Engineering Mathematics Aptitude Test

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 ⁄ s²-a²
c) a ⁄ s²-a²
d) Exists only if ‘t’ is complex
View Answer

Answer: c
Explanation: The Laplace transform of a function is given by
\(\{f(t)\}=F(s)=\int_0^{\infty} f(t)e^{-st}dt\)
put f(t) = sinhat
On solving, a ⁄ s²-a² is obtained.

2. If f(t) = coshat, its Laplace transform is given by?
a) s ⁄ s²-a²
b) s+a ⁄ s-a
c) Indeterminate
d) (sinh(at))²
View Answer

Answer: a
Explanation: The Laplace transform of a function is given by
\(\{f(t)\}=F(s)=\int_0^{\infty} f(t)e^{-st}dt\)
put f(t) = coshat
On solving, s ⁄ s²-a² is obtained.

3. If f(t) = e^at sin(bt), then its Laplace transform is given by?
a) s²-a² ⁄ (s – a)²
b) b ⁄ (s + a)² + b²
c) b ⁄ (s – a)² + b²
d) Indeterminate
View Answer

Answer: c
Explanation: The Laplace transform of a function is given by
\(\{f(t)\}=F(s)=\int_0^{\infty} f(t)e^{-st}dt\)
put f(t) = e^atsin(bt)
On solving, we get the b ⁄ (s – a)² + b².

4. If f(t) = e^at cos(bt), then its Laplace transform is?
a) 2a³ ⁄ (s² + a²)
b) s+a ⁄ s-a
c) Indeterminate
d) s-a ⁄ (s – a)² + b²
View Answer

Answer: d
Explanation: The Laplace transform of a function is given by
\(\{f(t)\}=F(s)=\int_0^{\infty} f(t)e^{-st}dt\)
put f(t) = e^atcos(bt)
Solve the above integral, to obtain s-a ⁄ (s – a)² + b².

5. If f(t) = e^at sinh(bt) then its Laplace transform is?
a) e^-as ⁄ s
b) s+a ⁄ (s – a)² + b²
c) b ⁄ (s – a)² – b²
d) Does not exist
View Answer

Answer: c
Explanation: The Laplace transform of a function is given by
\(\{f(t)\}=F(s)=\int_0^{\infty} f(t)e^{-st}dt\)
put f(t) = e^atsinh(bt)
On solving, we get the b ⁄ (s – a)² – b².

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6. If f(t) = ¹⁄_a sinh(at), then its Laplace transform is?
a) 1⁄s²-a²
b) 2a ⁄ (s – b)² + b²
c) n! ⁄ (s – a)^n-1
d) Does not exist
View Answer

Answer: a
Explanation: The Laplace transform of a function is given by
\(\{f(t)\}=F(s)=\int_0^{\infty} f(t)e^{-st}dt\)
put f(t) = f(t) = 1⁄a sinh(at)
On solving the above integral, we get the 1⁄s²-a².

7. If f(t) = tⁿ ⁄ n, then its Laplace transform is?
a) \(\frac{s+a}{(s-a)}{(s-a)^2+b^2}\)
b) \(\frac{b^2}{(s-a)}{(s-a)^2+b^2}\)
c) \(\frac{2a^3}{(s^2+a^2)}\)
d) \(\frac{(n-1)!}{s^{n+1}}\)
View Answer

Answer: d
Explanation: The Laplace transform of a function is given by
\(\{f(t)\}=F(s)=\int_0^{\infty} f(t)e^{-st}dt\)
put f(t) = tⁿ ⁄ n
On solving, we obtain the Laplace transform of the required function.

8. If f(t) = ¹ ⁄ _√Πt, then its Laplace transform is?
a) \(\frac{s^2-a^2}{(s-a)^2}\)
b) S^-1/2
c) \(\frac{n!}{(s-a)^{n-1}}\)
d) \(\frac{n!}{(s-a)^{n-1}}\)
View Answer

Answer: b
Explanation: The Laplace transform of a function is given by
\(\{f(t)\}=F(s)=\int_0^{\infty} f(t)e^{-st}dt\)
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 + b²
b) 2a ⁄ (s – b)² + b²
c) Indeterminate
d) s ⁄ (s² + a²)²
View Answer

Answer: d
Explanation: The Laplace transform of a function is given by
\(\{f(t)\}=F(s)=\int_0^{\infty} f(t)e^{-st}dt\)
put f(t) = t⁄2a sinat
Integrate to obtain, the required transform s ⁄ (s² + a²)².

10. If f(t) = δ(t), then its Laplace transform is?
a) s + a ⁄ (s – a)² + b²
b) a³ ⁄ (s² + a²)²
c) 1
d) Does not exist
View Answer

Answer: c
Explanation: The Laplace transform of a function is given by
\(\{f(t)\}=F(s)=\int_0^{\infty} f(t)e^{-st}dt\)
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?
a) \(\frac{1}{(s+a)^2}\)
b) \(\frac{2a}{(s-b)^2+b^2}\)
c) \(\frac{a^3}{s^2+a^2)^2}\)
d) Indeterminate
View Answer

Answer: a
Explanation: The Laplace transform of a function is given by
\(\{f(t)\}=F(s)=\int_0^{\infty} f(t)e^{-st}dt\)
put f(t) = te^-at
On solving, the required answer is obtained.

12. If f(t) = u(t), then its Laplace transform is?
a) \(\frac{scos(b)-asin(b)}{s^2+a^2}\)
b) 1/2
c) \(\frac{s}{s^2-a^2}\)
d) \(\frac{b}{(s-a)^2+b^2}\)
View Answer

Answer: b
Explanation: The Laplace transform of a function is given by
\(\{f(t)\}=F(s)=\int_0^{\infty} f(t)e^{-st}dt\)
put f(t) = u(t) to solve the problem.

13. f(t) = t, then its Laplace transform is?
a) \(\frac{(s)sin(b)+acos(b)}{s^2+a^2}\)
b) \(\frac{2as^2}{(s^2+a^2)^2}\)
c) \(\frac{\Gamma(p+1)}{s^{p+1}}\)
d) 1/s²
View Answer

Answer: d
Explanation: The Laplace transform of a function is given by
\(\{f(t)\}=F(s)=\int_0^{\infty} f(t)e^{-st}dt\)
put f(t) = t to solve the problem.

14. If f(t)=\(\frac{1}{b}e^{at}sinh(bt)\), then its Laplace transform is?
a) 1/s
b) Indeterminate
c) \(\frac{b}{(s-a)^2-b^2}\)
d) \(f(t)=\frac{1}{(s-a)^2-b^2}\)
View Answer

Answer: d
Explanation:The Laplace transform of a function is given by
\(\{f(t)\}=F(s)=\int_0^{\infty} f(t)e^{-st}dt\)
put f(t) = 1⁄b e^atsinh(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

Answer: b
Explanation: This is the Linearity property of Laplace transform.

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