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. $\frac{5+\sqrt{2} i}{1-\sqrt{2} i} \times \frac{1+\sqrt{2} i}{1+\sqrt{2} i}=\frac{5+5 \sqrt{2} i+\sqrt{2} i-2}{1+2}$
$=\frac{3+6 \sqrt{2} i}{3}=1+2 \sqrt{2} i$