Section Formula Using Complex Numbers

Q.

Let A = {1, 2, 3, 4, 5, 6}. Define a relation on set A by R = {(X, Y): y = x +1}

(i) Depict this relation using an arrow diagram.

(ii) Write down the domain, codomain and range of R.

Q. If three complex numbers are in A.P, then they lie on a .

1. straight line
2. circle
3. parabola

Q.

If three complex numbers are in A.P., then they lie on

1. A straight line in the complex plane

2. A circle in the complex plane

3. A parabola in the complex plane

4. None of these

Q. Let $z_1$ and $z_2$ be two distinct complex numbers and let $z=(1-t)z_1+t{z}_2$ for some real number $t$ with $0<t<1$. If arg$\left(\omega \right)$ denotes the principal argument of a non-zero complex number $\omega$, then

1. $|z-z_1|+|z+z_2|=|z_1-z_2|$
2. arg$(z-z_1)$ = arg$(z-z_2)$
3. $\left|\begin{array}{cc}z-z_1& \overline{z}-\overline{z_1}\\ z_2-z_1& \overline{z_2}-\overline{z_1}\end{array}\right|=0$
4. arg$(z-z_1)$ = arg$(z_2-z_1)$

Q. Let $z_1$ and $z_2$ be two distinct complex numbers and let $z=(1-t)z_1+t{z}_2$ for some real number $t$ with $0<t<1$. If $Arg\left(w\right)$ denotes the principal argument of a non-zero complex number $w$, then:

1. $|z-z_1|+|z-z_2|=|z_1-z_2|$
2. $Arg(z-z_1)=Arg(z-z_2)$
3. $\left|\begin{array}{cc}z-z_1& \overline{z}-{\overline{z}}_1\\ z_2-z_1& {\overline{z}}_2-{\overline{z}}_1\end{array}\right|=0$
4. $Arg(z-z_1)=Arg(z_2-z_1)$

Q. Let $z_1,z_2,z_3$ be three complex numbers and $a,b,c$ be real numbers not all zero, such that $a+b+c=0$ and $a{z}_1+b{z}_2+c{z}_3=0$, then

1. $z_1,z_2,z_3$ are vertices of a triangle.
2. $z_1,z_2,z_3$ lies on circumference of a circle
3. $z_1,z_2,z_3$ are collinear.
4. None of these.

Q.

If three complex numbers are in A.P., then they lie on

1. A circle in the complex plane

2. A straight line in the complex plane

3. None of these

4. A parabola in the complex plane