mts315109 exam paper - tqa.tas.gov.au · page 4 section a this section assesses criterion 3....

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: MTS315109

External Assessment

2013

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 3 Demonstrate an understanding of finite and infinite sequences and

series. Criterion 4 Demonstrate an understanding of matrices and linear

transformations. Criterion 5 Use differential calculus and apply integral calculus to areas and

volumes. Criterion 6 Use techniques of integration and solve differential equations. Criterion 7 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 ensure that you have addressed ALL of the externally assessed criteria on this examination paper. This examination paper has FIVE sections. You must answer ALL questions. It is suggested that you spend approximately 36 minutes on each section. The 2013 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 3. 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)

Write down the nth term of the sequence 43! 7

,!!" 167!11

,!! 3611!15

,!!– 6415!19

,!!...

Question 2 (3 marks)

For what values of x does the sequence ex !1( )n"#$

%&'

converge to 0?


Given that n3 !1= n !1( ) n2 + n +1( ), prove that the sequence n2 + n +1n3 +1

!"#

$#

%&#

'# converges to zero.

Question 4 (5 marks) A particular sequence tn{ } is defined over the positive integers, and it can be shown that

trr=1

n! =

n n+1( ) n2+1( )2 .

(a) Evaluate the first three terms of the sequence tn{ } .

(b) Evaluate trr=51

100! .

Section A continues opposite.

SECTION A – SEQUENCES AND SERIES


Section A (continued) Question 5 (7 marks)

Prove by mathematical induction that ! 1r(r + 2)

!=! 34!!! 2n + 32(n +1)(n + 2)r=1

n" .


For the series 81.3.5

+ 34+ 83.5.7

+ 12+ 85.7.9

+ 13+ 87.9.11

+ 29+ ...!, determine

(a) the sum to 2n terms, and (b) the sum to infinity. Your answers do not need to be fully simplified.



If X 12

!3!4

"#$

%&'= 4

10!2

"#$

%&', solve for matrix X.

Question 8 (3 marks) Find the image of the y-axis under the transformation T : (x, y)! (x + y, x " y). Question 9 (5 marks) Prove that for any 2 ! 2 matrices A and B, A B = AB . Question 10 (5 marks)

After the application of the transformation defined by the matrix 35

! 45

45

! 35

"

#

$$

%

&

'', the image of a certain

curve is given by the equation x2 ! 2xy ! y2 = 4. Determine the equation of the original curve.

Section B continues opposite.

SECTION B – MATRICES AND LINEAR TRANSFORMATIONS


Section B (continued) Question 11 (7 marks) The points 3,0,3( ),! 0,!2, 4( ) and 1,!8,1( ) satisfy the equation ax !+!by!+!cz!=!24. (a) Write down three equations involving the variables a, b and c, and 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

determine the values of a, b and c. Question 12 (7 marks) The circle with equation x2 + y2 ! 4x = 0 is rotated anticlockwise about the origin through !3 radians, then reflected in the line x + y = 0 and, finally, dilated by a factor of 2 in the direction of the y-axis. Determine the equation of the resulting curve.


This section assesses Criterion 5. 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)

Find dydx if


The straight line y = 32x +1 intersects the curve y = 2x at the points 0,1( ) and 2, 4( ) .

Determine the finite area enclosed between the line and the curve. Question 15 (5 marks)

Prove that the curves with equations y = exe and x = e

ye touch each other (that is, are tangential to

each other) at the point where x = e.

Section C continues opposite.

SECTION C – DIFFERENTIAL AND INTEGRAL CALCULUS

x2 !+!2x !+!y2 !=!4.


Section C (continued) Question 16 (5 marks) A family has installed solar panels on the roof of their house to generate electricity. The family also has a gauge that shows how much electric power is being generated by the panels at any given time. One sunny autumn day the family recorded the readings on the gauge every two hours from 7 am until 7 pm, as shown in the table below.

Time (hours since midnight) 7 9 11 13 15 17 19 Electric power generated (kilowatts) 0.38 1.61 2.67 3.05 2.58 1.42 0.28

Use the trapezoidal rule to estimate the total amount of electricity generated (in kilowatt-hours) during that twelve-hour period. Give your answer correct to two decimal places. Question 17 (7 marks) (a) If f (x)!=!sin!1 x, determine f '(x),! f ''(x) and f '''(x) without using your calculator. (b) Determine the Maclaurin series expansion for f (x) = sin!1 x , as far as the x3 term, and use your

series to obtain an approximate value of !sin!1 x !0

0.1

" dx.

Question 18 (7 marks) (a) For a >1, sketch the area enclosed by the x-axis, the curve y = loga x and the lines x = a and

x !=!a2. (b) Determine the volume generated when this area is rotated about the y-axis.



Solve the differential equation dydx

=1+ y2 , given that when x = !2

, y =1 .


Show that y = 2e3x is a solution to the differential equation d2ydx2

! 2 dydx

! 3y = 0.

Question 21 (5 marks) Determine ! x2 cos x!dx, without using your calculator. Question 22 (5 marks)

Without using your calculator, determine the exact value of 6x + 3(x +1)2(x ! 2)

!dx0

1

" .

Section D continues opposite.

SECTION D – TECHNIQUES OF INTEGRATION


Section D (continued) Question 23 (7 marks) An electrical circuit contains a battery pack supplying E volts, a resistor of R ohms, an inductor of H henries and a switch. When the switch is closed, a current of I amperes flows, where I is a function of time t.

The current flowing can be modelled by the differential equation H dIdt

+ RI = E.

If H = 4,!R =12 and E = 60 , determine I as a function of t, given that I < 5 and that I = 0 when t = 0. Question 24 (7 marks)

Solve the differential equation dydx

= x2 +!2y2

xy. Simplify your answer.



Without using your calculator, simplify !!(3! 3i)2

3+ 3i.

Question 26 (3 marks) Sketch the region of the Argand plane which satisfies the conditions that:

z + 2 ! 2 and !!2!!"!Arg(z + 2) < !

4.

Question 27 (5 marks) Determine the cube roots of i in the form a!+!ib.

Section E continues opposite.

SECTION E – COMPLEX NUMBERS


Section E (continued) Question 28 (5 marks) (a) Write down the smallest positive value of ! such that i = ei! .

(b) Using (a) or otherwise, show that a possible value for i i is e!

!2.

(c) Hence write (i i)i in the form a!+!ib. Question 29 (7 marks) (a) Show that 1! 2i is a root of the equation z4 !8z3 + 42z2 !80z +125 = 0. Your calculator may be used to simplify powers of a complex number. (b) Hence, without using your calculator, write the polynomial z4 !8z3 + 42z2 !80z +125 as a

product of linear factors. Question 30 (7 marks) Factorise P(z) = z5 ! z4 + z3 ! z2 + z !1 into: (a) linear factors; (b) real factors.


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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.

Mathematics Specialised

Subject Code: MTS315109

2013 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

!

" y2dxa

b# about y-axis

!

" 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

!

" > 0 ,

!

" N(#) such that

!

an " L < # $ n > N . The sequence

!

an{ } diverges to positive infinity if, for any

!

" > 0,

!

" N(#) such that

!

an >" # n > N . The sequence

!

an{ } diverges to negative infinity if, for any

!

" > 0 ,

!

" N(#) such that

!

an < "# $ 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 " #sin "sin " cos "$

% &

'

( ) ,

!

cos 2" sin 2"sin 2" #cos 2"$

% &

'

( ) .

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.

2013 External Examination Information Sheet

Page 1 of 3

Mathematics Methods

Subject Code: MTM315109

FUNCTION STUDY

Quadratic Formula: If 02 =++ cbxax , then a

acbbx

242 −±−

=

Graph Shapes: Quadratic Cubic Hyperbola Truncus

( ) khxay +−= 2 ( ) khxay +−= 3 khxay +−

= ( ) k

hxay +−

= 2

x

y

x

y

x

y

x

y

Square Root Absolute Value Exponential Logarithmic khxay +−= khxay +−= xay = xy alog=

x

y

x

y

x

y

x

y

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

( ) nn

nnn

nnnnnnnn yCyxCyxCyxCxCyx +++++=+ −−

−− 11

222

110 ...

1

(h, k)

(h, k)

(h, k) (h, k) 1

h

k

h

k

of 3

CIRCULAR FUNCTIONS

Conversion:

To convert from radians to degrees multiply by π180

To convert from degrees to radians multiply by 180π

Basic Identities:

1cossin 22 =+ xx xxx

cossintan =

xxtan1cot =

xxcos1sec =

xxsin1cosec =

Exact Values: Cast Diagram:

x 0 6π

4π

3π

2π π

23π

π2

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=

x-2π – 3π2

– π – π2

π2

π 3π2

2π

y

-1

1

x-2π – 3π2

– π – π2

π2

π 3π2

2π

y

-1

1

x-2π – 3π2

– π – π2

π2

π 3π2

2π

y

-4

-2

2

4

Graphical Transformation: The graph of ( ) cbxnay ++= sin or ( ) cbxnay ++= cos has: amplitude: |a|

period: nπ2

phase shift: b (shift of xy sin= or xy cos= b units to the left) vertical shift: c units upwards The graph of cbxnay ++= )(tan has: dilation: by factor a in the direction of the y axis

period: nπ

phase shift: b (shift of xy tan= b units to the left) vertical shift: c units upwards Trigonometric Equations: If ax =sin then ( ) anx n arcsin1−+π= , J∈n If ax =cos then anx arccos2 ±π= , J∈n If ax =tan then anx arctan+π= , J∈n

C

A S

T

of 3

CALCULUS Definition of Derivative: ( )

hxfhxfxf

h

)()(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)(

)(').()(').(xg

xgxfxfxg −

{ })(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

−=

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 ( ) ( )[ ]222 XEXE −=σ ( )pnp −=σ 12 ⎟⎟⎠

⎞⎜⎜⎝

⎛−−

⎟⎠⎞⎜

⎝⎛ −⎟⎠⎞⎜

⎝⎛=σ

112

NnN

ND

NnD

mts315109 exam paper - tqa.tas.gov.au · page 4 section a this section assesses criterion 3....

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