Maths Quest Specialist 12 Textbook 4E TI-Nspire CAS Companion

MATHS QUEST 12

Specialist Mathematics

4TH EDITION

TI-NSPIRE C AS C ALCULATOR COMPANION

VCE M AT H EM AT I CS U N I T S 3 & 4

MATHS QUEST 12

Specialist Mathematics RAYMOND ROZEN | PAULINE HOLLAND | BRIAN HODGSON HOWARD LISTON | JENNIFER NOLAN | GEOFF PHILLIPS

4TH EDITION

TI-NSPIRE C AS C ALCULATOR COMPANION

First published 2013 by John Wiley & Sons Australia, Ltd 42 McDougall Street, Milton, Qld 4064 Typeset in 10/12 pt Times LT Std © John Wiley & Sons Australia, Ltd 2013 The moral rights of the authors have been asserted. ISBN: 978 1 118 31811 9 978 1 118 31809 6 (flexisaver) Reproduction and communication for educational purposes The Australian Copyright Act 1968 (the Act) allows a maximum of one chapter or 10% of the pages of this work, whichever is the greater, to be reproduced and/or communicated by any educational institution for its educational purposes provided that the educational institution (or the body that administers it) has given a remuneration notice to Copyright Agency Limited (CAL). Reproduction and communication for other purposes Except as permitted under the Act (for example, a fair dealing for the purposes of study, research, criticism or review), no part of this book may be reproduced, stored in a retrieval system, communicated or transmitted in any form or by any means without prior written permission. All inquiries should be made to the publisher. Cover and internal design images: © vic&dd/Shutterstock.com Typeset in India by Aptara Illustrated by Aptara and Wiley Composition Services Printed in Singapore by Craft Print International Ltd 10 9 8 7 6 5 4 3 2 1 Acknowledgements The authors and publisher would like to thank the following copyright holders, organisations and individuals for their permission to reproduce copyright material in this book. Images Texas Instruments: Screenshots from TI-Nspire reproduced with permission of Texas Instruments Every effort has been made to trace the ownership of copyright material. Information that will enable the publisher to rectify any error or omission in subsequent editions will be welcome. In such cases, please contact the Permissions Section of John Wiley & Sons Australia, Ltd.

Contents Introduction

vi

CHAPTER 6

Integral calculus

CHAPTER 1

Coordinate geometry

1

15 21

Relations and regions of the complex plane 27 CHAPTER 5

29

53

CHAPTER 9

Vectors

CHAPTER 4

Differential calculus

CHAPTER 8

Kinematics

CHAPTER 3

Complex numbers

CHAPTER 7

Differential equations

CHAPTER 2

Circular functions

37

63

CHAPTER 10

Vector calculus CHAPTER 11

Mechanics

79

71

47

Introduction This booklet is designed as a companion to Maths Quest 12 Specialist Mathematics Fourth Edition. It contains worked examples from the student text that have been re-worked using the TI-Nspire CX CAS calculator with Operating System v3. The content of this booklet will be updated online as new operating systems are released by Texas Instruments. The companion is designed to assist students and teachers in making decisions about the judicious use of CAS technology in answering mathematical questions. The calculator companion booklet is also available as a PDF file on the eBookPLUS under the preliminary section of Maths Quest 12 Specialist Mathematics Fourth Edition.

vi

Introduction

Chapter 1

Coordinate geometry Worked example 4

Sketch the graph of y =

x2 + 2 including all asymptotes and intercepts. 3x

think 1

On a Graphs page, complete the entry line as: x2 + 2 f 1( x ) = 3x then press ENTER ·.

2

To determine the equation of the oblique asymptote,

Write/display

x2 + 2 into two functions. 3x To do this on a Calculator screen press: • MENU b • 2: Number 2 • 7: Fraction Tools 7 • 1: Proper Fraction 1 Complete the entry line as: propfrac(f1(x)) then press ENTER ·.

divide y =

3

Write the equations of the asymptotes.

The equations of the asymptotes are: 1 x = 0 and y = x . 3

Chapter 1 • Coordinate geometry

1

4

To determine the turning points, press: • MENU b • 3: Algebra 3 • 1: Solve 1 • MENU b • 4: Calculus 4 • 1: Derivative 1 Complete the entry line as:  d  solve  ( f 1( x )) = 0, x   dx  then press ENTER ·.

5

Solving

To find the y-coordinates of the stationary points by substitution, complete the entry lines as:

x=

f 1( 2 )

f 1( 2 ) Press ENTER · after each entry. Describe the nature and coordinates of the stationary points, as deduced from the graph. Sketch the graph of y =

−

2  or  x = 2.

The coordinates of the stationary points are:  2 2 Local minimum  2,   3 

−

6

d  x2 + 2   = 0 for x gives dx  3 x 

− −2 2 Local maximum  2,  3  

x2 + 2. 3x

y 10

x = 0 (Asymptote)

5

−10

−5

0 −5 −10

2

Maths Quest 12 Specialist Mathematics

5

1 y = – x (Asymptote) 3 x 10

Worked example 12

Sketch the graph of

( x − 1) 2 ( y − 2) 2 + = 1. 25 9

think 1

Write/display

Compare

( x − 1)2 ( y − 2)2 + = 1 with 25 9

( x − h) 2 ( y − k ) 2 + = 1. a2 b2

h = 1, k = 2 and so the centre is (1, 2). a2 = 25 b2 = 9 a=5 b=3

2

The major axis is parallel to the x-axis as a > b.

3

The extreme points (vertices) parallel to the x-axis for the ellipse are: (−a + h, k) (a + h, k)

Vertices are: (−5 + 1, 2) = (−4, 2)

(5 + 1, 2) = (6, 2)

The extreme points (vertices) parallel to the y-axis for the ellipse are: (h, −b + k) (h, b + k)

and (1, −3 + 2) = (1, −1)

(1, 3 + 2) = (1, 5)

4

5

Find the x- and y-intercepts. On a Calculator page, complete the entry lines as:  ( x − 1)2 ( y − 2)2  + = 1, x  y = 0 solve   25  9  ( x − 1)2 ( y − 2)2  + = 1, y  x = 0 solve   25  9 Press ENTER · after each entry.

The x-intercepts are: 3−5 5 3+5 5 ,  x = 3 3 The y-intercepts are: x=

y= 6

10 − 6 6 10 + 6 6 ,  y = 5 5

To sketch the graph of the ellipse, on a Graphs page press: • MENU b • 3: Graph Entry/Edit 3 • 2: Equation 2 • 4: Ellipse 4

( x − h )2 ( y − k )2

+ =11 a2 b2 Complete as shown: ( x − 1)2 ( y − 2 )2 + =1 52 32 Press ENTER · after the entry, the graph is shown.

• 1:

Chapter 1 • Coordinate geometry

3

7

Sketch the graph of the ellipse.

+ 6√6 0, 10 ––––––– 5

y 6 4

3 − 5√5 , –––––– 3

−6

0

(−4, 2)

−4 5

4

Maths Quest 12 Specialist Mathematics

(1, 2)

0 −2

(y − 2)

2

+ –––––– =1 9

(1, 5)

2

−2

− 6√6 0, 10 –––––––

(x − 1)2 –––––– 25

2

(6, 2)

4

6 x

(1, −1) 3 + 5√5 , –––––– 3

0

Worked example 13

Sketch the graph of

( x − 2) 2 ( y + 4) 2 + = 1. 9 16

think

Write/display

( x − 2)2 ( y + 4)2 + = 1 with 9 16 ( x − h) 2 ( y − k ) 2 + = 1. a2 b2

h = 2, k = −4 So the centre is (2, −4). a2 = 9 b2 = 16 a=3 b=4

1

Compare

2

The major axis is parallel to the y-axis as b > a.

3

The extreme points (vertices) parallel to the x-axis for the ellipse are: (−a + h, k) (a + h, k)

Vertices are: (−3 + 2, −4) = (−1, −4)

4

The extreme points (vertices) parallel to the y-axis for the ellipse are: (h, −b + k) (h, b + k)

and (2, −4 − 4) = (2, −8)

5

(3 + 2, −4) = (5, −4) (2, 4 − 4) = (2, 0)

Find the x- and y-intercepts. On a Calculator page, complete the entry lines as:  ( x − 2)2 ( y + 4)2  solve  + = 1, x  y = 0  9  16  ( x − 2)2 ( y + 4)2  + = 1, y  x = 0 solve   9  16 Press ENTER · after each entry.

The x-intercept is x = 2. The y-intercepts are: y= 6

− 12 − 4

3

5

,

y=

−

12 + 5 . 3

To sketch the graph of the ellipse, on a Graphs page press: • MENU b • 3: Graph Entry/Edit 3 • 2: Equation 2 • 4: Ellipse 4 • 1: Complete the entry line as:

( x − 2 )2

2 y − − 4) ( + =1

32 42 then press ENTER ·. Note that the viewing window has been changed.

Chapter 1 • Coordinate geometry

5

7

y

Sketch the graph of the ellipse.

2 −12 + 4√5

0, –––––––– 3

−1 −2 −4

−12 − 4√5 0, –––––––– 3

6

Maths Quest 12 Specialist Mathematics

−6

0 (2, 0) 123456 x (−1, −4) (2, −4)

−8 (2, −8) −10

(5, −4) (x − 2)2 –––––– 9

2

(y + 4)

+ –––––– =1 16

Worked example 14

Sketch the graph of 5 x2 + 9(y − 2)2 = 45. think

Write/display

5x2 + 9(y − 2)2 = 45

1

Rearrange and simplify by dividing both sides by 45 to make the RHS = 1.

2

Simplify by cancelling.

3

Compare

4

Major axis is parallel to the x-axis as a > b.

5

The extreme points (vertices) parallel to the x-axis for the ellipse are: (−a + h, k) (a + h, k)

Vertices are: (−3 + 0, 2) = (−3, 2)

The extreme points (vertices) parallel to the y-axis for the ellipse are: (h, −b + k) (h, b + k)

and (0, 5 + 2) or (0, 2 − 5 )

6

7

x 2 ( y − 2)2 + = 1 with 9 5 ( x − h) 2 ( y − k ) 2 + = 1. a2 b2

5 x 2 9( y − 2)2 45 + = 45 45 45 2 2 x ( y − 2) + =1 9 5 h = 0, k = 2 and so the centre is (0, 2). b2 = 5 as a, b > 0 a2 = 9 a=3 b= 5

(3 + 0, 2) = (3, 2)

−

(0, 5 + 2) (0, 2 + 5 )

≈ (0, −0.24)

≈ (0, 4.24)

Find the x-intercepts. On a Calculator page, complete the entry line as: solve(5 x 2 + 9( y − 2)2 = 45, x ) | y = 0 then press ENTER ·.

x= 8

−3

5

5

,

x=

3 5 5

To sketch the graph of the ellipse, on a Graphs page press: • MENU b • 3: Graph Entry/Edit 3 • 2: Equation 2 • 6: Conic 6 • 1: Complete the entry line as: 5 x 2 + 0 xy + 9 y 2 + 0 x + − 36 y + − 9 = 0 then press ENTER ·.

Chapter 1 • Coordinate geometry

7

9

Sketch the graph of the ellipse.

y 6 (0, 2 + 5 ) 5x2 + 9(y − 2)2 = 45 4 2 (0, 2)

(−3, 2) −4 −3 −2 −3√5 –––– , 5

8

Maths Quest 12 Specialist Mathematics

−1 0

(3, 2)

0 1

2

−2 (0, 2 − 5 )

3

4x

3√5 –––– , 5

0

Worked Example 15

Sketch the graph of the relation described by the rule: 25x2 + 150x + 4y2 − 8y + 129 = 0. Think 1

Write/Display

To locate the intercepts, on a Calculator page, complete the entry lines as: solve ( 25 x 2 + 150 x + 4 y 2 − 8 y + 129 = 0, x ) |y=0 solve ( 25 x 2 + 150 x + 4 y 2 − 8 y + 129 = 0, y ) |x=0 Make a record of the intercepts.

2

To sketch the graph of the ellipse, on a Graphs page press: • MENU b • 3: Graph Entry/Edit 3 • 2: Equation 2 • 6: Conic 6 • 1: Complete the entry line as: 25 x 2 + 0 xy + 4 y 2 + 0 x + − 8 y + 129 = 0 then press ENTER ·.

3

Write the x-intercepts.

x= x=

4

− 15 − 4

6

5 − 15

+4 6 5

Sketch the graph of the ellipse.

y (−3, 6)

(−5, 1) −15 − 4√6 , –––––––– 5

0

−5

(−3, 1)

6 (−1, 1)

−3 −1 0 −4 (−3, −4)

−15 + 4√6 , –––––––– 5

x 0

Chapter 1  •  Coordinate geometry  9

Worked Example 16

Determine the Cartesian equation of the curve with parametric equations x = 2 + 3 sin (t) and y = 1 − 2 cos (t ) where t ∈ R. Describe the graph and state its domain and range. Think

Write/Display

1

Use a CAS calculator to sketch the graph in a Graphs page, in parametric mode, by completing the entry line as:  x1(t ) = 2 + 3 sin (t )  y1(t ) = 1 − 2 cos (t )  Then press ENTER ·.

2

Rewrite the parameters by isolating cos (t) and sin (t).

y −1 x−2 = sin (t) and − = cos (t) 2 3

3

Square both sides of each equation then add.

( x − 2)2 ( y − 1)2 = sin2 (t) + cos2 (t) + 9 4 =1

4

Describe the relation.

This represents an ellipse with centre (2, 1).

5

The domain is the range of the parametric equation x = 2 + 3 sin (t).

Domain is [2 − 3, 2 + 3] = [−1, 5]

6

The range is the range of the parametric equation y = 1 – 2 cos (t).

Range is [1 − 2, 1 + 2] = [−1, 3]

10  Maths Quest 12 Specialist Mathematics

Worked Example 25

Express each of the following as partial fractions. a 

5 x 2 + 10 x − 52 2 x3 − 5 x2 + 3 x + 7   b  ( x − 2)( x + 4) x2 − x − 2

Think

a & b

1

Write/Display

On a Calculator page, press: • MENU b • 3: Algebra 3 • 3: Expand 3 Complete the entry lines as:  5 x 2 + 10 x − 52  expand   ( x − 2) × ( x + 4)   2 x 3 − 5x 2 + 3x + 7  expand    x2 − x − 2 Press ENTER · after each entry.

2

Write the answers.

a

5 x 2 + 10 x − 52 2 2 = − +5 ( x − 2) × ( x + 4) x + 4 x − 2

b

2 x 3 − 5x 2 + 3x + 7 1 3 + 2x − 3 = + x2 − x − 2 x +1 x − 2

Chapter 1  •  Coordinate geometry  11

Worked Example 27

Sketch the graph of the function y =

x 2 − 5x + 6 . x−4

Think 1

Use a CAS calculator to express the rational function as partial fractions by completing the following steps. Press: • MENU b • 2: Number 2 • 7: Fraction Tools 7 • 1: Proper Fraction 1 Complete the entry line as:  x 2 − 5x + 6  propFrac   x − 4  then press ENTER ·.

2

Express the function as partial fractions.

3

Sketch the graphs of y1 = x − 1 2 (asymptote) and y2 = on the same x − 4 axes.

Write/Display

y=

2 + x −1 x−4 y y1 = x − 1 2 y2 = x—— −4

0

x

1 2 3 4

−1

x=4 4

Determine any x-intercepts.

y = 0, x2 − 5x + 6 = 0 (x − 2)(x − 3) = 0 ⇒ x = 2 and x = 3

5

Determine the y-intercept.

x = 0, y =

6

Add the two graphs by addition of ordinates x 2 − 5x + 6 . to obtain the graph of y = x−4

6 4 −3 y= 2 −

y

(3, 0)

(2, 0)

y1 = x − 1 2 y2 = x—— −4

0 (0,

−1

− 3–2 )

1 2 3 4

2 y = x − 1 + x—— −4

x=4

12  Maths Quest 12 Specialist Mathematics

x

7

Open a Graphs page, and complete the entry lines as; f 1( x ) = x − 1 2 x−4 f 3 ( x ) = f 1( x ) + f 2 ( x ) f 2( x ) =

then press ENTER ·.

Chapter 1  •  Coordinate geometry  13

ChapTer 2

Circular functions Worked example 2

If cosec (x) = 43 and, 0 ≤ x ≤ 90°, find x (to the nearest tenth of a degree). Think 1

express the equation cosec (x) = in terms of sin (x).

2

On a Calculator page, press: • Menu b • 3: Algebra 3 • 1: Solve 1 Complete the entry line as

WriTe 4 3

cosec  ( x ) =

1 4 = sin ( x ) 3

4   1 solve  = , x | 0 ≤ x ≤ 90  sin( x ) 3  Then press enTeR ·. Alternatively, the three reciprocal functions are built into the TI-nspire. They can be accessed by the µ key, or through the catalogue, or you can simply use the letter keys and enter csc, sec or cot as needed. 3

Write the solution.

Solving cosec (x) = 43 for x ∈[0,90°], x = 48.5904°

4

Round off the answer to 1 decimal place.

x = 48.6°

ChapTer 2 • Circular functions

15

Worked Example 12

Solve cosec (x) = 1.8 over the interval 0 ≤ x ≤ 4π. Give your answer(s) correct to 2 decimal places. Think

Write

1

On a Calculator page, press: • Menu b • 3: Algebra 3 • 1: Solve 1 Complete the entry line as solve (csc(x) = 1.8, x) | 0 ≤ x ≤ 4π Then press ENTER ·.

2

Write the solution.

Solving cosec (x) = 1.8 over the interval 0 ≤ x ≤ 4π gives

3

Round the answers to 2 decimal places.

x = 0.59, 2.55, 6.87, 8.84

16  Maths Quest 12 Specialist Mathematics

Worked Example 13

a  Expand, and simplify where possible, each of the following.    i  sin (x − 2y)  ii  cos (x + 30°) b  Simplify the expression sin (2x) cos (y) + cos (2x) sin (y). Think

a

i On a Calculator page, press: & • MENU b ii • 3: Algebra 3

Write

a

i & ii

• B: Trigonometry B • 1: Expand 1 Complete then entry lines as shown, then press ENTER ·.

b

1

Write the appropriate compoundangle formula.

2

Substitute A = 2x and B = y to reveal the answer.

3

Alternatively, on a Calculator page, press: • Menu b • 3: Algebra 3 • B: Trigonometry B • 2: Collect 2 Complete the entry line as tCollect(sin(2x)cos(y) + cos(2x)sin(y)) Then press ENTER ·.

Write the solution.

b

sin (A) cos (B) + cos (A) sin (B) = sin (A + B) sin (2x) cos (y) + cos (2x) sin (y) = sin (2x + y)

sin (2x) cos (y) + cos (2x) sin (y) = sin (2x + y)

Chapter 2  •  Circular functions  17

Worked Example 15

Simplify: a  sin (270 − C °)       b  sec 

 π −θ . 2 

Think

a & b

1

On a Calculator page, press: • Menu b • 3: Algebra 3 • 3: Expand 3 Complete the entry lines as: expand(sin (270 − c), c)   π − θ  , θ expand  sec 2   

Write

a & b

Press ENTER · after each entry. Note: The calculator should be in degree mode for the first expansion above, and radian mode for the second. 2

Express the answer as a reciprocal function.

18  Maths Quest 12 Specialist Mathematics

π sec   − θ  = cosec(θ ) 2 

Worked example 16

5π  Find the exact value of cot   .  12  Think

WriTe

express

2

express cot in terms of its reciprocal,

3

use the appropriate compound-angle formula to expand the denominator.

4

5

5π  π π cot    = cot   +   4 6  12 

π π 5π as the sum of and . 6 4 12

1

express in simplest fraction form.

Simplify.

1 . tan

=

=

=

=

=

=

=

1 π π  tan  +  4 6 1 π  tan    + tan   π    4  6    π π   1 − tan     tan       4  6 

π π 1 − tan    tan     4  6 π π tan    + tan     4  6  1  3  1

1 − (1)  1+ 1− 1+

3

1 3 1 3

 3 −1  3   3 +1  3 

3 −1 3 +1

6

Rationalise the denominator.

=

( 3 − 1) ( 3 − 1) ( 3 + 1) ( 3 − 1)

7

Simplify.

=

3− 2 3 +1 3−1

4−2 3 2 = 2− 3 =

ChapTer 2 • Circular functions

19

Note: It is possible to check the answer using a calculator. On a Calculator page, complete the entry line as: 5π cot    12  then press enTeR ·.

20

Maths Quest 12 Specialist Mathematics

ChapTer 3

Complex numbers Worked example 1

Using the imaginary number i, write down an expression for: −16

a

b

−5

.

Think

a & b

WriTe

1

Change the document settings to Rectangular mode. To do this, press: • HOME c • 5: Settings 5 • 2: Settings 2 • 2: Document Settings 2 Tab down to Real or Complex and select Rectangular.

2

On a Calculator page, complete the entry lines as: −

a & b

16

−

5 Press ENTER · after each entry.

ChapTer 3 • Complex numbers

21

Worked Example 4

Simplify z = i4 − 2i2 + 1 and w = i6 − 3i4 + 3i2 − 1. Think

Write

1

On a Calculator page, complete the entry lines as: i4 − 2i2 + 1 i6 − 3i4 + 3i2 − 1 Press ENTER · after each entry.

2

Write the answer.

z=4 w = −8

Worked Example 5

Evaluate each of the following.  1 − 3 i − i 2 − i3  a   Re(7 + 6i)  b   Im(10)  c   Re(2 + i − 3i3)  d  Im    2 Think

Write

a , b , 1 On a Calculator page, press: • Menu b c & • 2: Number 2 d

• 9: Complex Number Tools 9 • 2: Real Part 2 or • 3: Imaginary Part 3 Complete the entry lines as: real(7 + 6i) imag(10) real(2 + i − 3i2)  1 − 3i − i 2 − i 3  imag    2 Press ENTER · after each entry. 2

Write the answers.

a  Re(7 + 6i) = 7 b  Im(10) = 0 c  Re(2 + i − 3i3) = 2

 1 − 3i − i 2 − i3  −  = 1  2

d  Im 

22  Maths Quest 12 Specialist Mathematics

Worked Example 10

Determine Re(z2w) + Im(zw2) for z = 4 + i and w = 3 − i. Think 1

On a Calculator page, complete the entry lines as: Define z = 4 + i Define w = 3 – i Press ENTER · after each entry. Then press: • Menu b • 2: Number 2 • 9: Complex Number Tools 9 • 2: Real Part 2 Complete the entry line as: real(z2 × w) + imag(z × w2) Then press ENTER ·. Note: The imaginary part can be found in the same menu as the real part.

2

Write the answer.

Write

Re(z2 w) + Im(zw2) = 37

Worked Example 12

Write down the conjugate of each of the following complex numbers. a  8 + 5i  b   −2 − 3i Think

a & b

1

On a Calculator page, press: • Menu b • 2: Number 2 • 9: Complex Number Tools 9 • 1: Complex Conjugate 1 Complete the entry lines as: conj (8 + 5i) conj (−2 − 3i) Press ENTER · after each entry.

2

Write the answers.

Write

a 8 − 5i b  −2 + 3i

Chapter 3  •  Complex numbers  23

Worked Example 16

If z = a + bi, find a and b such that Think 1

5 z − 15 = 4 − 3 i. z−1 Write

On a Calculator page, press: • Menu b • 3: Algebra 3 • C: Complex C • 1: Solve 1 Complete the entry line as:  5z − 15  = 4 − 3i,  z  cSolve    z −1 Then press ENTER ·.

2

a is the real part of z, b is the imaginary part.

a = 2, b = −3

Worked Example 17

Find the modulus of the complex number z = 8 − 6i. Think 1

On a Calculator page, press: • Menu b • 2: Number 2 • 9: Complex Number Tools 9 • 5: Magnitude 5 Complete the entry line as: |8 − 6i| Then press ENTER ·.

2

Write the answer.

24  Maths Quest 12 Specialist Mathematics

Write

| z | = |8 − 6i| = 10

Worked Example 23

Express each of the following in polar form, r cis (θ ), where θ = arg(z), −π < θ ≤ π. a   z = 1 + i  b  z = 1 − 3 i Think

a & b

1

On a Calculator page, complete the entry line as: 1+i Then press: • Menu b • 2: Number 2 • 9: Complex Number Tools 9 • 6: Convert to Polar 6 Then press ENTER ·.

2

Write the answer.

3

Use the relationship reiθ = r cos (θ ) + ir sin (θ ) to express the answer in the required form. The calculator always gives θ in principle valued form.

Write

a & b

π For a , 1 + i = 2  cis    4

Key in 1 − 3i and repeat the above procedure. 4

Write the answer.

For b , 1 − 3i = 2e

−i

π 3

−π  = 2 cis   3

Chapter 3  •  Complex numbers  25

Worked Example 36

a  If f (z) = z3 + 7z2 + 16z + 10, find all factors of f (z) over C. b  Factorise P(z) = z3 − (3 −i)z2 + 2z − 6 + 2i. Think

a & b

1

On a Calculator page, press: • Menu b • 3: Algebra 3 • C: Complex C • 2: Factor 2 Complete the entry lines as cFactor (z3 + 7z2 + 16z + 10, z) cFactor (z3 − (3 − i)z2 + 2z − 6 + 2i, z) Press ENTER · after each entry.

2

Write the answers in the required form.

26  Maths Quest 12 Specialist Mathematics

Write

a & b

For a, the three factors of P(z) are (z + 1), (z + 3 − i) and (z + 3 + i) For b, P( z ) = ( z − 3 + i)( z + 2i)( z − 2i)

CHAPTER 4

Relations and regions of the complex plane WORKED EXAMPLE 16

Express each of the following expressions in Cartesian form. a Re(z + 5) b Im(z − 2 − 3i) c | z − 4 + 2i | THINK

a , 1 On a Calculator page, complete the entry lines as: b Define z = x + yi & Then press ENTER ·. c To answer part a press:

2

• MENU b • 2: Number 2 • 9: Complex Number Tools 9 • 2: Real Part 2 or • 3: Imaginary Part 3 or • 5: Magnitude 5 Complete the entry line as: real(z + 5) imag(z − 2 − 3i) | z − 4 + 2i | Press ENTER · after each entry. Write the answers.

WRITE

a, b & c

For a , Re(z + 5) = x + 5. For b , Im(z − 2 − 3i) = y − 3. For c , | z − 4 + 2i |  = x 2 − 8 x + y 2 + 4 y + 20 .

CHAPTER 4 • Relations and regions of the complex plane

27

ChapTer 5

Differential calculus Worked example 1

Differentiate the following expressions with respect to x.  4x a y = tan (6 x) b y = 2 tan   3  Think

a & b

1

On a Calculator page, press: • Menu b • 4: Calculus 4 • 1: Derivative 1 Complete the entry lines as: d (tan (6 x )) dx

WriTe

a & b

d  4x   2 tan     dx 3 Press enTeR · after each entry.

2

Write the solutions.

For a,

d 6  [tan (6 x )] = dx [cos (6 x )]2

For b,

d dx

  2 tan

 4x  =    3  3  cos 

8  4x    3  

2

ChapTer 5 • Differential calculus

29

Worked Example 4

Find the equation of the tangent to the curve y = 3x + cos (2x) + tan (x) where x = Think 1

π . 4

Write

On a Calculator page, press: • Menu b • 4: Calculus 4 • 9: Tangent Line 9 Complete the entry lines as:

π tangentLine  3 x + cos (2 x ) + tan ( x ), x ,   4 Then press ENTER ·.

2

Write the solution.

30  Maths Quest 12 Specialist Mathematics

Equation of the tangent is y = 3x + 1

Worked example 6

Find, using calculus, f ″( x) if f ( x) is equal to: a ecos (2x) + loge (x)

b

sin ( x ) . x

Think

a & b

1

On a Calculator page, complete the entry line as: f (x) : = ecos (2x) + ln(x) Then press enTeR ·. Note: The syntax used here is another way of defining a function or variable. You can use the Define or Store methods if you prefer.

2

Complete the entry line as:

WriTe

a & b

d2 ( f ( x )) dx 2 Then press enTeR ·.

3

Write the solution.

The second derivative, f '' ( x ) = [4 sin 2 (2 x ) − 4 cos (2 x )]e cos (2 x ) −

4

1 x2

On a Calculator page, complete the entry line as: sin ( x ) f (x) : = x Complete the entry line as: d2 ( f ( x )) dx 2 Press enTeR · after each entry.

ChapTer 5 • Differential calculus

31

5

Write the solution.

6

You may rearrange the answer to a form similar to that given in the solution obtained manually as follows. Press: • Menu b • 3: Algebra 3 • 2: Factor 2 Then select and paste the previous answer to obtain the entry line:

The second derivative,   cos ( x ) 3 1   sin ( x ) − f ′′( x ) =  5 − 3 x  2  x2 4x

   3 1 cos ( x )    sin ( x ) − factor   5 − 3 x      4x 2 x2  Then press enTeR ·. 7

Write the solution.

The second derivative: (3 − 4 x 2 )  sin ( x ) − 4 x cos ( x ) f ′′( x ) = 5 4x 2

Worked example 16

Find the equation of the normal to the curve with equation: x y = 2 cos −1   at the point where x = 3 .  2 Think

32

1

On a calculator page, press: • Menu b • 4: Calculus 4 • A: normalLine A Complete the entry line as: x   normalLine  2 cos−1   , x , 3    2 Then press enTeR ·.

2

Write your solution in an appropriate form.

Maths Quest 12 Specialist Mathematics

WriTe

The equation of the normal is y =

x 3 π − + . 2 2 3

Worked Example 18

Find the antiderivative for each of the following expressions: −3 20 1    b   a     c   . 2 2 16 + x 2 25 − x 49 − x Think

Write

a , 1 On a Calculator page, press: • Menu b b • 4: Calculus 4 & • 3: Integral 3 c

Complete the entry lines as:

∫ ∫ ∫

  1   dx  2 25 − x  −3     dx  2 49 − x 

 20    dx  16 + x 2 

Press ENTER · after each entry. Note: The calculator finds the second form of the antiderivative in part b . Also, it does not include the constant. You will have to do that yourself. 2

Write your solutions, remembering to include the constant of integration.

a b c

∫ ∫ ∫

1 25 −

x2

x   dx = sin −1   + c  5

x2

x   dx = − 3 sin −1   + c  7

−3

49 −

20 x   dx = 5 tan −1   + c  4 16 + x 2

Chapter 5  •  Differential calculus  33

Worked Example 21

Differentiate the equation y2 + 3x2 = 4 to find Think 1

On a Calculator page, press: • Menu b • 4: Calculus 4 • E: Implicit Differentiation E Complete the entry line as: impDif (y2 + 3x2 = 4, x, y) Then press ENTER ·.

2

Substitute for y as in part a (which is preferable in this straightforward equation) or continue to use the calculator to make y the subject in the equation. Press: • Menu b • 3: Algebra 3 • 1: Solve 1 Complete the entry line as: Solve (y2 + 3x2 = 4, y) Then press ENTER ·.

3

Express the domain, 3x2 − 4 ≤ 0 shown in the screen in a more appropriate form. Take care to change ≤ to < as y is in the denominator in the solution.

4

Write your solution, remembering to include the domain.

34  Maths Quest 12 Specialist Mathematics

dy in terms of x. dx Write

−2

3 3