Would anyone be able to help me with a work through for the attached question as I have no idea where to start. I also have no idea on how to draw the circuit reponse for circuit, ie. how to draw the curve relative to the time line on the bottom axis. Thanks in advance.

 

This is a curious problem. In the absence of a circuit diagram for the question one would probably assume the statement "parallel RLC circuit" meant R in parallel with L in parallel with C in parallel with the DC supply.

If this were the case, the solution would be indeterminate. For instance - an infinite current would flow between the source and the capacitor at t=0. I wonder if it's meant to be a series RLC circuit?

Was there a circuit diagram with the question which you haven't included?

 

You should have some knowledge of differential equations in order to solve this problem. I assume these are test problems for a quiz or exam. You should be allowed a graphing calculator and with the proper time dependant voltage equation be able to produce the required sketch.

Assuming this is a source free circuit where the initial voltage is supplied by the capacitor the following equations apply.

For a critically damped case,
L=4*R^2*C
.010H=4*R^2*.0001F
R=sqrt(L/4C)
R=sqrt(.010H/.0004F)=5ohms

The equation for the response of a critically damped parallel RLC configuration is,
v(t)=(A1+A2*t)*e^(-alpha*t) where A1 and A2 are the roots of the differential equation and alpha being your damping factor.

The differential equation for a Source Free Parallel RLC circuit is as follows,
s^2+(1/RC)*s+1/(LC)=0 , where s^2 and s are the 2nd and 1st deriviates of V(t) , t=0.
The roots of this equation produce (A1 and A2). However because this is a critically damped case
A1=A2

The roots of the differential equation are as follows,
A(1&2)=-alpha(plus or minus)sqrt(alpha^2-omeganaught^2)
where alpha=1/(2*RC) and omeganaught=1/sqrt(LC)

Quality factor for parallel RLC equation is,
Q=R*sqrt(C/L)

Hopefully this gets ya on your way.