# Engineering Mathematics Questions and Answers – Existence and Laplace Transform of Elementary Functions – 1

This set of Engineering Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Existence and Laplace Transform of Elementary Functions – 1”.

1. If f(t) = 1, then its Laplace Transform is given by?
a) s
b) 1s
c) 1
d) Does not exist
View Answer

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 1s.

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
View Answer

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) 12
b) 1s
c) √π ⁄ 2√s
d) Does not exist
View Answer

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.
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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
View Answer

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
View Answer

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.
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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
View Answer

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
View Answer

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.
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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
View Answer

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
View Answer

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.
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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
View Answer

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) $$\frac{2as^2}{(s^2+a^2)^2}$$
View Answer

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
View Answer

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
View Answer

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) $$\frac{2a^3}{(s^2+a^2)}$$
View Answer

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
View Answer

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.

Sanfoundry Global Education & Learning Series – Engineering Mathematics.

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