Question

If $z_1,z_2$ are complex numbers then the correct match List- I to List II is:

List - I	List - II
A) arg z_1z_2	1) $arg{z}_1-arg{z}_2$
B) $arg{z}_1\overline{z_2}$	2) $arg{z}_1-arg{z}_2=\frac{\pi }{2}$
C) $|z_1+z_2|=|z_1-z_2|$	3) $arg{z}_1=arg{z}_2$
D) $|z_1+z_2{|}^2$	4) $arg{z}_1+arg{z}_2$
E) $|z_1+z_2|=|z_1|+|z_2|$	5) $r_1^2+r_2^2+2r_1{r}_{2}cos({\theta }_1-{\theta }_2)$

• A. A B C D E$\to$
  4 1 2 5 3
• B. A B C D E$\to$
  2 3 4 1 5
• C. A B C D E$\to$
  5 2 1 3 4
• D. A B C D E$\to$
  4 5 2 1 3

Answer 1

The correct option is A A B C D E$\to$

4 1 2 5 3

A) arg($z_1.z_2$) = arg$z_1$ + arg$z_2$ ...(fundamental rule)

B) arg($z_1.\overline{z_2}$) = arg$z_1$ + arg$\overline{z_2}$ - (1)

arg$\overline{z_2}$ = arg$\frac{\overline{z_2}.z_2}{z_2}$ ...(multiplying and dividing by $z_2$)

= arg$\frac{{|z_2|}^2}{z_2}$ = arg${|z_2|}^2$ -arg $z_2$ (since, arg$\frac{z_1}{z_2}$ = arg$z_1$ - arg$z_2$ )

= 0 - arg$z_2$ ...(since principalargument of a real number is 0) - (2)

Comparing (1) and (2),

arg($z_1.\overline{z_2}$) = arg$z_1$ - arg$z_2$

Hence B matches with 1.

$\therefore$ A is the correct option.