Higher Engineering Mathematics, Sixth Edition

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 − 4 t+5 (b) 3e−^2 t−4sin2t
   (c) 3cosh2t (d) 2t^4 e−^3 t
   (e) 5e^2 tcos3t (f) 2e^3 tsinh4t (16)


2. Find the inverse Laplace transforms of the follow-
   ing functions:

(a)

5

2 s+ 1

(b)

12

s^5

(c)

4 s

s^2 + 9

(d)

5

s^2 − 9

(e)

3

(s+ 2 )^4

(f)

s− 4

s^2 − 8 s− 20

(g)

8

s^2 − 4 s+ 3

(17)

3. Use partial fractions to determine the following:

(a) L−^1

{

5 s− 1

s^2 −s− 2

}

(b) L−^1

{

2 s^2 + 11 s− 9

s(s− 1 )(s+ 3 )

}

(c) L−^1

{

13 −s^2

s(s^2 + 4 s+ 13 )

}

(24)

4. In a galvanometer the deflectionθ satisfies the
   differential equation:

d^2 θ

dt^2

+ 2

dθ

dt

+θ= 4

Use Laplace transforms to solve the equation forθ

given that whent=0,θ=0and

dθ

dt

=0. (13)

5. Solve the following pair of simultaneous differen-
   tial equations:

3

dx

dt

= 3 x+ 2 y

2

dy

dt

+ 3 x= 6 y

given that whent=0,x=1andy=3. (20)

6. Determine the poles and zeros for the transfer func-
   tion:F(s)=

(s+ 2 )(s− 3 )

(s+ 3 )(s^2 + 2 s+ 5 )

and plot them on

a pole-zero diagram. (10)