LAPLACE TRANSFORMS

1. Find the value of L ( sin3 t )

3 3 3
1. 2 2. 2 3. 2 4. None of these 3
s +9 s −9 s +9

2. Find the value of L ( sinat ) =¿

a a a s
1. 2 4 2. 2 2 3. 2 4. 2
2 2
s +a s +a s +a s +a

3. Find the value of L ( sinkt ) =¿

k k k s
1. 2. 3. 4. 2
s + k4
2
s + k2
2 2
s +k s + k2
2

4. Find the value of L ( cosat )=¿

a a a s
1. 42 2. 2 2 3. 2 4. 2 2 4
s +a s +a s +a s +a

5. Find the value of L ( sinhat ) =¿

a a a s
1. 2. 3. 4. 2
s −a 4
2
s −a 2
2 2
s −a s −a 2
2

6. Find the value of L ( sinh7 t )=¿

7 7 7 s
1. 2 4 2. 2 2 3. 2 4. 22 2
s −7 s −7 s −7 s −7

7. Find the value of L ( e−kt )=¿

k 1 1 s
1. s +k 2. s−k 3. s +k 4. s +k 3

8. Find the value of L ( coshat )

a a a s
1. 2. 3. 2 4. 4
s −a 4
2 2
s −a 2
s −a s −a 2
2

9. Find the value of L ( e at )=¿

a 1 1 s
1. s +a 2. s−a 3. s +a 4. s +a 2
10. Find the value of L ( e−at ) =¿
a 1 1 s
1. s +a 2. s−a 3. s +a 4. s +a 3

11. . Find the value of L ( e 3 at )

3a 1 1 s
1. s +3 a 2. s +3 a 3. s−3 a 4. s +3 a 3

12. Find the value of L ( t n ) =¿

n! n! n n
1. n−1 2. n+1 3. n−1 4. n+1 2
s s s s

13. Find the value of L ( t 4 )

4! 4! 4 4
1. 2. 3. 4. 2
s 4−1 s 4 +1 s 4−1
s 4 +1

14.If L { F(t ) }=f ( s ) then L {e at f ( s) }=¿

1. f ¿ a) 2. f (s ) 3. f (s +a) 4. None of these 1

15.If L { F(t ) }=f ( s ) then L {e−at f (s) }=¿

1. f (s−a) 2. f (s ) 3. f (s +a) 4. None of these 3

16. If L { F(t ) }=f ( s ) then L { F(at) }=¿

1
1. a f ( a )
s 1
( s)
2. s f a 3. f ( a )
s
4. None of these 1

17.The Laplace transform of L ( t 5 e 3 t )=¿

120 120 120 120
1. 2. 3. 4. 3
( s+ 3 )6 ( s+ 5 )6 ( s−3 )6 ( s−5 )6

18. The Laplace transform of L ( t3 e−3 t ) =¿

6 6 6 6
1. ( s+3 )4 2. ( s+ 5 )4 3. ( s−3 ) 4 4. ( s−5 ) 4 1

19 The Laplace transform of L ( t 3 e 3 t )=¿

6 6 6 6
1. ( s+3 )4 2. ( s+ 5 )4 3. ( s−3 ) 4 4. ( s−5 ) 4 3
 {∫ }
t

20 If L { F(t ) }=f ( s ) then L F ( u ) du =¿

0

1 s f (s)
1. a f ( a ) 2. 3.f(s/a) 4. None of these 2
s

21 If L { F(t ) }=f ( s ) then L {( t n F (t )) }=¿ 1

n
n d n d
1.( −1 ) f ( s ) , n=1,2,3 … 2. (−1 ) ds f ( s ) , n=1,2,3 …
dsn
n
d
3. n f ( s ) , n=1,2,3 … 4. None of the above
ds

22. The Laplace transform of t . sinat 2

2as 2 as 2 as
1. 2. 2 2 3. 4.none of the above
s 2 + a2 ( s +a )
2
( s+ a )2

23. The Laplace transform of t coshat 3

s 2 +a2 2 ( s 2+ a2 ) s 2+ a2
1. 2 2 2. 2 3. 2 4.none of the above
s −a ( s2−a2 ) ( s 2−a 2 )

24. If L { F(t ) }=f ( s ) then L { } F(t )

t
=¿ 4

∞ ∞ s

1. ∫ f ( s ) ds 2.∫ f ( s ) ds 3. ∫ f ( s ) ds 4. None of the above

0 s 0

t
sinx
25. The Laplace transform of ∫ dx 2
0 x

tan −1 ( s ) cot−1 ( s )
1. 2. 3.cot−1 ( x ) 4. None of the above
s s
∞
sint
26.∫ dt=¿ 3
0 t

π π
1. 4 2. π 3. 2 4.2 π

∞ −6 t −3 t
e −e
27. ∫ dt=¿ 2
0 t
1. log 2 2 .log 2 () 1
3. log 2 () 3
4. None of the above

cosat
28.Laplace transform of t
is 4

1. cot
−1
( as ) 2. tan
−1
( as ) 3. tan−1 ( s ) 4. Does not exist

29.∫ t e−3 t sint dt=¿ 3

0

13 13 3
1. 15 2. 50 3. 50 4. None of the above

30. ∫ t 3 e−t sint dt=¿ 3

0

1. 1 2.−1 3.0 4. ∞

31.The Laplace transform of J o ( t ) is 1

1 s 1
1. 2. 3. 4. None of the above
√ s +1
2
√ s +12
√ s 2−1
32. The Laplace transform of J 1 ( t ) is 3
s s 1
1. 2. 1− 3. 1− 4. None of the above
√ s +1 2
√s +1
2
√ s 2+ 1
33. The Laplace transform of J o ( at ) is 1
1 1 a s
1. 2. 3. 4.
√s +a2 2
√ s +1 2
√s +a
2 2
√ s +a2
2

34. The Laplace transform of J 1 ( at ) is 1

1
1. a 1− [ s
√s +a
2 2 ] [ 1
2. a 1+ 2 2 3. a 1+ 2 2
√ s +a
s
√ s +a ] [ 1 1
] 4. ¿

x n
e d ( −x n )
35. If Ln x = n ! n e x then L ( Ln (t)) =¿
( ) 2
dx

( s+1 )n ( s−1 )n ( s−1 )n ( s+ 1 )n

1. n+1 2. n−1 3. n+1 4. n−1
s s s s

36. The Laplace transform of t sinhat 2

 2 2
2 as 2 as s +a 2
s +a
2
1. 2 2 2. 2 2 3. 2 2 4.
( s 2 +a ) ( s 2−a ) ( s 2−a ) 2
s −a
2

∞
sinx
37.If S ( t ) =∫ dx then L { S (t ) }= 2
t x

tan −1 ( s ) cot−1 ( s ) 1
1. 2. 3. s tans 4. None of the above
s s
∞
cosx
38. .If C ( t )=∫ dx then L { C ( t ) }=¿ 2
t x

1 1 1
1. 2 log ( s +1 ) 2. log ( s + 1 ) 4. None of the above
2 2
3.
2 s log ( s +1 ) 2 s
2

∞ −x
e
39. .If E ( t )=∫ dx then L { E ( t ) } =¿ 3
t x

1 1 1 1
1. 2 s log ( s+ 1 ) 2. 2 s log ( s−1 ) 3. s log ( s +1 ) 4. s log ( s−1 )

40. Find the value of L ( e 6 t )=¿ 2

1 1 s s
1. s +6 2. s−6 3. s +6 4. s−6

41. Find the value of L ( t 4 )=¿ 2

12 24 12 24
1. 2. 3. 4.
s4 s5 s5 s4

42. Find the value of L {t } = 1

n

n! n! n!
1. 2. 3. n 4. None
S n+1 S n−1 S
1 1
43.Find the value of L {1} = 41. 2 2.-1/2 3. s 4. s

44. Find the value of L { 2 } = 3

−2 2
1.s 2. s 3. s

45. Find the value of L { 100 } = 4

3 1 10 100
1. s 2. s 3. s 4. s
46. Find the value of L {t 2 } = 1
2! 3! 1! 0!
1. 3 2. 3 3. 3 4. 3
S S S S

47. Find the value of L {t 5 } = 2

42 120 24 24
1. 5 2. 6 3. 4.
S S S5 S4

48. Find the value of L {t }

7
= 4
6! 8! 5! 7!
1. 8 2. 6 3. 8 4. 8
S S S S

49. Find the value of L {t } =

0
3
1 −1 1
1. 2. 3. s 4.
2 s s

50. Find the value of L {e at } = 2

1 1 −1 −1
1. s +a 2.
s−a
3. s−a 4. s +a

51. Find the value of L {e−at } = 1

1 1 −1 −1
1. s +a 2.
s−a
3. s−a 4. s +a

52. Find the value of L {e 2 t } = 3

−1 −1 1 1
1. s +2 2. s−2 3.
s−2
4. s +2

53. Find the value of L {e−4 t} = 4

−1 1 −1 1
1. s +4 2.
s−4
3. s−4 4. s +4

54. Find the value of L {sin at } = 2

−a a s −s
1. 2 2
2. 2 2 3. 4.
s +a s +a s + a2
2
s 2 + a2

55. Find the value of L {cos at } = 3

 −a a s −s
1. 2. 3. 4.
s2 +a 2 s −a 2
2
s + a2
2
s 2+ a2

56. Find the value of L {sin 2 t } = 4

−2 s 4 2
1. 2 2. 2 3. 2 4. 2
s +4 s +4 s +4 s +4

57. Find the value of L {cos 3 t } = 2

9 s −s s
1. 2 2. 2 3. 4.
s +9 s +9 s 2+ 9 2
s −9

58 Find the value of L {sinh at } = 3

a s a s
1. 22 2. 2
2
3. 2 2 4. 22
s +a s −a s −a s +a

59. Find the value of L {cosh at } = 2

a s a s
1. 2 2
2. 2 2 3. 4.
s +a s −a s −a 2
2
s + a2
2

60. Find the value of L {t e } =

n at
3
n! ( n+1 ) ! n!
1. n+1 2. n+1 3. n+1 4. None
(s +a) ( s−a) ( s−a)