1
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
2
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
3
JEE Main 2024 (Online) 30th January Morning Shift
+4
-1

If $$z=x+i y, x y \neq 0$$, satisfies the equation $$z^2+i \bar{z}=0$$, then $$\left|z^2\right|$$ is equal to :

A
9
B
$$\frac{1}{4}$$
C
4
D
1
4
JEE Main 2024 (Online) 29th January Evening Shift
+4
-1

Let $$\mathrm{r}$$ and $$\theta$$ respectively be the modulus and amplitude of the complex number $$z=2-i\left(2 \tan \frac{5 \pi}{8}\right)$$, then $$(\mathrm{r}, \theta)$$ is equal to

A
$$\left(2 \sec \frac{11 \pi}{8}, \frac{11 \pi}{8}\right)$$
B
$$\left(2 \sec \frac{3 \pi}{8}, \frac{3 \pi}{8}\right)$$
C
$$\left(2 \sec \frac{5 \pi}{8}, \frac{3 \pi}{8}\right)$$
D
$$\left(2 \sec \frac{3 \pi}{8}, \frac{5 \pi}{8}\right)$$
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