Network Theory Multiple Choice Questions on “Operational Transforms”.
1. The Laplace transform of kf(t) is?
A. F(s)
B. kF(s)
C. F(s)/k
D. k2 F(s)
Answer: B
Clarification: Operational transforms indicate how mathematical operations performed in either f(t) or F(s) are converted into the opposite domain. Linearity property states that L (kf (t)) = kF (s).
2. The Laplace transform of f1 (t) + f2 (t) is?
A. F1(s) + F2(s)
B. F1(s) – F2(s)
C. F1(s) – 2F2(s)
D. F1(s) + 2F2(s)
Answer: A
Clarification: Addition or subtraction in time domain translates into addition or subtraction in frequency domain. L (f1 (t) + f2 (t)) = F1(s) + F2(s).
3. Find the Laplace transform of the function f (t) = 4t3 + t2 – 6t + 7.
A. 24/s4 + 2/s3 + 6/s2 + 7/s
B. 24/s4 – 2/s3 – 6/s2 + 7/s
C. 24/s4 + 2/s3 – 6/s2 + 7/s
D. 24/s4 – 2/s3 + 6/s2 + 7/s
Answer: C
Clarification: L (4t3 + T2 -6t +7) = 4L (t3) + L(t2)-6L (t) + 7L(1) = 4×3!/s4 + 2!/s3 – 6 (1!)/(s2)+71/s = 24/s4 + 2/s3 -6/s2 + 7/s.
4. Find the Laplace transform of the function f(t) = cos2t.
A. (2s2+4)/2s(s2-4)
B. (2s2-4)/2s(s2-4)
C. (2s2-4)/2s(s2+4)
D. (2s2+4)/2s(s2+4)
Answer: D
Clarification: The Laplace transform of the function f(t) = cos2t is L (cos2t) = L((1+cos2t)/2) = L(1/2)+L(cos2t/2) = 1/2[L(1)+L(cos2t)] = (2s2+4)/2s(s2+4).
5. Find the Laplace transform of the function f (t) = 3t4 – 2t3 + 4e-3t – 2sin5t + 3cos2t.
A. 72/s5 – 12/s4 + 4/(s+3)+10/(s2+25)+3s/(s2+4)
B. 72/s5 – 12/s4 + 4/(s+3)-10/(s2+25)+3s/(s2+4)
C. 72/s5 – 12/s4 – 4/(s+3)+10/(s2+25)+3s/(s2+4)
D. 72/s5 – 12/s4 – 4/(s+3)-10/(s2+25)+3s/(s2+4)
Answer: B
Clarification: L (3t4 -2t3+4e-3t – 2sin5t +3cos2t) = 3 L (t4)-2L (t3)+4L (e-3t)-2L (sin5t) + 3L (cos2t) = 72/s5) -12/s4 +4/(s+3)-10/(s2+25)+3s/(s2+4).
6. Find the Laplace transform of eatsinbt.
A. b/((s-A.2+b2)
B. b/((s+A.2+b2)
C. b/((s+A.2-b2)
D. b/((s-A.2-b2)
Answer: A
Clarification: The Laplace transform of sinbt is L(sinbt)=b/(s2+b2). So the Laplace transform of eatsinbt is L(exp(at) sinbt)=b/((s-A.2+b2).
7. Find the Laplace transform of (t + 2)2 et.
A. 2/(s-1)3 – 2/(s-1)2 + 4/(s-1)
B. 2/(s-1)3 – 2/(s-1)2 – 4/(s-1)
C. 2/(s-1)3 + 2/(s-1)2 + 4/(s-1)
D. 2/(s-1)3 + 2/(s-1)2 – 4/(s-1)
Answer: C
Clarification: The Laplace transform of t2+2t+4 is L(t2+2t+4)=2/(s)3 + 2/(s)2+4/s. So the Laplace transform of (t + 2)2 et is L((t + 2)2 et) = 2/(s-1)3 + 2/(s-1)2 + 4/(s-1).
8. Find the Laplace transform of ramp function r (t) = t.
A. 1/s
B. 1/s2
C. 1/s3
D. 1/s4
9.Find the Laplace transform of the function f (t) = tsin2t.
A. 4s/(s2+4)2
B. -4s/(s2+4)2
C. -4s/(s2-4)2
D. 4s/(s2-4)2
Answer: A
Clarification: The Laplace transform of the function of sin2t is L(sin2t)=2/(s2+4). So the Laplace transform of the function f (t) = tsin2t is L(tsin2t) = -d/ds [2/(s2+4)] = 4s/(s2+4)2.
10.If u (t) = 1 for t >= 0 and u (t) = 0 for t < 0, determine the Laplace transform of [u (t) – u (t – A.].
A. 1/s(1+e(-as))
B. 1/s(1-e(-as))
C. 1/s(1+eas)
D. 1/s(1-eas)
Answer: B
Clarification: As u (t) = 1 for t >= 0 and u (t) = 0 for t < 0, the Laplace transform of [u (t) – u (t – A.] is L[u (t)– u (t – A.] = 1/s-e(-as)1/s = 1/s (1-e(-as)).