1
- 2 weeks, 3 days ago
- 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
- 1
- 2 weeks, 3 days ago
- 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
- 1
- 2 weeks, 3 days ago
- 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?
- 1
- 2 weeks, 3 days ago
- Let $\alpha, \beta$ be the roots of the equation $x^{2}-\sqrt{2} x+2=0$. Then $\alpha^{14}+\beta^{14}$ is equal to
- 1
- 2 weeks, 3 days ago
- The complex number $\mathrm{z}=\frac{\mathrm{i}-1}{\cos \frac{\pi}{3}+\mathrm{i} \sin \frac{\pi}{3}}$ is equal to
- 1
- 2 weeks, 3 days ago
- 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 $\mathrm{x}+\mathrm{y}^{2}=0$,
- 1
- 2 weeks, 3 days ago
- For $\mathrm{a} \in \mathrm{C}$, let $\mathrm{A}=\{\mathrm{z} \in \mathrm{C}: \operatorname{Re}(\mathrm{a}+\overline{\mathrm{z}})>\operatorname{lm}(\overline{\mathrm{a}}+\mathrm{z})\}$
- 1
- 2 weeks, 3 days ago
- 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
- 1
- 2 weeks, 3 days ago
- 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
- 1
- 2 weeks, 3 days ago
- 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 $\operatorname{Re}\left(\mathrm{z}_{1}+\mathrm{z}_{2}\right)=0$,
- 1
- 2 weeks, 3 days ago
- Let $\mathrm{p}, \mathrm{q} \in \mathrm{R}$ and $(1-\sqrt{3} \mathrm{i})^{200}=2^{199}(\mathrm{p}+\mathrm{iq}), \mathrm{i}=\sqrt{-1}$.
- 1
- 2 weeks, 3 days ago
- 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
- 1
- 2 weeks, 3 days ago
- 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 \_\_\_\_.
- 1
- 2 weeks, 3 days ago
- 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
- 1
- 2 weeks, 3 days ago
- 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
- 1
- 2 weeks, 3 days ago
- 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
- 1
- 2 weeks, 3 days ago
- Let $\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}, \mathrm{x}_{4}$ be the solution of the equation $4 \mathrm{x}^{4}+8 \mathrm{x}^{3}-17 \mathrm{x}^{2}-12 \mathrm{x}+9=0$
- 1
- 2 weeks, 3 days ago
- 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
- 1
- 2 weeks, 3 days ago
- 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
- 1
- 2 weeks, 3 days ago
- 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,
- 1
- 2 weeks, 3 days ago
- Let $\alpha, \beta$ be the roots of the equation $x^{2}-\sqrt{6} x+3=0$ such that $\operatorname{Im}(\alpha)>\operatorname{Im}(\beta)$. Let a,
- 1
- 2 weeks, 3 days ago
- Let r and $\theta$ respectively be the modulus and amplitude of the complex number $\mathrm{z}=2-\mathrm{i}\left(2 \tan \frac{5 \pi}{8}\right)$, then $(\mathrm{r}, \theta)$ is equal to
- 1
- 2 weeks, 3 days ago
- 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
- 1
- 2 weeks, 3 days ago
- The number of complex numbers z, satisfying $|z|=1$ and $\left|\frac{z}{\bar{z}}+\frac{\bar{z}}{z}\right|=1$, is
- 1
- 2 weeks, 3 days ago
- 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 $P_{3}=-5 \sqrt{7} i, P_{4}=-3 \sqrt{7} i$,
- 1
- 2 weeks, 3 days ago
- If $\alpha$ and $\beta$ are theroots of theequation $2 z^{2}-3 z-2 i=0$, where $i=\sqrt{-1}$,
- 1
- 2 weeks, 3 days ago
- 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
- 1
- 2 weeks, 3 days ago
- 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
- 1
- 2 weeks, 3 days ago
- 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 $\mathrm{S}=\mathrm{A} \cap \mathrm{B}$.
- 1
- 2 weeks, 3 days ago
- 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 $\alpha+\mathrm{i} \beta, \mathrm{i}=\sqrt{-1}$.
- 1
- 2 weeks, 3 days ago
- 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,
- 1
- 2 weeks, 3 days ago
- Let $\mathrm{A}=\left\{\theta \in[0,2 \pi]: 1+10 \operatorname{Re}\left(\frac{2 \cos \theta+\mathrm{i} \sin \theta}{\cos \theta-3 \mathrm{i} \sin \theta}\right)=0\right\}$.
- 1
- 2 weeks, 3 days ago
- 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 $\arg \left(\mathrm{z}_{3}\right)=\frac{\pi}{4}$.
- 1
- 2 weeks, 3 days ago
- 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
- 1
- 2 weeks, 3 days ago
- Let $\mathrm{S}=\left\{\mathrm{z} \in \mathrm{C}: \mathrm{z}^{2}+\sqrt{6} \mathrm{i} \mathrm{z}-3=0\right\}$. Then $\sum_{\mathrm{z} \in \mathrm{S}} \mathrm{z}^{8}$ is equal to
- 1
- 2 weeks, 3 days ago
- Let $\mathrm{S}=\left\{\mathrm{z} \in \mathrm{C}: 4 \mathrm{z}^{2}+\overline{\mathrm{z}}=0\right\}$. Then $\sum_{\mathrm{z} \in \mathrm{S}}|\mathrm{z}|^{2}$ is equal to
- 1
- 2 weeks, 3 days ago
- If $\mathrm{z}=\frac{\sqrt{3}}{2}+\frac{\mathrm{i}}{2}, \mathrm{i}=\sqrt{-1}$, then $\left(\mathrm{z}^{201}-\mathrm{i}\right)^{8}$ is equal to
- 1
- 2 weeks, 3 days ago
- 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
- 1
- 2 weeks, 3 days ago
- Let x and y be real number such that $50\left(\frac{2 \mathrm{x}}{1+3 \mathrm{i}}-\frac{\mathrm{y}}{1-2 \mathrm{i}}\right)=31+17 \mathrm{i}, \mathrm{i}=\sqrt{-1}$. Then the value of 10(x -3y) is
- 1
- 2 weeks, 3 days ago
- Let z be a complex number such that $|z+2|=|z-2|$ and $\arg \left(\frac{z+3}{z-i}\right)=\frac{\pi}{4}$. Then $|z|^{2}$ is equal to
- 1
- 2 weeks, 3 days ago
- Let $\mathrm{S}=\left\{\mathrm{z} \in \mathrm{C}: \mathrm{z}^{2}+4 \mathrm{z}+16=0\right\}$. Then $\sum_{\mathrm{z} \in \mathrm{S}}|\mathrm{z}+\sqrt{3} \mathrm{i}|^{2}$ is equal to
- 1
- 2 weeks, 3 days ago
- Let $\mathrm{z}_{1}, \mathrm{z}_{2} \in \mathrm{C}$ be the distinct solutions of the equation $\mathrm{z}^{2}+4 \mathrm{z}-(1+12 \mathrm{i})=0$.
- 1
- 2 weeks, 3 days ago
- Let the set of all values of $\mathrm{k} \in \mathrm{R}$ such that the equation $\mathrm{z}(\overline{\mathrm{z}}+2+\mathrm{i})+\mathrm{k}(2+3 \mathrm{i})=0, \mathrm{z} \in \mathrm{C}$, has at least one solution,
- 1
- 2 weeks, 3 days ago
- Let x and y be real number such that $50\left(\frac{2 \mathrm{x}}{1+3 \mathrm{i}}-\frac{\mathrm{y}}{1-2 \mathrm{i}}\right)=31+17 \mathrm{i}, \mathrm{i}=\sqrt{-1}$. Then the value of 10(x -3y) is
- 1
- 1 month ago
- 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? [JEE (Main) 10 April 2023 S II]
- 1
- 1 month ago
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