Engineering Mathematics Multiple Choice Questions on “Existence and Laplace Transform of Elementary Functions – 1”.
1. If f(t) = 1, then its Laplace Transform is given by?
a) s
b) 1⁄s
c) 1
d) Does not exist
Answer: b
Explanation: The Laplace Transform of a functions is given by
(L{f(t)}=F(s)=int_0^{infty}f(t)e^{-st}dt)
put f(t) = 1
On simplifying, we get 1⁄s.
2. If f(t) = tn where, ‘n’ is an integer greater than zero, then its Laplace Transform is given by?
a) n!
b) tn+1
c) n! ⁄ sn+1
d) Does not exist
Answer: c
Explanation:The Laplace Transform of a functions is given by
(L{f(t)}=F(s)=int_0^{infty}f(t)e^{-st}dt)
f(t) = tn
On simplifying, we get n! ⁄ sn+1.
3. If f(t)=√t, then its Laplace Transform is given by?
a) 1⁄2
b) 1⁄s
c) √π ⁄ 2√s
d) Does not exist
Answer: c
Explanation:The Laplace Transform of a functions is given by
(L{f(t)}=F(s)=int_0^{infty}f(t)e^{-st}dt)
Put f(t)=√t
On Solving, we get √π ⁄ 2√s.
4. If f(t) = sin(at), then its Laplace Transform is given by?
a) cos(at)
b) 1 ⁄ asin(at)
c) Indeterminate
d) a ⁄ s2+a2
Answer: d
Explanation:The Laplace Transform of a functions is given by
(L{f(t)}=F(s)=int_0^{infty}f(t)e^{-st}dt)
Put f(t) = sin(at)
On solving, we get a ⁄ s2+a2.
5. If f(t) = tsin(at) then its Laplace Transform is given by?
a) 2as ⁄ (s2+a2)2
b) a ⁄ s2+a2
c) Indeterminate
d) √π ⁄ 2√s
Answer: a
Explanation:The Laplace Transform of a functions is given by
(L{f(t)}=F(s)=int_0^{infty}f(t)e^{-st}dt)
Put f(t) = tsin(at)
On Solving, we get 2as ⁄ (s2+a2)2.
6. If f(t) = eat, its Laplace Transform is given by?
a) a ⁄ s2+a2
b) √π ⁄ 2√s
c) 1 ⁄ s-a
d) Does not exist
Answer: c
Explanation: The Laplace Transform of a functions is given by
(L{f(t)}=F(s)=int_0^{infty}f(t)e^{-st}dt)
Put f(t) = eat
On solving the above integral, we obtain 1 ⁄ s-a.
7. If f(t) = tp where p > – 1, its Laplace Transform is given by?
a) √π ⁄ 2√s
b) f(t) = tsin(at)
c) γ(p+1) ⁄ sp+1
d) Does not exist
Answer: d
Explanation:The Laplace Transform of a functions is given by
(L{f(t)}=F(s)=int_0^{infty}f(t)e^{-st}dt)
Put f(t) = tp
On Solving, we get γ(p+1) ⁄ sp+1.
8. If f(t) = cos(at), its Laplace transform is given by?
a) s ⁄ s2+a2
b) a ⁄ s2+a2
c) √π ⁄ 2√s
d) Does not exist
Answer: a
Explanation:The Laplace Transform of a functions is given by
(L{f(t)}=F(s)=int_0^{infty}f(t)e^{-st}dt)
Put f(t) = cos(at)
On solving the above integral, we get s ⁄ s2+a2.
9. If f(t) = tcos(at), its Laplace transform is given by?
a) 1 ⁄ s-a
b) s2 – a2 ⁄ (s2+a2)2
c) Indeterminate
d) s2at
Answer: b
Explanation:The Laplace Transform of a functions is given by
(L{f(t)}=F(s)=int_0^{infty}f(t)e^{-st}dt)
Put f(t) = tcos(at)
On solving the above integral, using suitable rules of integration we get the answer s2 – a2 ⁄ (s2+a2)2.
10. If f(t) = sin(at) – atcos(at), then its Laplace transform is given by?
a) Indeterminate form is encountered
b) a3 ⁄ (s2 + a2)2
c) 2a3 ⁄ (s2 – a2)2
d) 2a3 ⁄ (s2 + a2)2
Answer: d
Explanation:The Laplace Transform of a functions is given by
(L{f(t)}=F(s)=int_0^{infty}f(t)e^{-st}dt)
Put f(t) = sin(at) – atcos(at)
On solving the above integral, we obtain the answer2 a3 ⁄ (s2 + a2)2.
11. If f(t) = sin(at) – atcos(at), then its Laplace transform is given by?
a) (frac{s(s^2-a^2)}{(s^2+a^2)^2})
b) (frac{s(s^2-3a^2)}{(s^2+a^2)^2})
c) Indeterminate
d) (frac{2as^2}{(s^2+a^2)^2})
Answer: d
Explanation:The Laplace Transform of a functions is given by
(L{f(t)}=F(s)=int_0^{infty}f(t)e^{-st}dt)
Put f(t) = sin(at) – atcos(at)
On solving, we obtain 2as2 ⁄ (s2+a2)2
12. If f(t) = cos(at) – atsin(at), then its Laplace transform is given by?
a) sinat2
b) (frac{s(s^2-a^2)}{(s^2+a^2)^2})
c) (frac{Gamma(p+1)}{s^{p+1}})
d) Does not exist
Answer: b
Explanation:The Laplace Transform of a functions is given by
(L{f(t)}=F(s)=int_0^{infty}f(t)e^{-st}dt)
Put f(t) = cos(at) – atsin(at)
On solving, we obtain a3 ⁄ (s2 + a2)2.
13. If f(t) = cos(at) + atsin(at), its Laplace transform is given by?
a) (frac{s+a}{s-a})
b) (frac{a^3}{(s^2+a^2)^2})
c) (frac{s(s^2+3a^2)}{(s^2+a^2)^2})
d) Does not exist
Answer: c
Explanation:The Laplace Transform of a functions is given by
(L{f(t)}=F(s)=int_0^{infty}f(t)e^{-st}dt)
Put f(t) = cos(at) + atsin(at) to solve the problem.
14. If f(t) = sin(at + b), its Laplace transform is given by?
a) Indeterminate
b) (frac{(s)sin(b)+acos(b)}{s^2+a^2})
c) (frac{s^2-a^2}{(s-a)^2})
d) (frac{2a^3}{(s^2+a^2)})
Answer: b
Explanation:The Laplace Transform of a functions is given by
(L{f(t)}=F(s)=int_0^{infty}f(t)e^{-st}dt)
Put f(t) = sin(at + b) to solve the problem.
15. If f(t) = cos(at + b), its Laplace transform is given by?
a) (frac{a}{s^2+a^2})
b) (frac{2as}{(s^2+a^2)^2})
c) (frac{scos(b)-asin(b)}{s^2+a^2})
d) Does not exist
Answer: c
Explanation:The Laplace Transform of a functions is given by
(L{f(t)}=F(s)=int_0^{infty}f(t)e^{-st}dt)
Put f(t) = cos(at + b) to solve the problem.