1
EE/ECE Batch : Hinglish
Practice sheet-01
Subject- Signals & Systems
Chapter -Laplace Transform
1. The Laplace transform of signal x(t) shown in figure is 5. The Laplace transform and its ROC of the signal
x (t ) = e
−a t
is
2a
(a) − , −a Re ( s ) a
s − a2
2
2a
(b) , Re ( s ) a
s − a2
2
1
1 1 1 (c) , −a Re ( s ) a
(a) [e−2 s − e−5s ] (b) + s − a2
2
s 2
( ) ( 5)
s + 2 s +
2s
1 −2 s (d) , −a Re ( s ) a
(c) [e − e−5 s ] (d) (e−2 s − e−5s ) s − a2
2
s
6. The region of convergence for the Laplace transform of
2. The Laplace transform of a signal
a signal x(t) = e–t u(t) + e–t t u(–t) is
x(t) = – exp (–at) u(–t) is
(a) < Re (s) <
1
(a) , ROC : Re ( s ) −a (b) – < Re (s) < –
s+a
−1 (c) Re (s) > max (,)
(b) , ROC : Re ( s ) −a
s+a (d) Re (s) > max (–, –)
1
(c) , ROC : Re ( s ) −a
s+a 7. Match List I (continuous Time Signal) with List II
−1 (Laplace transform) and choose the correct answer
(d) , ROC : Re ( s ) −a using the codes given below.
s+a
List-I List-II
s
3. The ROC of LT of the continuous time signal P. sin (2t) 1.
x(t) = 3e–2t u(t) + 4etu(–t) is s +4
2
(a) –1 < Re (s) < 2 (b) –2 < Re (s) < 1 2
Q. cos (2t) 2.
(c) 1 < Re (s) < 2 (d) 0 < Re (s) < 2 s +4
2
s
R. sin h (2t) 3.
4. A CT signal is given as x(t) = [P + Qe–bt] u(t) where P s −4
2
and Q are constant. The ROC of Laplace transform of 2
R. cos h (2t) 4.
x(t) is s −4
2
(a) Re (s) > max (–b, 0) Codes:
(b) Re (s) > max (b, 0) P Q R S
(c) 0 < Re (s) < b (a) 2 1 3 4
(d) Re (s) > min (–b, 0) (b) 1 2 3 4
(c) 2 1 4 3
(d) 1 2 4 3
2
8. Which of the following plot represents correct ROC of 10. Let x(t) be a CT signal and X(s) be its Laplace
a signal x(t) = e2t u(–t) –e–5t u(t)? transform given
5
X ( s) =
(a) (s 2
+ 3s − 4 )
Match inverse Laplace transform L1, L2, L3 with region
of convergence R1, R2, R3.
(b) Inverse Laplace Transform ROC
L1 : x(t ) = −e−4t u (t ) − et u (−t ) R1 : Re(s) 1
(c) L2 : x(t ) = −e−4t u(t ) + et u(t ) R2 : Re(s) −4
L3 : x(t ) = −e−4t u (−t ) + et u (−t ) R3 : −4 Re(s) 1
(d) (a) ( L1, R3 ) , ( L2 , R2 ) , ( L3 , R1 )
(b) ( L1, R3 ) , ( L2 , R1 ) , ( L3 , R2 )
9. The pole zero diagram of a continuous time system is
(c) ( L1, R1 ) , ( L2 , R2 ) , ( L3 , R3 )
shown in the figure (d) ( L1, R2 ) , ( L2 , R1 ) , ( L3 , R3 )
11. The Laplace transform of a signal x(t) shown in figure
is
Match List I (ROC of system function) with List II
(Stability and causality) and choose the correct answer
using the codes given below
List-I List-II
P. Re(s) < –2 1. Unstable and non-causal
Q. –2 < Re(s) < –1 2. Unstable and causal
1 − st0
R. –1 < Re(s) < 1 3. Unstable and anti-causal (a) e
s
S. Re(s) > 1 4. Stable and non-causal
Codes: 1 − st0
(b) e
P Q R S s2
(a) 1 4 2 3
(b) 2 4 1 3 (c)
1
t0 s 2 (
1 − e− st0 )
(c) 3 1 4 2
(d) 4 1 3 2 (d)
1
s 2 (
1 − e− st0 )
3
12. Consider two signals x(t) and y(t) shown in figure and 15. Consider a signal x(t ) = e−2t u (t ) and its LT pair
their Laplace transform pairs x(t ) LX ( s) and L
x(t ) ⎯⎯
→ X (s)
y (t ) LY ( s )
X (s)
The inverse Laplace transform of is
s
(a) 2e−2t u(t )
(b) 1 + e−4t u (t )
(c) 2e−4t u(t )
1 − e −2t
(d) u (t )
1 2
If X ( s ) = 5 − 10e−2 s + 5e−4 s , then Y(s) will be
s
(a) 5 − 10e−2 s + 5e−4 s 16. If X(s) = 4/(s2 – s) is the Laplace transform of x(t), then
1 its final value will be
(b) 2
5 − 10e−2 s + 5e−4 s
2s (a) –4
(c) s 5 − 10e−2 s + 5e−4 s (b) 0
(c) 4
1
(d) 5 − 10e−2 s + 5e−4 s (d) Does not exist
s2
L
−2t 17. For a Laplace transform pair x(t ) ⎯⎯ X ( s) which of
13. Consider a signal x(t ) = e u (t ) and its LT pair
the following statement is not true?
L
x(t ) ⎯⎯
→ X (s)
(a) If x(t) is an even function then X(s) = X(–s)
The inverse Laplace transform of X (s/2) is (b) If x(t) is an odd function then X(s) = – X (–s)
1 −t
(a) e u (t ) (b) 2e−4t u(t ) (c) For an even x(t), ROC must be chosen to
2 correspond to a two sided signal.
1 −4t
(c) e−2t u (t − 2) (d) e u (t ) (d) none of these
2
18. An electric network is shown in the figure below
14. Consider a signal x(t ) = e−2t u (t ) and its LT pair
L
x(t ) ⎯⎯
→ X (t )
The inverse Laplace Transform of {X(s – 2) + X(s +
2)} is
(a) ( e−4t + e4t ) u(t ) Which of the following statement regarding the
stability of the system is true?
(b) ( e−2t + e−6t ) u(t ) (a) The system is BIBO stable.
(c) (1 + e−4t ) u(t ) (b) The system is BIBO unstable
(c) The system stability depends on the value of R and
(d) (1 + e−2t ) u(t ) L
(d) none of these
4
19. If the system is stable, the impulse response h(t) of the 23. The one-sided Laplace transform of signal sin 5t is
system will be
5
1 2t 1 (a)
e u (t ) − e−t u (t ) s +5
2
(a)
3 3
s
1 1 (b)
(b) − e−2t u ( −t ) + e−t u ( −t ) s +5
2
3 3
1 1 5
(c) − e 2t u ( −t ) − e −t u ( t ) (c)
3 3 s + 25
2
1 2t 1
(d) e u ( t ) + e −t u ( t ) (d)
s
3 3 s + 25
2
20. The block diagram representation of a continuous time
system is shown in the figure. 24. The Laplace transform of (t + 1) is
(a) es, Re(s) > 0
(b) es, Re(s) < 0
(c) es, all s
(d) None of above
Assume all initial conditions are zero. For an input x(t)
= u(t), output y(t) will be 25. The Laplace transform of e–tu(t + 2) is
e −4t u ( t )
e 2( s +1)
(a)
(a) , Re( s ) −1
s +1
(b)
1
4
( )
1 − e−4t u ( t )
1
(b) , Re( s) −1
(c)
1
4
( )
1 − t 2 u (t ) 1+ s
e 2( s +1)
(d)
1
4
( )
1 + e−4t u ( t )
(c)
s +1
, Re( s ) −1
21. The Laplace transform of signal u(t – 2) is 1
(d) , Re( s) −1
1+ s
−e −2s e −2s
(a) (b)
s s
e −2 s 26. The Laplace transform of u(– t + 3) is
(c) (d) 0
1+ s − e −3t
(a) , Re( s ) 0
s
22. The one sided Laplace transform of signal e2t u(−t + 2)
− e −3 s
is (b) , Re( s ) 0
s
e ( ) −1
2 s −2
e −2 s
(a)
s−2
(b)
s+2 1 − 6−3s
(c) , Re( s ) 0
s
1− e ( )
−2 s − 2
e −2 s
(c) (d)
s−2 s−2 − e −3 s
(d) , Re( s ) 0
s
5
27. The unilateral Laplace transform of a signal x(t) is 29. The Laplace transform of e−3t +6u (t + 3) is
3s + 2
X (s) = e3 s
s + 2s + 10
2 (a) , Re( s ) 3
s −3
The time signal x(t) is
e3 s
−t 1 −t (b) , Re( s ) 3
(a) 3e cos3t − e sin 3t u ( t ) s −3
3
e3( s −1)
−t 1 −t (c) , Re( s ) 3
(b) 3e sin 3t − e cos3t u ( t ) s−3
3
e3( s −1)
(c) ( )
3e−t cos3t − e−t sin 3t u ( t ) (d)
s−3
, Re( s ) 3
(d) (3e−t sin 3t + 3e−t cos3t ) u (t ) 30. Consider the Laplace transform pair given below.
cos ( 2t ) u ( t ) X (s)
28. The Laplace transform of a signal x(t) is
The time signal corresponding to (s + 1) X (s) is
1 e− s
X ( s) = −
s
− e−2 s , ROC : Re( s) 0
(a) cos2t − 2sin 2t u (t )
s
sin 2t
The signal x(t) is (b) cos 2t + u (t )
2
(a) u(–t) + u(–t + 1) + (t + 2)
(b) u(–t) – u(–t + 1) – (t + 2)
(c) cos2t + 2sin 2t u(t)
(c) u(–t) – u(–t + 1) – (t + 2) sin 2t
(d) cos 2t − u (t )
(d) –u(–t) + u(–t + 1) + (t + 2) 2
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Answer Key
1. (c) 16. (d)
2. (c) 17. (d)
3. (b) 18. (a)
4. (a) 19. (c)
5. (a) 20. (b)
6. (c) 21. (b)
7. (c) 22. (c)
8. (c) 23. (c)
9. (c) 24. (c)
10. (b) 25. (a)
11. (c) 26. (b)
12. (b) 27. (a)
13. (b) 28. (d)
14. (c) 29. (c)
15. (d) 30. (a)
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