Question

Assertion :If $3+i{x}^y$ & $x^2+y+4i$ are conjugate of each other, then $x^2+y^2=3$. Reason: If sum & product of two complex numbers are real, then they are conjugates of each other.

• A. Both (A) & (R) are individually true & (R) is correct explanation of (A).
• B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
• C. (A) is true (R) is false.
• D. (A) is false (R) is true.

Answer 1

The correct option is D (A) is false (R) is true.

If they are conjugate to one another, then
$z=3+i{x}^y$
$w=x^2+y+4i$
$\overline{z}=w$
That means
$x^2+y=3$ and $x^y=-4$
$x^y=-4$
$x^y=(2i{)}^2$
$x=2i$ and $y=2$
However $-4+2\ne 3$
Hence the assertion is false.