Q.1 What is the Laplace transform of 1?
1/s
s
e^s
0
Explanation - The Laplace transform of a constant 1 is ∫₀^∞ e^(-st) dt = 1/s.
Correct answer is: 1/s
Q.2 The Laplace transform is defined as:
L{f(t)} = ∫₀^∞ f(t) dt
L{f(t)} = ∫₀^∞ e^(-st) f(t) dt
L{f(t)} = ∫₀^∞ e^(st) f(t) dt
L{f(t)} = ∫₀^∞ f(t) e^(t) dt
Explanation - By definition, the Laplace transform is L{f(t)} = ∫₀^∞ e^(-st) f(t) dt.
Correct answer is: L{f(t)} = ∫₀^∞ e^(-st) f(t) dt
Q.3 What is the Laplace transform of e^(at)?
1/(s+a)
1/(s-a)
s/(s-a)
a/(s-a)
Explanation - L{e^(at)} = ∫₀^∞ e^(-st) e^(at) dt = 1/(s-a).
Correct answer is: 1/(s-a)
Q.4 The Laplace transform of sin(at) is:
a/(s^2 + a^2)
s/(s^2 + a^2)
1/(s+a)
s/(s-a)
Explanation - Using standard formula: L{sin(at)} = a/(s^2 + a^2).
Correct answer is: a/(s^2 + a^2)
Q.5 The Fourier transform of δ(t) is:
1
δ(ω)
∞
0
Explanation - The delta function has a Fourier transform equal to 1 for all frequencies.
Correct answer is: 1
Q.6 What is the Laplace transform of cos(at)?
s/(s^2 + a^2)
a/(s^2 + a^2)
1/(s-a)
s/(s-a)
Explanation - Standard result: L{cos(at)} = s/(s^2 + a^2).
Correct answer is: s/(s^2 + a^2)
Q.7 Laplace transform is useful for solving:
Differential equations
Algebraic equations
Geometry problems
Probability
Explanation - Laplace transform simplifies solving differential equations by converting them to algebraic equations.
Correct answer is: Differential equations
Q.8 The Fourier transform of e^(-at)u(t), a>0 is:
1/(a+jω)
1/(s-a)
δ(ω-a)
a/(s^2+a^2)
Explanation - FT{e^(-at)u(t)} = ∫₀^∞ e^(-at) e^(-jωt) dt = 1/(a+jω).
Correct answer is: 1/(a+jω)
Q.9 The Fourier transform pair is given by:
F(ω)=∫ f(t)e^(jωt)dt
F(ω)=∫ f(t)e^(-jωt)dt
F(ω)=∫ f(t)dt
F(ω)=∫ f(t)e^t dt
Explanation - By definition, Fourier transform is F(ω) = ∫ f(t)e^(-jωt)dt.
Correct answer is: F(ω)=∫ f(t)e^(-jωt)dt
Q.10 Which of the following relates Laplace and Fourier transforms?
Laplace is a generalization of Fourier
Fourier is a generalization of Laplace
They are unrelated
Both are the same
Explanation - Fourier is a special case of Laplace with s = jω.
Correct answer is: Laplace is a generalization of Fourier
Q.11 Region of convergence (ROC) in Laplace transform refers to:
Values of t for which Laplace exists
Values of s for which Laplace exists
Values of a for which Laplace exists
Frequency values only
Explanation - ROC defines the set of s in the complex plane for which the Laplace transform converges.
Correct answer is: Values of s for which Laplace exists
Q.12 The inverse Laplace transform is computed using:
Residue theorem
Taylor series
Differentiation
Integration by parts
Explanation - Inverse Laplace is often computed using complex integration and the residue theorem.
Correct answer is: Residue theorem
Q.13 Laplace transform of t^n is:
n!/s^(n+1)
s^n/n!
1/(s+n)
s^n
Explanation - L{t^n} = ∫₀^∞ e^(-st) t^n dt = n!/s^(n+1).
Correct answer is: n!/s^(n+1)
Q.14 Fourier transform is mainly used in:
Time-frequency analysis
Algebra
Geometry
Linear equations
Explanation - Fourier transform expresses signals in terms of frequency components.
Correct answer is: Time-frequency analysis
Q.15 What is the Laplace transform of δ(t)?
1
s
δ(s)
0
Explanation - Since δ(t) picks the value at t=0, L{δ(t)} = 1.
Correct answer is: 1
Q.16 The Fourier transform of cos(ω₀t) is:
δ(ω-ω₀)+δ(ω+ω₀)
1/(jω)
δ(ω)
∞
Explanation - FT{cos(ω₀t)} = π[δ(ω-ω₀) + δ(ω+ω₀)].
Correct answer is: δ(ω-ω₀)+δ(ω+ω₀)
Q.17 Laplace transform is applied to:
Causal signals
Non-causal signals
Both
Neither
Explanation - Laplace transform can be applied to both causal and non-causal signals depending on ROC.
Correct answer is: Both
Q.18 The bilateral Laplace transform is defined over:
t>0
t<0
-∞<t<∞
0<t<1
Explanation - Bilateral Laplace transform integrates over the entire real line.
Correct answer is: -∞<t<∞
Q.19 The Fourier transform of rect(t) is:
sinc(ω)
δ(ω)
cos(ω)
1
Explanation - Rectangular pulse transforms into sinc function in frequency domain.
Correct answer is: sinc(ω)
Q.20 The initial value theorem in Laplace transform relates:
f(0+) to lim(sF(s))
f(∞) to lim(sF(s))
f(0-) to lim(sF(s))
f(0) to F(s)
Explanation - Initial value theorem: f(0+) = lim_{s→∞} sF(s).
Correct answer is: f(0+) to lim(sF(s))
Q.21 Which function's Laplace transform gives 1/s^2?
t
1
e^t
δ(t)
Explanation - L{t} = 1/s^2.
Correct answer is: t
Q.22 Convolution in time domain corresponds to what in Laplace domain?
Multiplication
Addition
Division
Differentiation
Explanation - Convolution theorem: convolution in time domain = multiplication in Laplace domain.
Correct answer is: Multiplication
Q.23 Fourier transform of an even function is:
Real and even
Imaginary and odd
Complex
Zero
Explanation - Even functions produce real, even Fourier transforms.
Correct answer is: Real and even
Q.24 Laplace transform of u(t) (unit step function) is:
1/s
s
1
δ(s)
Explanation - L{u(t)} = ∫₀^∞ e^(-st) dt = 1/s.
Correct answer is: 1/s