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EE/EC Batch : Hinglish

Signal and System Mega DPP – 03

Topic : Laplace Transform

1. The ROC of LT of the continuous time signal 4. Let x(t) be a CT signal and its Laplace transform is
x(t ) = 3e−2t u (t ) + 4et u (−t ) is s( s + 5)
X (s) = 2
( s + 16)
(a) –1 < Re(s) < 2 (b) –2 < Re(s) < 1
If x(t) = 5 cos 4t – 4 sin 4t + x3 (t), then x3(t) will be
(c) 1 < Re(s) < 2 (d) 0 < Re(s) < 2 (a) an impulse function
(b) a step function
(c) a ramp function
2. A CT signal is given as x(t ) =  P + Qe−bt  u (t )
  (d) a sinusoidal function
where P and Q are constant. The ROC of Laplace
5. Which of the following pole-zero plot corresponds
transform of x(t) is
to an even function of time?
(a) Re(s) > max (– b, 0)
(b) Re(s) > max (b, 0)
(c) 0 < Re(s) < b
(a)
(d) Re(s) > min (–b, 0)

3. The pole zero diagram of a continuous time system

is shown in the figure

(b)

(c)
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 (d)
R. –1 < Re(s) < 1 3. Unstable and anti-causal
S. Re(s) > 1 4. Stable and non-causal
Codes: d  −2t
6. The Laplace transform of et e u (−t )  is
P Q R S dt  
(a) 1 4 2 3 1− s 1− s
(a) , Re( s )  −1 (b) , Re( s )  −1
(b) 2 4 1 3 s +1 s +1
s −1 s −1
(c) 3 1 4 2 (c) , Re( s )  −1 (d) , Re( s )  −1
s +1 s +1
(d) 4 1 3 2
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7. The Laplace transform of signal t

d −t
dt
 
e cos t u ( t ) is
11. In the figure below, one cycle of a sinusoid for 0  t
  is shown. Let the Laplace transform of x(t) is

(a)
−( s 2 + 4s + 2)
(b)
( s 2 + 4 s + 2)
X (s) =
( )
10 1 − e − ks
( s + 2s + 2) ( s + 2 s + 2)
( s2 + 4)
2 2 2 2

( s 2 + 2 s + 2) −( s 2 + 2s + 2)
(c) (d) What is the value of k in the expression?
( s 2 + 4 s + 2) 2 ( s 2 + 4s + 2) 2
Common Data for Q.No.-8 and 9
Consider the one-sided Laplace transform pair given
below.
2s
x ( t ) ⎯→
L
s +2 2

Determine the Laplace transform Y(s) of the given

time signal in question and choose correct option.
12. The impulse response h(t) of the system H(s) of an
dx(t )
8. The Laplace transform of x(t ) * is LTI system:
dt
1. When the input to the system is x(t) = e2t and
3
4s 4 1
(a) (b) the output is e 2t .
( s2 + 2) ( s2 + 2)
2 2
6
2. When h(t) satisfies the differential equation,
−4s3 4 dh(t )
(c) (d) + 2h(t ) = e −4t u (t ) − bu (t ) where ‘b’ is an
(s ) ( s2 + 2)
2 2
2
+2 dt
unknown constant. Value of ‘b’ is ______.
9. The Laplace transform of 2tx(t) is
2e+5 s
8 − 4s 2
4s − 8
2 13. If H ( s ) = ,   −1 then system is
(a) (b) ( s + 3)( s + 2)( s + 1)
( s2 + 2) ( s2 + 2)
2 2
(a) stable, causal
(b) unstable, causal
4s 2 s2
(c) (d) 2 (c) stable, non-causal
s2 + 1 s +1
(d) unstable, non-causal
10. A differential equation for a LTI system with
specified input and initial conditions is
14. If H(s) is transfer function of a system
d 3 y (t ) d 2 y (t ) dy (t )
+4 +3 = x (t ) s 2 + 2s + 1
dt 3 dt 2 dt H (s) =
( s − 1)( s + 1)
All initial condition are zero, x ( t ) = 10e −2t The
Then the rms value of output of the system when
system output is input is 2 cos (1000t + 70°) is ________.
5 −t −2t 5 −3t 
(a)  3 + 5e − 5e + 3 e  u (t )
15. Consider a signal y(t) which is related to two signals
5 −t −2t 5 −3t  x1(t) and x2(t) by
(b)  3 − 5e + 5e − 3 e  u (t ) y(t ) = x1(t − 2) * x2 (−t + 3)

(c)
5 5
u (t ) − 5u (t − 1) + 5u (t − 2) + u (t − 3) Where, x1(t ) = e−2t u (t ) and x2 (t ) = e−3t u (t ) then
3 3 y(t) at t = 5 is _______.
5 5
(d) u (t ) + 5u (t − 1) − 5u (t − 2) + u (t − 3)
3 3
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16. The unilateral Laplace transform of a signal x(t) is 19. A system is described by the differential equation
2s 2 + 11s + 16 + e −2 s d2y dy
X (s) = +5 + 6 y(t ) = x(t )
(s 2
+ 5s + 6 ) dt 2 dt
Let x(t) be a rectangular pulse given by
The time signal x(t) is
1 0  t  2
(a) 2(t ) + 3e−2t − 2e−3t  u (t − 2) x(t ) = 
  0 otherwise
(b) 2(t ) +  2e−2t − 2e−3t + e−2(t −2) + e−3(t −2)  u (t ) Assuming that y ( 0 ) = 0,
dy
= 0 at t = 0, the Laplace
 
dt
2(t ) +  2e −2t − e −3t  u (t ) transform of y(t) is
 
(c)
+ e −2t − e −3t  u ( t − 2 ) e−2 s
  (a)
s ( s + 2 )( s + 3)
2(t ) +  2e−2t − e −3t  u (t )
  1 − e−2 s
(d) (b)
+ e−2(t −2) − e−3(t −2)  u (t − 2) s ( s + 2 )( s + 3)
 
17. The Laplace transform of the waveform shown in e−2 s
(c)
the figure is ( s + 2 )( s + 3)
1 − e−2 s
(d)
( s + 2 )( s + 3)
K
20. If sF ( s) = then, lim f (t ) is given by
( s + 1)( s 2 + 4) t →

K
(a) (b) Zero
1  4
2
1 + Ae− s + Be−4s + Ce−6s + De−8s 
s  (c) Infinite (d) Undefined
What is the value of D?  27 s + 97 
21. If  2  is the Laplace transform of f(t), then
(a) – 0.5 (b) – 1.5  s + 33s 
(c) 0.5 (d) 2.0 ( )
f 0+ is
18. The impulse response of the system shown in figure
(a) Zero (b) 97/33
is
(c) 27 (d) Infinity
22. The transfer function H(s) of a stable system is
s 2 + 5s − 9
H (s) =
( s + 1) ( s 2 − 2s + 10 )
The impulse response is
1
(a) u (t ) − (t − ) (a) −e−t u (t ) + et sin 3t + 2et cos3t  u (t )
  
1
u (t ) − u (t − ) (b) −e−t u (t ) − et sin 3t + 2et cos3t  u ( −t )
(b)
  

1 (c) −e−t u (t ) − et sin 3t + 2et cos3t  u (t )

(c) u (t − )  

(d) −e−t u (t ) + et sin 3t + 2et cos3t  u ( −t )
1  
 (  ) −  ( t −  )

(d)
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23. A causal LTI system with impulse response h(t) 25. Given that h(t ) = 10e−10t u (t ) , and e(t) = sin (10t)u(t),
satisfies the following properties:
the Laplace transform of the signal
1. When the input of the system is x(t ) = e2t then t
f (t ) =  h(t − )e()d  is given by
1
output is y (t ) = e2t . 0
6
10
(a)
2. h(t) satisfies the differential equation ( s + 10)( s 2 + 100)
d
h(t ) + 2h(t ) = e −4t u (t ) + bu (t ) 10( s + 10)
dt (b)
( s 2 + 100)
Determine the impulse response of the system.
100
1 (c)
(a) 1 + e −4t  u ( t ) ( s + 10)( s 2 + 100)
2  
1
1 (d)
(b) 1 − e −4t − 2e −2t  u ( t ) ( s + 10)( s 2 + 100)
2 
26. Let x(t) be the input and y(t) be the output of a
1
1 − e −4t  u ( t )
continuous time LTI system described by following
(c)
2  differential equation,
1 d 2 y (t ) 9 y(t ) dx(t )
(d) 1 − e −4t + 2e −2t  u ( t ) + + 2 y(t ) = 5 + 2 x(t )
2  dt 2 dt dt

1 The transfer function of the system is

24. If impulse response is given by h(t ) = . Which 1
t +1 2 2
(s )
(a) (b)
of the following sketch is correct for output ( s + 9) 2
+ 9s + 2
response y (t ) = h(t )  u (t ) .
5s 2 ( 5s + 2 )
(s )
(c) (d)
( 9s + 2 ) 2
+ 9s + 2
(a)
27. Let x(t) be a CT signal and X(s) be its Laplace
transform given as
5
X (s) =
(s 2
+ 3s − 4 )
(b) Match inverse Laplace transform L1, L2, L3 with
Region of convergence R1, R2, R3.
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 ) R 2 : Re( s )  −4
L3 : x(t ) = e −4t u (−t ) − et u (−t ) R3 : −4  Re( s )  1
(a) ( L1, R3 ) , ( L2 , R2 ) , ( L3, R1 )
(b) ( L1, R3 ) , ( L2 , R1 ) , ( L3, R2 )
(d) (c) ( L1, R1 ) , ( L2 , R2 ) , ( L3, R3 )
(d) ( L1, R2 ) , ( L2 , R1 ) , ( L3, R3 )
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 31. A stable linear time invariant (LTI) system has a

28. Given the Laplace transform, V ( s) =  e− st v(t )dt .
0 1
transfer function H (s) = 2 . To make this
The inverse transform v(t) is s + s −6
+ j system causal it needs to be cascaded with another
(a)  V ( s)ds LTI system having a transfer function H1(s). A
− j
correct choice for H1(s) among the following option
+ j is
1
(b) 
2j − j
e stV ( s )ds (a) s + 3 (b) s – 2
(c) s – 6 (d) s + 1

1 32. Input x(t) and output y(t) of an LTI system are related
(c) 
2j 0
e stV ( s )ds
by the differential equation y(t) - y(t) – 6y(t) = x(t).
+ j If the system is neither causal not stable, the impulse
1
e− stV ( s )ds
2j −j
(d) response h(t) of the system is
1 3t 1
(a) e u ( −t ) + e −2t u ( −t )
29. A system is described by the following differential 5 5
equation, where u(t) is the output to the system and 1 1
(b) − e3t u ( −t ) + e −2t u ( −t )
y(t) is the output of the system, 5 5
y '(t ) + 5 y (t ) = u (t ) 1 3t 1
(c) e u ( −t ) − e−2t u ( t )
When y(0) = 1 and u(t) is a unit step function, y(t) is 5 5
(a) 0.2 + 0.8e−5t (d)
1 1
− e3t u ( −t ) − e −2t u ( t )
5 5
(b) 0.2 − 0.2e−5t
33. The integrator given by
(c) 0.8 + 0.2e−5t t
y (t ) =  x () d
(d) 0.8 − 0.8e5t
−
1
30. The unilateral Laplace transform of f(t) is (a) has no finite singularities in its double sided
s + s +1 2
Laplace transform H(s).
Which one of the following is the unilateral Laplace
(b) produces a bounded output for every causal
transform of g(t) = t. f(t)?
bounded input.
−s −(2 s + 1)
(a) (b) (c) produces a bounded output for every anti
( s + s + 1)
(s )
2 2 2
2
+ s +1 causal bounded input.
(d) has no finite zeros in its double sided Laplace
s 2s + 1
(c) (d) transform H(s).
( s2 + s + 1) ( s2 + s + 1)
2 2
t
34. The Laplace transform of f ( t ) = 2 is s −3/2 .

The Laplace transform of g ( t ) = 1/ t is
3s −5/2
(a) (b) s −1/2
2
(c) s1/2 (d) s3/2
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Answer Key
1. (b) 18. (b)
2. (a) 19. (b)
3. (c) 20. (d)
4. (a) 21. (c)
5. (c) 22. (b)
6. (a) 23. (c)
7. (a) 24. (b)
8. (a) 25. (c)
9. (b) 26. (d)
10. (b) 27. (b)
11. (3.14) 28. (b)
12. (-1) 29. (a)
13. (c) 30. (d)
14. (1.414) 31. (b)
15. (0.2) 32. (b)
16. (d) 33. (d)
17. (a) 34. (b)

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