MTS315114 Exam Paper · (a) Represent these equations in an augmented matrix. (b) Use a series of...

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

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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.


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


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∑ .



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


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.


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


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.



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


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.


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


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.


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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 ...

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

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∫

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.

MTS315114 Exam Paper · (a) Represent these equations in an augmented matrix. (b) Use a series of...

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