1
JEE Main 2023 (Online) 1st February Evening Shift
+4
-1

Let $$a,b$$ be two real numbers such that $$ab < 0$$. IF the complex number $$\frac{1+ai}{b+i}$$ is of unit modulus and $$a+ib$$ lies on the circle $$|z-1|=|2z|$$, then a possible value of $$\frac{1+[a]}{4b}$$, where $$[t]$$ is greatest integer function, is :

A
$\left(\frac{1+\sqrt{7}}{4}\right)$
B
$$\frac{1}{2}$$
C
0
D
$$-$$1
2
JEE Main 2023 (Online) 1st February Morning Shift
+4
-1

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 :

A
12
B
10
C
11
D
9
3
JEE Main 2023 (Online) 31st January Evening Shift
+4
-1
The complex number $z=\frac{i-1}{\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}}$ is equal to :
A
$\cos \frac{\pi}{12}-i \sin \frac{\pi}{12}$
B
$\sqrt{2}\left(\cos \frac{\pi}{12}+i \sin \frac{\pi}{12}\right)$
C
$\sqrt{2} i\left(\cos \frac{5 \pi}{12}-i \sin \frac{5 \pi}{12}\right)$
D
$\sqrt{2}\left(\cos \frac{5 \pi}{12}+i \sin \frac{5 \pi}{12}\right)$
4
JEE Main 2023 (Online) 31st January Morning Shift
+4
-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 :

A
the curves $$C_{1}$$ and $$C_{2}$$ intersect at 4 points
B
the curve $$C_{2}$$ lies inside $$C_{1}$$
C
the curve $$C_{1}$$ lies inside $$C_{2}$$
D
the curves $$C_{1}$$ and $$C_{2}$$ intersect at 2 points
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