JEE Main 2023 (Online) 1st February Morning Shift | Complex Numbers Question 39 | Mathematics

If the center and radius of the circle $$\left| {{{z - 2} \over {z - 3}}} \right| = 2$$ are respectively $$(\alpha,\beta)$$ and $$\gamma$$, then $$3(\alpha+\beta+\gamma)$$ is equal to :

JEE Main 2023 (Online) 31st January Evening Shift

MCQ (Single Correct Answer)

-1

The complex number $z=\frac{i-1}{\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}}$ is equal to :

$\cos \frac{\pi}{12}-i \sin \frac{\pi}{12}$

$\sqrt{2}\left(\cos \frac{\pi}{12}+i \sin \frac{\pi}{12}\right)$

$\sqrt{2} i\left(\cos \frac{5 \pi}{12}-i \sin \frac{5 \pi}{12}\right)$

$\sqrt{2}\left(\cos \frac{5 \pi}{12}+i \sin \frac{5 \pi}{12}\right)$

JEE Main 2023 (Online) 31st January Morning Shift

MCQ (Single Correct Answer)

-1

For all $$z \in C$$ on the curve $$C_{1}:|z|=4$$, let the locus of the point $$z+\frac{1}{z}$$ be the curve $$\mathrm{C}_{2}$$. Then :

the curves $$C_{1}$$ and $$C_{2}$$ intersect at 4 points

the curve $$C_{2}$$ lies inside $$C_{1}$$

the curve $$C_{1}$$ lies inside $$C_{2}$$

the curves $$C_{1}$$ and $$C_{2}$$ intersect at 2 points

JEE Main 2023 (Online) 29th January Morning Shift

MCQ (Single Correct Answer)

-1

For two non-zero complex numbers $$z_{1}$$ and $$z_{2}$$, if $$\operatorname{Re}\left(z_{1} z_{2}\right)=0$$ and $$\operatorname{Re}\left(z_{1}+z_{2}\right)=0$$, then which of the following are possible?

A. $$\operatorname{Im}\left(z_{1}\right)>0$$ and $$\operatorname{Im}\left(z_{2}\right) > 0$$

B. $$\operatorname{Im}\left(z_{1}\right) < 0$$ and $$\operatorname{Im}\left(z_{2}\right) > 0$$

C. $$\operatorname{Im}\left(z_{1}\right) > 0$$ and $$\operatorname{Im}\left(z_{2}\right) < 0$$

D. $$\operatorname{Im}\left(z_{1}\right) < 0$$ and $$\operatorname{Im}\left(z_{2}\right) < 0$$

Choose the correct answer from the options given below :

A and C

A and B

B and D

B and C

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