PYQs Based Test - 22 : Analytic Function and C-R Equations

An analytic function f (z) of complex variable z = x + iy may be written as f (z) = u (x, y) + iv (x, y). Then, u (x, y) and v (x, y) must satisfy

If f(z) = (x² + ay²) + i bxy is a complex analytic function of z = x + iy, where $i=\sqrt{-1}$, then

Which one of the following functions is analytic in the region |z| ≤ 1?

Consider the function $f\left(z\right)=z+z^{\ast }$ where is a complex variable and denotes its complex conjugate. Which one of the following is TRUE?

Consider the complex valued function $\mathrm{f}(\mathrm{z})=2{\mathrm{z}}^3+b|z{|}^3$ where $\mathrm{z}$ is a complex variable.

The value of $\mathrm{b}$ for which the function $\mathrm{f}\left(\mathrm{z}\right)$ is analytic is ________