Answer
See the detailed answer below.
Work Step by Step
$$\color{blue}{\bf [a]}$$
According to Lenz's loop, the induced current must create an opposite flux that fights the increasing flux due to the area increase.
Hence, the induced flux must be into the page which means that the induced current is $\bf counterclockwise$.
$$\color{blue}{\bf [b]}$$
We have here a loop and a changing flux due to the change in the area that enters the uniform flux, so we must have an induced current and an induced emf.
We know that the induced current is given by
$$I_{\rm loop}=\dfrac{\varepsilon_{\rm loop }}{R_{\rm loop}}\tag 1$$
We know that the induced emf $\varepsilon$ is given by
$$\varepsilon_{\rm loop}=\left|\dfrac{d\Phi_{\rm }}{dt}\right|\tag 2$$
where $ \Phi =\vec A\cdot \vec B =AB\cos\theta=AB $ since $\theta=0^\circ$, so for
$$d\Phi =dA B$$
Plug into (2),
$$\varepsilon_{\rm loop}=B\left|\dfrac{dA}{dt}\right| \tag 3$$
where $dA$ is changing uniformly as the $\rm L$-shaped conductor moves.
At $t=0$ s, $A_0=0$, and the moving of the $\rm L$-shaped conductor creates a square that has equal length.
Let's assume that the length of one side is $x$, so the area is
$$A=x^2$$
where $x=v_xt=v\cos45^\circ t$, so
$$A=(v\cos45^\circ t)^2$$
Plug into (3)
$$\varepsilon_{\rm loop}=B\left|\dfrac{d}{dt}v^2\cos^2 45^\circ t^2\right| $$
$$\varepsilon_{\rm loop}=Bv^2\cos^2 45^\circ\left|\dfrac{d}{dt} t^2\right| $$
$$\varepsilon_{\rm loop}=2Bv^2\cos^2 45^\circ t\tag 4 $$
Plug the known;
$$\varepsilon_{\rm loop}=2(0.1)(10)^2\cos^2 45^\circ t $$
$$\boxed{\varepsilon_{\rm loop}=10 t} $$
Plug into (1),
$$I_{\rm loop}=\dfrac{10 t}{R_{\rm loop}} $$
where the total length of the loop is $L=4x$ and we are given the resistance per meter. So, $R_{\rm loop}=0.01L=0.01 (4x)=0.04v\cos45^\circ t$
$$I_{\rm loop}=\dfrac{10 t}{0.04 v\cos45^\circ t}=\dfrac{250}{\cos45^\circ v} $$
Plug the known;
$$I_{\rm loop} =\dfrac{250}{10\cos45^\circ} $$
$$I_{\rm loop}=\color{red}{\bf 35.4}\;\rm A$$
$$\color{blue}{\bf [c]}$$
It is obvious that the induced current is constant, so we need to find the magnitude of the induced emf at $t=0.10$ s.
$$\varepsilon_{\rm loop}=10 (0.1) =\color{red}{\bf 1.0}\;\rm V $$
