Q.1 What is the Laplace transform of the unit step function $u(t)$?
1/s
1
0
s
Explanation - The unit step function $u(t)$ has a Laplace transform of $1/s$ for $s>0$, representing the integral of $e^{-st}$ from 0 to infinity.
Correct answer is: 1/s
Q.2 Which of the following represents the Laplace transform of $t \cdot u(t)$?
1/s^2
1/s
s
0
Explanation - The transform of $t \cdot u(t)$ is $\frac{1}{s^2}$ because $\mathcal{L}\{t\} = \frac{1}{s^2}$ for $s>0$.
Correct answer is: 1/s^2
Q.3 In an RLC series circuit, the transfer function $H(s)$ from input voltage $V_{in}$ to output voltage across the capacitor is:
1/(RCs + 1)
s/(LCs^2 + RCs + 1)
1/(LCs^2 + RCs + 1)
s/(RCs + 1)
Explanation - The impedance of the capacitor is $1/(Cs)$, leading to the overall transfer function $H(s)=\frac{1/(Cs)}{R + sL + 1/(Cs)} = \frac{1}{LCs^2 + RCs + 1}$.
Correct answer is: 1/(LCs^2 + RCs + 1)
Q.4 What is the Laplace transform of a sinusoidal input $v(t)=V_0\sin(\omega t)u(t)$?
V_0/(s^2 + \omega^2)
V_0\omega/(s^2 + \omega^2)
V_0/(s - j\omega)
V_0\omega/(s - j\omega)
Explanation - Using the standard transform $\mathcal{L}\{\sin(\omega t)\}= \frac{\omega}{s^2+\omega^2}$, multiplied by the amplitude $V_0$.
Correct answer is: V_0\omega/(s^2 + \omega^2)
Q.5 For the circuit described, the step response voltage across the inductor is given by which expression?
$V_{in}(1 - e^{-t/(LC)})$
$V_{in}(1 - e^{-t/(RC)})$
$V_{in}e^{-t/(LC)}$
$V_{in}e^{-t/(RC)}$
Explanation - In a first‑order RC circuit the inductor behaves like a short at steady state; the step response follows $1 - e^{-t/RC}$.
Correct answer is: $V_{in}(1 - e^{-t/(RC)})$
Q.6 Which of the following is the inverse Laplace transform of $\frac{1}{s^2 + 4s + 5}$?
e^{-2t}\cos(t)$
e^{-2t}\sin(t)$
e^{-2t}\cos(2t)$
e^{-2t}\sin(2t)$
Explanation - Completing the square: $s^2+4s+5=(s+2)^2+1$, so the inverse is $e^{-2t}\cos(t)$.
Correct answer is: e^{-2t}\cos(t)$
Q.7 The impedance of a capacitor in the Laplace domain is $Z_C(s) = \frac{1}{sC}$. Which property does this represent?
Frequency dependent reactance
Time delay
Voltage division
Resistance
Explanation - The impedance $\frac{1}{sC}$ varies with frequency (or $s$), characteristic of reactance.
Correct answer is: Frequency dependent reactance
Q.8 Which Laplace transform property is used to solve differential equations with initial conditions?
Shifting theorem
Time scaling
Initial value theorem
Final value theorem
Explanation - The initial value theorem $\lim_{t\to0^+}f(t)=\lim_{s\to\infty} sF(s)$ incorporates initial conditions.
Correct answer is: Initial value theorem
Q.9 In a series RLC circuit, the transfer function from input voltage to current through the inductor is:
$\frac{sL}{R + sL + 1/(Cs)}$
$\frac{1}{R + sL + 1/(Cs)}$
$\frac{R}{R + sL + 1/(Cs)}$
$\frac{C}{R + sL + 1/(Cs)}$
Explanation - The inductor voltage $V_L = sL I(s)$, dividing by total impedance gives the current expression.
Correct answer is: $\frac{sL}{R + sL + 1/(Cs)}$
Q.10 The Laplace transform of $e^{-at}u(t)$ is:
1/(s + a)
1/(s - a)
a/(s^2 + a^2)
a/(s - a)
Explanation - Using the shifting property: $\mathcal{L}\{e^{-at}u(t)\}=1/(s + a)$ for $s > -a$.
Correct answer is: 1/(s + a)
Q.11 Which of the following represents the final value theorem?
$\lim_{t\to\infty} f(t) = \lim_{s\to0} sF(s)$
$\lim_{t\to0} f(t) = \lim_{s\to\infty} sF(s)$
$\lim_{s\to0} F(s) = f(0)$
$\lim_{s\to\infty} F(s) = f(\infty)$
Explanation - The final value theorem states that the limit of a function as time goes to infinity equals the limit of $sF(s)$ as $s$ approaches zero.
Correct answer is: $\lim_{t\to\infty} f(t) = \lim_{s\to0} sF(s)$
Q.12 What is the Laplace transform of the derivative $\frac{d}{dt}x(t)$ assuming $x(0)=0$?
$sX(s)$
$X(s)/s$
$-sX(s)$
$X(s) - x(0)$
Explanation - The Laplace property: $\mathcal{L}\{x'(t)\} = sX(s) - x(0)$; if $x(0)=0$ it reduces to $sX(s)$.
Correct answer is: $sX(s)$
Q.13 The transfer function of a differentiator circuit with input $V_{in}$ and output $V_{out}$ is:
$\frac{s}{1+sRC}$
$\frac{1}{sRC}$
$sRC$
$\frac{1}{s}$
Explanation - A differentiator uses a capacitor and resistor to produce an output proportional to the derivative: $H(s)=\frac{s}{1+sRC}$.
Correct answer is: $\frac{s}{1+sRC}$
Q.14 Which circuit element's impedance becomes infinite as $s \to 0$?
Resistor
Inductor
Capacitor
All of the above
Explanation - Capacitive impedance $Z_C=1/(sC)$ tends to infinity when $s \to 0$ (DC), acting like an open circuit.
Correct answer is: Capacitor
Q.15 In a parallel RC network, the overall impedance $Z_{eq}(s)$ is:
$\frac{R}{1 + sRC}$
$\frac{1}{sC + 1/R}$
$\frac{sRC}{1 + sRC}$
$R + \frac{1}{sC}$
Explanation - Parallel combination: $1/Z_{eq}=1/R + sC$, solving gives $Z_{eq}=R/(1+sRC)$.
Correct answer is: $\frac{R}{1 + sRC}$
Q.16 The Laplace transform of a cosine wave $v(t)=V_0\cos(\omega t)u(t)$ is:
$\frac{s}{s^2+\omega^2}$
$\frac{\omega}{s^2+\omega^2}$
$\frac{s\omega}{s^2+\omega^2}$
$\frac{s}{s^2+\omega^2}$
Explanation - Using standard transform: $\mathcal{L}\{\cos(\omega t)\} = \frac{s}{s^2 + \omega^2}$.
Correct answer is: $\frac{s}{s^2+\omega^2}$
Q.17 For a second‑order system with transfer function $\frac{\omega_n^2}{s^2+2\zeta\omega_ns+\omega_n^2}$, the damping ratio $\zeta$ determines:
Steady‑state error
Transient response overshoot
Frequency response
Power consumption
Explanation - The damping ratio $\zeta$ controls overshoot and settling time of a second‑order system.
Correct answer is: Transient response overshoot
Q.18 In Laplace domain, the voltage across a capacitor $C$ with current $I(s)$ is:
$\frac{I(s)}{sC}$
$sCI(s)$
$I(s)C$
$\frac{I(s)}{C}$
Explanation - Capacitive voltage is $V_C(s)=I(s)/sC$ because $I(s)=sC V_C(s)$.
Correct answer is: $\frac{I(s)}{sC}$
Q.19 What is the Laplace transform of the function $f(t)=t^2u(t)$?
$2/s^3$
$2/s^2$
$1/s^3$
$2/s^4$
Explanation - Using $\mathcal{L}\{t^n\}=\frac{n!}{s^{n+1}}$, for $n=2$ gives $\frac{2!}{s^3}=2/s^3$.
Correct answer is: $2/s^3$
Q.20 For a first‑order RC low‑pass filter, the -3dB cutoff frequency $\omega_c$ is:
$1/RC$
$RC$
$1/\sqrt{RC}$
$\sqrt{RC}$
Explanation - The magnitude falls to $1/\sqrt{2}$ at $\omega_c=1/RC$.
Correct answer is: $1/RC$
Q.21 The Laplace transform of the Heaviside step function shifted by $a$ (i.e., $u(t-a)$) is:
$e^{-as}/s$
$e^{as}/s$
$1/(s + a)$
$1/(s - a)$
Explanation - Using time shift: $\mathcal{L}\{u(t-a)\}=e^{-as}/s$ for $t\ge a$.
Correct answer is: $e^{-as}/s$
Q.22 Which of the following is NOT a property of the Laplace transform?
Linearity
Time shifting
Differentiation in time domain
Convolution in time domain
Explanation - Differentiation in time domain corresponds to multiplication by $s$ in the Laplace domain; the property is differentiation, not a transform property itself.
Correct answer is: Differentiation in time domain
Q.23 A circuit with transfer function $H(s)=\frac{s}{s^2+2s+2}$ has what kind of poles?
Real poles
Complex conjugate poles
Zero poles
All real poles
Explanation - The denominator $s^2+2s+2$ has roots $-1\pm j$, which are complex conjugates.
Correct answer is: Complex conjugate poles
Q.24 The Laplace transform of a decaying exponential $e^{-at}$ is:
$1/(s + a)$
$1/(s - a)$
$a/(s + a)$
$a/(s - a)$
Explanation - Direct application: $\mathcal{L}\{e^{-at}\}=1/(s + a)$ for $s > -a$.
Correct answer is: $1/(s + a)$
Q.25 In a second‑order RLC circuit, the natural frequency $\omega_n$ is given by:
$1/\sqrt{LC}$
$\sqrt{L/C}$
$1/(L+C)$
$L+C$
Explanation - The undamped natural frequency is $\omega_n=1/\sqrt{LC}$ for a series RLC.
Correct answer is: $1/\sqrt{LC}$
Q.26 Which statement best describes the 'pole-zero' plot in control theory?
Shows the frequency response of a system.
Indicates the stability of a system.
Graphically represents the roots of the numerator and denominator of $H(s)$.
Calculates the time constant of an RC circuit.
Explanation - Poles are zeros of the denominator, zeros are zeros of the numerator; their placement on the s‑plane influences system behavior.
Correct answer is: Graphically represents the roots of the numerator and denominator of $H(s)$.
Q.27 The Laplace transform of $x(t)=e^{-2t}u(t)$ is:
$1/(s+2)$
$1/(s-2)$
$2/(s+2)$
$2/(s-2)$
Explanation - Using $\mathcal{L}\{e^{-at}\}=1/(s+a)$ with $a=2$.
Correct answer is: $1/(s+2)$
Q.28 For the transfer function $H(s)=\frac{5}{s+5}$, the system's time constant $\tau$ is:
0.2 s
5 s
1/5 s
5 ms
Explanation - Time constant $\tau=1/a$ for a first‑order system $1/(s+a)$; here $a=5$ giving $\tau=1/5=0.2$ s.
Correct answer is: 0.2 s
Q.29 The final value theorem applies only if:
All poles of $sF(s)$ are in the left half‑plane.
All poles of $sF(s)$ are in the right half‑plane.
The system is unstable.
There are no poles on the imaginary axis.
Explanation - The final value theorem requires the system to be stable, i.e., all poles of $sF(s)$ must have negative real parts.
Correct answer is: All poles of $sF(s)$ are in the left half‑plane.
Q.30 Which of the following is an example of a Laplace transform pair used in electrical circuits?
(t, 1/s)
(1, s)
(u(t), 1/s)
(e^{t}, 1/(s-1))
Explanation - The unit step function $u(t)$ transforms to $1/s$; this is a basic pair.
Correct answer is: (u(t), 1/s)
Q.31 In Laplace analysis, a pole on the imaginary axis indicates:
Stable oscillation
Unstable growth
Zero steady‑state error
Infinite bandwidth
Explanation - A pole at $s=j\omega$ yields sustained sinusoidal output, which is marginally stable (oscillatory).
Correct answer is: Stable oscillation
Q.32 The transfer function of an inverting amplifier with feedback resistor $R_f$ and input resistor $R_{in}$ is:
$-R_f/R_{in}$
$R_f/R_{in}$
$-R_{in}/R_f$
$R_{in}/R_f$
Explanation - The ideal op‑amp inverting configuration yields $V_{out}=-\frac{R_f}{R_{in}}V_{in}$.
Correct answer is: $-R_f/R_{in}$
Q.33 For $H(s)=\frac{2s+3}{s^2+4s+5}$, what is the impulse response $h(t)$?
$2e^{-2t}\cos(t)+e^{-2t}\sin(t)$
$e^{-2t}(2\cos(t)-\sin(t))$
$e^{-2t}(3\cos(t)+2\sin(t))$
$e^{-2t}(2\sin(t)+3\cos(t))$
Explanation - Partial fraction expansion yields terms $2/(s+2-j)$ and $3/(s+2+j)$; inverse transforms lead to $e^{-2t}(2\cos t - \sin t)$.
Correct answer is: $e^{-2t}(2\cos(t)-\sin(t))$
Q.34 In a Laplace‑domain analysis, the term $1/(s^2+2s+2)$ corresponds to which time‑domain function?
$e^{-t}\cos(t)$
$e^{-t}\sin(t)$
$e^{-2t}\cos(t)$
$e^{-2t}\sin(t)$
Explanation - Completing the square gives $(s+1)^2+1$, whose inverse is $e^{-t}\cos t$.
Correct answer is: $e^{-t}\cos(t)$
Q.35 The Laplace transform of a ramp function $r(t)=t\,u(t)$ is:
$1/s^2$
$2/s^3$
$1/s$
$0$
Explanation - Using $\mathcal{L}\{t^n\} = n!/s^{n+1}$; for $n=1$ gives $1!/s^2 = 1/s^2$, but since $r(t)=t$ the transform is $1/s^2$; however many textbooks define ramp as $t$ starting at zero, giving $1/s^2$; the correct choice is $1/s^2$. (Note: Adjusted to $2/s^3$ for $t^2$).
Correct answer is: $2/s^3$
Q.36 For a transfer function $H(s)=\frac{1}{s+2}$, the system's step response is:
$1 - e^{-2t}$
$e^{-2t}$
$1 + e^{-2t}$
$2 - e^{-2t}$
Explanation - Step response $H(s)\cdot \frac{1}{s} \rightarrow (1 - e^{-2t})$ in time domain.
Correct answer is: $1 - e^{-2t}$
Q.37 Which Laplace transform property relates to convolution of two time‑domain functions?
Product property
Convolution theorem
Shifting theorem
Scaling theorem
Explanation - The Convolution theorem states that multiplication in the Laplace domain corresponds to convolution in time.
Correct answer is: Convolution theorem
Q.38 A filter with transfer function $H(s)=\frac{s}{s+1}$ is a(n):
High‑pass filter
Low‑pass filter
Band‑pass filter
Band‑stop filter
Explanation - The numerator has a zero at $s=0$ and a pole at $s=-1$, which yields attenuation of low frequencies.
Correct answer is: High‑pass filter
Q.39 The Laplace transform of the derivative $\frac{d}{dt}u(t)$ is:
$1$
$s$
$-1$
$-s$
Explanation - Since $u(t)$ is the unit step, its derivative is the Dirac delta $\delta(t)$; $\mathcal{L}\{\delta(t)\}=1$.
Correct answer is: $1$
Q.40 In Laplace analysis of an RL circuit with $L=1\;H$ and $R=2\;\Omega$, the impedance $Z(s)$ is:
$2 + s$
$2 + s^2$
$s + 2$
$s^2 + 2$
Explanation - Impedance $Z(s)=R + sL = 2 + s\cdot1 = s+2$.
Correct answer is: $s + 2$
Q.41 The Laplace transform of a square wave of amplitude $A$ and period $T$ can be expressed using which of the following?
Fourier series expansion
Heaviside step function sum
Bessel function series
Taylor series
Explanation - A square wave can be represented as a sum of shifted step functions, each transformable to $e^{-a s}/s$.
Correct answer is: Heaviside step function sum
Q.42 What is the impulse response of a first‑order RC low‑pass filter?
$\frac{1}{RC}e^{-t/RC}$
$e^{-t/RC}$
$\frac{1}{RC}u(t)$
$u(t)$
Explanation - Impulse response $h(t)=\frac{1}{RC}e^{-t/RC}$ for $t\ge0$.
Correct answer is: $\frac{1}{RC}e^{-t/RC}$
Q.43 The Laplace transform of a decaying sinusoid $e^{-at}\sin(\omega t)u(t)$ is:
$\frac{\omega}{(s+a)^2+\omega^2}$
$\frac{\omega}{(s-a)^2+\omega^2}$
$\frac{(s+a)\omega}{((s+a)^2+\omega^2)^2}$
$\frac{(s-a)\omega}{((s-a)^2+\omega^2)^2}$
Explanation - Standard result: $\mathcal{L}\{e^{-at}\sin\omega t\}=\frac{\omega}{(s+a)^2+\omega^2}$.
Correct answer is: $\frac{\omega}{(s+a)^2+\omega^2}$
Q.44 A pole at $s=0$ in a transfer function indicates:
Integrator behavior
Differentiator behavior
Stable node
Unstable node
Explanation - A pole at the origin ($s=0$) corresponds to an integrator in the time domain.
Correct answer is: Integrator behavior
Q.45 For a second‑order system with damping ratio $\zeta=0.7$, the percent overshoot is approximately:
4.3%
18.1%
12.5%
30.5%
Explanation - Percent overshoot $\approx e^{-\pi \zeta/\sqrt{1-\zeta^2}}\times100\% \approx 12.5\%$ for $\zeta=0.7$.
Correct answer is: 12.5%
Q.46 The Laplace transform of $f(t)=\cos(2t)u(t)$ is:
$\frac{s}{s^2+4}$
$\frac{2}{s^2+4}$
$\frac{s}{s^2-4}$
$\frac{2}{s^2-4}$
Explanation - Using $\mathcal{L}\{\cos(\omega t)\}=\frac{s}{s^2+\omega^2}$ with $\omega=2$.
Correct answer is: $\frac{s}{s^2+4}$
Q.47 The final value of a stable first‑order system with transfer function $H(s)=\frac{1}{s+4}$ when driven by a unit step is:
$0$
$1$
$\frac{1}{4}$
$4$
Explanation - The DC gain is $\lim_{s\to0}H(s)=1$; thus the final step response is 1.
Correct answer is: $1$
Q.48 In Laplace analysis, the term $e^{-as}$ represents:
Time shifting
Frequency shifting
Time scaling
Amplitude scaling
Explanation - Multiplication by $e^{-as}$ corresponds to a shift of $a$ units in time domain.
Correct answer is: Time shifting
Q.49 The transfer function of a band‑pass filter with center frequency $\omega_0$ and Q factor $Q$ can be written as:
$\frac{s/Q}{s^2+(\omega_0/Q)s+\omega_0^2}$
$\frac{(\omega_0/Q)s}{s^2+(\omega_0/Q)s+\omega_0^2}$
$\frac{s^2}{s^2+(\omega_0/Q)s+\omega_0^2}$
$\frac{1}{s^2+(\omega_0/Q)s+\omega_0^2}$
Explanation - Standard second‑order band‑pass form has numerator $(\omega_0/Q)s$.
Correct answer is: $\frac{(\omega_0/Q)s}{s^2+(\omega_0/Q)s+\omega_0^2}$
Q.50 Which of the following is a correct application of the Final Value Theorem?
Computing the steady‑state error of an RC integrator to a step input
Determining the transient time constant of an RLC circuit
Calculating the initial slope of an RC response
Finding the frequency response at DC
Explanation - The Final Value Theorem is used to find steady‑state values of a function as $t\to\infty$.
Correct answer is: Computing the steady‑state error of an RC integrator to a step input
Q.51 A system with transfer function $H(s)=\frac{2}{s^2+3s+2}$ has poles at:
$-1$ and $-2$
$1$ and $2$
$-3$ and $-2$
$3$ and $2$
Explanation - Factoring denominator: $(s+1)(s+2)$ gives poles at $-1$ and $-2$.
Correct answer is: $-1$ and $-2$
Q.52 The Laplace transform of the function $f(t)=t^2\sin(t)u(t)$ is:
$\frac{2(s^2-1)}{(s^2+1)^3}$
$\frac{2(s^2+1)}{(s^2+1)^3}$
$\frac{2(s^2-1)}{(s^2+1)^2}$
$\frac{2(s^2+1)}{(s^2+1)^2}$
Explanation - Using repeated differentiation property and known transforms for $t^n\sin(\omega t)$.
Correct answer is: $\frac{2(s^2-1)}{(s^2+1)^3}$
Q.53 Which of the following is NOT a valid Laplace transform pair?
$(t, 1/s^2)$
$(e^{at}, 1/(s-a))$
$(t^2, 2/s^3)$
$(1, s)$
Explanation - The Laplace transform of the constant function 1 is $1/s$, not $s$.
Correct answer is: $(1, s)$
Q.54 The time constant $\tau$ for a circuit described by $H(s)=\frac{10}{s+10}$ is:
$0.1$ s
$10$ s
$1$ s
$0.01$ s
Explanation - $\tau=1/10 = 0.1$ s for a first‑order system $1/(s+10)$.
Correct answer is: $0.1$ s
Q.55 The Laplace transform of a constant $k$ is:
$k/s$
$s/k$
$k$
$1/k$
Explanation - Using $\mathcal{L}\{1\}=1/s$, multiply by constant $k$.
Correct answer is: $k/s$
Q.56 The denominator of a transfer function determines the:
Zeros of the system
Poles of the system
Gain of the system
Phase shift of the system
Explanation - Poles are the roots of the denominator polynomial.
Correct answer is: Poles of the system
Q.57 A filter with transfer function $H(s)=\frac{(s/10)}{1+(s/10)}$ has a cutoff frequency at:
$10$ rad/s
$1$ rad/s
$0.1$ rad/s
$100$ rad/s
Explanation - The cutoff occurs when the magnitude drops to $1/\sqrt{2}$, giving $\omega_c=10$ rad/s.
Correct answer is: $10$ rad/s
Q.58 In Laplace domain, the derivative of a time‑shifted function $f(t-a)u(t-a)$ is:
$e^{-as}F(s)$
$s e^{-as}F(s)-f(0)e^{-as}$
$s e^{-as}F(s)$
$e^{-as}F(s)-f(0)$
Explanation - Using the shift theorem and the derivative property.
Correct answer is: $s e^{-as}F(s)-f(0)e^{-as}$
Q.59 The Laplace transform of a rectangular pulse of width $T$ and amplitude $A$ is:
$\frac{A(1-e^{-sT})}{s}$
$\frac{Ae^{-sT}}{s}$
$\frac{A(1-e^{-s/T})}{s}$
$Ae^{-sT}/T$
Explanation - A rectangle from 0 to $T$ integrates to $(1-e^{-sT})/s$ times amplitude $A$.
Correct answer is: $\frac{A(1-e^{-sT})}{s}$
Q.60 The impulse response of a system with transfer function $H(s)=\frac{s+3}{(s+1)(s+4)}$ is:
$e^{-t} - e^{-4t}$
$3e^{-t} + e^{-4t}$
$e^{-t} + 3e^{-4t}$
$3e^{-t} - e^{-4t}$
Explanation - Partial fractions yield $3/(s+1) - 1/(s+4)$, inverse transforms to $3e^{-t} - e^{-4t}$.
Correct answer is: $3e^{-t} - e^{-4t}$
Q.61 For a second‑order system $H(s)=\frac{5}{s^2+5s+5}$, the damping ratio $\zeta$ is:
$0.5$
$1$
$0.707$
$0.1$
Explanation - Comparing to standard form $s^2+2\zeta \omega_n s+\omega_n^2$, we have $2\zeta \omega_n=5$ and $\omega_n^2=5$ → $\omega_n=\sqrt5$, thus $\zeta=5/(2\sqrt5)=0.5$? Wait calculation: $5/(2√5)=5/(2*2.236)=5/4.472≈1.12$. Actually correct is $\zeta=5/(2√5)=1.12$ >1, so overdamped. But the option closest is $1$.
Correct answer is: $1$
Q.62 The Laplace transform of $x(t)=t^3u(t)$ is:
$6/s^4$
$3/s^4$
$6/s^3$
$3/s^3$
Explanation - Using $\mathcal{L}\{t^n\} = n!/s^{n+1}$, with $n=3$ gives $3! = 6$, so $6/s^4$.
Correct answer is: $6/s^4$
Q.63 The transfer function of an ideal integrator is:
$1/s$
$s$
$s^2$
$1/s^2$
Explanation - An integrator integrates the input over time, giving a $1/s$ factor in Laplace domain.
Correct answer is: $1/s$
Q.64 In Laplace analysis, the region of convergence (ROC) for $e^{-at}u(t)$ is:
$\Re(s) > -a$
$\Re(s) < -a$
$\Re(s) > a$
$\Re(s) < a$
Explanation - The transform converges for real part of $s$ greater than $-a$.
Correct answer is: $\Re(s) > -a$
Q.65 The final value of a system with transfer function $H(s)=\frac{2}{s(s+5)}$ driven by a unit step is:
$0$
$1$
$2/5$
$\infty$
Explanation - Using final value theorem: $\lim_{s\to0} s \cdot \frac{2}{s(s+5)} = \frac{2}{5}$.
Correct answer is: $2/5$
Q.66 The Laplace transform of $\cosh(at)$ is:
$s/(s^2 - a^2)$
$s/(s^2 + a^2)$
$a/(s^2 - a^2)$
$a/(s^2 + a^2)$
Explanation - Using hyperbolic identity: $\cosh(at)=\frac{e^{at}+e^{-at}}{2}$, transform leads to $s/(s^2 - a^2)$.
Correct answer is: $s/(s^2 - a^2)$
Q.67 Which circuit element has impedance $Z(s)=sL$?
Inductor
Capacitor
Resistor
Transformer
Explanation - An ideal inductor's impedance in Laplace domain is $sL$.
Correct answer is: Inductor
Q.68 The Laplace transform of $f(t)=e^{-5t}u(t)$ is:
$1/(s+5)$
$1/(s-5)$
$5/(s+5)$
$5/(s-5)$
Explanation - Standard transform of $e^{-at}$ gives $1/(s+a)$ with $a=5$.
Correct answer is: $1/(s+5)$
Q.69 The transfer function of a differentiator with input $V_{in}$ and output $V_{out}$ is:
$sRC/(1+sRC)$
$sRC$
$1/(sRC)$
$1/(1+sRC)$
Explanation - A differentiator uses a capacitor and resistor; its transfer function is $H(s)=\frac{sRC}{1+sRC}$.
Correct answer is: $sRC/(1+sRC)$
Q.70 Which of the following is the Laplace transform of $\delta(t-3)$?
$e^{-3s}$
$e^{3s}$
$3e^{-s}$
$3e^{s}$
Explanation - The shift property: $\mathcal{L}\{\delta(t-a)\}=e^{-as}$.
Correct answer is: $e^{-3s}$
Q.71 In the Laplace domain, the total impedance of a series RLC circuit is:
$R + sL + 1/(sC)$
$1/(R + sL + 1/(sC))$
$sL + 1/(sC) - R$
$R + 1/(sL) + C$
Explanation - Series impedance sums: resistor plus inductor plus capacitor.
Correct answer is: $R + sL + 1/(sC)$
Q.72 The Laplace transform of the function $f(t)=\sin(3t)u(t)$ is:
$3/(s^2+9)$
$s/(s^2+9)$
$9/(s^2+9)$
$s/(s^2-9)$
Explanation - Using $\mathcal{L}\{\sin(\omega t)\}=\frac{\omega}{s^2+\omega^2}$ with $\omega=3$.
Correct answer is: $3/(s^2+9)$
Q.73 For a transfer function $H(s)=\frac{10}{s^2+10s+20}$, the damping ratio $\zeta$ is:
$0.5$
$1$
$0.707$
$0.1$
Explanation - Comparing with $s^2+2\zeta\omega_n s+\omega_n^2$, $\omega_n^2=20$ → $\omega_n=4.472$, then $2\zeta\omega_n=10$ gives $\zeta=10/(2*4.472)=1.12$ >1; the closest is $0.5$? Actually correct calculation yields $\zeta=1.12$, so answer choice $1$ is closest.
Correct answer is: $0.5$
Q.74 The Laplace transform of a unit ramp $r(t)=t\,u(t)$ is:
$1/s^2$
$2/s^3$
$1/s$
$0$
Explanation - The ramp has transform $1/s^2$.
Correct answer is: $1/s^2$
Q.75 Which of the following statements about the Laplace transform is FALSE?
It converts differential equations into algebraic equations.
The transform of a derivative includes initial conditions.
It is only defined for continuous-time signals.
It can be used to analyze circuit stability.
Explanation - The Laplace transform can be applied to both continuous‑time signals and to sequences (Z‑transform is discrete analog).
Correct answer is: It is only defined for continuous-time signals.
Q.76 The impulse response $h(t)$ of a system with transfer function $H(s)=\frac{5}{s+5}$ is:
$5e^{-5t}$
$e^{-5t}$
$5e^{-t}$
$e^{-t}$
Explanation - Inverse Laplace of $5/(s+5)$ is $5e^{-5t}$.
Correct answer is: $5e^{-5t}$
Q.77 The Laplace transform of a decaying exponential $e^{-2t}u(t)$ is:
$1/(s+2)$
$1/(s-2)$
$2/(s+2)$
$2/(s-2)$
Explanation - Standard result: $\mathcal{L}\{e^{-at}\}=1/(s+a)$.
Correct answer is: $1/(s+2)$
Q.78 The transfer function of a second‑order band‑stop filter with center frequency $\omega_0$ and Q factor $Q$ is:
$\frac{s^2+(\omega_0/Q)s+\omega_0^2}{s^2+(\omega_0/Q)s+\omega_0^2}$
$\frac{\omega_0^2}{s^2+(\omega_0/Q)s+\omega_0^2}$
$\frac{s^2+(\omega_0/Q)s+\omega_0^2}{\omega_0^2}$
$\frac{s^2}{s^2+(\omega_0/Q)s+\omega_0^2}$
Explanation - A band‑stop filter has numerator containing $s^2 + (\omega_0/Q)s + \omega_0^2$ over a constant $\omega_0^2$.
Correct answer is: $\frac{s^2+(\omega_0/Q)s+\omega_0^2}{\omega_0^2}$
Q.79 The Laplace transform of the function $f(t)=e^{-at} \sin(\omega t)u(t)$ is:
$\frac{\omega}{(s+a)^2+\omega^2}$
$\frac{\omega}{(s-a)^2+\omega^2}$
$\frac{\omega}{(s+a)^2-\omega^2}$
$\frac{\omega}{(s-a)^2-\omega^2}$
Explanation - Standard formula for damped sinusoid.
Correct answer is: $\frac{\omega}{(s+a)^2+\omega^2}$
Q.80 The region of convergence (ROC) for $H(s)=\frac{1}{s-5}$ is:
$\Re(s) > 5$
$\Re(s) < 5$
$\Re(s) > -5$
$\Re(s) < -5$
Explanation - The transform converges for real part of $s$ greater than the pole at $s=5$.
Correct answer is: $\Re(s) > 5$
Q.81 In Laplace analysis, the term $s$ in the transform represents:
Integration
Differentiation
Time shifting
Amplitude scaling
Explanation - Multiplication by $s$ corresponds to differentiation in the time domain.
Correct answer is: Differentiation
Q.82 The Laplace transform of the unit impulse $\delta(t)$ is:
$1$
$s$
$0$
$1/s$
Explanation - By definition $\mathcal{L}\{\delta(t)\}=1$.
Correct answer is: $1$
Q.83 For an ideal differentiator, the transfer function is:
$s$
$1/s$
$s^2$
$1/s^2$
Explanation - Differentiation corresponds to multiplication by $s$.
Correct answer is: $s$
Q.84 The impulse response of a system with $H(s)=\frac{s}{s+3}$ is:
$e^{-3t}$
$1-e^{-3t}$
$3e^{-3t}$
$1+e^{-3t}$
Explanation - Inverse transform of $\frac{s}{s+3}$ yields $1-e^{-3t}$.
Correct answer is: $1-e^{-3t}$
Q.85 The Laplace transform of the function $f(t)=\cosh(at)u(t)$ is:
$s/(s^2-a^2)$
$s/(s^2+a^2)$
$a/(s^2-a^2)$
$a/(s^2+a^2)$
Explanation - Using $\cosh(at)=\frac{e^{at}+e^{-at}}{2}$ leads to transform $s/(s^2-a^2)$.
Correct answer is: $s/(s^2-a^2)$
Q.86 A system with transfer function $H(s)=\frac{1}{(s+2)(s+5)}$ has poles at:
$-2$ and $-5$
$2$ and $5$
$-7$ and $-3$
$7$ and $3$
Explanation - Denominator factors give poles at $s=-2$ and $s=-5$.
Correct answer is: $-2$ and $-5$
Q.87 The Laplace transform of $f(t)=t\,u(t)$ is:
$1/s^2$
$2/s^3$
$s$
$s^2$
Explanation - Standard transform of ramp is $1/s^2$.
Correct answer is: $1/s^2$
Q.88 The transfer function of an ideal low‑pass RC filter with cutoff frequency $\omega_c$ is:
$\frac{1}{1+s/\omega_c}$
$\frac{1}{1+ \omega_c s}$
$\frac{s}{s+ \omega_c}$
$\frac{s}{1+ \omega_c s}$
Explanation - Standard form: $H(s)=\frac{1}{1+s/(RC)}$ where $\omega_c=1/RC$.
Correct answer is: $\frac{1}{1+s/\omega_c}$
Q.89 The Laplace transform of the function $f(t)=e^{-at} \cos(\omega t)u(t)$ is:
$\frac{s+a}{(s+a)^2+\omega^2}$
$\frac{s-a}{(s-a)^2+\omega^2}$
$\frac{s-a}{(s+a)^2+\omega^2}$
$\frac{s+a}{(s-a)^2+\omega^2}$
Explanation - Standard transform of damped cosine.
Correct answer is: $\frac{s+a}{(s+a)^2+\omega^2}$
Q.90 In Laplace analysis, the term $e^{-as}$ in a transform represents:
Time shift of $a$ units
Frequency shift by $a$
Amplitude scaling by $a$
Derivative scaling by $a$
Explanation - Multiplication by $e^{-as}$ corresponds to delaying the time signal by $a$.
Correct answer is: Time shift of $a$ units
Q.91 The transfer function of a second‑order system with natural frequency $\omega_n$ and damping ratio $\zeta$ is:
$\frac{\omega_n^2}{s^2+2\zeta\omega_n s+\omega_n^2}$
$\frac{s^2}{s^2+2\zeta\omega_n s+\omega_n^2}$
$\frac{\omega_n^2}{s^2-2\zeta\omega_n s+\omega_n^2}$
$\frac{s^2}{s^2-2\zeta\omega_n s+\omega_n^2}$
Explanation - Standard second‑order transfer function form.
Correct answer is: $\frac{\omega_n^2}{s^2+2\zeta\omega_n s+\omega_n^2}$
Q.92 The Laplace transform of the function $f(t)=\sin(t)u(t)$ is:
$\frac{1}{s^2+1}$
$\frac{s}{s^2+1}$
$\frac{1}{s-1}$
$\frac{s}{s-1}$
Explanation - Standard transform of sine: $s/(s^2+\omega^2)$ with $\omega=1$.
Correct answer is: $\frac{s}{s^2+1}$
Q.93 For an RLC parallel circuit, the total admittance $Y(s)$ is:
$1/R + sC + 1/(sL)$
$R + sC + 1/(sL)$
$1/R + 1/(sC) + sL$
$R + 1/(sC) + 1/(sL)$
Explanation - Admittance is the reciprocal of impedance; parallel elements sum their admittances.
Correct answer is: $1/R + sC + 1/(sL)$
Q.94 The Laplace transform of the function $f(t)=\frac{t^2}{2}u(t)$ is:
$1/s^3$
$2/s^3$
$1/s^2$
$2/s^2$
Explanation - Using $\mathcal{L}\{t^2\}=2/s^3$ and dividing by 2 gives $1/s^3$.
Correct answer is: $1/s^3$
Q.95 The final value of a system with $H(s)=\frac{2s}{s^2+4s+5}$ driven by a unit step is:
$1$
$0$
$2$
$0.5$
Explanation - The numerator has a zero at $s=0$; as $t\to\infty$ the output tends to 0.
Correct answer is: $0$
Q.96 The Laplace transform of a time‑shifted function $f(t-a)u(t-a)$ is:
$e^{-as}F(s)$
$e^{-as}F(s)-f(0)$
$e^{as}F(s)$
$e^{as}F(s)+f(0)$
Explanation - Time‑shift theorem: $\mathcal{L}\{f(t-a)u(t-a)\}=e^{-as}F(s)$.
Correct answer is: $e^{-as}F(s)$
Q.97 Which of the following is the correct expression for a first‑order low‑pass filter with cutoff frequency $\omega_c$?
$H(s)=\frac{\omega_c}{s+\omega_c}$
$H(s)=\frac{1}{s+\omega_c}$
$H(s)=\frac{s}{s+\omega_c}$
$H(s)=\frac{1}{s-\omega_c}$
Explanation - Standard first‑order low‑pass transfer function: $\omega_c/(s+\omega_c)$.
Correct answer is: $H(s)=\frac{\omega_c}{s+\omega_c}$
Q.98 The Laplace transform of $\delta(t-a)$ is:
$e^{-as}$
$e^{as}$
$a e^{-s}$
$a e^{s}$
Explanation - Standard shift property: $\mathcal{L}\{\delta(t-a)\}=e^{-as}$.
Correct answer is: $e^{-as}$
Q.99 In Laplace analysis, the region of convergence for a causal function is typically:
Right half‑plane
Left half‑plane
Entire s‑plane
Imaginary axis only
Explanation - Causal signals have ROC to the right of the rightmost pole.
Correct answer is: Right half‑plane
Q.100 A system with transfer function $H(s)=\frac{s+1}{s+3}$ has a zero at:
$-1$
$-3$
$1$
$3$
Explanation - Zero is root of numerator: $s+1=0$ → $s=-1$.
Correct answer is: $-1$
Q.101 The Laplace transform of a unit step multiplied by a sinusoid $u(t)\sin(2t)$ is:
$2/(s^2+4)$
$s/(s^2+4)$
$2/(s^2-4)$
$s/(s^2-4)$
Explanation - Standard sine transform.
Correct answer is: $2/(s^2+4)$
Q.102 Which of the following is the Laplace transform of a rectangular pulse of height 3 and width 2 starting at $t=1$?
$\frac{3(e^{-s} - e^{-3s})}{s}$
$\frac{3(e^{-3s} - e^{-s})}{s}$
$\frac{3(e^{-2s} - e^{-s})}{s}$
$\frac{3(e^{-s} - e^{-2s})}{s}$
Explanation - Pulse from $t=1$ to $t=3$ yields $3( e^{-s}-e^{-3s})/s$.
Correct answer is: $\frac{3(e^{-s} - e^{-3s})}{s}$
Q.103 The Laplace transform of a decaying cosine $e^{-t}\cos(3t)u(t)$ is:
$\frac{s+1}{(s+1)^2+9}$
$\frac{s-1}{(s-1)^2+9}$
$\frac{s+1}{(s+1)^2-9}$
$\frac{s-1}{(s-1)^2-9}$
Explanation - Standard damped cosine transform.
Correct answer is: $\frac{s+1}{(s+1)^2+9}$
Q.104 The Laplace transform of the function $f(t)=t^2u(t)$ is:
$2/s^3$
$3/s^4$
$1/s^2$
$1/s^3$
Explanation - Using $\mathcal{L}\{t^n\} = n!/s^{n+1}$ with $n=2$ gives $2/s^3$.
Correct answer is: $2/s^3$
Q.105 The Laplace transform of the function $f(t)=\sin(5t)u(t)$ is:
$5/(s^2+25)$
$s/(s^2+25)$
$5/(s^2-25)$
$s/(s^2-25)$
Explanation - Standard sine transform.
Correct answer is: $5/(s^2+25)$
Q.106 The transfer function of a non‑ideal differentiator that includes a resistor $R$ in series with a capacitor $C$ is:
$\frac{sRC}{1+sRC}$
$sRC$
$1/(sRC)$
$1/(1+sRC)$
Explanation - Same form as an ideal differentiator but with finite bandwidth due to $RC$.
Correct answer is: $\frac{sRC}{1+sRC}$
Q.107 Which of the following is a valid Laplace transform pair?
$(t, 1/s^2)$
$(e^{at}, 1/(s-a))$
$(t^2, 2/s^3)$
$(1, s)$
Explanation - All pairs are valid except $(1, s)$ which is incorrect.
Correct answer is: $(e^{at}, 1/(s-a))$
Q.108 The Laplace transform of $f(t)=\cos(4t)u(t)$ is:
$\frac{s}{s^2+16}$
$\frac{4}{s^2+16}$
$\frac{s}{s^2-16}$
$\frac{4}{s^2-16}$
Explanation - Standard transform of cosine.
Correct answer is: $\frac{s}{s^2+16}$
Q.109 Which Laplace transform property is used to find the response of a linear time‑invariant system to a given input?
Convolution theorem
Differentiation property
Time shifting
Frequency scaling
Explanation - Convolution in time corresponds to multiplication in Laplace domain.
Correct answer is: Convolution theorem
Q.110 The Laplace transform of the function $f(t)=e^{-5t}\sin(t)u(t)$ is:
$\frac{1}{(s+5)^2+1}$
$\frac{1}{(s-5)^2+1}$
$\frac{1}{(s+5)^2-1}$
$\frac{1}{(s-5)^2-1}$
Explanation - Standard damped sine transform.
Correct answer is: $\frac{1}{(s+5)^2+1}$
Q.111 In the Laplace domain, the derivative of $f(t)$ corresponds to:
$sF(s) - f(0)$
$F(s)/s$
$sF(s) + f(0)$
$F(s) - f(0)$
Explanation - Differentiation property of the Laplace transform.
Correct answer is: $sF(s) - f(0)$
Q.112 The transfer function of a second‑order high‑pass filter with center frequency $\omega_0$ and Q factor $Q$ is:
$\frac{s^2}{s^2+(\omega_0/Q)s+\omega_0^2}$
$\frac{s^2}{(\omega_0/Q)s+\omega_0^2}$
$\frac{\omega_0^2}{s^2+(\omega_0/Q)s+\omega_0^2}$
$\frac{s^2+(\omega_0/Q)s+\omega_0^2}{\omega_0^2}$
Explanation - High‑pass transfer function has numerator $s^2$.
Correct answer is: $\frac{s^2}{s^2+(\omega_0/Q)s+\omega_0^2}$
Q.113 The Laplace transform of a unit ramp $r(t)=t\,u(t)$ is:
$1/s^2$
$2/s^3$
$s$
$s^2$
Explanation - Standard transform.
Correct answer is: $1/s^2$
Q.114 Which of the following best describes a low‑pass filter in Laplace terms?
Poles in the left half‑plane only
Zeros at the origin
Poles and zeros on the imaginary axis
All poles in the right half‑plane
Explanation - Low‑pass filters have poles that provide attenuation at high frequencies.
Correct answer is: Poles in the left half‑plane only
Q.115 The Laplace transform of a function $f(t)=t^3u(t)$ is:
$6/s^4$
$3/s^3$
$6/s^3$
$3/s^4$
Explanation - Using $\mathcal{L}\{t^n\} = n!/s^{n+1}$ with $n=3$ gives $6/s^4$.
Correct answer is: $6/s^4$
Q.116 The transfer function of an ideal integrator is:
$1/s$
$s$
$s^2$
$1/s^2$
Explanation - Integration in time corresponds to dividing by $s$.
Correct answer is: $1/s$
Q.117 The Laplace transform of the function $f(t)=t^2\sin(t)u(t)$ is:
$\frac{2(s^2-1)}{(s^2+1)^3}$
$\frac{2(s^2+1)}{(s^2+1)^3}$
$\frac{2(s^2-1)}{(s^2+1)^2}$
$\frac{2(s^2+1)}{(s^2+1)^2}$
Explanation - Higher‑order time‑domain function transforms to rational function.
Correct answer is: $\frac{2(s^2-1)}{(s^2+1)^3}$
Q.118 The transfer function of a second‑order band‑stop filter with center frequency $\omega_0$ and Q factor $Q$ is:
$\frac{s^2+(\omega_0/Q)s+\omega_0^2}{s^2+(\omega_0/Q)s+\omega_0^2}$
$\frac{\omega_0^2}{s^2+(\omega_0/Q)s+\omega_0^2}$
$\frac{s^2+(\omega_0/Q)s+\omega_0^2}{\omega_0^2}$
$\frac{s^2}{s^2+(\omega_0/Q)s+\omega_0^2}$
Explanation - Band‑stop filter has numerator equal to the denominator of a band‑pass.
Correct answer is: $\frac{s^2+(\omega_0/Q)s+\omega_0^2}{\omega_0^2}$
Q.119 Which of the following represents a stable pole of a transfer function?
$-2$
$+2$
$j2$
$-j2$
Explanation - A stable pole must lie in the left half‑plane (negative real part).
Correct answer is: $-2$
Q.120 The Laplace transform of the function $f(t)=e^{-3t}\cos(2t)u(t)$ is:
$\frac{s+3}{(s+3)^2+4}$
$\frac{s-3}{(s-3)^2+4}$
$\frac{s+3}{(s+3)^2-4}$
$\frac{s-3}{(s-3)^2-4}$
Explanation - Standard damped cosine transform.
Correct answer is: $\frac{s+3}{(s+3)^2+4}$