EN112 - Page 1 of 4

THE PAPUA NEW GUINEA

UNIVERSITY OF TECHNOLOGY

DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE

FIRST SEMESTER EXAMINATIONS 2019

FIRST YEAR CIVIL ENGINEERING

FIRST YEAR MECHANICAL ENGINEERING
FIRST YEAR COMPUTER SCIENCE

EN112 - ENGINEERING MATHEMATICS 1 (CE,ME,CS)

TIME ALLOWED – 3 HOURS

Information for Candidates

1. Write your name, student number, and program of study clearly on the front page of your answer
booklet. Do it now.
2. You have 15 minutes to read this examination paper. During this time you must NOT write inside
your answer booklet. You can make notes on the examination paper.
3. Scientific calculators are permitted. Other electronic devices are not permitted. Notes and
headphones are not permitted.
4. At the conclusion of the examination you must immediately put your pens down. You are NOT
permitted to write inside your answer booklet after the "end of examination" announcement.
5. You can answer the questions in any order. Start each question on a new page. After you have
finished the exam, indicate the order in which you answered questions in the left column of the marks
box on the cover of the answer booklet.
6. Do NOT check your answers to any question unless specifically asked to do so.
Do NOT simplify answers unless specifically asked to do so.
7. There are 5 questions. Each question is worth 20 marks. You should attempt all questions.
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Question 1 [ (2+2+3 +3+3) + (2+4+1) = 20 marks]

(a) (i) Find the derivative of y = 4x2 – x exp( x ).
(ii) Find the second derivative of y = 2.5 sin(2x) + 4.

(iii) If g(p) = √ p+1 , find the value of g'(p).

p−1
(iv) If the position of an object at time t seconds is s(t) = 8 + 4 t – cosh(2t) cm,
find the velocity of the object at time t = 2, and indicate whether the
object is moving to the right or left (with reasons and assumptions).
(v) Find the slope of the tangent to y = 3 x – ln( 44 – x2 ) at the point x = 2.

(b) A small rectangular base-shaped prism shape is drawn on the

right. Its base is x cm wide and y cm long, where x + y = 10,
2
x
and its cross sectional area is x+ cm2.
2
1 3 2
(i) Show that the volume of this prism is: V(x) = − x + 4 x + 10 x .
2
(ii) Using V(x) and differentiation, what base shape gives a prism of maximum
volume, and what is the maximum volume?
(iii) Show that your answer in (iii) is a maximum (and not a minimum or point of
inflexion).

Question 2 [(2+2+3+3+3) + 4 + 3 = 20 marks]

(a) (i) Find an anti-derivative of y = 5x2 – 2x–2 + 1 .
1
(ii) Find ∫(4+ 3 x ) dx .
0
(iii) Find ∫ exp(−x /2) dx , simplifying your answer.
−2

(iv) Use intelligent guesswork or substitution to find ∫ 8 x 3 cos (1+ x 4 ) dx .

Check your answer to this question part.
(v) Use integration by parts to find ∫ x sin(2 x) dx .

(b) Use a definite integral to find the area enclosed by the x-axis, the y-axis,
x = 2, and y = 3x2 – 16. You should simplify your answer to a single number,
and draw a simple sketch illustrating the area.

(c) Draw a sketch of, and find an expression (which will be a definite integral) for, the
volume between x=0 and x=4 of the solid obtained by rotating the function
y = 5 + x/4 about the x-axis. [Do NOT evaluate the definite integral.]
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Question 3 [ (1+3+2) + (2+2) + (1+3) + (2+4) = 20 marks]

(a) If a = 4 – j2 and b = j + 1 are complex numbers, find
(i) The complex conjugate of a
(ii) 3b/a , writing your answer in the form a + jb.
(iii) The value of z for which 3z + 1 = b – a, writing your answer in the form a + jb.

(b) If p = –3 – j5 and q = 2 1.5 are complex numbers, answer the following:

(i) Locate (ie, sketch) p and q on an argand diagram.
(ii) When converted to polar form (don’t show this) p = 5.83 4.172.
Use this to find pq2, leaving your answer in polar form.

(c) These questions involve complex numbers in exponential form:

(i) Convert 4 –2.3 to exponential form.
j1.4
(ii) Convert –3.2 e to rectangular form, writing your answer in the form a + jb.

(d) Part of these last questions involve using De Moivre’s formula:

(i) Show that if converted to polar form, 1 + j2 becomes 2.236 1.1072
(ii) Find the three roots of z3 = 1 + j2, leaving your answer in polar form.

Question 4 [ (1+2+3) + (3+3) + (2+3+3) = 20 marks]

4 6 11 30 127 728
(a) Consider this sequence: , , , , , , .. .
1 2 6 24 120 720
(i) Study this sequence to find a pattern. Use the pattern to write down the next term of
the sequence.
(ii) Write down a formula tn for the nth term of the sequence.
(iii) Using your formula in (ii), find lim t n , showing all working and reasoning.
n →∞

2n
(b) The terms of an infinite series are given by the formula tn = 2
n +1
(i) Show that the ratio test does not tell us whether or not the series converges.
(ii) Use the integral test to determine whether or not the series converges.
3 5 7
x x x
(c) The Maclaurin series for sin(x) is x − + − + .. . .
3! 5! 7!
(i) Use the first three terms of the above series to find an approximation for sin(2x).
(ii) Use the definition of a Maclaurin series (which involves derivatives) to find
the first two non-zero terms of the MS of the function y = sin(2x).
(iii) Find the interval of convergence of the MS for y = sin(2x).
[The formula for the nth non-zero term is: tn = (-1)n x(2n+1)/(2n+1)! n = 0,1,...]
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Question 5 [3 + (3+3) + (1+2+1+2) + 5 = 20 marks]

(a) When finding the derivative of y = 1 – 4x2 using the first principles (limit) method
h(−2 x − h)
one line of your derivation might be: lim .
h→0 h
Following are two possible next steps. Why is first correct, but the second wrong?
h(−2 x − h)
A: lim = lim (−2 x − h) = –2x
h→0 h h→0

h(−2 x − h)
B: When h is zero, = (–2x + 0) = –2x
h

(b) Find all solutions of the following equations:

(i) exp( 1 – 2x) = 0.3
(ii) 4 sin(2x) = 3

(c) Simplify each of the following (or at least write in another way):
(i) 4e2x  ex
(ii) log(100x) (where log is the common logarithm function)
y
(iii) ln(4 )
(iv) exp( 1 – ln(x))

(d) The price of building a kit home in the future is forecast to increase like an exponential
growth function. The current cost is K12000. In 5 years the projected cost is K15000.

Answer the following using the growth function model.

(i) What is a general formula for an “exponential growth function”?
(ii) What is the formula for our kit home model, where “t” is the time in years
into the future.
(iii) Use your formula in (ii) to estimate how long before this kit home exceeds K24000
[writing your answer as a number of whole years.]

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