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Pages: 16 Questions: 30 ©Copyright for part(s) of this examination may be held by individuals and/or organisations other than the Tasmanian Qualifications Authority.
Tasmanian Certificate of Education
MATHEMATICS SPECIALISED
Senior Secondary
Subject Code: MTS315114
External Assessment
2014
Writing Time: Three hours
On the basis of your performance in this examination, the examiners will provide results on each of the following criteria taken from the course statement: Criterion 4 Demonstrate an understanding of finite and infinite sequences and
series. Criterion 5 Demonstrate an understanding of matrices and linear
transformations. Criterion 6 Use differential calculus and apply integral calculus to areas and
volumes. Criterion 7 Use techniques of integration and solve differential equations. Criterion 8 Demonstrate an understanding of complex numbers. T
AS
MA
NIA
N Q
UA
LIF
ICA
TIO
NS
AU
TH
OR
ITY
PLACE LABEL HERE
Mathematics – Specialised
Page 2
BLANK PAGE
Mathematics – Specialised
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CANDIDATE INSTRUCTIONS You MUST make sure that your responses to the questions in this examination paper will show your achievement in the criteria being assessed. This examination paper has FIVE sections. You must answer ALL questions. It is suggested that you spend approximately 36 minutes on each section. The 2014 Information Sheets for Mathematics Specialised and Mathematics Methods can be used throughout the examination (provided with the paper). No other written material is allowed into the examination. Markers will look at your presentation of answers and at the arguments leading to answers when determining your result on each criterion. You must show the method used to solve a question. If you show only your answers you will get few, if any, marks. You are permitted to bring into the examination room: pens, pencils, highlighters, erasers, sharpeners, rulers, a protractor, set-squares, aids for curve sketching and an approved scientific or Graphics or CAS calculator (memory may be retained). Unless instructed otherwise, your calculator may be used to its full capacity when undertaking this examination. Answer each section in a separate answer booklet. All written responses must be in English.
Mathematics – Specialised
Page 4
This section assesses Criterion 4. Markers will look at your presentation of answers and at the arguments leading to answers when determining your result on each criterion. You must show the method used to solve a question. If you show only your answers you will get few, if any, marks. Use a separate answer booklet for this section. Question 1 (3 marks)
Find the minimum value of n such that 3× 4r−1( ) >109r=1
n∑ .
Question 2 (3 marks)
Determine whether or not 2748 is a member of the sequence 3n+15n+1{ }.
Question 3 (5 marks) Determine the sum to 3n terms of the series 12 + 22 −32 + 42 + 52 − 62 + 72 +82 − 92 +... Simplify your answer. Question 4 (5 marks)
Prove that the sequence n2+1n2+n+1{ } converges to 1.
Section A continues.
SECTION A – SEQUENCES AND SERIES
Mathematics – Specialised
Page 5
Section A (continued) Question 5 (7 marks)
For the series 83.6.9 +
84.7.10 +
85.8.11+
86.9.12 +... , determine
(a) the sum to n terms, and (b) the sum to infinity. Your answers do not need to be fully simplified. Question 6 (7 marks)
Determine rr=k
n∑"
#$$
%
&''
k=1
n∑ .
Mathematics – Specialised
Page 6
This section assesses Criterion 5. Markers will look at your presentation of answers and at the arguments leading to answers when determining your result on each criterion. You must show the method used to solve a question. If you show only your answers you will get few, if any, marks. Use a separate answer booklet for this section. Question 7 (3 marks)
Given A= 1 −2−3 6
04
"
#$
%
&' and B = 2 3
−1 1
"
#$
%
&' , state, giving reasons, whether or not each of the
matrices X, Y and Z exists. It is not necessary to carry out any calculations. (a) X = AB (b) Y = BA (c) Z = A2 – B2 Question 8 (3 marks) The circle with equation x2 + y2 =16 undergoes the transformation T : (x, y)→ (x + y, x − y). What is the area enclosed by the resulting curve? Question 9 (5 marks) Given that P and Q are 2×2 matrices, expand and simplify (P +Q)2 − (P −Q)2. Question 10 (5 marks) After undergoing the transformation which rotates the plane anti-clockwise through 2π3 radians, the
image of a certain curve is the parabola 4y = x2 . Determine the equation of the original curve.
Section B continues.
SECTION B – MATRICES AND LINEAR TRANSFORMATIONS
Mathematics – Specialised
Page 7
Section B (continued) Question 11 (7 marks) Consider the following system of equations: 2x + y− 6z = 0 x −3z = 2 4x + y−12z = 4 (a) Represent these equations in an augmented matrix. (b) Use a series of annotated steps to reduce the matrix to reduced row-echelon form and hence solve
the system of equations. (c) Interpret the result. Question 12 (7 marks) Consider Ellipse A and Ellipse B in the diagram at right.
Ellipse A has equation (x+4)2
16 + (y−2)2
4 =1. (a) Determine a linear transformation under which
Ellipse A is transformed into Ellipse B. (b) Use this transformation to determine the
equation of Ellipse B.
Mathematics – Specialised
Page 8
This section assesses Criterion 6. Markers will look at your presentation of answers and at the statement of arguments leading to answers when determining your result on each criterion. You must show the method used to solve a question. If you show only your answers you will get few, if any, marks. Use a separate answer booklet for this section. Question 13 (3 marks) The area between the x-axis and the curve 𝑦 = sec𝑥 on the interval 0, π4
!"
#$ is rotated about the x-axis.
Find the volume generated. Question 14 (3 marks) Without using your calculator, determine the derivative with respect to x of y = cos−1(sin(x)) , given that 0 < x < π
2 . Express your answer in simplest form. Question 15 (5 marks) Given that ddθ cotθ = −cosec
2θ and that cot2 A+1= cosec2A , prove that ddx cot−1 x = − 1
1+x2.
Question 16 (5 marks) The curves y = 2x and y = 4x − 2 intersect when x = a. Show that the area between the curves on the interval 0,a[ ] is 2− 1
2 ln2 .
Section C continues.
SECTION C – DIFFERENTIAL CALCULUS, AREAS AND VOLUMES
Mathematics – Specialised
Page 9
Section C (continued) Question 17 (7 marks) The curve with equation x2 − 4xy+ y2 +3x − 2y−3= 0 has two y-intercepts. (a) Determine these y-intercepts. (b) Determine the equation of the tangent at each y-intercept. (c) Determine the point of intersection of these tangents. Question 18 (7 marks) Let f (x) = cos(ex ) for x ≤ 2. (a) Determine f '(x) and f ''(x) without using your calculator. (b) Determine and classify the stationary points of f (x). Exact values are required, and your
calculator should not be used. (c) What happens to f (x) as x→−∞? (d) Sketch the graph of the function y = f (x). Exact values of the zeros, y-intercept and end point
should be shown. You are not required to determine the inflection points of the function.
Mathematics – Specialised
Page 10
This section assesses Criterion 7. Markers will look at your presentation of answers and at the arguments leading to answers when determining your result on each criterion. You must show the method used to solve a question. If you show only your answers you will get few, if any, marks. Use a separate answer booklet for this section. Question 19 (3 marks)
Without using your calculator, determine the exact value of 2xx2−1
2
3∫ dx.
Question 20 (3 marks)
Show that y = 12 ex2 − 12 is a solution to the differential equation dydx − 2xy = x.
Question 21 (5 marks) Solve the differential equation dydx = (1+ x
2)(1+ y2) , given that when x =1, y = 0. Question 22 (5 marks) Solve the differential equation dydx =
xy +
yx , given that when x =1, y = 2.
Section D continues.
SECTION D – INTEGRAL CALCULUS
Mathematics – Specialised
Page 11
Section D (continued) Question 23 (7 marks)
Using partial fractions, or otherwise, find the exact value of 4x2+4x+4(1+x2 )2
−1
1∫ dx.
All working must be shown. Question 24 (7 marks) Without using your calculator, determine eax cos∫ bx dx , given that a and b are constants.
Mathematics – Specialised
Page 12
This section assesses Criterion 8. Markers will look at your presentation of answers and at the arguments leading to answers when determining your result on each criterion. You must show the method used to solve a question. If you show only your answers you will get few, if any, marks. Use a separate answer booklet for this section. Question 25 (3 marks) If z = −3+ i 3, determine z and Arg(z) without using your calculator. Question 26 (3 marks) If w = 2cis π3 , show on the Argand plane the location of the complex numbers w, w2 and Arg(w). Question 27 (5 marks) Define the set of complex numbers z represented by the shaded region of the Argand plane shown at right.
Section E continues.
SECTION E – COMPLEX NUMBERS
Mathematics – Specialised
Page 13
Section E (continued) Question 28 (5 marks) (a) Show that (5+ 2i)2 = 21+ 20i. (b) Without using your calculator, solve z2 − z− (5+ 5i) = 0 for complex numbers z. Question 29 (7 marks) (a) Show that 2+ i is a root of the equation z4 − 4z3 + 9z2 −16z+ 20 = 0. Your calculator may be
used to simplify powers of a complex number. (b) Hence, without using your calculator, write the polynomial 𝑧! − 4𝑧! + 9𝑧! − 16𝑧 + 20 as a
product of linear factors. Question 30 (7 marks) Factorise P(z) = z5 +16z into both real and linear factors, with complex numbers written in the form a + ib.
Mathematics – Specialised
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Mathematics – Specialised
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Mathematics – Specialised
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This question paper and any materials associated with this examination (including answer booklets, cover sheets, rough note paper, or information sheets) remain the property of the Tasmanian Qualifications Authority.
2014 External Examination Information Sheet
Page 1 of 3
Mathematics Methods
Subject Code: MTM315114
FUNCTION STUDY
Quadratic Formula: If 02 =++ cbxax , then a
acbbx
242 −±−
=
Graph Shapes:
Quadratic Cubic Hyperbola Truncus
( ) khxay +−= 2 ( ) khxay +−= 3 khxay +−
= ( )
khxay +−
= 2
Square Root Circle Exponential Logarithmic khxay +−= ( ) ( ) 222 rkyhx =−+− kbay x +×= ( ) khxay n +−= log
Graphical Transformations: The graph of:
)(xfy −= is a reflection of the graph of )(xfy= in the x axis
)( xfy −= is a reflection of the graph of )(xfy= in the y axis
)(xfay= is a dilation of the graph of )(xfy= by factor a in the direction of the y axis
)(axfy= is a dilation of the graph of )(xfy= by factor a1 in the direction of the x axis
)( bxfy += is a translation of the graph of )(xfy= by b units to the left
bxfy += )( is a translation of the graph of )(xfy= by b units upwards Index Laws
yxyx aaa +=× yxyx aaa −=÷
( ) yxyx aa ×=
( ) yy aa =1
( ) y xyx
aa =
Log Laws yxyx aaa logloglog +=
yxyx
aaa logloglog −=⎟⎟⎠
⎞⎜⎜⎝
⎛
xnx an
a loglog =
axx
b
ba log
loglog =
Useful log results Definition: If xay= then
xya =log 01log =a
01ln = 1log =aa
1ln =e
Inverse Functions
( ){ } ( ){ } xxffxff == −− 11 Binomial Expansion ( ) n
nnn
nnnnnnnnn yCyxCyxCyxCxCyx +++++=+ −
−−− 1
122
21
10 ...
Page 2 of 3
CIRCULAR FUNCTIONS
Conversion:
To convert from radians to degrees multiply by π
180
To convert from degrees to radians multiply by 180π
Basic Identities:
1cossin 22 =+ xx xxx
cossintan =
xxtan1cot =
xxcos1sec =
xxsin1cosec =
Multiple Angle Formulae:
( ) BABABA sincoscossinsin +=+ ( ) BABABA sincoscossinsin −=−
( ) BABABA sinsincoscoscos −=+ ( ) BABABA sinsincoscoscos +=−
AAA cossin22sin = AAAAA 2222 sin211cos2sincos2cos −=−=−=
( )BABABA
tantan1tantantan
−
+=+
( )
BABABA
tantan1tantantan
+
−=−
AAA 2tan1
tan22tan−
=
Exact Values: Cast Diagram:
x 0 6π
4π
3π
2π π
23π 2π
xsin 0 21
22
23 1 0 -1 0
xcos 1 23
22
21 0 -1 0 1
xtan 0 33 1 3 undefined 0 undefined 0
Trigonometric Graphs:
xy sin= xy cos= xy tan=
Graphical Transformation: The graph of
( ) cbxnay ++= sin or ( ) cbxnay ++= cos has: The graph of
cbxnay ++= )(tan has: amplitude: |a|
period: nπ2
phase shift: b (shift of b units to the left) vertical shift: c units upwards
dilation: by factor a in the direction of the y axis
period: nπ
phase shift: b (shift of b units to the left) vertical shift: c units upwards
C
A S
T
Page 3 of 3
Trigonometric Equations:
If ax =sin then ( ) anx n arcsin1−+= π , Z∈n If ax =cos then anx arccos2 ±= π , Z∈n If ax =tan then anx arctan+= π , Z∈n CALCULUS
Definition of Derivative: ( )h
xfhxfxfh
)()(lim0
' −+=
→
Differentiation and Integration
Differentiation Formulae Function Derivative
nx 1−nxn
xsin xcos
xcos xsin−
xtan x
x 22
cos1orsec
xe xe
xxe lnorlog x1
)().( xgxf )().(')(').( xgxfxgxf +
)()(xgxf
{ }2)()(').()(').(
xgxgxfxfxg −
{ })(xfg { } )('.)(' xfxfg
Integration Formulae
Function Integral
a cax+
nx cnxn
++
+
1
1
( )nbax+ ( ) cnabax n
++
+ +
)1(
1
xe cex +
x1 cx +ln
xsin cx+− cos
xcos cx+sin
PROBABILITY DISTRIBUTIONS
Combinations: ( )!!!rnr
nCrn
−= 123)2)(1(! ××−−= !nnnn
Discrete Random Distribution Binomial Distribution Hypergeometric
Distribution
( )x=XPr as table ( ) ( ) xnxx
n ppCx −−== 1XPr ( ) ( )( )n
Nxn
DNx
D
CCCx −
−
==XPr
Expected Value ( ) ( )( )∑ == xx XPr.XE np=µ NnD
=µ
Variance ( ) ( )[ ]22 XEXE 2 −=σ ( )pnp −= 12σ ⎟⎟⎠
⎞⎜⎜⎝
⎛
−
−⎟⎠
⎞⎜⎝
⎛ −⎟⎠
⎞⎜⎝
⎛=1
12
NnN
ND
NnD
σ
Standard Normal:
σµ−
=xz
Mathematics Specialised
Subject Code: MTS315114
2014 External Examination Information Sheet
Page 1 of 2
TRIGONOMETRY:
€
sin2 A + cos2 A = 1
€
1 + tan2 A = sec2 A
€
1 + cot2 A = cosec2 A
€
sin(A + B) = sinA cosB + cosA sinB
€
cos(A + B) = cosA cosB − sinA sinB
€
sin(A − B) = sinA cosB − cosA sinB
€
cos(A − B) = cosA cosB + sinA sinB
€
tan(A + B) =tanA + tanB1 − tanA tanB
€
tan(A − B) =tanA − tanB1 + tanA tanB
€
sin2A = 2sinA cosA
€
tan2A =2 tanA1 − tan2 A
€
cos2A = cos2 A − sin2 A
€
cos2A = 2 cos2 A − 1
€
cos2A = 1 − 2 sin2 A
2 sin A cos B = sin (A + B) + sin (A – B) sin C + sin D = 2 sin
€
C + D2
cosC −D2
2 cos A sin B = sin (A + B) – sin (A – B) sin C – sin D = 2 cos
€
C + D2
sinC −D2
2 cos A cos B = cos (A + B) + cos (A – B) cos C + cos D = 2 cos
€
C + D2
cosC −D2
2 sin A sin B = cos (A – B) – cos (A + B) cos C – cos D = 2 sin
€
C + D2
sinD−C2
CALCULUS: d sin-1xdx
= 1
1− x2 d cos-1x
dx= − 1
1− x2 d tan-1x
dx= 1
1+ x2
1
a2 − x2∫ dx = sin-1 x
a+ c or −cos-1 x
a+ c 1
a2 + x2dx∫ = 1
atan−1 x
a+ c
€
dax
dx= ax lna
€
axdx =∫ax
lna+ c
€
d loga xdx
=1
x lna
€
loga∫ xdx =x ln x − xlna
+ c
€
f (x) " g (x)dx = f (x)g(x) − " f (x)g(x)dx + c∫∫ Volumes of solids of revolution:
about x-axis
€
π y2dxa
b∫ about y-axis
€
π x2dya
b∫
Page 2 of 2
SEQUENCES AND SERIES: Arithmetic Series:
€
Un = a + (n−1)d (often denoted by
€
l the last term)
€
Sn =n2(2a + (n −1)d) or
€
n2(a + l)
Geometric Series:
€
Un = arn −1
€
Sn =a(1− rn )
1− r if r ≠1 or na when r = 1
€
S∞ =a
1− r if r <1
€
r = n n +1( )
2r =1
n∑
€
r2 = n n +1( ) 2n +1( )
6r =1
n∑
€
r3 = n2 n +1( )2
4r =1
n∑
The sequence
€
an{ } converges to a finite limit L if, for any
€
ε > 0 ,
€
∃ N(ε) such that
€
an − L < ε ∀ n > N . The sequence
€
an{ } diverges to positive infinity if, for any
€
κ > 0,
€
∃ N(κ) such that
€
an >κ ∀ n > N . The sequence
€
an{ } diverges to negative infinity if, for any
€
κ > 0 ,
€
∃ N(κ) such that
€
an < −κ ∀ n > N . MacLaurin’s series for f(x) is:
€
f (x) = f (0) + " f (0).x + " " f (0). x2
2!+ " " " f (0). x
3
3!+...+ f (n)(0) xn
n!+ ....
MATRICES:
Some important transformations are described by the matrices: Dilation Matrices: Shear Matrices:
€
a 00 1"
# $
%
& ' and
€
1 00 a"
# $
%
& ' ,
€
1 a0 1"
# $
%
& ' and
€
1 0a 1"
# $
%
& ' .
Rotation Matrix: Reflection Matrix:
€
cos θ −sin θsin θ cos θ$
% &
'
( ) ,
€
cos 2θ sin 2θsin 2θ −cos 2θ$
% &
'
( ) .
Equation of circle centre (h, k) and radius r is
€
(x − h)2 + (y − k)2 = r2
Equation of ellipse centre (h, k) and horizontal semi-axis of length a and vertical semi-axis of
length b is
€
(x − h)2
a2+(y − k)2
b2=1.