Question

Consider the following statements
I Additive inverse of $(1-i)$ is equal to $-1+i$
II. If $z_1$ and $z_2$ are two complex numbers, then $z_1-z_2$ represents a complex number which is sum of $z_1$ and additive inverse of $z_2$
III. Simplest form of $\frac{5+\sqrt{2}i}{1-\sqrt{2}i}$ is $1+2\sqrt{2}$ i.
Choose the correct option

• A. Only I and II are correct
• B. Only II and III are correct
• C. I, II and III are correct
• D. I, II and III are incorrect

Answer 1

Answer: (C)

I. Additive inverse of $(1-i)=-(1-i)=-1+i$
II. Since, difference of two complex numbers is a complex number and $z_1-z_2$ can be written as $\left(z_1\right)+\left(-z_2\right)$ which is sum of $z_1$ and addit inverse of $z_2$
III. $\frac{5+\sqrt{2}i}{1-\sqrt{2}i}\times \frac{1+\sqrt{2}i}{1+\sqrt{2}i}=\frac{5+5\sqrt{2}i+\sqrt{2}i-2}{1+2}$
$=\frac{3+6\sqrt{2}i}{3}=1+2\sqrt{2}i$