MATHEMATICS SPECIALISED - Department of …...is a circle, with centre (1,2) and radius 2. (4 marks)...

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Tasmanian Certificate of Education External Assessment 2019

MATHEMATICS SPECIALISED (MTS415118)

Time allowed for this paper Working time: 3 hours

Plus 15 minutes recommended reading time

On the basis of your performance in this examination, the examiners will provide results on each of the following criteria taken from the course document:

Criterion 4 Solve problems and use techniques involving finite and infinite sequences and series. Criterion 5 Solve problems and use techniques involving matrices and linear algebra. 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 Solve problems and use techniques involving of complex numbers.

PLACE YOUR CANDIDATE

LABEL HERE

Pages: 16 Questions: 30 Attachments: Information Sheet

Candidate Instructions

1. You MUST make sure that your responses to the questions in this examination paper will show

your achievement in the criteria being assessed.

2. There are FIVE sections to this paper.

3. You must answer ALL questions.

4. Answer each section in a SEPARATE answer booklet.

5. It is recommended that you spend approximately 36 minutes on each section.

6. The Information Sheet for Mathematics Specialised can be used throughout the examination (provided with the paper).

7. No other written material is allowed into the examination.

8. All written responses must be in English.

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Additional Instructions for Candidates

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.

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Answer ALL questions in this section.

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.

This section assesses Criterion 4.

Question 1 (4 marks)

A series nu has sum to n termsnS . If 1 1u and 1 3n nS S

for 1n , find nS in terms of n .

Question 2

Find the values of x for which the series

22 21 1

1 ...2 2

x x

has a finite sum. (4 marks)

Question 3

A series is given by 1 2 3 4 ....nS to n terms.

(a) Find the thn term. (2 marks)

(b) Find nS in factorised form. (4 marks)

Question 4

Prove that 2

2

2lim 2

1n

n n

n

. (6 marks)

Section A continues.

SECTION A

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Section A (continued)

Question 5

Consider the series nS where 22 1! 5 2! 10 3! .... 1 !nS n n for n Z .

Prove using mathematical induction that 1 !nS n n . (7 marks)

Question 6

(a) Show that 2

1 2 1 2

1 1 1r r r r r

. (2 marks)

(b) Using this result, show that 3

2

1 1 1

4 2 1

n

r r r n n

. (4 marks)

(c) Evaluate 3

10

1

r r r

. (3 marks)

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any, marks.



Question 7

Matrix 0 1

1 3

A .

Show that 1 3 A I A , where

1 0

0 1

I . (3 marks)

Question 8

A transformation T is defined by : ( , ) (2 , )x y y x T .

(a) Find the matrix of T . (2 marks)

(b) T is a combination of a rotation R clockwise through 90° followed by another

transformation U .

Find the matrix of U and describe its effect. (3 marks)

Question 9

Use the Gauss-Jordan method to solve the following system of equations for x , y and z . (6 marks)

2 8

2 3 1

3 7 4 10

x y z

x y z

x y z

Section B continues.

SECTION B

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Section B (continued)

Question 10

Matrix2 1

1 0

A represents a linear transformation.

(a) Find the equation of the image of the straight line 2 1y x under A . (3 marks)

(b) A figure is transformed under A and its image has the equation 2 24 5 9x xy y .

Find the equation of the original figure, and hence the area of the image. (4 marks)

Question 11

The transformation A is a reflection in the line y x .

The transformation B is a rotation counter clockwise through an obtuse angle q where 3

sin5

.

(a) Find matrices for A and B. (3 marks)

(b) The point (10,5) is mapped to the point P by A followed by B.

Find the coordinates of P. (2 marks)

(c) Find the set of points that is left unchanged by applying A followed by B. (3 marks)

Question 12

The equation of a plane P is given by 11 7 12 13x y z .

(a) Find the equation of the plane containing the point (-1,0,1) that is parallel to P. (2 marks)

(b) Find the parametric equation L of the straight line passing through the points (1,2,-2) and (2,-3,0).

(2 marks)

(c) Show that L is parallel to P. (3 marks)

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any, marks.



Question 13

Show that sin cosy a kx b kx with , ,a b k R is a solution of the equation

22

20

d yk y

dx . (3 marks)

Question 14

Given 2 arctan2

x xg x

, evaluate 2'g . (4 marks)

Question 15

A curve is defined implicitly by lnln

xy

x for 0x and 0y .

Find the equation of the normal to the curve at the point with x e . (5 marks)

Section C continues.

SECTION C

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Section C (continued)

Question 16

Given the function arcsin cos y x

(a) Find dy

dx. (3 marks)

(b) Sketch a labelled graph of the curve arcsin cos 0y x x . (3 marks)

Question 17

Given the function 2 23

x k

yx k

where 0k

(a) Find and classify all stationary points on the graph of the function. (4 marks)

(b) Show that the graph has no points of inflection. (3 marks)

(c) Draw a labelled sketch of the curve. (2 marks)

Question 18

The region R is defined by the area with 0y bounded by the curve lny x , the x -axis and the line

x e .

(a) Calculate the area of R. (2 marks)

(b) Given that 2 2

ln 2 ln 2 lnd

x x x x x xdx

, find the volume formed when R is

rotated about the x - axis. (3 marks)

(c) Find the volume formed when R is rotated about the y - axis. (4 marks)

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any, marks.



Question 19

Find 1

x

x

edx

e . (3 marks)

Question 20

Find

1

2

0

xxe dx . (4 marks)

Question 21

Using partial fractions, find 2

1

dx

x x . (5 marks)

Question 22

(a) Given 1 sin

ln1 sin

xy

x

, find

dy

dx. (4 marks)

(b) Hence, evaluate 6

0

cos

dx

x

. (3 marks)

Section D continues.

SECTION D

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Section D (continued)

Question 23

By making the substitution y ux , or otherwise, solve the differential equation (8 marks)

2dy

x x y ydx

given that 0x , and when 1, 0x y .

Give an explicit solution for y in terms of x .

Question 24

A pond is 20 cm deep, and the temperature is such that ice is forming on the surface. The water

continues to freeze, and at any time the rate at which the thickness of the ice,T, is changing is

proportional to the depth of water remaining below the ice.

(a) Set up a differential equation connecting the thickness T and time t (in hours). (3 marks)

At the start of measurement, the ice is just beginning to form.

After 1 hour the thickness is 5 cm.

(b) Determine T in terms of t . (4 marks)

Fish can only survive in the pond if the depth of water remaining is at least 5 cm.

(c) How long from the start will it take to reach this level? (2 marks)

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any, marks.



Question 25

Find z if 1

1 2 1 2

z

i i

. (3 marks)

Give your answer in the form z a bi where a b R, .

Question 26

For complex numbers w and z , show that wz w z . (3 marks)

Question 27

Given 3z i , P is the point z , Q is iz and R is z iz .

(a) Show P, Q and R on an Argand diagram. (3 marks)

(b) Find the area of the quadrilateral OPRQ. (3 marks)

Question 28

(a) Show that z i is a factor of 4 3P z z iz z i . (2 marks)

(b) Hence solve P z 0 , giving answers in Cartesian (rectangular) form. (5 marks)

Section E continues.

SECTION E

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Section E (continued)

Question 29

(a) Show that the set of points such that 1 2z z i is a circle, with centre (1,2) and radius 2.

(4 marks)

(b) Show on an Argand diagram the set of points given by

z z z i z z: 1 2 :Re( ) 0 .

Show all significant points on your diagram. (4 marks)

Question 30

Given z =cosq +isinq

(a) Find zz

1 1 in rectangular form in terms of q , and hence find z z 1 . (2 marks)

(b) Using de Moivre’s theorem, show that 3 3 32cosz z . (3 marks)

(c) Hence, using a binomial expansion of z z3

1 , show that 3cos3 4cos 3cos . (4 marks)

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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.

MATHEMATICS SPECIALISED - Department of …...is a circle, with centre (1,2) and radius 2. (4 marks)...

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