Distance Formula Using Complex Numbers

Q.

Let the complex numbers z₁, z₂ and z₃ be the vertices of an equilateral triangle. Let z₀ be the circumcentre of the triangle, then $z_1^2+z_2^2+z_3^2$=

1. 

2. 

3. 

4. 

Q. If A, B, C are represented by 3 + 4i, 5 - 2i , -1 + 16i, then A, B, C are
[RPET 1986]

1. Collinear

2. Vertices of right angled triangle

3. Vertices of equilateral triangle

4. Vertices of isosceles triangle

Q. Length of the line segment joining the points -1 -i and 2 + 3i is

1. – 5

2. 15

3. 5

4. 25

Q. For all complex numbers $z_1,z_2$ satisfying $|z_1|=12and|z_2-3-4i|$ = 5, the minimum value of $|z_1-z_2|$ is
[IIT Screening 2002]

1. 0

2. 7

3. 2

4. 17

Q. If $z=3+2i$, then value of $3z^3-13z^2+9z+65$ is

1. $0$
2. $1$
3. $2$
4. $3$

Q. If $a$ and $b$ are positive integers such that $N=(a+ib{)}^3-107i$ is a positive integer then $\frac{N}{6}$ is

Q. If $b+ic=(1+a)z$ and $a^2+b^2+c^2=1,$ where $a,b,c\in R,$ then $\frac{1+iz}{1-iz}=\frac{a+ib}{k}$ where $k$ is equal to

1. $1+a$
2. $1+b$
3. $1+c$
4. $2+b$

Q. If $(3-5i{)}^3$ can be expressed in the form $a+ib$ , then the ordered pair $(a,b)$ is

1. $(-198,10)$
2. $(-198,-10)$
3. $(198,-10)$
4. $(198,10)$

Q. If the expression $\frac{1-i\sin \alpha }{1+2i\sin \alpha }$ is purely real, which of following is/are correct?

1. $Re\left(z\right)=-\frac{1}{5}$
2. $Re\left(z\right)=1$
3. $\alpha =n\pi ,n\in Z$
4. $\alpha =\frac{n\pi }{2},n\in Z$

Q. If $z=x+iy,z^{1/3}=a-ib$ and $\frac{x}{a}-\frac{y}{b}=k(a^2-b^2),$ then $k=$