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2007 SPECIALIST SPECI ALIST MA MATHEMA THEMATICS TICS FOR OFFICE USE ONLY
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Friday 16 November: 9 a.m.
Pages: 41 Questions: 16
Time: 3 hours Examination material: one 41-page question booklet one SACE registration number label
Approved dictionaries, notes, calculators, and computer software may be used. Instrctions to Stdents
1.
You will have 10 minutes to read the paper. paper. You must not write in your question booklet or use a calculator during this reading time but you may make notes on the scribbling paper provided.
2.
This paper consists of three sections: Section A (Questions 1 to 10) Answer all questions in Section A.
75 marks
Section B (Questions 11 to 14) Answer all questions in Section B.
60 marks
Section C (Questions 15 and 16) Answer one question from Section C.
15 marks
3.
Writ riteyouransw eyouranswers ers inthespac inthespacesprov esprovide idedin din thi thisquest squestion ion boo bookle klet.Ther t.Thereis eis noneed tollallthespac tollallthespace e provided. You You may write on page 40 if you need more space, making sure to label each answer clearly.
4.
Appropriate steps of logic and correct answers are required required for full marks. marks.
5.
Show all working in this booklet. (Y (You ou are strongly advised consider incorrect should be crossed out with a single line.)
6.
Use only black or blue pens for all work other than graphs and diagrams, for which you may may use a sharp dark pencil.
7.
Statealla State allanswer nswerscor scorrectt recttothre othreesig esignica nicantgu ntgures, res,unles unlessoth sotherwis erwisesta estatedo tedoras rasapprop appropriate riate..
8.
Diagrams, where given, are not necessarily drawn to scale.
9.
The list of mathematical formulae is on page page 41. You may remove the page page from this booklet before the examination begins.
10.
Complete the box on the top right-hand side of this page with information about the electronic technology you are using in this examination.
11.
Attach your SACE registration number label label to the box at the top of this page.
not to
use scribbling paper. Work that you
SECTION A (Questions 1 to 10)
(75 marks)
Answer all questions in this section.
QuESTION 1
(5 marks)
(a) Find in parametric form the equation equation of the line through A(2, 6,
-2)
and B(5, 0, 7).
(2 marks)
(b) Find where the line in part (a) intersects the plane x + 3 y + 2 z = 14 .
(3 marks)
2
QuESTION 2
(5 marks) 2
Consider the function f ( x) = e x sin 3 x. π
∫
2
3 (a) Find an approximate value for e x sin x dx. π
2
(1 mark)
(b) Show that f ( x x) is an odd function.
(2 marks)
(c) (c) Henc Hencendanexa endanexact ct valu valueof eof k such that k
π
π
π
∫− f ( x)dx = − ∫ f ( x)dx, explaining your reasoning. 2
(2 marks) 3
PLEASE TURN OVER
(6 marks)
QuESTION 3
In Figure 1 points A, B, C, and D lie on the circumference of a circle and AB is parallel to DC . The line AT is a tangent to the circle and AD bisects ∠TAC .
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(a) Prove that AC bisects ∠ DAB.
(3 marks)
4
(b) Prove that AD = DC = BC .
(3 marks)
5
PLEASE TURN OVER
QuESTION 4
(6 marks)
(a) Prove that if the line
x − k
then Aa + Bb + Cc = 0 .
a
=
y − l b
=
z
−m c
is parallel to the plane Ax + By + C z = D ,
(2 marks) (b) For the line x − 3 3
=
y+4 4
=
z
−m c
and the plane 2 x − y + z = 18:
(i) nd c if the line is parallel to the plane.
(2 marks) (ii) (i i) n nd d m if the line is in the plane.
(2 marks) 6
QuESTION 5
(5 marks)
(a) If y = ln ( sin x ) , where 0 < x < π , show that
d y d x
= cot x.
(2 marks)
π
2
(b) Hence, Hence, orotherwis orotherwise,nd e,nd anexact valuefor valuefor
∫ cot x dx.
π
6
(3 marks)
7
PLEASE TURN OVER
QuESTION 6
(a) Given that w =
(9 marks) 3 2
+
3i 2
, write w in the form r cis q , with exact values for r and q .
(2 marks)
(b) Given the set of complex numbers z such that | z − 2i | = 1: (i) sketch | z − 2i | = 1 on the Argand diagram in Figure 2. � �
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�
8
(3 marks)
(ii) show that arg z = arg w is a tangent to | z − 2i | = 1.
(2 marks) (iii)ndthe larg largest estposs possiblevalue iblevalue of arg z z. Give reasons for your answer.
(2 marks)
9
PLEASE TURN OVER
(9 marks)
QuESTION 7
A damaged oil tanker is leaking oil into the sea. A constant current pushes the spreading oil spill into the shape of a sector with a radius of r metres and a sector angle of q radians (as shown in Figure 3), where r and q change with time. Let the area of the sector be A.
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Source: International MARSAC, Source: www.marsac.nl/REF.htm
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(a) Show that
dθ dt
2 1 d A dr = − θ . dt r r dt
(3 marks)
10
(b) The radius of the oil spill is increasing at a constant rate of 2 metres per second and and the area of the oil spill is increasing at a constant rate of 2 p square metres per second. Consider the oil spill when it has a radius of 6 metres. (i) Show that the area of the oil spill is A
=
6 p square metres at this instant.
(2 marks) (ii) (ii ) Hen Hencend cend the exa exactvalue ctvalue ofthesector ang angle, le, q , at this instant.
(2 marks) (iii)Hencendtherateofchangeof q at this instant.
(2 marks)
11
PLEASE TURN OVER
(10 marks)
QuESTION 8
(a) Let f ( x) = x
+ 7. f ( x) − f ( 2 ) 3
Show that
has a factor of x − 2 .
(2 marks)
(b) If p( x polynomia mial l ofdegree≥ ofdegree≥ 1,prove 1,prove that that p( x) − p( k ) has a factor of x − k . x) isany polyno
(2 marks)
(c) T ( x x) is a real cubic polynomial with a zero of 1
+
2i.
(i) Find a real quadratic factor of T ( x x).
(3 marks) 12
(ii) Find T ( x x), given that T ( x) − 25 has a factor of x − 2 and that T ( x) − 12 has a factor of x − 1.
(3 marks)
13
PLEASE TURN OVER
QuESTION 9
(10 marks)
Points A(-1, 1), B(1, 2), and C (2, (2, -1)arexedpointsintheplanewhichdeterminethe simultaneous motion of points P , Q, and R so that
OP = [ 2t − 1, t + 1 ] OQ = [t + 1,
− 3t + 2 ]
OR = − t 2 + 4t − 1,
− 4t 2 + 2t + 1
where 0 ≤ t ≤ 1 is the time for which the points are in motion. The graph in Figure 4 represents this situation at some time t .
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(a) Graph the paths of P , Q, and R on the axes in Figure 4.
14
(3 marks)
(b) Find PR and PQ in terms of t and hence give a vector proof that P , R, and Q are collinear.
(3 marks)
(c) (i) Draw the vectors AP , AR, PQ, RC , and QC on Figure 4. (ii) Using the triangle inequality, inequality, show that
AP+ PQ+ QC ≥
(1 mark)
AR+ RC.
(3 marks)
15
PLEASE TURN OVER
QuESTION 10
(10 marks)
The velocity, v, of a small raindrop may be found by solving the differential equation dv dt
= g − cv
where g is the gravitational constant, c is a positive constant that can be found experimentally experimentally,, and time, t , is measured in seconds.
(a) By solving the differential equation with the initial condition of v(0) = 0 , show that the velocity of a small raindrop is
v(t ) =
g
1 − e− ct ) units per second. ( c
Source: Photograph by Altrendo Nature, from Source: Getty Images, http://creative http://creative.gettyimage .gettyimages.com s.com
(5 marks) 16
(b) The limiting value of the velocity as time increases is called the terminal velocity. velocity. Find the terminal velocity of a small raindrop using the solution from part (a), given that g = 9.8 and c = 16.
(1 mark)
(c) On the axes in Figure 5, sketch the velocity curve curve of a small raindrop raindrop using the values of g and c given in part (b). Indicate the terminal velocity on the sketch.
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�
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(3 marks)
(d) Small raindrops approach terminal velocity velocity rapidly. rapidly. Byassuming Byassumin g tha that t the vel veloci ocityof tyof a sma smallraind llraindrop rop iscons isconstan tant,nd t,nd anappro anapproxim ximate ate value for the time taken for a small raindrop to fall a distance of 10 units.
(1 mark) 17
PLEASE TURN OVER
SECTION B (Questions 11 to 14) (60 marks)
Answer all questions in this section.
(15 marks)
QuESTION 11
Figure 6 shows skew lines l 1 and l 2 connected by vector NM . Line l 1 is parallel to vector equation
v
�
�
= [1, 1, 4] and has
l1 : [ x, y, z ] = [1, 6 , − 9 ] + s [1, 1, 4 ]. Line l 2 is parallel to w = [ 2, − 1, 2 ] and has parametric equations
l 2
x = 2 + 2t : y = 1 − t z = 1 + 2t .
(a) (i) Calculate
v
�
× w .
(2 marks) (ii) Show that point P (4, 9, 3) lies on l 1 and point Q(10, − 3, 9) lies on l 2.
(2 marks) 18
(iii) Calculate the length of the projection of vector PQ on
v
× w .
(2 marks)
(b) As shown in Figure 6, M is a point on l 1 and N is a point on l 2. Show that NM = [ s − 2t − 1, s + t + 5, 4 s − 2t − 10 ].
(1 mark)
(c) (i) If NM = k [ 2, 2, − 1] , show that s, t , and k are related by the system of equations
s − 2t − 2k = 1 s + t − 2k = −5 4 s − 2t + k = 10 .
(1 mark)
19
PLEASE TURN OVER
(ii) Solve the system of equations equations from part (c)(i) for s, t , and k .
(1 mark) (iii)Hence (iii)Henc e ndthe coord coordinates inates of M and N .
(2 marks) (iv) Find NM and calculate the length of this vector.
(2 marks)
(d) (i) Comment on your your answers answers to part (a)(iii) (a)(iii) and part (c)(iv). (c)(iv).
(1 mark)
20
(ii) Explain why NM found in part (c)(iv) is the shortest distance between l 1 and l 2.
(1 mark)
21
PLEASE TURN OVER
(16 marks)
QuESTION 12
(a) (i) Solve
z
6
= − 64, giving the roots in the form
r cis q .
Illustrate the roots on the Argand diagram in Figure 7.
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(5 marks) (ii) Show that
z
2
+4
is a factor of z 6
+ 64 .
(2 marks)
22
(iii) Using parts (a)(i) and (ii), or otherwise, solve answers in the form r r cis cis q .
z
4
−
4z 2
+
16 = 0 , giving your
(2 marks)
(b) (i) Show that if p p(( z) = ( z − z1 )( z − z2 ) is a quadratic polynomial with zeros then the coefcient of z in p( p( z) is − ( z1 + z2 ).
z
1
and z2,
(1 mark) (ii) Let
(pz) = ( z −
z
3
If p If p(( z) has zeros z
2
in p( p( z) is
−
)( z2 +
z+
1 , z2 ,
and
z
( z1 +
z + z
2
)bbe a cubic polynomial, where
a
3
z
3
a and b are constants.
then, using part (b)(i), show that the coefcient of
).
(1 mark)
23
PLEASE TURN OVER
(iii) Use an inductive argument to show that if p( z) = zn + a zn −1 + b zn − 2 + . . . is a polynomial of degree n with zeros z1 , z2 , z3 , z4 , . . . , zn ,thenthecoefcientof z
n −1
in p( z) is
− ( z1 + z2 + z3 +
...
+ zn ).
(3 marks)
(c) If (i)
1 , z2 , z3 , z4 , z5 ,
z
z
1
and z6 are the roots of z 6
+ z2 + z3 + z4 + z5 +
= − 64frompart(a)(i),nd:
6.
z
(1 mark) (ii)
1 2 3 4 5 6.
z z z z z z
(1 mark) 24
QuESTION 13
(14 marks)
(a) A mathematical model for the growth of a population P = P(t ) offruitiesina laboratory is given by d P dt
=
1000 − P P 500 1000 49
wheretimeismeasuredindaysandthemaximumnumberoffruitiesabletobe sustained in the laboratory is 1000. Initiallythereare100fruitiespresent.Thatis P (0) = 100 . (i) Fig Figure ure 8 sho showsthe wsthe slo slopeeldfor peeldfor the dif differ ferent entialequa ialequatio tion n giv givenabove enabove.. Drawthesolutioncurveontheslopeeld. P P 1200 1200
1000
1000
800 800
600 600
400 400
200 200
t
t 10 10
20 20
30 30
Figre 8
(ii) Show that
1
P
+
1 1000 − P
=
1000
P(1000 − P)
.
40 40
50 50
(3 marks)
(1 mark) 25
PLEASE TURN OVER
(iii) Solve the differential equation
d P dt
=
P(t ) =
1000 − P to show that P 500 1000 49
1000 098t 1 + 9e − 0.098
.
(5 marks) (iv)Howmanydaysdoesittakeforthepopulationoffruitiestoreach500?
(1 mark) 26
(b) More More genera generally lly, , consid considera era modelfor modelfor the growth growth ofa popula populatio tion n offruit ies tobe d P dt
=
kP(1000 − P ) 1000
, where P (0) = 100 and k > 0 is a constant.
Let T rep repres resent ent the time tak takenforthe enforthe pop popula ulatio tion n toreach 500 fru fruities. ities. The approximate relationship between T and k is T ≈
2.20 k
days.
(i) To what level of accuracy does this relationship hold hold for your answer to part (a)(i (a)(iv)? v)?
(1 mark) (ii) Suggest a value of k for which T will be less than your answer to part (a)(iv).
(1 mark)
(c) Afterthirt Afterthirtydays600 ydays600 fruitiesare fruitiesare remove removed d from from the popula populatio tion n descri described bed in part (a)(iii). Using the value of k from part (b)(ii), state a new differential equation with a new initial condition that could be used to model the growth of the remaining population.
(2 marks)
27
PLEASE TURN OVER
QuESTION 14
(15 marks) The curve x = sin 2 t , y = cos t
While designing an animated advertisement for Fable 8 Fantasy Bookstores, a computer graphics specialist uses a moving point controlled by the parametric equations
x = sin 2t , y = cos t
�
where 0 ≤ t ≤ 2π is the time taken for the point to complete one circuit of the curve shown in Figure 9.
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(a) Find the velocity vector for the moving point.
(2 marks)
(b) (i) Hence show that s(t ), ), the speed of the moving point at time t , is given by
s(t ) = 16 sin 4 t − 15 sin 2 t + 4.
(3 marks) 28
(ii) Graph s(t ) on the axes in Figure 10. �
�
� �
�
�
(c) (i) Show that s' (t ) =
sin 2t ( 32 sin 2 t − 15 ) 2 s (t )
(3 marks)
.
(4 marks)
29
PLEASE TURN OVER
(ii) (ii ) Hencend Hencend exa exactvalue ctvaluesfor sfor the max maximu imum m spe speedand edand min minimu imum m spe speedof edof the moving point as it completes one circuit of the curve shown in Figure 9.
(3 marks) 30
SECTION C (Questions 15 and 16) (15 marks)
Answer one question from this section,
31
either Question
15
or Question
16.
PLEASEPLEASE TURN OV ER OVER TURN
Answer either Question 15
or Question
16.
(15 marks)
QuESTION 15
x' = x + y Figure 11 show shows s the theslope slope eld eldforthe forthe diff differenti erential al syste system m where 0 ≤ t ≤ 2π . y' = −2 x − y y y 22
11
--33
--22
1 --1
1 1
22
33
x x
-1 -1
-2 -2
Figure 11 Figre 11
(a) On Figure 11 draw the solution curve that passes through the point (1, 0).
(b) Show that
d y d x
=
(3 marks)
−2 x − y . x + y
(1 mark)
32
d y For a function y = f ( x ) obeying the differential equation d x method are
= xn −1 + h yn = yn −1 + hf ′( xn −1 )
xn
= f ′( x) , the equations for Euler ’s
where hisanumberofsufcientlysmallsize.
In the current situation the equations for Euler’s method can be adapted to
xn
= xn −1 + h
yn
−2 x − y = yn −1 + h n −1 n −1 . xn −1 + yn −1
With h = − 0.1,theseequationscanbeusedtondanestimateforthepositive y y-intercept of the solution curve that you drew in part (a).
(c) Comple Completethe tethe last last column column ofthe tablebelow tablebelow tond anestimate anestimate for the positiv positivee -interceptof thesolut the solution ioncurve curve. . Onlythe results results forthe rstthree calculations calculations andthe and the y-interceptof last last threecalcu threecalculati lations ons are needed needed.Y .Youdo oudo not need need toll inany oftheshaded oftheshaded cells, cells, but you may use them if you wish.
n
xn−1
yn −1
h
−2 xn −1 − yn −1 xn −1 + yn −1
yn
1
1.0
0
-0.1
-2.0000
0.2000
2
0.9
0.2000
-0.1
3
8 9 10 (4 marks)
33
PLEASE TURN OVER
(d) It can be shown that that the given differential system
x' = x + y y' = −2 x − y has a solution of the form
where 0 ≤ t ≤ 2π
x(t ) = A cos t + B sin t y (t ) = C cos t + D sin t
where A, B, C , and D are constants.
( Note Note: You do not have to prove this.) The solution curve that you drew in part (a) has initial conditions x(0) = 1, y( 0) = 0 . (i) Find the values for A, B, C , and D.
(3 marks) (ii) (ii ) Hen Hencend cend the exa exactvalue ctvalue ofthe pos positi itive ve y-intercept of the solution curve.
(2 marks)
34
(iii) Find the Cartesian equation for the solution curve.
(2 marks)
35
PLEASE TURN OVER
Answer either Question 15
QuESTION 16
or Question
16.
(15 marks)
2
(a) Find ( 2 − i ) .
(1 mark)
(b) Consider the complex iteration
z
c 1 → z + . 2 z
(i) Complete the the table of iterates below with
z
n
z
0
1
0
= 1 and c = 3 − 4i .
n
1 2 3 4 5 6 7 (2 marks)
36
(ii) Complete the table of iterates below with
z
n
z
0
i
0
= i and c = 3 − 4i .
n
1 2 3 4 5 6 7 (2 marks)
(c) (i) Show that, in general, the complex iteration z
=±
c.
z
1 c → z + has invariant points 2 z
(2 marks) (ii) Hence explain the results obtained for part (a) and parts (b)(i) (b)(i) and (ii).
(1 mark)
37
PLEASE TURN OVER
(d) (i) State the invariant points for the iteration
z
1 4 → z + . 2 z
(1 mark) (ii) Complete the table of iterates for the iteration
n
z
0
1
z
1 4 → z + with z0 = 1. 2 z
n
1 2 3 4 5 6 7 (2 marks) (iii) For the iteration small size. Show that
z
n +1
z
1 4 → z + , let zn = 2 + a , where 2 z
= 2+
a2 4 + 2a
a is a non-zero number of
.
(2 marks) 38
(iv) Hence explain that if zn
then ≠ 2, then
z
n +1
≠ 2.
(1 mark) (v) Explain any apparent contradiction contradiction in your answers to parts (d)(i), (ii), and (iv).
(1 mark)
39
PLEASE TURN OVER
You may write on this page if you need more space to nish your answers. Make sure to label each answer carefully (e.g. ‘Question 8(c)(ii) continued’).
© Senior Secondary Assessment Board of South Australia 2007 40
You may remove this page from the booklet by tearing along the perforations. LIST OF MATHEMATICAL FORMULAE FOR USE IN STAGE 2 SPECIALIST MATHEMATICS
Properties of Derivatives
Circular Functions
sin 2 A cos 2 A
tan 2 A 1
d
1
d x
sec 2 A
2
cosec A
sin( A r B )
® ¾ d x ¯ g ( x) ¿
sin A cos B r cos A sin B
d
cos( A r B) cos A cos B B sin A sin B
d x
>g ( x)@2
f ( g ( x)) f c( g ( x)) g c( x )
tan A r tan B
tan( A r B )
1 B tan A tan B
Quadratic Equations
sin 2 A 2 si sin Acos A 2
cos 2 A cos
f c( x) g ( x) f ( x ) g c( x )
d - f ( x) ½
2
1 cot A
{ f ( x ) g ( x )} f c( x ) g ( x ) f ( x) g c( x )
A sin
2
If ax
2
bx c
0 then x
2
br
b 2a
A
2
2 cos A 1 Distance from a Point to a Plane
2
1 2 si sin A
The distance from ( x1 , y1 , z1 ) to
2tan A
tan tan 2 A
Ax By Cz D
1 tan 2 A
2 sin A cos B 2 cos A cos B 2 sin A sin B
sin A sin B
Ax1
sin( A B ) sin( A B )
A
cos( A B) cos( A B )
By1
2
B
Cz1
2
0 is given by
D
.
2
C
cos( A B) cos( A B) Mensuration
2 sin 12 ( A B) cos 12 ( AB B) 1 2
cos A cos B 2 cos ( A
1 2
B) cos ( A
2 sin 12 ( A
cos A cos B
Area of sector =
B)
1 2
r 2T
Arc length = r (where is in radians)
B) sin 12 ( A B)
In any triangle ABC triangle ABC : A
Matrices and Determinants
If A
A
1
ªa bº « » ¬ c d ¼
c
then det
A
A
ad bc
C
1 ª d b º « ». a¼ A ¬ c
B
Area of triangle = a
Derivatives
f ( x) x
y
n
x
ln x
f c( x ) nx
e
e
sin x cos x tan x
log e x
b
and
b
d y
sin A
d x
a2
n 1
x
1 x cos x sin x sec 2 x
41
b2
sin B c2
a
1 2
ab sin C
c
sin C
2bc cos A
4ac
.
