Answer
a) ${\bf 5}\;\rm mV$
b) ${\bf 10}\;\rm mV$
Work Step by Step
We have here a loop and a changing flux, so we must have an induced current and an induced emf.
We know that the induced emf $\varepsilon$ is given by
$$\varepsilon=\left|\dfrac{d\Phi}{dt}\right|\tag 1$$
where $\Phi =\vec A\cdot \vec B $ and here $A$ is constant and its vector is in the $z$-directino since the coil lies on the $x-y$ plane.
So that
$$\Phi =(0.1^2\;\hat k)\cdot (0.30t\;\hat i+0.50t^2\;\hat k) $$
Recall that $\;\hat k\cdot \;\hat i=0$, and $\;\hat k\cdot \;\hat k=1$.
Thus,
$$\Phi= 0.005t^2 $$
Plug into (1),
$$\varepsilon=\left|\dfrac{d (0.005t^2)}{dt}\right|$$
$$\varepsilon=0.01 t \tag 2$$
Now we just need to plug $t$s into (2):
$$\color{blue}{\bf [a]}$$
At $t=0.5$ s
$$\varepsilon=0.01(0.5)$$
$$\varepsilon=\color{red}{\bf 5}\;\rm mV$$
$$\color{blue}{\bf [a]}$$
At $t=1$ s
$$\varepsilon=0.01(1)$$
$$\varepsilon=\color{red}{\bf 10}\;\rm mV$$
