The Copperbelt University
School of Mathematics and Natural Science
Mathematics Department
MA310 - Engineering Mathematics II
2022 - 2023 Academic Year
30 th March 2023
Total Marks: 60
Test 1
Duration: 2 Hours
INSTRUCTIONS
1. Clearly write your SIN and Program.
2. Read all questions carefully before attempting.
3. There are Three (3) questions in this paper, answer all.
4. Distribution of marks is shown for all the parts of the questions.
5. Show all your working. No marks will be awarded for the final answer only.
6. This is a closed book examination. No course material is allowed in the examination
room.
7. If two or more different solutions are provided for a given question, no marks shall
be awarded.
8. No Integral Tables, Calculators or Formulas should be Provided in this examination.
Page 1 of 4
QUESTION ONE
a. Use the definition of Laplace Transform to obtain:
4 sin 2t
[5 Marks]
b. Find the following Laplace transforms
i. Lt 3 cosh 4t
[5 Marks]
ii. L t 
t sin 

0
d
[5 Marks]
iii.
t
L  2 sin 2 cos 7d .
0
[5 Marks]
[TOTAL  20 MARKS]
Page 2 of 4
QUESTION TWO
a. Find the Laplace transform of the periodic function
[5 Marks]
b. Solve the differential eqaution


y t  2 y t  yt  4 cos 2t

subject to the initial conditions y  0, y  2 at t  0.
[5 Marks]
c. Find the following inverse Laplace transform:
L 1
27  12s

s 4s 2  9
[5 Marks]
d. Using Convolution theorem find the following inverse Laplace transforms:
L 1
1
s  1 2 s  2 3
[5 Marks]
[TOTAL  20 MARKS]
Page 3 of 4
QUESTION THREE
a. Find ft if Fs is defined as
Fs 
21  e 3s 
31  e 3s 

s3
s2
[4 marks]
b. Find the following inverse Laplace transform
L 1 ln
ss 2  a 2 
s 2  b 2 
[5 marks]
c. Obtain a solution to the system of differential equations, using Laplace transform
method

x

y
 2x  y
 4x  3y
subject x0  1 and y0  2.
[6 Marks]
d. Show that

0
cos 2t  cos 14t dt  ln 7
t
[5 Marks]
[TOTAL  20 Marks]
© THE COPPERBELT UNIVERSITY 2023
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