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Unformatted text preview: AUTUMN Term Examination 2001 Candidate Name: (BLOCK LETTERS): Candidate Number: Unit Code: 84137 Unit Name: TECHNOLOGY MATHEMATICS OfferMode: Internal D External D Both a Duration: THREE (3) Hours Perusal Time: TEN (10) Minutes Lecturer: MILTON FULLER Telephone Number: (07) 4930 9455
Moderator: FAE MARTIN (07) 4940 7451 Release Examination to candidate: YES a NO D Questions to be answered on Examination Paper: YES D NO a Open Book: YES D NO a Calculators: Type Any YES El NO D Dictionaries: Type Language dictionary YES NO D
Materials Supplied by lecturer/Authorised Materials: Table of Normal Probabilities (attached), Formula Sheet (attached) Instructions to Candidate: 1. This is a CLOSED book examination.
Attempt FIVE questions from Section A.
Attempt TWO questions from Section B.
All questions clearly show the mark allocated to each part of the question. Show your working for all questions attempted. Materials to be supplied by Examinations Section (Please Specify): Examination Booklets: YES Optimal Mark Reader Cards: N/A 55 D 100 D Graph Paper: N/A Sheets per Candidate: Other: N/A Central Queensland University considers improper conduct in examinations to be a serious offence Penalties for cheating are exclusion from the
University and cancellation with academic penalty from the course concerned. AUTUMN TERM EXAMINATION 2000 Page 1 of 5 Technology Mathematics (84137) SECTION A Attempt FIVE questions from this Section
(All questions are of equal value) Question 1 (a) Solve the equations for x
x +1 x ‘ 3 + — = —— — —l (1) 2 4 (x ) (ii) (x—a)2—4=0 (iii) Check that your solution for (i) is correct. (2,2,1) (b) A skateboarder walks up an inclined path at 40 metres per minute. Without pausing he then
skates down the path at a constant speed of 130 metres per minute. The total time taken to walk up and skate down the path is 9 minutes. Calculate the length of the path. (4)
Question 2
1 1 3
(a) If (3a — b) =1, show that a = —b—1. (3)
+ (b) A straight line, L, is parallel to the straight line y—2x—7 = O and passes through the point (— l, — 4). Find the point where the line, L, crosses the x—axis. (3)
(c) If I = W express R2 in terms of V, R] and y . (3)
R1 R2
Question 3
(a) (i) Solve the equations
1
3x = — +
2 y
4x — y = 0 (ii) Write down one more linear equation which has the same solution as the equations in (i).
(2,2) Question 3 continued overleaf Page 2 of 5 AUTUMN TERM EXAMINATION 2000 Technology Mathematics (84137) Question 3 continued (b) A remote controlled toy aeroplane is launched and follows a path where its height above level
ground, s, in metres, t seconds after launch, is given by s(t) = —3t2 + 8t +16 .
(i) Write down an interpretation for 5(0).
(ii) How long after launch is the aeroplane 13 metres above the ground?
(iii) Calculate the greatest height reached by the aeroplane. (1,2,2)
Question 4
(a) If sin6 = —0.5 and tan 0 < 0 calculate the size ofthe angle 0 for 0 S (9 S 360° . (2)
(b) A hot air balloon is ascending vertically. When it is 120 metres above the ground the angle of
depression of a mark on the level ground is 35°. Several minutes later the angle of depression
of the same mark is 55°. What is the height of the balloon at this time? (3)
(C)
Two tug boats A and B exert forces on a ship, V, as
follows:
Boat A pulls with a force of 8 units in a direction which
is 57° to the positive direction of the horizontal whilst B
pulls with a force of 5.5 units in a direction 322° to the
horizontal as shown in the diagram. What is the resultant
force on V?
(4)
Question 5
(a) Solve the equations for x
(i) 10 4  1
8x 2
(ii) log(x + 7)— log(x — 2)— 1 = 0 (2,3)
(b) (i) The amount of radioactive material, A(t), remaining after 1‘ years is given by A(t) = A0 (0.5)! where A0 is the original amount of radioactive material. How long will
it take for the material to be reduced to one ﬁfth of the original amount? (ii) On a set of suitable axes give a sketch graph of A(t) taking A0 = 10. (3,1) Page 3 of5
Question 6
(a) (i) Write the equations x +§ = 5
x + i = 6
3
z
+—=9
y 3
in matrix form AX = B.
(ii) Show that the inverse of A is given by
2 l :1
4 4 4
A“ z _3_ :_3_ 2
4 4
;9 _9_ 2
4 4 4
(iii) Hence solve the given equations by a matrix method.
2 sin 6 0 cos 6 O
(b) If =
0 1 0 1
(i) Show that 2tan 6 = 1 . (ii) AUTUMN TERM EXAMINATION 2000 Find a value for 6 such that 0 < 6 < 72. Technology Mathematics (84137) (1, 2, 3) (2,1) Page 4 of 5 AUTUMN TERM EXAMINATION 2000 Technology Mathematics (84137) RE“— S E C T I O N B
Attempt TWO questions only from this Section (all questions are of equal value) Question 7
(a) The population ofa state is given by P(t) = 2.52e’” where P(t) is the population (in millions) at
time tyears measured from 1990 (t = 0 ).
(i) If the population was 2.96 million in 2000 show that k = 0.016 (to 3 dp).
(ii) In which year is the population predicted to reach 3.77 million? List any assumptions
which are made in obtaining your answer. (3,3)
(b) An engine, initially valued at A0 dollars, depreciates so that its value at the end of each year is
80% of its value at the beginning of that year.
(i) Develop an exponential model which can be used to determine the value of the engine
after H years.
(ii) The engine is to be replaced when its value is 25% of its original value, A0. If the
engine was new at the beginning of 2000 in what year should it be replaced? (3,3)
Question 8
(a) A function is said to be an odd function if f (— x): —— f (x) for all x. An even function is such
that f(— x): f(x).
(i) Show that the derivative of an odd function is an even function.
(ii) Use the function g(x) = 3x3 to illustrate your result in (i). (3,2)
(b) (i) If y is directly proportional to x and y = 2 when x = 32 ﬁnd a value for x when y = 10.
(ii) The volume rate of ﬂow of blood, V, through an artery varies directly as the square of
the radius of the artery and inversely as the length of the artery. A person has a
defective artery with an effective radius of two millimetres and a length of 20
millimetres. After surgery the radius is doubled and the artery shortened by 4
millimetres. Show that the volume rate of ﬂow, is now ﬁve times what it was before
surgery. (1,4)
(0) The probability P(x) of an event x happening is given by P(x)= x_1 . What is the domain 2x
of this function. (2) AUTUMN TERM EXAMINATION 2000 Page 5 of 5 Technology Mathematics (84137)
Question 9
(a) (i) Write down the equation of a circle which has its centre at (0, — 2) and has a radius of 2 (b) (0) units. (ii) Show that this circle passes through the origin. (1,1) A goat is tethered (tied) to a stake, by an inelastic rope, in a rectangular ﬁeld. The adjacent
sides of the ﬁeld can be assumed to be a set of rectangular axes. The goat can move along a path given by x2 + y2 — 4x — 6y + 9 = 0 when the rope is fully extended. Question 10 (a) (b) (i) Determine the coordinates of the stake. (ii) Calculate the area of the ﬁeld that the goat can access. (2,1) If f(x)= (2x —3)2 (i) Write down the derivative, f ’(x) (ii) Determine the equation of the tangent line to this function which has a slope (gradient) of 4. (iii) Show that jf(x)dx = %(2x — 3)3 + c .
3 (iv) Give an interpretation for If (1,3,2,1)
2 For the function f (t) = 5 sin(7zt — write down: (i) The period. (ii) The phase shift. (iii) Show that the function has a maximum value when t = %. (iv) Write down an expression for the rate of change of f (t) at any time t. (1,1,2,2) (i) Motors used in laboratory vacuum pumps have a mean operating time of 500 hours with a standard deviation of 100 hours. Once a motor reaches an operating time of 700
hours it has to be replaced. Assuming the operating hours for these motors are
normally distributed calculate the percentage of motors which have to be replaced. (ii) What proportion of the motors can be expected to have an operating time between 420
hours and 600 hours? (3,3) Technology Mathematics 84137 FORMULA SHEET Areas of a Triangle, % (base x perpendicular height) Circle, 7rr2 (r is the radius) 2 Sector of Circle, %6 (6 in radians) Trigonometry 27: radians is equivalent to 360° Length of arc of circle = r0 Sinusoidal Function Period is 2—” [2 Phase shift is — E b Quadratic Functions
For f(x)= ax2 +bx+c The solution of f(x) = 0 —bi\/b2 —4ac 2a isx= . — b
Equation of axis of symmetry 15 x = 2——
a Circles The equation of a circle with centre (a, b)
and radius r units is (x——a)2 +(y—b)2 =r2 Rule of differentiation If = ax” then f'(x) = nax"'l If f (x) = u(x)v(x) then
f '(x) = u’(x)v(x) + u(x)v’(x)
If f(x)= f(u) where u = g(x)
f ’(x) = f ’(u)g'(x) Integration
If f(x)=ax", n¢—1,
n+1
If(x)dx = ax + c n+1 If Ig(x)dx = F (x) then gm): gm TABLE OF NORMAL PROBABILITIES ummmmamm
121mm
m
mum:
mm
mm
m
m
um
um
m
In] .4192 .4207 .4222 .4236 .4265 .4279 .4292 .4319
IE] .4452 .4463 .4474
mm .4686 .4693: .4699
mm .4726 4732
m
m ...
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This note was uploaded on 05/10/2008 for the course MATH MATH11160 taught by Professor Mcdougall during the Spring '08 term at A.T. Still University.
 Spring '08
 McDougall
 Math

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