1
JEE Main 2024 (Online) 1st February Evening Shift
+4
-1
If $z$ is a complex number such that $|z| \leqslant 1$, then the minimum value of $\left|z+\frac{1}{2}(3+4 i)\right|$ is :
A
2
B
$\frac{5}{2}$
C
$\frac{3}{2}$
D
3
2
JEE Main 2024 (Online) 1st February Morning Shift
+4
-1
Let $\mathrm{S}=|\mathrm{z} \in \mathrm{C}:| z-1 \mid=1$ and $(\sqrt{2}-1)(z+\bar{z})-i(z-\bar{z})=2 \sqrt{2} \mid$. Let $z_1, z_2 \in \mathrm{S}$ be such that $\left|z_1\right|=\max\limits_{z \in s}|z|$ and $\left|z_2\right|=\min\limits _{z \in S}|z|$. Then $\left|\sqrt{2} z_1-z_2\right|^2$ equals :
A
1
B
4
C
3
D
2
3
JEE Main 2024 (Online) 31st January Evening Shift
+4
-1

Let $$z_1$$ and $$z_2$$ be two complex numbers such that $$z_1+z_2=5$$ and $$z_1^3+z_2^3=20+15 i$$ Then, $$\left|z_1^4+z_2^4\right|$$ equals -

A
$$15 \sqrt{15}$$
B
$$30 \sqrt{3}$$
C
$$25 \sqrt{3}$$
D
75
4
JEE Main 2024 (Online) 30th January Evening Shift
+4
-1

If $$z$$ is a complex number, then the number of common roots of the equations $$z^{1985}+z^{100}+1=0$$ and $$z^3+2 z^2+2 z+1=0$$, is equal to

A
0
B
2
C
1
D
3
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