MAT3700 EXAM PACK 2022

MAT3700
EXAM PACK




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,UNIVERSITY EXAMINATIONS




September-December 2021

MAT3700

Mathematics III (Engineering)

Duration: 3 hours Marks: 100
Examiners:
First: Ms LE Greyling
Second: Mr S Blose
External: Dr JN Mwambakana
Use of a non-programmable pocket calculator is permissible.

This is a closed book examination and will be IRIS invigilated.
This online paper is the property of UNISA and may not be distributed electronically.

This examination question paper consists of 3 pages including this cover page plus
Formulae sheets (pages 4 to 8) plus
A table of integrals (pages 9 and 10) plus
A table of Laplace transforms (page 11).

Examination rules:
1. Students must upload their answer scripts in a single PDF file (answer scripts must not be password
protected or uploaded as “read only” files).
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5. Incorrect file format and uncollated answer scripts will not be considered.
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and no opportunity will be granted for resubmission.
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will be granted.
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resubmission will be granted.
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Submissions made after the official examination time will be rejected by the examination regulations and
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, -2- MAT3700
September-December 2020

QUESTION 1
Solve the following differential equations:

1.1 1  x  dy
2

dx
 1  y  . 2
(4)

1.2 x 2
 
 y 2 dx  x 2  xy dy  0 Hint: Let y  vx  . (7)
dy
1.3  y cot x  cos x. (7)
dx
[18]

QUESTION 2
Find the general solution of the following differential equation using the method of
d 2y dy
undetermined coefficients: 3 2  2  y  2x  3 . (10)
dx dx
[10]

QUESTION 3
Find the general solution of the following differential equations using D-operator methods:

3.1 D 2

 3D  2 y  sin3 x . (8)

3.2 D 2
 6D  9  y  e 2 x
cosh 2 x . (6)
[14]

QUESTION 4
Solve for x and y by using D-operator methods in the following set of simultaneous
equations:
 D  1 y  x  4et
. (10)
y   D  3  x  1
[10]

QUESTION 5
Determine the following:

5.1 
L et cos 2t .  (2)
 8se 2s 
5.2 L1  2 . (4)
 s  9 
[6]
[TURN OVER]

, -3- MAT3700
September-December 2020

QUESTION 6
Given y " y  sin t
Use Laplace transforms to solve the equation if the initial values for the equation are
y  0   1 and y '  0   0 . (8)
[8]


QUESTION 7
The equation of motion of a system is
d 2x dx
2
5  4 x  3  t  2 
dt dt

If x  0   2 and x '  0   2 find an expression for the displacement x in terms of t.
(12)
[12]


QUESTION 8
2
The period, T, of natural vibrations of a building is given by T  where  is an

 2 1 
eigenvalue of matrix A . Find the period(s) if A   . (5)
 1 2 
[5]

QUESTION 9

Find all the eigenvalues of matrix A and an eigenvector corresponding to .
2 0 0
A   4 1 0  . (7)
 1 2 1
[7]

QUESTION 10

A function f(x) is defined by
 2 x 0
f x   .
 0x2
Determine the Fourier series expansion of the periodic function f(x) with period 4.
(10)
[10]

Full marks = 100
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