In a semi-conductor factory, an engineer is required to analyse the functionality of electric circuit boards. The circuit consists of a switch, an electromotive force E (usually supplied by a battery or generator), a resistor R, an inductor L, and a capacitor C. If the charge (Q) on the capacitor at time t is Q(t), then the current (I) is the rate of change of charge with respect to t, i.e., I(t)= dt/dQ. The electric circuits can be represented as second-order linear differential equation with constant coefficients as follow : L d^2Q/dt^2 + R dq/dt + 1/c Q = E(t). A series circuit is given to the engineer to do the analysis. Given that the circuit contains a resistor with R=24Ohm(Ω), an inductor with L=2Henry(H), and a capacitor with C= 0.005 Farad (F). The engineer needs to determine a) the charge at time t,Q(t) when the switch if off and without battery supply. b) the charge at time t,Q(t) when the switch if on and with a 12-Volt battery supply i. using the method of Undetermined Coefficients; and ii. using the method of Variation of Parameters. c) the current at time t,I(t) based on the charge with battery supply in question (b) above. d) the current at time t,I(t) if given that when the electric circuit has initial charge with Q=0.001 Coulomb (C) and the initial current with I=0 Ampere (A). (5 marks