List of top Engineering Mathematics Questions on Cauchy-Riemann equations asked in GATE XE

Let $f(z)$ be an analytic function such that \[ Re(f'(z)) = 3x^2 - 4y - 3y^2, f(i)=0, f'(0)=0, \] where $i = \sqrt{-1}$. Then the value of $f(1)$ is:

Let \[ P=\begin{bmatrix}4 & -2 & 2\\ 6 & -3 & 4\\ 3 & -2 & 3\end{bmatrix}, \qquad Q=\begin{bmatrix}3 & -2 & 2\\ 4 & -4 & 6\\ 2 & -3 & 5\end{bmatrix}. \] The eigenvalues of both $P$ and $Q$ are $1,1,2$. Which one of the following statements is TRUE?