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Correct ANSWER is (a) i = e^-37.5t(c1cos1290t + c2sin1290t) + 0.7cos(500t + π/4 + 88.5⁰)

The best I can EXPLAIN: The complete solution is the SUM of the complementary function and the particular integral. SOI = e^-37.5t(c1cos1290t + c2sin1290t) + 0.7cos(500t + π/4 + 88.5⁰).

The correct option is (d) ip = 0.7cos(500T + π/4 + 88.5⁰)

BEST EXPLANATION: PARTICULAR SOLUTION is ip = V/√(R^2+(1/ωC-ωL)^2) cos⁡(ωt+θ+tan^-1)⁡((1/ωC-ωL)/R)). ip = 0.7cos(500t + π/4 + 88.5⁰).

Correct answer is (a) IC = e^-37.5t(c1cos1290t + c2sin1290t)

For explanation I WOULD say: The roots of the charactesistic equation are D1 = -37.5+j1290 and D2 = -37.5-j1290. The COMPLEMENTARY current obtained is ic =e^-37.5t(c1cos1290t + c2sin1290t).

Right answer is (a) IC = c1 E^(K1+K2)t + c1 e^(K1-K2)t

Easiest explanation: From the R-L circuit, we get the characteristic equation as

(D^2+R/L D+1/LC)=0. The complementary function of the solution i isic = c1 e^(K1+K2)t + c1 e^(K1-K2)t.

Correct OPTION is (B) ip = V/√(R^2+(1/ωC-ωL)^2) cos⁡(ωt+θ+tan^-1)⁡((1/ωC-ωL)/R))

To EXPLAIN I would say: The CHARACTERISTIC equation consists of two parts, viz. complementary function and particular integral. The particular integral is ip =V/√(R^2+(1/ωC-ωL)^2) cos⁡(ωt+θ+tan^-1)⁡((1/ωC-ωL)/R)).

Correct option is (a) C = (3.53-4.99×10^-3) cos⁡(π/4+89.94^o)

EASIEST explanation: At t = 0, the current flowing through the CIRCUIT is 3.53A. So, c = (3.53-4.99×10^-3)cos⁡(π/4+89.94^o).

The CORRECT OPTION is (b) i = c exp (-t/10^-5) + (4.99×10^-3) cos⁡(100t+π/2+89.94^o)

EASY explanation: The COMPLETE solution for the current is the sum of the complementary function and the particular integral. The complete solution for the current becomesi = c exp (-t/10^-5) + (4.99×10^-3)cos⁡(100t+π/2+89.94^o).

The CORRECT option is (C) 3.53

The best I can explain: At t = 0 that is initially CURRENT flowing through the circuit is i = V/R cosθ = (50/10)COS(π/4) =3.53A.

The correct answer is (d) ip = (4.99×10^-3) cos⁡(100t+π/4+89.94^o)

For explanation I would say: ASSUMING particular integral as ip = A cos (ωt + θ) + B sin (ωt + θ)

we GET ip = V/√(R^2+(1/ωC)^2) cos⁡(ωt+θ-tan^-1(1/ωRC))

where ω = 1000 rad/sec, θ = π/4, C = 1µF, R = 10Ω. On SUBSTITUTING, we get ip = (4.99×10^-3) cos⁡(100t+π/4+89.94^o).

The CORRECT choice is (c) ic = c exp (-t/10^-5)

To elaborate: By APPLYING Kirchhoff’s voltage LAW to the CIRCUIT, we have

(D+1/10^-5) )i=-500sin⁡(1000t+π/4). The complementary FUNCTION is ic = c exp (-t/10^-5).

The correct choice is (d) c = V/R cosθ-V/√(R^2+(1/(ωC))^2) cos⁡(θ+tan^-1(1/ωRC))

The best I can explain: Since the CAPACITOR does not allow sudden CHANGES in VOLTAGES, at t = 0, i = V/R cosθ. So, c = V/R cosθ-V/√(R^2+(1/(ωC))^2) cos⁡(θ+tan^-1(1/ωRC)).

Correct answer is (a) ip = V/√(R^2+(1/ωC)^2) cos⁡(ωt+θ+tan^-1(1/ωRC))

The best EXPLANATION: The characteristic equation CONSISTS of two parts, VIZ. COMPLEMENTARY function and particular integral. The particular integral is ip = V/√(R^2+(1/ωC)^2) cos⁡(ωt+θ+tan^-1(1/ωRC)).

The CORRECT option is (a) IC = ce^-t/RC

The EXPLANATION is: From the R-c circuit, we get the CHARACTERISTIC equation as (D+1/RC)i=-Vω/R sin⁡(ωt+θ). The complementary FUNCTION of the solution i is ic = ce^-t/RC.

Right option is (a) c = -0.98cos⁡(π/2-78.6^o)

To explain: At t = 0, the CURRENT flowing through the circuit is ZERO. Placing i = 0 in the current EQUATION we get c = -0.98cos⁡(π/2-78.6^o).

Correct answer is (a) i = [-0.98 cos⁡(π/2-78.6^o)] exp⁡(-200t)+0.98cos⁡(1000t+π/2-78.6^o)

EASY EXPLANATION: The complete SOLUTION for the CURRENT is the sum of the COMPLEMENTARY function and the particular integral.

So, i = [-0.98 cos⁡(π/2-78.6^o)] exp⁡(-200t)+0.98cos⁡(1000t+π/2-78.6^o).

Right choice is (b) i = ce^-200t + 0.98cos⁡(1000t+π/2-78.6^o)

Easiest EXPLANATION: The COMPLETE solution for the CURRENT is the sum of the COMPLEMENTARY function and the particular integral. The complete solution for the current becomesi = ce^-200t + 0.98cos⁡(1000t+π/2-78.6^o).

The correct OPTION is (a) ip = 0.98cos⁡(1000t+π/2-78.6^o)

For EXPLANATION: Assuming particular integral as ip = A COS (ωt + θ) + B sin(ωt + θ). We get ip = V/√(R^2+(ωL)^2) cos⁡(ωt+θ-tan^-1(ωL/R)) where ω = 1000 rad/sec, V = 100V, θ =π/2, L = 0.1H, R = 20Ω. On SUBSTITUTING, we get ip = 0.98cos⁡(1000t+π/2-78.6^o).

Right option is (c) ic = ce^-200t

For EXPLANATION: By applying KIRCHHOFF’s VOLTAGE law to the CIRCUIT, we have 20i+0.1di/dt=100cos⁡(10^3 t+π/2) => (D+200)i=1000cos⁡(1000t+π/2). The complementary function is ic = ce^-200t.

Correct OPTION is (b) c = -V/√(R^2+(ωL)^2) cos⁡(θ-tan^-1(ωL/R))

The best I can explain: SINCE the inductor does not allow sudden CHANGES in currents, at t = 0, i = 0. So,c = -V/√(R^2+(ωL)^2) cos⁡(θ-tan^-1(ωL/R)).

Correct CHOICE is (a) ic = ce^-t(R/L)

To EXPLAIN: From the R-L circuit, we get the characteristic equation as (D+R/L)i=V/L cos⁡(ωt+θ). The complementary FUNCTION of the solution i is ic = ce^-t(R/L).

Right choice is (B) -200±j979.8

The best explanation: Let the roots of the characteristic equation are denoted by D1, D2. So on differentiating the equation 100 = 20i + 0.05 \(\frac{di}{dt} + \frac{1}{20 \times 10^{-6}} \INT idt\), we get D1 = -200+j979.8, D2 = -200-j979.8.

Correct choice is (c) CRITICALLY damped

The EXPLANATION: If the roots of an equation are REAL and EQUAL, then the response will be critically damped response. For a critically damped system, the system returns to equilibrium as quickly as possible without OSCILLATING.

The correct choice is (d) under damped

Easy explanation: If the roots of an equation are complex CONJUGATE, then the response will be under damped response. Damping is an influence WITHIN or upon an oscillatory SYSTEM that has the EFFECT of reducing, RESTRICTING or preventing its oscillations.

Right CHOICE is (c) over damped

The explanation: If the roots of an EQUATION are real and unequal, then the RESPONSE will be over damped response. Over damped response of a system is defined as the system returns (EXPONENTIALLY decays) to equilibrium without OSCILLATING.

Right CHOICE is (a) VC = 60(1-e^-t)V

To explain: The expression of voltage ACROSS capacitor in the circuit VC = V(1-e^-t/RC) = 20(1-e^-t)V.

The correct answer is (b) 2

For EXPLANATION I would SAY: At t = 0, switch S is CLOSED. Since the capacitor does not ALLOW sudden changes in voltage, the current in the circuit is i = V/R = 20/10 = 2A. At t = 0, i = 2A.

The correct choice is (d) di/dt+i=0

The EXPLANATION: By APPLYING KIRCHHOFF’s law, we GET

Differentiating with RESPECT to t, we get 10 di/dt+i/0.1=0 => di/dt+i=0.

Correct choice is (d) 5

To EXPLAIN I would say: After five time constants, the transient part of the response reaches more than 99 percent of its final value.

The CORRECT OPTION is (a) RC

To explain I would say: The time constant of an R-C CIRCUIT is RC and it is denoted by τ and the VALUE of τ in DC response of R-C circuit is RC sec.

Correct option is (b) decays with time

Easy explanation: In a R-C CIRCUIT, when the SWITCH is closed, the RESPONSE decays with time that is the response V/R decreases with increase in time.

Right option is (a) V/R

To elaborate: The CAPACITOR never ALLOWS sudden CHANGES in voltage, it will act as a SHORT circuit at t = 0^+. So the current in the circuit at t = 0^+ is V/R.

The correct ANSWER is (c) VL = 60(e^-2t)V

Easy explanation: VOLTAGE across the inductor VL = Ldi/dt. On substituting the EXPRESSION of current we get voltage across the inductor = 15×(d/dt)(2(1-e^-2t)))=60(e^-2t)V.

Correct choice is (B) VR = 60(1-e^-2t)V

To EXPLAIN: VOLTAGE across the resistor VR = IR. On substituting the expression of CURRENT we get voltage across resistor = (2(1-e^-2t))×30=60(1-e^-2t)V.

The correct ANSWER is (a) i=2(1-e^-2t)A

For explanation: At t = 0^+ the current in the circuit is zero. Therefore at t = 0^+, i = 0 => 0 = C + 2 =>c = -2. Substituting the value of ‘c’ in the current EQUATION, we have i = 2(1-e^-2t)A.

Correct ANSWER is (d) 0

For explanation: Since the inductor never allows sudden CHANGES in currents. At t = 0^+ that just after the INITIAL state the current in the circuit is zero.

The correct answer is (c) di/dt+2i=4

To explain: Let the i be the current FLOWING through the CIRCUIT. By applying KIRCHHOFF’s voltage law, we GET 15 di/dt+30i=60 => di/dt+2i=4.

Correct option is (d) 5

To EXPLAIN: After five time constants, the TRANSIENT part of the response reaches more than 99 PERCENT of its final VALUE.

Right option is (a) L/R

Easy explanation: The TIME constant of a function (V/R)e^-(R/L)t is the time at which the exponent of e is unity where e is the BASE of the natural LOGARITHMS. The term L/R is called the time constant and is denoted by ‘τ’.

Right choice is (b) (V/R)(-exp⁡((R/L)t))

Explanation: The expression of current in the R-L circuit has the TRANSIENT PART as

(V/R)(-exp⁡((R/L)t)). The transition period is defined as the time taken for the current to reach its final or steady STATE value from its INITIAL value.