2015
HIGHER SCHOOL CERTIFICATE
EXAMINATION
Mathematics Extension 2
General Instructions
•Readingtime–5minutes
•Workingtime–3hours
•Writeusingblackpen
•Board-approvedcalculatorsmay
beused
•Atableofstandardintegralsis
providedatthebackofthispaper
•InQuestions11–16,show
relevantmathematicalreasoning
and/orcalculations
2610
Total marks – 100
Section I
Pages 2–6
10 marks
•AttemptQuestions1–10
•Allowabout15minutesforthissection
Section II
Pages 7–18
90 marks
•AttemptQuestions11–16
•Allowabout2hoursand45minutesforthis
section
Section I
10 marks
Attempt Questions 1–10
Allow about 15 minutes for this section
Usethemultiple-choiceanswersheetforQuestions1–10.
1
Whichconichaseccentricity
(A)
(B)
(C)
(D)
2
13
?
3
x 2 y2
+
=1
3
2
x2
32
+
y2
22
=1
x 2 y2
−
=1
3
2
x2
32
–
y2
22
=1
Whatvalueofzsatisfies z2 = 7 –24i ?
(A) 4−3i
(B) −4−3i
(C) 3−4i
(D) −3−4i
–2–
3
Whichgraphbestrepresentsthecurve y = ( x − 1)2 ( x + 3)5 ?
y
(A)
–3
O
1
x
y
(C)
–3
O
y
(B)
–1 O
x
–1 O
4
Thepolynomial x3 + x2 −5x +3 hasadoublerootat x = a.
Whatisthevalueofa ?
(A) −
5
3
(B) − 1
(C) 1
(D)
5
3
–3–
x
3
x
y
(D)
1
3
5
Given that z = 1 − i, which expression is equal to z3 ?
(A)
−3π ⎞
−3π ⎞ ⎞
⎛
2 ⎜ cos ⎛
+ i sin ⎛
⎝ 4 ⎠
⎝ 4 ⎠ ⎟⎠
⎝
−3π ⎞
−3π ⎞ ⎞
⎛
(B) 2 2 ⎜ cos ⎛
+ i sin ⎛
⎝ 4 ⎠
⎝ 4 ⎠ ⎟⎠
⎝
(C)
3π
3π ⎞
⎛
2 ⎜ cos ⎛ ⎞ + i sin ⎛ ⎞ ⎟
⎝
⎠
⎝
⎝
4
4 ⎠⎠
3π
3π ⎞
⎛
(D) 2 2 ⎜ cos ⎛ ⎞ + i sin ⎛ ⎞ ⎟
⎝ 4 ⎠
⎝ 4 ⎠⎠
⎝
6
⌠
Whichexpressionisequalto ⎮ x 2 sin x dx ?
⌡
⌠
(A) −x 2 cos x − ⎮ 2x cos x dx
⌡
⌠
(B) −2x cos x + ⎮ x 2 cos x dx
⌡
⌠
(C) −x 2 cos x + ⎮ 2x cos x dx
⌡
⌠
(D) −2x cos x − ⎮ x 2 cos x dx
⌡
7
The numbers 1, 2, . . . n, for n ≥ 4, are randomly arranged in a row.
What is the probability that the number 1 is somewhere to the left of the number 2?
(A)
1
2
(B)
1
n
(C)
1
2 ( n − 2 )!
(D)
1
2 ( n − 1)!
–4–
8
The graph of the function y = ƒ ( x ) is shown.
y
O
a
b
x
A second graph is obtained from the function y = ƒ ( x ) .
y
O
a
Whichequationbestrepresentsthesecondgraph?
(A) y 2 = ƒ ( x )
(B)
y2 = ƒ ( x )
(C)
y=
(D) y = f
ƒ (x)
( x)
–5–
b
x
9
Thecomplexnumberzsatisfies z − 1 = 1 .
WhatisthegreatestdistancethatzcanbefromthepointiontheArganddiagram?
(A) 1
(B)
5
(C) 2 2
(D)
10
2 +1
Consider the expansion of
(1 + x + x 2 + ! + x n )(1 + 2x + 3x 2 + ! + ( n + 1) x n ) .
Whatisthecoefficientof x nwhenn = 100?
(A) 4950
(B) 5050
(C) 5151
(D) 5253
–6–
Section II
90 marks
Attempt Questions 11–16
Allow about 2 hours and 45 minutes for this section
AnswereachquestioninaSEPARATEwritingbooklet.Extrawritingbookletsareavailable.
In Questions 11–16, your responses should include relevant mathematical reasoning and/or
calculations.
Question 11 (15marks)UseaSEPARATEwritingbooklet.
(a)
Express
4 + 3i
intheform x + iy, wherexandyarereal.
2−i
2
π
π
(b) Considerthecomplexnumbers z = − 3 + i and w = 3 ⎛ cos + i sin ⎞ .
⎝
7
7⎠
(i) Evaluate z .
1
(ii) Evaluatearg(z).
1
(iii) Findtheargumentof
(c)
FindA,BandCsuchthat
(d) Sketch
(e)
(f)
z
.
w
(
1
x x2 + 2
1
)
=
A
x
+
Bx + C
x2 + 2
.
x 2 y2
+
= 1 indicatingthecoordinatesofthefoci.
25 16
Findthevalueof
dy
atthepoint(2,–1)onthecurve x + x2y3 = –2.
dx
2
2
3
θ
(i) Showthat cot θ + cosec θ = cot ⎛ ⎞ .
⎝ 2⎠
2
⌠
(ii) Hence,orotherwise,find ⎮ ( cot θ + cosec θ ) dθ .
⌡
1
–7–
Question 12 (15 marks) Use a SEPARATE writing booklet.
π
.
4
(a)
Thecomplexnumberzissuchthat z = 2 and arg ( z ) =
Plot each of the following complex numbers on the same half-page Argand
diagram.
(i) z
1
(ii) u = z2
1
(iii) v = z 2 − z
1
(b) The polynomial P ( x ) = x 4 − 4x 3 + 11x 2 − 14x + 10 has roots a + ib and
a + 2ibwhereaandbarerealand b ≠0.
(i) Byevaluatingaandb,findalltherootsofP (x).
3
(ii) Hence,orotherwise,findonequadraticpolynomialwithrealcoefficients
thatisafactorofP (x).
1
(c)
(i) Bywriting
( x − 2 )( x − 5)
x −1
oftheobliqueasymptoteof y =
a
,findtheequation
x −1
2
, clearly indicating all
2
intheform mx + b +
(ii) Hence sketch the graph y =
( x − 2 )( x − 5)
x −1
( x − 2 )( x − 5)
interceptsandasymptotes.
x −1
.
Question 12 continues on page 9
–8–
Question 12 (continued)
(d) Thediagramshowsthegraph y = x + 1 for 0≤ x ≤3. Theshadedregionis
rotatedabouttheline x =3 toformasolid.
y
1
3
O
x
Use the method of cylindrical shells to find the volume of the solid.
End of Question 12
–9–
4
Question 13 (15 marks) Use a SEPARATE writing booklet.
(a)
The hyperbolas H1 :
diagram.
x2
a2
−
y2
b2
= 1 and H 2 :
x2
a2
−
y2
b2
= −1 are shown in the
LetP (a secq,b tanq ) lieonH1asshownonthediagram.
LetQbethepoint(a tanq,b secq ).
H2
y
H1
Q
P
x
O
H1
H2
(i) Verify that the coordinates of Q (a tanq, b secq ) satisfy the equation
forH2.
1
(ii) ShowthattheequationofthelinePQis bx + ay = ab (tanq + secq ).
2
(iii) ProvethattheareaofrOPQisindependentofq.
3
Question 13 continues on page 11
– 10 –
Question 13 (continued)
(b) Two quarter cylinders, each of radius a, intersect at right angles to form the
shaded solid.
a
a
A horizontal slice ABCD of the solid is taken at height h from the base. You may
assume that ABCD is a square, and is parallel to the base.
B
h
C
A
D
(i) Showthat AB = a 2 − h 2 .
1
(ii) Findthevolumeofthesolid.
2
Question 13 continues on page 12
– 11 –
Question 13 (continued)
(c)
Asmallsphericalballoonisreleasedandrisesintotheair.Attimetseconds,
ithasradiusrcm,surfacearea S =4pr 2 andvolume V =
4 3
pr .
3
As the balloon rises it expands, causing its surface area to increase at a rate
1
4π ⎞ 3
of ⎛
cm2 s−1.Astheballoonexpandsitmaintainsasphericalshape.
⎝ 3 ⎠
1
dr
1 ⎛4 ⎞3
π .
(i) Byconsideringthesurfacearea,showthat
=
dt 8π r ⎝ 3 ⎠
dV 1 3
(ii) Showthat
= V .
dt
2
(iii) Whentheballoonisreleaseditsvolumeis8000 cm3. Whenthevolume
oftheballoonreaches64 000 cm3itwillburst.
2
1
2
Howlongafteritisreleasedwilltheballoonburst?
End of Question 13
– 12 –
2
Question 14 (15 marks) Use a SEPARATE writing booklet.
(a)
(i) Differentiatesinn –1q cosq,expressingtheresultintermsofsinqonly.
π
π
( n − 1) ⌠ 2 n−2
⌠2
(ii) Hence,orotherwise,deducethat ⎮ sin n θ dθ =
sin θ dθ ,
n ⎮
⌡0
⌡0
forn > 1.
π
⌠2
(iii) Find ⎮ sin 4 θ dθ .
⌡0
2
2
1
(b) Thecubicequation x3 – px + q = 0 hasroots a,bandg.
Itisgiventhat a2 + b 2 + g 2 = 16 and a3 + b 3 + g 3 = –9.
(i) Showthat p =8.
1
(ii) Findthevalueofq.
2
(iii) Findthevalueof a4 + b 4 + g 4.
2
Question 14 continues on page 14
– 13 –
Question 14 (continued)
(c)
A car of mass m is driven at speed v around a circular track of radius r. The
π
track is banked at a constant angle q to the horizontal, where 0 < θ < . At
2
thespeedvthereisatendencyforthecartoslideupthetrack.Thisisopposed
byafrictionalforcemN,whereNisthenormalreactionbetweenthecarandthe
track,andm > 0.Theaccelerationduetogravityisg.
N
mN
q
mg
(ii) AttheparticularspeedV,where V 2 = rg,thereisstillatendencyforthe
cartoslideupthetrack.
Usingtheresultfrompart(i),orotherwise,showthat m < 1.
End of Question 14
– 14 –
2
Question 15 (15 marks) Use a SEPARATE writing booklet.
(a)
A particle A of unit mass travels horizontally through a viscous medium. When
t = 0, the particle is at point O with initial speed u. The resistance on particle A
due to the medium is kv 2, where v is the velocity of the particle at time t and k
is a positive constant.
When t = 0, a second particle B of equal mass is projected vertically upwards
from O with the same initial speed u through the same medium. It experiences
both a gravitational force and a resistance due to the medium. The resistance
on particle B is kw 2, where w is the velocity of the particle B at time t. The
acceleration due to gravity is g.
1
1
= kt + .
u
v
(i) ShowthatthevelocityvofparticleAisgivenby
(ii) ByconsideringthevelocitywofparticleB,showthat
t=
1
gk
3
⎛
k⎞
k ⎞⎞
−1 ⎛
−1 ⎛
.
tan
u
w
−
tan
⎜
⎜⎝ g ⎟⎠
⎜⎝
g ⎟⎠ ⎟⎠
⎝
(iii) ShowthatthevelocityVofparticleAwhenparticleBisatrestisgiven
by
1 1
= +
V u
2
1
⎛ k⎞
k
.
tan −1 ⎜ u
g
⎝ g ⎟⎠
(iv) Hence,ifuisverylarge,explainwhy V ≈
2
π
g
.
k
Question 15 continues on page 16
– 15 –
1
Question 15 (continued)
(b) Supposethat x ≥ 0 andnisapositiveinteger.
1
≤ 1.
1+ x
(i) Showthat 1 − x ≤
(ii) Hence,orotherwise,showthat 1 −
1 n
(iii) Hence,explainwhy lim ⎛ 1 + ⎞ = e.
n→∞ ⎝
n⎠
(c)
Forpositiverealnumbersx and y,
(i) Prove
(ii) Prove
andd.
xy ≤
4
2
xy ≤
1
1
≤ n ln ⎛ 1 + ⎞ ≤ 1.
⎝
2n
n⎠
1
x+y
. (DoNOTprovethis.)
2
x 2 + y2
,forpositiverealnumbersxandy.
2
abcd ≤
2
a2 + b2 + c2 + d 2
, for positive real numbers a, b, c
4
End of Question 15
– 16 –
1
2
Question 16 (15 marks) Use a SEPARATE writing booklet.
(a)
(i) A table has 3 rows and 5 columns, creating 15 cells as shown.
Counters are to be placed randomly on the table so that there is one
counter in each cell. There are 5 identical black counters and 10 identical
white counters.
Show that the probability that there is exactly one black counter in each
81
column is .
1001
(ii) The table is extended to have n rows and q columns. There are
nq counters, where q are identical black counters and the remainder are
identical white counters. The counters are placed randomly on the table
with one counter in each cell.
Let Pn be the probability that each column contains exactly one black
counter.
Showthat Pn =
2
2
nq
.
⎛ nq ⎞
⎜⎝ q ⎟⎠
2
(iii) Find lim Pn .
n→∞
Question 16 continues on page 18
– 17 –
Question16(continued)
(b) Letnbeapositiveinteger.
(i) Byconsidering ( cos α + i sin α )2n ,showthat
2
⎛ 2n⎞
⎛ 2n⎞
cos (2n α ) = cos2n α − ⎜ ⎟ cos2n−2 α sin 2 α + ⎜ ⎟ cos2n−4 α sin 4 α − !
⎝ 2⎠
⎝ 4⎠
⎛ 2n ⎞
+ ! + (−1)n −1 ⎜
cos2 α sin 2n−2 α + (−1)n sin 2n α .
⎟
⎝ 2n − 2⎠
(
)
Let T2n ( x ) = cos 2n cos−1 x ,for −1 ≤ x ≤1.
2
(ii) Showthat
⎛ 2n⎞
⎛ 2n⎞
T2n ( x ) = x 2 n − ⎜ ⎟ x 2n − 2 1 − x 2 + ⎜ ⎟ x 2n−4 1 − x 2
⎝ 2⎠
⎝ 4⎠
(
)
(
)
2
(
+ ! + (−1)n 1 − x 2
(iii) ByconsideringtherootsofT2n(x),findthevalueof
)
n
.
3
3π
π
⎛ (4n − 1) π ⎞
cos ⎛ ⎞ cos ⎛ ⎞ ! cos ⎜
⎟.
⎝ 4n ⎠
⎝ 4n ⎠
⎝
4n ⎠
2
(iv) Provethat
nπ
⎛ 2n⎞ ⎛ 2n⎞ ⎛ 2n⎞
⎛ 2n⎞
1 − ⎜ ⎟ + ⎜ ⎟ − ⎜ ⎟ + ! + (−1)n ⎜ ⎟ = 2n cos ⎛ ⎞ .
⎝ 2 ⎠
⎝ 2⎠ ⎝ 4⎠ ⎝ 6⎠
⎝ 2n⎠
End of paper
– 18 –
BLANK PAGE
– 19 –
STANDARD INTEGRALS
⌠ n
⎮ x dx
⌡
=
⌠ 1
⎮ x dx
⌡
= ln x , x > 0
⌠ ax
⎮ e dx
⌡
=
1 ax
e , a≠0
a
⌠
⎮ cos ax dx
⌡
=
1
sin ax , a ≠ 0
a
⌠
⎮ sin ax dx
⌡
1
= − cos ax , a ≠ 0
a
⌠
2
⎮ sec ax dx
⌡
=
1 n+1
x , n ≠ −1; x ≠ 0, if n < 0
n +1
1
tan ax , a ≠ 0
a
⌠
1
⎮ sec ax tan ax dx = a sec ax , a ≠ 0
⌡
⌠
1
dx
⎮ 2
⌡ a + x2
=
⌠
⎮
⌡
dx
x
= sin −1 , a > 0, − a < x < a
a
dx
= ln x + x 2 − a 2 , x > a > 0
dx
= ln x + x 2 + a 2
⌠
⎮
⌡
⌠
⎮
⌡
1
a2 − x 2
1
2
x −a
2
1
2
x +a
2
1
x
tan −1 , a ≠ 0
a
a
(
)
(
)
NOTE : ln x = loge x , x > 0
– 20 –
© 2015 Board of Studies, Teaching and Educational Standards NSW