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Unformatted text preview: Exam Cover Page germ: 2002 Autumn Term
ession: T2  Autumn Session
central Queensland Academic Institution: Central Queensland University
U N I V E R S I T Y Academic Group: Faculty of Info & Comm
Eganmi. Where Students Come First. éﬁgﬂeﬂﬁe‘faree" gngeirgrrgduate
Candidate Name (BLOCK LETTERS):
Candidate Number:
Course: 002788  Technology Mathematics
Subject Area: MATH
Catalog Number: 11160
Paper Number: 1
Component: All
Duration: 180 minutes Open/Closed Book: Closed Book
Perusal Time: 10 minutes
Lecturer: Milton Fuller Contact Number: 07 4923 2108
Moderator: David Ruxton Contact Number: 07 4930 9488 Release Exam Paper to Candidate ? Yes Instructor Authorised/Allowed Materials
Any calculator including programmable or graphics
Any unannotated bilingual translation dictionary Special Instructions to Candidates: Examination Office Supplied Materials
2x Examination booklets 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 2002 Page 1 of 6 Technology Mathematics (MATH 11160) SECTION A Attempt FIVE (5) questions from this Section. All questions are of equal value.
Marks allocated to each part of the question are shown. Question 1 (a) Solve the equations:
(i) (:—1)2 —9 =0.
m+1 m
" ——2 ——1 = ———. 2,
(11) 4 (m ) 3 3 ( 3) (b) (i) Uranium decays at a rate of approximately 0.0000000039% each day.
Write this percentage in scientiﬁc notation. (ii) Write the number L74 in scientiﬁc notation. (1, 1)
4.23 x 10 (c) If t = L: _1b express L in term of t, a and b. (3)
Question 2
(a) The expression 5 y + 3x — l 12 = 0 represents the cost relationship between x items at $3 each and y items at $5 each. . —3x 112
(1) Show that y = T "i" '—5'. (ii) Knowing that both x and y can only take values which are positive
integers, calculate two values of x and y which satisfy the cost
relationship. (1, 4) (b) A ship departs Port A at 8am on Monday. This ship travels directly toward Port B
at a speed of 28 kilometres per hour. Two hours later a second ship departs from
Port B and travels toward Port A at 35 kilometres per hour. If the distance between the two ports is 2230 kilometres, at what time will the ships pass each
other? (5) Question 3 overleaf AUTUMN TERM EXAMINATION 2002 Page 2 of 6 Technology Mathematics (MATH 11160)
Question 3
(a) A straight line, L, is parallel to the straight line, y—2x—3 = 0. L also passes through the point (—1, 4).
(i) Find the equation for the line L. (ii) At what point does L cross the xaXis? (2, 1)
(b) A projectile is launched from a vertical platform. The path of the projectile is
modelled by S(t) =10 +18t —z‘2 , where s(t) represents the height, in metres, of
the projectile above the ground (assumed horizontal) 1 seconds after being
launched.
(i) How high above the ground is the launching platform?
(ii) How long will it take the projectile to reach a height of 87m?
(iii) When the projectile lands, how far will it be from a point on the ground
directly below the launching platform?
(iv) Calculate the maximum height reached by the projectile?
(v) Using the information from (i) to (iv) draw a sketch graph of the path of
the projectile). (1, 2, 1, 2, 1)
Question 4
(a) Solve the equation, 5 sin 6 — 2 = 0 for 0 S (9 S 72'. (2)
(b) A vertical pole has a straight wire of length 20m secured from the top of the pole
to a point on the level ground which is 12m from the base of the pole.
(i) Calculate the height of the pole.
(ii) What is the size of the angle between the wire and the pole? (2, 2)
(c) The intensity, I of a sound source is directly proportional to the power, P, of the source and inversely proportional to the square of the distance, d, from the source.
If the power is doubled and the distance is halved, calculate the change to the
intensity. (4) Question 5 overleaf AUTUMN TERM EXAMINATION 2002 Page 3 of 6 Technology Mathematics (MATH 11160) Question 5 (a) (b) Solve the equations for x:
(i) log2x+log2(x+2)=3. (ii) logx 27 + 3 = 0. (2, 2) Research reveals that 1.5% of lead entering the bloodstream each day is removed
by natural means. For a child weighing between 23kg and 27kg there should not
be more than 0.03 milligrams of lead per kilogram in the bloodstream (otherwise
treatment for lead poisoning is required). A child weighing 25kg is found to have 2 milligrams of lead in her bloodstream.
(i) Show that the amount of lead, A(t) , remaining in her bloodstream after t days is given by the model, A(t) = 2(0.985)t. (ii) Use the model to calculate how long it will take for the child to reach an
acceptable level of lead in her bloodstream. (3, 3) Question 6 overleaf AUTUMN TERM EXAMINATION 2002 Page 4 of 6 Technology Mathematics (MATH 11160)
Question 6
(a) (i) The matrix
1 —1 0
A = 0 1 2
2 1 —1
is such that its determinant, det A, is non—zero. What can be said about
the inverse, A’l?
(ii) Demonstrate that
3 1 2
B = ~1— —4 1 2 is the inverse ofA.
7 2 3 —1
(iii) Hence, use a matrix method to solve the set of equations:
a —b = —4
b = —1 — 2c
2a+b=c+3. (1,2,3)
(b) A crate is hanging freely supported by two ropes. One rope is in a direction of 15° from the vertical. The tension, T1, in this rope is 99ON . The other rope is at an angle of 25° with the vertical and has a tension, T2 , 620N (see diagram).
Calculate the weight, W, of the crate. (4) Section B overleaf AUTUMN TERM EXAMINATION 2002 Page 5 of 6 Technology Mathematics (MATH 11160) SECTION B Attempt TWO (2) questions only from this Section. All questions are of equal value. Question 7 (a) A space vehicle is used to measure the radius, R, of a planet. When the vehicle is
h kilometres above the surface of the planet the angle of depression of the horizon
of the planet is x degrees. Knowing that the angle between a tangent to a circle
and the radius of the circle is a right angle, list any assumptions that you make and develop the model, R = hcos x . (4)
l—cosx
3 1
(b) f(x)=(x+1) ——,x¢0.
x
(i) Write down the derivative, f ’( x) .
(ii) Explain why f (x) is an increasing function over its domain.
(iii) Find the equation of the tangent line at x =1 . (2, 2, 4)
Question 8
(a) The charge, q, at any time, t, in a circuit which has an initial charge, qo, a
capacitance, C, and resistance, R, is given by In q + RLC = ln qo. Use this equation
to establish a model which gives q as a function of time, t. (qo, C and R are
constants.) (4)
(b) A meteor travelling toward the earth has a velocity, V, which is inversely proportional to the square root of the distance, x, of the meteor from the earth’s
centre. . k .
(1) Show that v = £x_ = —. (k is a constant.)
dt J;
—k2
(ii) Hence, establish that the acceleration of the meteor is given by 2 2 .
x
(iii) Express the acceleration of the meteor as a proportionality statement.
(2, 4, 2) Question 9 overleaf AUTUMN TERM EXAMINATION 2002 Page 6 of 6 Technology Mathematics (MATH 11160)
Question 9
x6 — x4 —1
(a) For the function F (x) = —4, x at 0.
x
. . . , 2
(1) Show that the derlvatrve F (x) '2 2[x+ 7].
x
(ii) Is F (x) an increasing or decreasing function on the interval
—10$xS—1? Why?
(iii) Find an expression for 2I(x + Esjdx. (2, 2, 3)
x
(b) The graph which is shown is a velocity/time graph with v =16 — t2. Question 10 (a) (b) (i)
(ii) Write down a physical interpretation for the shaded region on the graph.
Find a value for the shaded region. (2, 3) The model for the height of a tidal wave for a 24 hour period is given by
H(t)=1.25+0.85cos0.498(t—1), where H(t) is the height of the tide in metres at a time t measured in hours from midnight. (i)
(ii)
(iii) (0 (ii) What is the period of the wave?
What is the greatest height of the wave? At what time will high tide ﬁrst occur? (the wave attains its maximum
height) (19 19 2) Several samples of the measurements of the lengths of steel pegs cut by a
machine are taken. For one sample of 25 pegs the mean length was 3.94
centimetres. For another sample of 30 pegs the mean was 4.02cm. If 5
extra pegs are measured with lengths of 3.93, 3.99, 4.01, 4.03 and 4.09
(all centimetres) calculate the mean length of all of the pegs. The lengths of the pegs are assumed to be normally distributed with a
mean of 3.99 centimetres and a variance of 0.003. If a peg is longer than 4.05 centimetres it has to be rejected. What
proportion of the pegs should be rejected? (4, 4) Technology Mathematics MATH 11160 FORMULA SHEET l_.___
Areas Triangle, :— (base X perpendicular height) Circle, 7r r2 (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 sin
For the functions, f (x) = a }(bx + c),
cos
27:
the period 1s —b—, and
c the phase shift is — l; Quadratic Functions
For f(x)= ax2 +bx+c The solution of f(x) = 0
—bi\/b2 —4ac is x = —, providing b2 2 4ac
2a
The equation of the axis of symmetry is
x = :11
2a
Rules of Logarithms loga MN = loga M + log“ N loga(A—;—) = log, M —loga N loga M’r = xloga M Circles The equation of a circle with centre (a, b)
and radius r units is (xa)2+(yb)2= Rules of differentiation If f(x)=ax" then f’(x)=nax"‘1 If f (x): ux( )v(x) then
f(x )= u ’(x)v(x)+u(x)v’(x)
If f(x)= f(u ) where u =g(x) f’=(x):—fg <> Rules of Integration If f(x)=ax", rut—1, TABLE OF NORMAL PROBABILITIES mumnmﬁlﬂm
W
W
W .1772 was .1844
.2123 .2257
W
.2734 .2764: .2793
W.—
Inn—W .4265 .4279? .4292 .4306
.4370
.4474 .4434 .4495; .4505. .4515; .4525 , .4535; .4545
.4573 .4532 .4591; .4599; .4603; .4616 .4625} .4633 I L30 4641 .4649, .4656 5; .4699: .4706
Ian—— .4738 .4744; .4959 .4756! .4?61 g .4767
.4772 .4733 .4793; .4793 .4803; .4808 .4312
.4842 .4846: .4850 .4854 .4357
.4904 .4906 6909 .491:
.4929 .4931 .4932 .4934 .4936
.4959 .4960 69611—4962
.4963; .4969; .4970; .4971 i .4992
.4973 g .4979 g .4979 5 .4980
.4932 .4933 .4984 .4934 .4985; .4935 .4986 .4936 m .4992 .5992 .4992 .330 .4995 .4995 .4995 .4996 —.4996 .4996 —.4996 —.4996 _.4996 no
.4998
W ...
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