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) eat
b) s ⁄ s2-a2
c) a ⁄ s2-a2
d) Exists only if ‘t’ is complex
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
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) = eat 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
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) = eatsin(bt)
On solving, we get the b ⁄ (s – a)2 + b2.
4. If f(t) = eat 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
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) = eatcos(bt)
Solve the above integral, to obtain s-a ⁄ (s – a)2 + b2.
5. If f(t) = eat 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
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) = eatsinh(bt)
On solving, we get the b ⁄ (s – a)2 – b2.
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
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}})
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}})
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
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