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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 Office of Tasmanian Assessment, Standards and Certification.

Tasmanian Certificate of Education

MATHEMATICS SPECIALISED

Senior Secondary

Subject Code: MTS415114

External Assessment

2015

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.

PLACE LABEL HERE

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 2015 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 any approved scientific or graphics or CAS calculator (memory may be retained). Unless instructed otherwise, calculators may be used to their full capacity when undertaking this examination. Answer each section in a separate answer booklet. All written responses must be in English.


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)

A geometric series converges to 5 (i.e. S∞ = 5 ). Given that the first term of the series is 3, determine the sum of the first ten terms. Question 2 (3 marks)

Determine under which conditions placed on x the series 1x2

−1"

#$

%

&'

r

r=1

n∑ converges.

Question 3 (5 marks)

Prove that r2 + r −1r r +1( )r=1

n∑ =

n2

n+1.

Note: You may use the result r2 + r −1r r +1( )

=r2

r +1−r −1( )

2

r.


Prove that the sequence 1− n+ n2

n2"#$

%$

&'$

($ converges to 1.

Section A continues.

SECTION A – SEQUENCES AND SERIES


Section A (continued) Question 5 (6 marks) Prove by mathematical induction that 210 + 29 + 28 + ...+ 211−n = 211− 211−n for all integers n ≥1. Question 6 (8 marks) The following is the sum of the first ten terms of a series:

12 −1+ 22 + 2+32 −3+ 42 + 4+52 −5+ ... .

Develop an expression for the sum of first 2n terms of the series, where n is odd, and use your expression to find the sum of the first 222 terms.



A= a bc d

!

"##

$

%&& and B = −1 0

0 2

"

#$$

%

&'' . Find A given that AB = A+ 2B .


Give a geometrical description of the transformation T : x , y( )→ −x + y 32

, x 3+ y2

#

$%%

&

'(( .


Prove that for all non-singular 2×2 matrices A, A−1 = 1A

.


Given M = 2 00 2

!

"##

$

%&& :

(a) Show that M 2 −5M +6I =O , where I = 1 00 1

!

"##

$

%&&and O = 0 0

0 0

!

"##

$

%&& ;

(b) Find all matrix solutions to M 2 −5M +6I =O .

Section B continues.

SECTION B – MATRICES AND LINEAR TRANSFORMATIONS


Section B (continued) Question 11 (7 marks) Consider the following system of equations:

3x + 4y + 2z =1 x + y + 4z = 7 4x +5y +7z =10 (a) Represent these equations in an augmented matrix. (b) Use a full Gauss-Jordan process showing a series of annotated steps to reduce the matrix to

reduced row-echelon form and hence solve the system of equations. (c) Give a geometrical interpretation of the result. Question 12 (7 marks) A linear transformation T : x , y( )→ x , − 2x − y( ) is a combination of three linear transformations:

A (a reflection in the line =y x ); followed by

B; followed by

C (a rotation through π2

radians clockwise).

(a) Find the matrix B. (b) The image under T of a function is given by 2x +3y = 6 . Find the equation of the function.


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) Prove that the curve with equation y = x + 2x is everywhere concave up. Question 14 (4 marks)

Show that the line with equation y =1 is a tangent to the curve with implicit equation x + y( )2= 4x at a

point with x-coordinate 1. Question 15 (5 marks) Show that 1, 0( ) is the only point of inflection of the graph of the function y = arcsin 1− x( ) . You may use your calculator to evaluate derivatives for different x values. Question 16 (6 marks) The graph of the function y = x2 +1( )ex has no x-intercepts and the y-intercept is at y =1. Find exact

values of all stationary points and points of inflection and hence provide a sketch of the graph. You may use your calculator to evaluate any derivatives, if necessary.

Section C continues.

SECTION C – DIFFERENTIAL CALCULUS, AREAS AND VOLUMES


Section C (continued) Question 17 (6 marks) A spherical container of radius 2 metres is partly filled with water, as shown below. The depth of the water in the container is 1 metre.

A cross-section of the container is represented by the graph with equation x2 + y2 = 4 . Calculate the exact volume of water in the container algebraically and hence show that the proportion of water in the

container is 532

of the volume of the container.

Question 18 (6 marks) A tangent line passing through the origin is drawn to the graph with equation y = x3 +16 . Show that the area between this tangent line and the graph with equationy = x3 +16 is 108 square units.

1metre

0

0



Evaluate sin x cos4 xdx0

π

∫ .


By employing an appropriate substitution or otherwise, show that x1+ x

dx0

1∫ =1− ln 2 .


Solve for y the differential equation x dydx

= x + y given that y = 0 when x = 2 .


Given the standard identity cos2x = 2cos2 x −1, show that 11+ cos x

dxπ3

π2∫ =1− 1

3.

Section D continues.

SECTION D – INTEGRAL CALCULUS


Section D (continued) Question 23 (7 marks)

(a) Use integration by parts to verify that ln x( )ndx∫ = x ln x( )

n− n ln x( )

n−1dx∫ .

(b) Hence, or by employing some other algebraic means, find ln x( )3dx∫ .

Question 24 (7 marks) The rate at which a disease spreads throughout a community of 1000 people is directly proportional to the product of x, the number of people who have the disease at time t months, and the number of people in the community who do not have the disease at that time.

This is described by the differential equation dxdt= kx 1000− x( ) , for some constant k.

Given that there were initially 250 people who had the disease, and that there were 500 people with the disease after one month, solve this differential equation and hence determine the number of people with the disease after three months.



Given that 1w=11− i

+1i

, express w in rectangular Cartesian form.

Question 26 (3 marks) For any complex number z = x + iy with complex conjugate z , find Arg z + i z( ) . Question 27 (5 marks)

If z = rcisθ , find all values of Arg z( ) such that z2 is real. Question 28 (5 marks) You are given F (z) = z5 − 4z4 +7z3 − 4z2 +6z . Show that z = i is a solution to F (z) = 0 , and hence find all real factors of F (z) .

Section E continues.

SECTION E – COMPLEX NUMBERS


Section E (continued) Question 29 (6 marks) Show the region in the Argand plane defined by:

z :1< z ≤ 2{ }∩ z : 0 ≤ z + z ≤ 2{ }∩ z : Arg z( ) ≥ π3$%&

'()

.

Intersection points on the boundary of the region need to be shown. Question 30 (8 marks)

(a) Solve z6 − z4 + z2 −1= 0 .

(b) Hence solve z6 − 4z4 +16z2 −64 = 0, giving answers in rectangular Cartesian form.


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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 Office of Tasmanian Assessment, Standards and Certification.

2015 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 ...

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 xx

xcossintan =

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

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: MTS415114

2015 External Examination Information Sheet

Page 1 of 2

TRIGONOMETRY:

€

sin2 A + cos2 A = 1

€

1 + tan2 A = sec2 A

€

1 + cot2 A = cosec2 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:

21

1arcsin

xdxxd

−=

21

1arccos

xdxxd

−−=

21

1arctan

xdxxd

+=

caxdx

xa+=∫

−arcsin

221 or c

ax+− arccos c

ax

adx

xa+=

+∫ arctan1122

€

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∫∫ Trapezoidal Rule:

[ ]∫ +++≈ −

b

a nn xfxfxfxfxdxxf )()(2)(2)(2

)( 110 …δ , where

nabx −

=δ

Volumes of solids of revolution:

about x-axis

€

π y2dxa

b∫ about y-axis

€

π x2dya

b∫

of 2

SEQUENCES AND SERIES: Arithmetic Series: 1)d(nanu −+=

€

Sn =n2(2a + (n −1)d) or

€

n2(a + l), where dna )1( −+=ℓ

is the last term Geometric Series: 1narnu

−=

€

Sn =a(1− rn )

1− r if r ≠1 or na when r = 1

€

S∞ =a

1− r if r <1

( )∑=

+=

n

r

nnr1 2

1 ( )( )∑=

++=

n

r

nnnr1 6

121 2 ( )∑=

+=

n

r

nnr1 4

212 3

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:

....!

)0()(...+!3

3)0(

2!

2)0()0()0()( +×+×ʹ′ʹ′ʹ′+×ʹ′ʹ′+×ʹ′+=

n

nxnfxfxfxffxf

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.

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