Question

Statement I If $z_1=a+i b$ and $z_2=c+i d$ be any two complex numbers, then the product $z_1 z_2-(a c-b d)+i(a d+b c)$.
Statement II Product of two complex numbers is also a complex number.
Statement III $(3+i 5)(2+i 6)=-24+i 28$.

• A. Statement I is true
• B. Statement II is true
• C. All are true
• D. Neither I nor III is true

Answer 1

Answer: (C)

I. If $z_1=a+i b$ and $z_2=c+i d$, then
$z_1 z_2 =(a+i b)(c+i d) $
$ =a c+i a d+i b c+i^2 b d $
$ =(a c-b d)+i(a d+b c) \left(\because i^2=-1\right)$
Hence, Statement I is correct.
II. Since, product of two complex numbers is also a complex number. This is called closure law of multiplication. i.e.,
If $z_1$ and $z_2$ are two complex numbers, then $z_1 \cdot z_2$ is also a complex numbers.
III. $(3+5 i)(2+i 6)=(3 \times 2-5 \times 6)+i(3 \times 6+5 \times 2)$
$=-24+i 28$
Hence, all the statements are correct.