Revision Test 18
This Revision Test covers the material contained in Chapters 61 to 65. The marks for each question are shown in
brackets at the end of each question.
1. Find the Laplace transforms of the following
functions:
(a) 2t 3 − 4t + 5 (b) 3e −2t − 4 sin2t
(c) 3 cosh 2t
(d) 2t 4 e −3t
(e) 5e 2t cos 3t
(f) 2e 3t sinh 4t
(16)
2. Find the inverse Laplace transforms of the following functions:
(a)
5
2s + 1
(b)
12
s 5
(c)
4s
s 2 + 9
(d)
5
s 2 − 9
(e)
3
(s + 2) 4
(f)
s − 4
s 2 − 8s − 20
(g)
8
s 2 − 4s + 3
(17)
3. Use partial fractions to determine the following:
(a) L
−1
5s − 1
s 2 − s − 2
(b) L
−1
2s
2
+ 11s − 9
s(s − 1)(s + 3)
(c) L
−1
13 − s 2
s(s 2 + 4s + 13)
(24)
4. In a galvanometer the deflection θ satisfies the
differential equation:
d 2 θ
dt 2 + 2
dθ
dt
+ θ = 4
Use Laplace transforms to solve the equation for θ
given that when t = 0, θ = 0 and
dθ
dt
= 0. (13)
5. Solve the following pair of simultaneous differential equations:
3
dx
dt
= 3x + 2y
2
d y
dt
+ 3x = 6y
given that when t = 0, x = 1 and y = 3.
(20)
6. Determine the poles and zeros for the transfer function: F(s) =
(s + 2)(s − 3)
(s + 3)(s 2 + 2s + 5)
and plot them on
a pole-zero diagram.
(10)