This set of Engineering Mathematics Questions and Answers for Aptitude test focuses on “Existence and Laplace Transform of Elementary Functions – 2”.

1. If f(t) = sinhat, then its Laplace transform is

a) e^{at}

b) s ⁄ s^{2}-a^{2}

c) a ⁄ s^{2}-a^{2}

d) Exists only if ‘t’ is complex

View Answer

Explanation: The Laplace transform of a function is given by

put f(t) = sinhat

On solving, a ⁄ s

^{2}-a

^{2}is obtained.

2. If f(t) = coshat, its Laplace transform is given by

a) s ⁄ s^{2}-a^{2}

b) s+a ⁄ s-a

c) Indeterminate

d) (sinh(at))^{2}

View Answer

Explanation: The Laplace transform of a function is given by

put f(t) = coshat

On solving, s ⁄ s

^{2}-a

^{2}is obtained.

3. If f(t) = e^{at} sin(bt), then its Laplace transform is given by

a) s^{2}-a^{2} ⁄ (s – a)^{2}

b) b ⁄ (s + a)^{2} + b^{2}

c) b ⁄ (s – a)^{2} + b^{2}

d) Indeterminate

View Answer

Explanation: The Laplace transform of a function is given by

put f(t) = e

^{at}sin(bt)

On solving, we get the b ⁄ (s – a)

^{2}+ b

^{2}.

4. If f(t) = e^{at} cos(bt), then its Laplace transform is

a) 2a^{3} ⁄ (s^{2} + a^{2})

b) s+a ⁄ s-a

c) Indeterminate

d) s-a ⁄ (s – a)^{2} + b^{2}

View Answer

Explanation: The Laplace transform of a function is given by

put f(t) = e

^{at}cos(bt)

Solve the above integral, to obtain s-a ⁄ (s – a)

^{2}+ b

^{2}.

5. If f(t) = e^{at} sinh(bt) then its Laplace transform is

a) e^{-as} ⁄ s

b) s+a ⁄ (s – a)^{2} + b^{2}

c) b ⁄ (s – a)^{2} – b^{2}

d) Does not exist

View Answer

Explanation: The Laplace transform of a function is given by

put f(t) = e

^{at}sinh(bt)

On solving, we get the b ⁄ (s – a)

^{2}– b

^{2}.

6. If f(t) = ^{1}⁄_{a} sinh(at), then its Laplace transform is

a) 1⁄s^{2}-a^{2}

b) 2a ⁄ (s – b)^{2} + b^{2}

c) n! ⁄ (s – a)^{n-1}

d) Does not exist

View Answer

Explanation: The Laplace transform of a function is given by

put f(t) = f(t) = 1⁄a sinh(at)

On solving the above integral, we get the 1⁄s

^{2}-a

^{2}.

7. If f(t) = t^{n} ⁄ n, then its Laplace transform is

View Answer

Explanation: The Laplace transform of a function is given by

put f(t) = t

^{n}⁄ n

On solving, we obtain the Laplace transform of the required function.

8. If f(t) = ^{1} ⁄ _{√Πt}, then its Laplace transform is

View Answer

Explanation: The Laplace transform of a function is given by

put f(t) = 1 ⁄ √Πt

The solution for the above question is obtained by solving the above integral.

9. If f(t) = ^{t}⁄_{2} a sinat, then its Laplace transform is

a) b ⁄ (s + a)^{2} + b^{2}

b) 2a ⁄ (s – b)^{2} + b^{2}

c) Indeterminate

d) s ⁄ (s^{2} + a^{2})^{2}

View Answer

Explanation: The Laplace transform of a function is given by

put f(t) = t⁄2a sinat

Integrate to obtain, the required transform s ⁄ (s

^{2}+ a

^{2})

^{2}.

10. If f(t) = δ(t), then its Laplace transform is

a) s + a ⁄ (s – a)^{2} + b^{2}

b) a^{3} ⁄ (s^{2} + a^{2})^{2}

c) 1

d) Does not exist

View Answer

Explanation: The Laplace transform of a function is given by

put f(t) = δ(t)

Solve the above integral to obtain 1 as RHS.

11. If f(t) = te^{-at}, then its Laplace transform is

View Answer

Explanation: The Laplace transform of a function is given by

put f(t) = te

^{-at}

On solving, the required answer is obtained.

12. If f(t) = u(t), then its Laplace transform is

View Answer

Explanation: The Laplace transform of a function is given by

put f(t) = u(t) to solve the problem.

13. f(t) = t, then its Laplace transform is

View Answer

Explanation: The Laplace transform of a function is given by

put f(t) = t to solve the problem.

View Answer

Explanation:The Laplace transform of a function is given by

put f(t) = 1⁄b e

^{at}sinh(bt) to solve the problem.

15. If L { f(t) } = F(s), then L { kf(t) } =

a) F(s)

b) k F(s)

c) Does not exist

d) F(^{s}⁄_{k})

View Answer

Explanation: This is the Linearity property of Laplace transform.

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