You don't just need the CR equations to hold, you need the partials to be continuous there! Here is question a. worked out:
$u(x,y)=yx$, $v(x,y)=y^2$. Thus,
$$\partial_xu=y\\
\partial_yu=x\\
\partial_xv=0\\
\partial_yv=2y
$$
The partials are continuous everywhere. The Cauchy Riemann equations give:
$$\partial_xu=\partial_yv\Leftrightarrow y=2y\Leftrightarrow y=0
\\
\partial_yu=-\partial_xv\Leftrightarrow x=0
$$
$f$ is only complex differentiable at $(x,y)=0$ and $f^{\prime}(z)=0$
For b:
$u(x,y)=x^2$, $v(x,y)=y^2$. Thus,
$$\partial_xu=2x\\
\partial_yu=0\\
\partial_xv=0\\
\partial_yv=2y
$$
The partials are continuous everywhere. The Cauchy Riemann equations give:
$$\partial_xu=\partial_yv\Leftrightarrow 2x=2y\Leftrightarrow x=y
\\
\partial_yu=-\partial_xv\Leftrightarrow 0=0
$$
$f$ is only complex differentiable at $S=\left\{(x,x)\in \mathbb{C}:x\in \mathbb{R}\right\}$