MA272: Engineering Mathematics IV
Faculty of Science, Technology & Environment
School of Computing, Information & Mathematical Sciences
Final Examination
Semester II, 2019
Mode: Face to Face
Duration of Exam: 3 hours + 10 minutes
Reading Time: 10 minutes
Writing Time: 3 hours
Total marks: 100
INSTRUCTIONS:
1. There are 8 questions and all are compulsory.
2. There are 7 pages in this exam paper (including this cover page).
3. Write your answers in the answer booklet provided.
4. Start each question on a new page.
5. Show all necessary working. Partial marks will be awarded for partially correct answers.
6. Only non-programmable calculators are allowed.
7. Formulas and statistical table are provided on pages 5-7.
8. This exam is worth 50% of the overall mark. The minimum exam mark is 40/100.
MA272 Final Exam, Semester 2, 2019
Question 1
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[4+3+4=11 marks]
(a) The time to failure of the cathode ray tube has the following probability density function
f (t )  et , t  0
where   0 is a constant known as failure rate. Find the probability that the tube lives at least 100
hours.
(4 marks)
(b) An electronic product contains 40 integrated circuits. The probability that any integrated circuit is
defective is 0.01, and the integrated circuits are independent. The product operates only if there are
no defective integrated circuits. What is the probability that the product operates?
(3 marks)
(c) The fill volume of an automated filling machine used for filling cans of carbonated beverage is
normally distributed with mean of 12.4 fluid ounces and a standard deviation of 0.1 fluid ounce. What
is the probability a fill volume is less than 12 fluid ounces?
Question 2
(a) Evaluate
Start on a new page
(4 marks)
[4+2+4=10 marks]
1  3i
and express your answer in rectangular form.
1 i
z2 1
(b) Evaluate lim
.
z i z  i
(c) Show that f ( z)  z 3 is analytic on its domain.
Question 3
Start on a new page
[4+3+7=14 marks]
(a) Let u  (1,1, 2) and v  (1, 3, 2). Find
(i) The length of u and v.
(ii) d (u, v).
(2+2=4 marks)
(b) Explain why the set ( x, y ), x  0, y  R with standard operations not a vector space.
1 3 2 
(c) Consider the matrix A  
 . Find
4 2 1
(i) A basis for the column space of A.
(ii) The rank of A.
(iii) The nullity of A.
(3+2+2=7 marks)
Page 2 of 7
MA272 Final Exam, Semester 2, 2019
Question 4
Start on a new page
[4+4+8= 16 marks]
(a) Find the standard matrix for T1 T2 and T2 T1 , where
T1 : R 2  R 2 , T1 ( x, y )  ( x  2 y, 2 x  3 y ).
T2 : R 2  R 2 , T2 ( x, y )  (2 x, x  y ).
(b) Determine whether the linear transformation given by
1 1 1 
A   2 1 3 
 2 1 1
is one-to-one, onto or neither.
(c) Express the second order differential equation
x ''(t )  2 x '(t )  2 x(t )  0
as a system of first order differential equations and find the solution for x(t ).
Question 5
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[12 marks]
A periodic function is defined by
1,    x  0
f ( x)  
and f ( x)  f ( x  2 ).
 1, 0  x  
(a) Sketch the graph of f ( x) for x  3 to x  3 .
(b) Find the Fourier series expansion of f ( x).
1 1 1

(c) Show that 1      .
3 5 7
4
(3+5+4=12 marks)
Question 6
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(a) Find the Fourier transform f (t )  e
t
[6+4+5= 15 marks]
and use the inverse Fourier transform formula to show that



dw
 .
1  w2
(b) Find the Laplace transform of f (t )  (t  3) H (t 1).
(c) Use convolutions and Laplace transforms to solve the integral equation
t
y (t )  t   y ( )sin(t   )d .
0
Page 3 of 7
MA272 Final Exam, Semester 2, 2019
Question 7
Start on a new page
[4+6= 10 marks]
(a) Consider the equation sin 2 x  cos x  0. Starting with x0  1.5, do two iteration of the Newton’s
method to approximate the root of the given equation.
(b) Consider the equation f ( x)  x  cos x  0.
(i) Show that f has a root in the interval [0.7, 0.8].
(ii) If the secant method is used on this function f with x0  0.7 and x1  0.8, what is the value of
x2 ?
(2+4=6 marks)
Question 8
(a)
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[6+6= 12 marks]
Use Runge-Kutta method of order 2 with h  0.1 to approximate the solution of initial value problem
(IVP):
y '  y  x, 1  x  1.2,
(b)
y(1)  2.
Consider the IVP
dy
 x 2  y 2 , y (1)  0.
dx
Use the Euler-trapezoidal method, with one correction at each time step to solve the IVP on the
interval [1, 2] with h  0.2.
________ END OF EXAM _______
Page 4 of 7
MA272 Final Exam, Semester 2, 2019
FORMULAE
b
1.
P(a  X  b)   f ( x)dx
2.
n
P( X  x)    p x q n  x
 x
3.
z
4.
Cauchy-Riemann Equation: ux  v y and u y  vx
5.
Y  eat C1  U cos bt  V sin bt   C2  U sin bt  V cos bt  
6.
Integrals:
a
x

kx
 e dx 
ekx
C
k
 sin kxdx  
 cos kxdx 
7.
cos kx
C
k
sin kx
C
k
Fourier Series:

f ( x)  a0    an cos nx  bn sin nx , where
n 1
8.

a0 
1
2
  f ( x)dx
an 
1
f ( x) cos nxdx
 

bn 
1




  f ( x) sin nxdx

Fourier transforms:

( f (t ))  fˆ ( w)   f (t )eiwt dt

9.
Inverse Fourier transforms:
f (t ) 
1
2



fˆ ( w)eiwt dw
10. Laplace transforms:

( f (t ))  F ( s)   e st f (t )dt
0
 f (t  c) H (t  c)   e  cs F ( s )
Page 5 of 7
MA272 Final Exam, Semester 2, 2019
11. Convolution theorem:
( f  g )(t ) ,
F ( s )G ( s ) 
where
t
( f  g )(t )   f ( ) g (t   )d .
0
12. Newton iteration formula:
xn 1  xn 
f ( xn )
f '( xn )
13. Secant iteration formula:
xn 1  xn 
f ( xn )( xn  xn 1 )
f ( xn )  f ( xn 1 )
14. RK2 method:
yi 1  yi 
1
 K1  K 2  , where
2
K1  hf ( xi , yi ) and
K 2  hf  xi  h, yi  K1 
15. The Euler-trapezoidal method:
yi(1)
1  yi  hf  xi , yi  and
yi(k11)  yi 
h
 f  xi , yi   f  xi 1 , yi(k1)  

2
Page 6 of 7
MA272 Final Exam, Semester 2, 2019
Areas under Standard Normal Probability Curve (Source: Eton Table)
Page 7 of 7