IBSL REVIEW GUIDE

SEQUENCES AND SERIES

An arithmetic sequenceis a sequence in which each termdiffers

from the previous one by the same fixed number.

fung isarithmetic , un+1¡un =d for all positive integers

n wheredis aconstant(thecommon difference).

Ageometric sequenceis a sequence where the next member is obtained from the previous one bymultiplyingby a fixed number.

fungisgeometric , un

+1 un

=r for all positive integersn

whereris aconstant(thecommon ratio). Aseriesis the addition of the terms of a sequence,

i.e., u1+u2+u3+::::+un is a series.

Thesumof a series is the result when all terms of the series are added.

Sn=u1+u2+u3+::::+un is thesumof the firstnterms.

ARITHMETIC SEQUENCES AND SERIES

For anarithmetic sequencewithfirst termu1andcommon differ-encedthegeneral term(ornthterm) is un =u1+ (n¡1)d.

Sn = n₂(u1+un) or Sn = n₂(2u1+ (n¡1)d)

For ageometric sequencewithfirst termu1 andcommon ratior, thegeneral term(ornthterm) is un =u1rn¡1.

For r6= 1, Sn = u

1(rn ¡1)

r¡1 or Sn =

u1(1¡rn) 1¡r . SUM TO INFINITY OF A GEOMETRIC SERIES

S1 = ₁u_¡1

r for jrj<1. SIGMA NOTATION

u1+u2+u3+u4+::::+un can be written more compactly using

sigma notation.

P

, which is calledsigma, is a Greek letter. We write u1+u2+u3+u4+::::+un as

n

P

r=1

ur. So,

n

P

r=1

ur reads “thesum of all numbersof the formur where

r= 1,2,3, ...., up ton”.

INDICES

Ifnis a positive integer, then an is the product ofnfactors ofa

i.e., an =

_|

a£a£a£

_{z

a£::::£a

_}

nfactors

Index laws

am£an=am+n To multiply numbers with the same base, keep the base andaddthe indices.

am an =a

m¡n _{Todividenumbers with the same base, keep} the base andsubtractthe indices.

(am)n=am£n Whenraisingapowerto apower, keep the base andmultiplythe indices.

(ab)n=anbn The power of a product is the product of the powers.

³

_a

b

´

n =a

n

bn The power of a quotient is the quotient of the

powers.

ao= 1, a6= 0 Any non-zero number raised to the power of zero is1.

a¡n= 1

an and

1

a¡n =a

n _{and in particular a}¡1₌ 1

a: a

1

n

= pn_{a For example, a}12 ₌p_{a and a}13 ₌p3_a:

a m

n ₌ pn_am _{or a}mn _{= (}pn_a)m LAWS OF LOGARITHMS

For f(x) = 10x the inverse function is f¡1(x) = log₁₀x. log₁₀x is read the logarithm ofxin base10.

Note: ² If f(x) = 10x, then f¡1(x) = log₁₀x.

² If f(x) = 2x, then f¡1(x) = log₂x.

² If f(x) =bx, then f¡1(x) = log_bx.

LOGARITHMS IN BASEb

In general, if A=bn we say thatnis the logarithm ofA, in baseband that A=bn , n= log_bA, A >0. So A=bn , n= log_bA.

We say that A =bn and n= log_bA are equivalent or inter-changeable.

For example, if 8 = 23 we can immediately say that 3 = log₂8 and vice versa.

Note: a= 10loga for anya >0 and log 10x =x LAWS OF LOGARITHMS IN BASE 10

logA+ logB = log(AB) logA¡logB = log

³

_A

B

´

, B6= 0 nlogA= log (An₎

LAWS OF LOGARITHMS IN BASEe lnA+ lnB = ln(AB) lnA¡lnB = ln

³

_A

B

´

, B6= 0 nlnA = ln (An)

THE CHANGE OF BASE RULE log_bA= logcA

log_cb EXPONENTIAL EQUATIONS

² If ax =ak then x=k,

i.e., if the base numbers are the same, we can equate indices.

² If the bases cannot easily be made the same we can use logarithms. For example,

if 2x = 30 log(2x) = log 30

xlog 2 = log 30

x= log 30 log 2

x+ 4:91

² To solve ex _{=a, we use x= lna.} log a is written lna.

TOPIC 1 ALGEBRA

THE BINOMIAL EXPANSION

(a+b)n

=

¡

n₀

¢

an+

¡

n₁

¢

an¡1b+

¡

n₂

¢

an¡2b2+:::::+

¡

_nn_¡1

¢

abn¡1+

¡

n_n

¢

bn

where

¡

n_r

¢

is thebinomial coefficientof an¡rbr and

r= 0,1,2,3, ... ,n¡1,n.

Thegeneral term((r+ 1)th term) is Tr+1=

¡

n_r

¢

an¡rbr Note:

¡

n_r

¢

=C_rn

1 Find the20th term of:

a 51,45,39,33, ... b 0:125,0:5,2,8, ...

2 If the10th and15th terms of an arithmetic sequence are18 and63respectively, find the25th term.

3 Find the first term in the arithmetic sequence 100,130,160, 190, ... to exceed1200.

4 If the5th and 8th terms of a geometric sequence are18and 486respectively, find the12th term.

5 Maria invested800euros on Jan1st1996.

a If her investment earns interest of7%per annum, what was it worth on Jan1st2004?

b How many years will it take for her investment to reach 4000euros?

6 Ying is training to run a marathon. In one week he ran10km on the first day and increased the distance by 10% on each subsequent day.

a How far did he run on the seventh day?

b What was the total distance he ran during the week?

7 Find the sum of the series:

a 10 + 14 + 18 + 22 +::::::+ 138

b 6¡12 + 24¡48 + 96¡ ::::::+ 1536

8 Find the sum of the first20terms of: 18 + 151₂+ 13 + 101₂+ 8 +:::::: 9 Find the10th term of:

a x2,3x2,5x2, ... b x¡12,x,x212, ...

10 Simplify:

a

¡

5x2

¢

3£

³

_x 4

´

2

£ 8

25x5 b

24a3b8

15 (a2b)3 ¥ 5ab3

12a6b

c 8x

¡2_y3 3 (xy2)0£

9x0y¡1

4x¡3 d

4x12£x¡112 8x2 e p5a3£pa5

11 Without using a calculator, find:

a 16¡ 1

2 _{b 81}14 _{c 32}25

d 27¡ 4

3 _e

¡

1

9

¢

3

2 _{f (0:008)}¡53 12 Write in the form logA:

a log 2 + log 12 b log 36¡log 12

c 3 log 5 + 2 log 3 d 1₄log 81

13 Solve for x, giving answers correct to 3 significant figures where necessary:

a 3x= 243 b 5£2x= 160 c ex= 27

d (1:25)x= 10 e 7ex= 100

14 If A= log₁₀P, B= log₁₀Q and C= log₁₀R, express in terms ofA,BandC:

a log₁₀

¡

P2QpR

¢

b log

·

R3

(P Q2)4

¸

15 If log₅2 = X and log₅3 = Y, express the following in terms ofX andY:

a log₅12 b log₅(10:125) c log₅(4p3)

16 Solve fort: (correct to3s.f.)

a 200£e t

4

= 1500 b 0:15e0:012t= 2:18

c 20e0:2t= 250

17 Find the constant term in the expansion of

³

x+ 3

x2

´

9 .

18 Find the coefficient of x4y9 in the expansion of

¡

x+ 2y3

¢

7.

19 Find the coefficient of thex2term in the expansion of

³

3x¡ 2 x2

´

8 .

20 If A = 125e¡kt and A = 200 when t = 3, find the value ofk.

21 Twins, Pierre and Francesca were each given100 dollars on their15th birthday. They immediately put their money into their individual money boxes. Each week throughout the next year they added a portion of their weekly pocket money. Pierre added $10each week. Francesca added 50 cents the first week, $1the next, $1:50the next, and so on, adding an extra50cents each subsequent week.

a Find the amount that each added to their money boxes after 8weeks.

b How much did Francesca add to her money box on her16th birthday?

c Calculate the total amount they each had in their money boxes after one year.

22 Hayley and Patrick are training for a road cycling race. During the first week Hayley cycled60km and Patrick did likewise. Hayley then cycled an additional20km each subsequent week whereas Patrick increased his distance by20%each subsequent week.

a How far did each of them cycle in the5th week of training?

b Who was the first to cycle200km in one week?

c What total distance did each cycle in training after12weeks?

23 Kapil invested2000Rupees in a bank account on Jan1st1998. Each year thereafter, he invested another 2000 Rupees into the same account which pays8:25%per annum compounded annually.

a Find the total value of his investment immediately after he invested2000Rupees on Jan1st2005.

b Would it have been a better option for Kapil to invest his money each year into an account paying 9% per annum simple interest? Justify your answer.

24 At the beginning of1995the number of koalas on Koala Island was2400. The numbers steadily increased each year according to N(t) = 2400 + 250t wheretis the number of years since 1995.

The population of kangaroos on the island is given by

K(t) = 3200£(0:85)t.

a What was the kangaroo population at the beginning of1995?

b How many kangaroos were on the island at the beginning of2000?

c How many koalas were on the island at the beginning of 2000?

d After how many years will the kangaroo population fall below1000?

e When will the number of koalas exceed the number of kan-garoos?

² Arelationis any set of points in the Cartesian plane.

² Afunctionis a relation in which no two different ordered pairs have the samex-coordinates.

² The ‘vertical line test’

A relation isnota function if a vertical line can be drawn that cuts the graph more than once.

This is not a function.

² Thedomainof a relation is the set of permissable values that

xmay have.

Note: I lnxexists if x >0 I pxexists if x>0 I 1_x exists if x6= 0

² Therangeof a relation is the set of permissable values that

ymay have.

FUNCTION NOTATION

f: x`! f(x) represents a function which convertsxintof(x).

² If f: x`! f(x) and g: x`! g(x),

then f±g represents thecomposite functionoff andg

such that f±g: x`! f(g(x)).

² Theinverse functionof y=f(x) is denoted by f¡1(x).

y=f¡1(x) is the reflection of y=f(x) in the line y=x. For example,

f(x) = 2x+ 3 has f¡1, x= 2y+ 3 i.e., x¡3

2 = y

So, f¡1(x) = x¡3

2 or f

¡1_{: x`!} x¡3 2

LINEAR FUNCTIONS f: x`! ax+b, a6= 0 or f(x) =ax+b, a6= 0

or y =ax+b, a6= 0

all represent a linear function with slopeaandy-interceptb.

² Perpendicularlines have slopes which are negative reciprocals of each other.

² Theequationof a straight line with slope A

B has form Ax¡By=::::::

with slope ¡A

B has form Ax+By=::::::

where the constant term on the RHS of the equation (::::::) is obtained by substituting the coordinates of a point which lies on the line into the LHS form.

QUADRATIC FUNCTIONS f: x`! ax2+bx+c, a6= 0 or f(x) = ax2+bx+c, a6= 0

or y = ax2+bx+c, a6= 0

all represent a quadratic function wherexis the variable.

Note:

² All quadratic functions areparabolicin shape.

If ¡1< a <1, a6= 0 the graph is wider than y=x2. If a <¡1 or a >1 the graph is narrower than y=x2.

² They are symmetric about a vertical line called theaxis of symmetrywhich has equation x= ¡b

2a.

² They have a turning point called thevertexwhich has coordinates

³

_¡b 2a,f(

¡b

2a)

´

.

² They cut they-axis at they-interceptwhich is at f(0) orc.

² Thex-intercepts are

¡b§pb2¡4ac

2a

if ¢ =b2¡4ac>0.

QUADRATIC FORMS (a6= 0) y=a(x¡p)(x¡q)

x-intercepts arepandq

axis of symmetry is

x= p+q 2

y=a(x¡p)2 touchesx-axis atp

vertex is (p,0) axis of symmetry is

x=p

y=a(x¡h)2+k

vertex is (h,k) axis of symmetry is

x=h

y=ax2+bx+c

axis of symmetry is

x= ¡b 2a x-intercepts are

¡b§p¢

2a where

¢ =b2¡4ac is>0 yX=x

y

x

y

x

3

3

y x= y=¦-1( )x ¦( )=2!+3x

a>0 a<0

x y

vertex

axis

of

symmetry

zero zero

y-intercept parabola

minimum

x=h

V ( , )h k

- -b p¢

x

x=-b

2a

p

¢

- +b

2a

2a

p q

x=p+q

2

x=p

x

V ( , 0)p

TOPIC 2 FUNCTIONS AND EQUATIONS

Thediscriminant¢, helps us to decide (discriminate) between the possibilities of:

² not cutting thex-axis (¢<0)

² touching thex-axis (¢ = 0)

² cutting thex-axis (¢>0).

QUADRATIC EQUATIONS

Aquadratic equation, with variablex, is an equation of the form

ax2+bx+c= 0, a6= 0.

Theroots(orsolutions) of ax2+bx+c= 0 are the values ofx

which satisfy the equation (i.e., make it true). To solve quadratic equations we can:

² factorise the quadratic equation and then use theNull Factor Lawwhich states that “if ab= 0 then a= 0 or b= 0”

² complete the square ² use thequadratic formula

If ax2+bx+c= 0 then x=¡b§

p b2¡4ac

2a ² usetechnology.

Note:

Delta,¢, which is under the square root sign helps us decide on the nature of the roots of the equation.

¢ Nature of roots

<0 no real roots = 0 a repeated real root

>0 two distinct real roots >0 two real roots

ASYMPTOTES

Ahorizontal asymptoteis a horizontal line which a graph approaches for large values ofx(either positive or negative).

EXPONENTIAL AND LOGARITHMIC FUNCTIONS

y=ex where e¼2:718 281 83 is called theexponential func-tionin basee.

y= log_ex or y= lnx is called thelogarithmic function. These functions are inverses of each other.

For f: x`! ex, i.e., y=ex, the inverse function is

x=ey, i.e., y= lnx. So, f¡1: x`! lnx y=ex has a horizontal

asymptote y= 0

y= lnx has a vertical asymptote x= 0.

y= lnx is the reflection of y=ex in the line y=x.

GRAPHICAL TRANSFORMATIONS ON y=f(x)

² Translations

I y=f(x) +b The graph of y=f(x) istranslated verticallyupwardsbunits (b >0) and downwards if b <0.

I y=f(x¡a) The graph of y=f(x) istranslated horizontallyaunits to the right (a >0) or left if a <0.

² Stretches

I y=p f(x) The graph of y=f(x) isstretched verticallyby a scale factor ofp(in the

y-direction).

If p >1, a vertical stretch, and if 0< p <1, a vertical compression. I y=f(kx) The graph of y=f(x) isstretched

horizontallyby a scale factor of 1_k in thex-direction.

If k >1, horizontal compression, and if 0< k <1, horizontal stretch.

² Reflections

I y=¡f(x) The graph of y=f(x) isreflectedin thex-axis.

I y=f(¡x) The graph of y=f(x) isreflectedin they-axis.

1 Which of these relations are functions?

a b

c d

e f

2 State the domain and range of these functions:

a f: x`! px+ 3 b f(x) =x2+ 4

c f(x) =x2¡6x d f: x`! ln(x¡2)

e f(x) = p 1

x¡4

3 Use technology to find the range of these functions:

a f: x`! 25¡7x¡5x2

b y=esinx c f(x) = 2x+2

x 4 Given that G(x) = 3¡x¡2x2, find:

a G(¡1) b G(3)

c G(a3) d G(a¡2)

5 Given f: x`! 5x¡2 and

g: x`! 2x+ 7, find in simplest form:

a (f±g)(x) b (g±f)(x) c (g¡1±f)(x)

A is a

vertical line which a graph approaches for large values of (either positive or negative).

vertical asymptote

y

y

x

1

1

y x= y e= x

y=lnx

REVISION QUESTIONS (Topic 2)

y

x

y

x

y

x

y

x

y

x

y

x y

x

vertical asymptote

y

x

6 For the following functions, find f¡1(x):

a f: x`! 4x+ 12 b f(x) = ln(2x+ 9)

c f(x) = 10¡3x 4

d f: x`! (x¡4)2+ 9, x>4

7 Copy each of these graphs of y = f(x) and include the graphs of y=x and y=f¡1(x).

a b

8 For f : x`! lnx and g: x`! x3, find the value of:

a (f±g)(2) b (g±f)(2)

c (f¡1±g)(1:2) d (g±f)¡1(8)

9 Consider f(x) =x2+ 2x¡5.

a Sketch the graph of y=f(x).

b Explain whyfhas an inverse function f¡1 if x6¡1.

c Find the equation off¡1.

d Sketch the graph off¡1.

10 If f(x) =e2x+3, find the defining equation for f¡1(x).

11 g: x`! e4x and h: x`! 3x+ 2 Find the defining equations of:

a (h±g)(x) b (g¡1±h)(x)

12 Consider f: x`! ln(x+ 2)¡5, x >¡2.

a Find the defining equation off¡1.

b On the same set of axes, sketch the graphs off andf¡1.

c State the domain and range off andf¡1.

13 Copy this graph of y=f(x).

a y=f(x+ 2)

b y= 2f(x)¡3

c y= 4¡f(x)

14 Copy this graph of

y=g(x).

a y=g(¡x)

b y=g(2x)

15 Find the equations of these straight lines:

a passing through (¡7,2) and (3,¡3)

b passing through (4,¡1) and with a slope of 3₅

c passing through (¡2,8) and perpendicular to the line with equation 8x+ 3y= 10

d passing through (4,7) and parallel to they-axis.

16 On separate axes, sketch the graphs of these functions:

a y= 5¡2x

b f(x) = 2(x¡4)(x+ 2)

c f: x`! 3(x¡2)2+ 5

d y=¡4(x+ 1)2

17 For each of the following functions:

i find thex-intercepts

ii find the equation of the axis of symmetry

iii find the coordinates of the vertex

iv state they-intercept

v sketch the graph

a y=¡4x(x+ 3) b y= 1₂(x+ 6)(x¡4)

c y=¡3(x¡2)2 d y= 2(x+ 5)2¡4

18 For each of these quadratic functions, find the equation in the form y=f(x).

a b

c

19 Write these quadratics in the form y=a(x¡h)2+k. Hence, sketch each function, clearly showing the coordinates of its vertex.

a y=x2¡4x+ 9 b y=x2¡6x¡2

c y= 4x2+ 16x+ 11 d y=¡3x2+ 12x¡10

20 Use the quadratic formula to find the exact roots of:

a x2+ 12x+ 8 = 0 b 3x2¡4x¡1 = 0

21 Use technology to solve to5decimal places:

a 7x2¡10x+ 2 = 0 b 2x2= 11x+ 15

22 For the quadratic y= 2x2¡9x+ 3, find the:

a equation of its axis of symmetry

b coordinates of the vertex

c axis intercepts. Hence sketch the graph.

23 Determine the discriminant of x2+ 8x+k= 0. Hence, find the values ofkfor which the equation has:

a no real roots b two distinct real roots.

24 For what values ofmwould the graph of y=mx2+ 4x+ 6 lie entirely above thex-axis?

25 Consider the function f(x) = 4¡ 1

x¡2.

a State the domain of y=f(x).

b State the range of y=f(x).

c What are the axis intercepts of y=f(x)?

d Write down the equations of the asymptotes of y=f(x).

26 Use technology to find the asymptotes of f: x`! 6¡2x x+ 4 . Sketch the graph of the function, clearly showing the asymp-totes and the axis intercepts.

27 Use technology to find the asymptotes of:

a f(x) = ₂2x+ 9

¡ ¡ b f(x) =

2

p ¡ 2 On the grid, also

draw graphs of:

y

x y=¦( )x

On the grid, also draw graphs of:

y

x y g x= ( )

y

x -20

-1 5

y

x -8 -4

y

x

V ,(5 ¡-12)

38 y

x

8

3

y

x

)

0

(x>

2

2

TERMINOLOGY

The wave oscillates about a horizontal line called theprincipal axis

(ormean line).

Amaximum pointoccurs at the top of a crest and aminimum point

at the bottom of a trough.

Theamplitudeis the distance between a maximum (or minimum) point and the principal axis.

PERIODICITY

Aperiodic function is one which repeats itself over and over in a horizontal direction.

Theperiodof a periodic function is the length of one repetition or cycle.

RADIAN MEASURE

DEGREE-RADIAN CONVERSIONS

A full revolution is measured as360o using degrees and2¼ when using radians.

So,2¼radians is equivalent to360oand consequently

¼radians is equivalent to180o.

The following diagram is useful for converting from one system of measure to the other:

SECTORS OF CIRCLES

Forµ indegrees

² arc AB=

¡

₃₆₀µ

¢

£2¼r ² area OAB=

¡

₃₆₀µ

¢

£¼r2. Forµinradians ² arc AB=rµ

² area OAB= 1₂r2µ.

SOLUTION OF TRIANGLES

² Area of¢ABC A= 1₂absinC

² Sine Rule a

sinA = b

sinB = c

sinC

or sinA

a =

sinB

b =

sinC c ² Cosine Rule a2=b2+c2¡2bccosA

or cosA=b

2_+c2_¡a2 2bc SINE AND COSINE FROM THE UNIT CIRCLE

Theunit circleis the circle, centre (0,0) with radius1unit. If P(x,y) moves around the unit

circle such that OP makes an angle ofµwith the positivex-axis then:

thex-coordinate of P iscosµand they-coordinate of P issinµ:

From the unit circle we can see that

² cos2µ+ sin2µ= 1

² ¡16cosµ61 and ¡16sinµ61 for allµ ² tanµ= sinµ

cosµ (provided thatcosµ6= 0). ANGLE MEASUREMENT

Suppose P lies anywhere on the unit circle and A is (1,0).

Letµ be the angle measured from OA, on the positivex-axis.

µis

positivefor anticlockwise rotations and

negativefor clockwise rotations.

MULTIPLES OF45o OR ¼₄

MULTIPLES OF30o OR ¼₆

TOPIC 3 CIRCULAR FUNCTIONS AND TRIGONOMETRY

KEY FACTS AND CONCEPTS

period

amplitude

minimum point maximum point

principal axis

x y

R

R R

1 radian

One is the angle

swept out by one radius unit around a circle and is approx-imately equivalent to .

radian

57 3: o

180

180

£

£

¼

¼

degrees radians

q r

A

O B

B

A

C

c b

a

y

x

q

P

A 1 positive direction

negative direction x

q

1 1 1

-1 -1

y

y

x P( , )x y

N

x y

(0, 1)

(1, 0)

(0, 1) -( 1, 0)

-³

1

p

2; 1

p

2

´

³

1

p

2;¡ 1

p

2

´ ³

¡_p1 2;¡

1

p

2

´ ³

¡_p1 2;

1

p

2

´

45° 135° 135°

225°

315° 315° 1

p 2

1 p 2

45°

45° 1

x y

(0, 1)

(1, 0)

(0, 1) -(-1, 0)

³ 1 2;

p

3 2

´

³p

3 2 ;

1 2 ´

³p

3 2 ;¡

1 2 ´

³ 1 2;¡

p₃

2 ´ ³

¡1 2;¡

p₃

2 ´ ³

¡p3 2 ;¡

1 2 ´ ³

¡p3 2;

1 2 ´ ³

¡1 2;

p

3 2

´

120° 120°

210° 210°

1 2 1 2

p

3 2 30°

60°

Summary:

² Ifµis amultiple of 90o, the coordinates of the points on the unit circle involve0and§1.

² Ifµis amultiple of 45o, (but not a multiple of90o), the coordinates involve§p1

2:

² Ifµis amultiple of 30o, (but not a multiple of90o), the coordinates involve§1₂ and§p₂3.

TRANSFORMING y= sinx

² In y=Asinx, Aaffects the amplitude and the amplitude isjAj:

If A= 2, vertical distances from thex-axis to y= sinx

are doubled to get the graph of y= 2 sinx.

This is avertical dilation.

² In y= sinBx, B >0, the sine function completes a cycle for 06Bx62¼,

i.e., for 06x6 2¼

B, soB affects the period and

the period is 2¼

B:

If B= 2, the period is halved. If B= 1₂, the period is doubled.

Here we have ahorizontal dilationof factor 1

B: ² y= sin(x¡C) is ahorizontal translationof y= sinx

throughCunits.

² y= sinx+D is avertical translationof y= sinx

throughDunits.

² y= sin(x¡C) +D is atranslationof y= sinx

through [C,D].

THE GENERAL SINE FUNCTION y=Asin (B(x¡C))+D

affects

amplitude

affects

period

affects

horizontal translation

affects

vertical translation

THE COSINE FUNCTION

The graph of y= cosx is identical in shape to y= sinx and is a horizontal translation of it through¡¼₂ (¼₂ to the left). So the graph of y= cosx is:

THE TANGENT FUNCTION

The graph of y= tanx is:

Notice that y= sinx

cosx is undefined when

cosx= 0 i.e., when x= ¼₂+k¼:

This shows on the graph at vertical asymptotes.

DUPLICATION (OR DOUBLE ANGLE) FORMULAE

Theduplication(ordouble angle) formulae are:

sin2A= 2sinAcosA and cos2A=

8

<

:

cos2A¡sin2A

2cos2A¡1 1¡2sin2A

Notice also that sinA = 2sin

¡

A₂

¢

cos

¡

A₂

¢

and cosA =

8

>

>

>

<

>

>

>

:

cos2

¡

A₂

¢

¡sin2

¡

A₂

¢

2cos2

¡

A₂

¢

¡1 1¡2sin2

¡

A₂

¢

SOLVING TRIGONOMETRIC EQUATIONS Examples:

² Solve 2 sinx= cosx, 06x62¼. 2 sinx = cosx

) _{2 cos}2 sinx_x = cosx 2 cosx

) tanx = 1₂

) x =tan¡1(0:5) ) x + 0:464,3:605

² Solve sin 2x= cosx, 06x62¼. sin 2x= cosx

) 2 sinxcosx= cosx

) 2 sinxcosx¡cosx= 0 ) cosx(2 sinx¡1) = 0 ) cosx= 0 or sinx= 1₂

) x= ¼₂, 3₂¼, ¼₆, 5₆¼

² Solve sin2x= cosx¡1, 06x62¼. sin2x = cosx¡1 1¡cos2x = cosx¡1 cos2x+ cosx¡2 = 0 (cosx+ 2)(cosx¡1) = 0

cosx = 1 as ¡16cosx61

x = 0,2¼

y

x y=2 ¡sinx

y= ¡sinx

y

x y= ¡2sin x y= ¡sinx

is called thegeneral sine function.

y

x

-p_–– 2

-p_–– 2

-p_–– 2

p–_{2 ––}3p

2 p

2p

y¡ ¡= cos¡x y¡ ¡ ¡= sinx

y

3

-3

x -p

-p -–2-–2pp p–p–22 pp ––––3p3p22 2p2p ––––5p5p22

(0, 1)

-Note: Trigonometric equations can also be solved by graphical methods using a graphics calculator.

For example, sin(2x) = 0:4, 06x6¼.

The solutions are: x+0:206,1:365.

Note: If a straight line which passes

through the originmakes an angle ofµowith the positive

x-axis, then its slope istanµ. The equation of this line is

y= [tanµ]x.

1 Convert these into radians, leaving¼in your answer:

a 60o b 210o c ¡135o

2 Convert these to degrees:

a 2₃¼ b 5₄¼ c ¡11₆¼

3 Convert these radian measures to degrees (to the nearest tenth of a degree):

a 0:815 b 2:7

4 Convert these to radians, correct to3decimal places:

a 48o b 203o

5 Find the perimeter and area of a sector of:

a radius8cm and angle40o

b radius12cm and angle2:4radians.

6 O is the centre of a circle of radius32cm. Chord AB is 50cm long. Find the area of the shaded segment.

7 On separate grids, sketch the graphs of:

a y= 2 sin(x¡¼₃) for 06x62¼ b y= sinx+ 2 for ¡¼6x6¼

c y= 3 cos(2x) for 06x62¼ d y= tanx for ¡¼6x6¼ e y= cos(x₂)¡1 for 06x62¼ 8 For each of the following state:

i the period ii the amplitude (where applicable)

a y= sin(2x) + 2 b y= 4 cosx c y= 5 tan(3x) d y= 10¡6 sin(3x)

9 Find the area of:

a b

10 Findain:

a b

11 Solve forxif 06x62¼:

a sinx= 0:785 b 2 cosx= 5 sinx c tan(3x) = 0:9

12 Find the exact solutions of these equations if 06x62¼:

a p2 cosx+ 1 = 0 b 2 sinx=p3

c 2 sin2x+ 3 cosx= 3 d sin(2x) + sinx= 0

13 Solve fortif 06t612 and

a 5 sin(2t¡3) = 2

b 15 + 3 cos(₂t + 1) = 14:5 Give answers correct to3dec. places.

14

The above graph is of y=Asin(Bx)+C for 06x6¼. Find the values ofA,BandC.

15 The diagrams below show two triangles both satisfying the conditions AB= 30cm, AC= 12cm and ]ABC= 21o.

Diagram 1

Diagram 2

a Calculate the size of]ACB in each diagram.

b Calculate the area of¢ABC in diagram2.

16 The depth, d metres, of water in a tidal river t hours after midnight may be represented by a function of the form

d=A+Bsin

¡

2_k¼t

¢

whereA,B andkare constants. The water is at a maximum depth of22:5m at3am and again at3pm. It is at a minimum depth of8:7m at9am and9pm. Find the values ofA,Bandk.

17 µis an acute angle and cosµ= 3₈. Find the exact values of:

a sinµ b sin 2µ

18 ®is an obtuse angle and sin®= 2₃. Find the exact values of:

a cos® b cos 2®

19 Given that®is acute and cos 2®= ₁₃5, find the exact value ofsin®.

20 Given that sinµ=¡1₂ and cosµ=¡p₂3 where 0o< µ <360o, find the exact value of:

a µ b tanµ

21 Find, to3significant figures, the value ofxin:

a b

x

0.206 1.365

y=0 4.

y=sin(2x)

1

-1 y

REVISION QUESTIONS (Topic 3)

y

x q

O A

B

18 cm

12 cm

27° 5.4 cm

125° 7.8 cm

16 cm

acm

37°

29° _{100 m} am

120 m 31°

21° 30 cm

12 cm

B C

A

21° 30 cm

12 cm

B C

A

xm

80 m 40°

33°

xm

11 m 5

10 15 20 25

p p p

p

2 3 2 4

c

O

T

P

y

x

60°

y

x

30°

4

2 5

q x

z x

y

a b

q a

b q

c

2 4 6 8

2p p

p

2 3 2

x y

p

75° 25 cm

xcm

22 A triangle has sides of length12cm,16cm and21cm. Find the size of its largest angle.

23 Sketch the graph of y= sinx for 06x610.

24 In the diagram O is the cen-tre of the circle and PT is a tangent to the circle at T. If the circle has radius9cm and OP= 30cm, find the area of the shaded region.

25 Solve these equations for 0o6 µ 6360o, giving answers to the nearest degree:

a 4 cos2µ= 1 b 4 sin2µ= cos2µ 26 Sketch these graphs on ¡¼6x6¼:

a y= 2¡3 cosx b y= cos(3x) + 2

27

Part of the graph of y=A+Bsinx is drawn above. Find the value of: a A b B

28 Solve this equation on 06x62¼: sin(2x) = sin(x

3 ¡2)

29 An aeroplane averaging 400km h¡1 travelled directly north from an airport for45minutes before changing its course to north-east for90minutes. What is the direct distance between the plane’s present position and the airport?

30 Find the equations of these straight lines:

a b

31 a State the domain and range for f(x) = 1 sinx. b What is the graphical significance of your answers ina?

32 Consider f(x) = 1¡3 sin(2x).

a Find the period of y=f(x).

b Find the amplitude of y=f(x).

c Sketch the graph of y= 1¡3 sin(2x) for ¡¼₂ 6x6¼.

33 a What quadrant containsµ if tanµ >0 and cosµ <0?

b Find the exact value ofcosµif tanµ= 2 and 180o< µ <270o.

34 What transformation maps:

a y= sinx onto y=¡sinx b y= cosx onto y= 2 cosx c y= cosx onto y= cos(2x)

d y= sinx onto y= sin(x+¼₆) + 2?

35 Findcosµif sinµ=¡4₅ and tanµ=¡4₃.

36

Show that µ= arctan

³

_a

c

´

+ arctan

³

_b

c

´

.

37 Expressµin terms ofxin:

38 Findxif (sinx+ cosx)2= sin2x+ cos2x, 06x6¼.

39 Simplify log(sinµ)¡log(cosµ).

40 A ship leaves a port at2:00pm and sails in the direction034o at a speed of16kmph. Another ship leaves the port at2:30 pm and sails in the direction107oat a speed of19kmph. How far apart are the ships at4:00pm?

41 Simplify without using a calculator:

a tan(¡¼₆) b

p

2 sin(3₄¼)

42 Sketch graphs of:

a f(x) =p3 cosx for ¡2¼6x62¼ b f(x) =¡2 sin(2x) for ¡2¼6x62¼

43 At what points does the graph of y= 2¡xsin(2x) cut the

x-axis for 06x63¼?

44 Prove that tant=

p

1¡cos2t

cost , 0< t < ¼. 45 Find the solutions of sinx

x2+ 1= 0:3 to3decimal places.

46

Find a+b

in terms of

x,y,z and µ.

47 Solve forx:

a 2 cos(2x) + 1 = 0, 06x62¼ b cos(2x) = 2 cosx, 06x63¼ 48 Solve forx: arcsin(3x2¡2x) = 5₆¼

49 Findµ if 0< µ < ¼₂ and 1¡cos(2µ) sin(2µ) =

p

3.

50 Solve forxgiven that 2¡xsinx= 1₂x¡1 and x >0.

51 If 0< µ < ¼₂, findµgiven that tanµ= 2 cosµ.

52 Show that sin 2_µ

1 + cosµ = 1¡cosµ for allµ except

where cosµ=¡1.

53 Findµ where ¡2¼6µ62¼ and:

a 2 sinµ6p3 b p2 cosµ6¡1

Amatrix is a rectangular array of numbers arranged inrows and

columns.

It is usual to put square or round brackets around a matrix.

In

2

6

4

2 1 6 1

3

7

5

we have4rows and1column and we say that this is a4£1column matrix

orcolumn vector.

"

_{6 1 2}

9 2 3 10 3 4

#

has3rows and3columns and is called a3£3square matrix.

This element,3, is in row3, column2.

£

3 0 ¡1 2

¤

has1row and4columns and is called a1£4row matrixorrow vector.

Note: ² An m£n matrix hasmrows andncolumns.

" "

rows columns

² m£n specifies theorderof a matrix.

EQUALITY

Two matrices areequalif they have exactly the same shape (order) and elements in corresponding positions are equal.

For example, if

·

a b c d

¸

=

·

w x

y z

¸

then a=w,b=x,

c=y and d=z. Notice that

·

1 2 0

3 4 0

¸

6

=

·

1 2 3 4

¸

:

ADDITION AND SUBTRACTION

Toaddtwo matrices they must be of thesame order and then we simplyadd corresponding elements.

Tosubtractmatrices they must be of thesame order and then we simplysubtract corresponding elements.

SCALAR MULTIPLES

If a scalart is multiplied by a matrix A the result is matrix tA

obtained by multiplying every element ofAbyt.

ZERO MATRICES

Azero matrixis a matrix in which all elements are zero.

For example, the 2£2 zero matrix is

·

0 0 0 0

¸

,

the 2£3 zero matrix is

·

0 0 0

0 0 0

¸

:

Zero matrices have the property that:

IfAis a matrix of any order andOis the correspondingzero matrix, then _A+O=O+A=A.

NEGATIVE MATRICES

ThenegativematrixA, denoted¡Ais actually¡1A.

¡Ais obtained fromAby reversing the sign of each element ofA.

Note: A+(¡A)=(¡A)+A=O.

TOPIC 4 MATRICES

KEY FACTS AND CONCEPTS

MATRIX ALGEBRA FOR ADDITION

² IfAandBare matrices thenA+Bis also a matrix.

² A+B=B+A

² (A+B)+C=A+(B+C)

² A+O=O+A=A ² A+(¡A)=(¡A)+A=O ² a half ofAis 1₂A (not A

2) (Dividing a matrix by a real number has no meaning in

matrix algebra.)

MATRIX PRODUCTS

Theproductof an m£n matrix A with an n£p matrix B, is the m£p matrix (calledAB) in which the element in therth row andcth column is the sum of the products of the elements in therth row ofA with the corresponding elements in thecth column ofB. For example, if A=

·

a b c d

¸

and B=

·

p q r s

¸

,

then AB=

·

ap+br aq+bs cp+dr cq+ds

¸

,

and if C=

·

a b c d e f

¸

2£3

and D=

"

_x

y z

#

3£1 ,

then CD=

·

ax+by+cz dx+ey+f z

¸

.

2£1

Note:

² IfAandBare matrices that can be multiplied thenAB

is also a matrix.

² In general AB6=BA.

² IfOis a zero matrix thenAO=OA=O for allA.

² A(B+C)=AB+AC ² If I=

·

1 0 0 1

¸

then AI=IA=A for all2£2matricesA.

² An for n>2 can be determined provided thatAis a square andnis an integer.

DETERMINANTS

The determinant of the2£2matrix A=

·

a b c d

¸

is detA=jAj=ad¡bc.

The determinant of the3£3matrix A=

"

_{a b c}

d e f

g h k

#

is

detA=jAj=a

¯¯

¯¯

e f h k

¯¯

¯¯

¡b

¯¯

¯¯

d f g k

¯¯

¯¯

+c

¯¯

¯¯

d e g h

¯¯

¯¯

.

INVERSES OF MATRICES

Only square matrices can have inverses and not all square matrices do have inverses.

If a matrixAhas an inverse, calledA¡1, then A¡1A=AA¡1=I.

2£2 MATRIX INVERSES

2£2 matrix inverses can be found by using:

² if A=

·

a b c d

¸

, then A¡1= 1

ad¡bc

·

d ¡b ¡c a

¸

Notice that: ² IfA¡1exists, detA6= 0 and vice versa.

² If detA6= 0, Ais said to beinvertible. (ad¡bc= detA=jAj)

3£3 MATRIX INVERSES

3£3 inverses can be found by using the key of a graphics calculator.

SOLUTION OF SYSTEMS OF LINEAR EQUATIONS

If we wish to solve the matrix equation AX=B

we premultiply each side byA¡1 (if it exists). ) A¡1(AX) = A¡1B

) A¡1AX = A¡1B

) IX = A¡1B

) X = A¡1B

i.e., if AX=B then X=A¡1B.

Note: If XA=B then X=BA¡1. fpost multiplying byA¡1g

1 Given A=

·

2 ¡1

3 4

¸

and B=

·

¡3 2 6 ¡5

¸

, find:

a A+ 2B b 3A¡6I c AB

d detB e A¡1 f A2

2 Given P=

·

¡3 2

6 8

¸

and Q=

"

_{7 ¡2 3}

4 2 1

0 ¡3 5

#

, find:

a detP b detQ c P¡1 d Q¡1

3 Given M=

"

_{7 2 1}

¡3 4 4

2 1 6

#

and N=

"

_{2 8 ¡3}

1 2 1

4 ¡2 5

#

,

find a 3M¡2N b MN

4 Find the inverse of

·

3 2 5 4

¸

.

Hence, solve the system of equations 3x+ 2y = 14 5x+ 4y = 22

5 For S=

·

7 1 5

2 ¡3 3

¸

and T=

"

_{8 ¡2} 4 ¡1

3 2

#

, findST.

6 A=

·

7 2

4 ¡1

¸

, B=

·

8 2

3 ¡4

¸

and X=

·

a 2

1 ¡a

¸

.

Findaif 2A+X= 3B.

7 Use technology to find the inverse of

"

_{3 1 2}

2 4 1

1 ¡2 1

#

.

Hence solve the system of equations

3x+y+ 2z = 18 2x+ 4y+z = 8

x¡2y+z = 9

8 Erica, Wu and Yasmina went shopping together. Erica bought 10bread rolls,8apples and4bags of chips for a total cost of $17:20. Wu purchased5bread rolls, 12apples and1 bag of chips for a total cost of $9:80. Yasmina paid $13:60for12 bread rolls,6apples and2bags of chips.

Represent this information with a system of linear equations, explaining carefully what each variable represents.

Use an inverse matrix to find the individual cost of the three items the girls purchased.

9 a

Let A=

·

1 2 1

3 ¡2 0

¸

and B=

"

_{1 2}

¡1 1

0 2

#

FindCifAB¡2C=

·

¡1 ¡4

3 8

¸

:

b Let D=

·

0 2

¡2 0

¸

Show that ¡1₄D2=I, whereIis the identity matrix. Hence stateD¡1in terms ofD.

10 a If A=

·

2 4 3 2

¸

, findA¡1,

and use it to solve A

·

x y

¸

=

·

15

¡5

¸

.

b Let N=

·

a b

a2 b2

¸

, wherea,b6= 0.

Determine the relationship betweena andbif Ndoesnot have an inverse.

11 If A=

·

a 4

2 a+ 2

¸

, for what values ofadoes the inverse matrixA¡1exist?

12 ComputeABwhere:

A=

"

_{2 1 ¡1}

0 1 0

0 0 1

#

and B=

"

_{0 1 1}

1 1 0

1 2 1

#

.

Hence, or otherwise, obtain the matrix A10B, giving reasons.

13 Let A=

·

4 1 3 2

¸

and B=

·

2 0 7 1

¸

.

a Find A¡1. b Use technology to find det(AB)¡1: 14 Findaandbif:

3

·

a 4 1 b

¸

=

·

a 1 0 b

¸ ·

1 2

¡1 1

¸

+

·

3 9

2 ¡2

¸

15 Given A=

·

2 ¡1

¡4 3

¸

, findA¡1:

Hence, solve the system:

·

2 ¡1

¡4 3

¸ ·

x y

¸

=

·

6

¡2

¸

:

16 Find the real numberskfor which the matrix product

·

k k

4 2k

¸ ·

2 k

4 3

¸

has no inverse.

17 Show that the matrix A=

·

0 ¡2

¡1 1

¸

satisfies the equation A2=A+ 2I whereIis the multiplica-tive identity for2£2 matrices. Use this equation to show, without evaluatingA4, that A4= 5A+ 6I.

18 a i LetA=

"

_{1 2 4}

4 3 8

¡4 ¡4 ¡9

#

. Calculate

(1) B=A+ 4I, whereIis the identity matrix

(2) AB. Hence deduce the matrixA¡1.

ii Find all the values ofkfor which the determinant

¯¯

¯¯

k 2

2 k¡3

¯¯

¯¯

= 0:

x

-1

b Find the solution of the system of equations:

x¡2y+ 3z = 3 3x¡8y+ 5z = ¡5

¡2x+ 3y¡2z = 2:

19 Let S=

·

k 1 0

3 4 2

¸

and T=

"

_{2 1}

0 5 4 3

#

:

a CalculateST. b If det(ST)= 18, findk.

20 Let A=

·

3 1

¡2 2

¸

and I=

·

1 0 0 1

¸

.

a Verify that A2+ 8I= 5A.

b Hence, or otherwise, show that A¡1=1₈(5I¡A):

21 a If P=

·

5 ¡1

3 2

¸

and Q=

·

1 2 0 3

¸

,

calculate the determinant of the matrixPQ.

b For what values ofkdoes the system of equations not have a unique solution? kx+ 2y = 4

x+ (k¡1)y = 2

Ifk does not have those values, find the unique solution. For each value ofkfor which no unique solution exists, de-termine whether or not any solution to the system of equa-tions exists.

22 If A=

"

_{1 2 1}

5 2 ¡3

¡1 2 3

#

and B=

"

₁

1 1

#

,

find a real numberksuch that AB=kB.

23 Let A=

·

1 5 1 4

¸

, B=

·

1 ¡2

0 1

¸

,

C=

·

3 2

¡1 1

¸

, I=

·

1 0 0 1

¸

.

a Show, by direct calculation, that (AB)¡1=B¡1A¡1: b Verify thatCsatisfies the equation C2¡4C=¡5I.

c Hence, or otherwise, show that C¡1=1₅(4I¡C): 24 a Write the equations in the form AX=B.

x¡y = 3 2x+cy = 6

For what values ofcdoesA¡1 exist? For these values of

c, findA¡1and use it to find the unique solution of these equations.

b If A=

"

_{¡8 ¡6 ¡6}

2 0 2

8 8 6

#

, find:

i A2 and hence ii A¡1

25 a i Consider the matrix M=

·

n¡1 4

3 n¡2

¸

FinddetM, and find two distinct values ofnsuch that detM= 0. When doesM¡1exist?

ii Given P=

"

_{¡4 2} 1 ¡3

¡2 6

#

and Q=

·

3 ¡2 1 ¡4

¸

findPQ, and explain whyQPcannot be found.

b LetA=

"

_{2 0 0}

0 3 1

0 1 3

#

. FindA2, and show that

A2¡6A=¡8I. Henceverify that A¡1=1₈(6I¡A):

26 For what values of the constantkis the determinant, detA=

¯¯

¯¯

¯

1 2 k

2 ¡1 1

k ¡1 2

¯¯

¯¯

¯

, equal to zero?

27 If A=

"

_{1 1 1}

x 2 3

x2 4 9

#

, find allxsuch that detA= 0.

28 a Find, without technology, A¡1 if A=

·

3 5 1 7

¸

.

Hence, find det(A¡1).

b Represent the system of equations 3x¡4y= 1,

¡2x+ 5y= 4 in the form AX=B.

Find the inverse of matrixAand use it to solve the given system of equations.

29 Find the inverse of the matrix

·

2 4

¡1 ¡3

¸

.

Hence solve 2x+ 4y = 2

¡x¡3y = 1:

30 A=

"

_{2 0 ¡1}

0 2 1

¡1 1 2

#

, B=

"

_{15 ¡1 ¡4}

¡1 15 4

¡4 4 16

#

.

Write down the matrix (B¡6A) and show that

A(B¡6A)=kI wherekis some number andIis the identity matrix. Hence find the inverseA¡1:

31 a For which real values ofxdoes the matrix

·

x 2

¡1 x+ 2

¸

have an inverse?

b LetA=

"

_{1 4}

2 5

1 ¡3

#

and B=

·

2 ¡1 1 8 ¡5 2

¸

.

FindABandBA. IsBthe inverse ofA? Give reasons for your answer.

32 Let A=

·

3 4

¡2 ¡3

¸

.

a Find A¡1 and A2.

b Hence, or otherwise, simplify A3+ 4A2¡2A¡3I, and show that this matrix has no inverse.

33 a Solve the system of equations ¡x+y¡z = 1 3x¡2y+z = ¡2

3x+y = ¡4: b The cubic polynomial f(t) =at3+bt2+ct+d has the

following properties:

f(0) = 0 f(¡1) = 1 f0(¡1) =¡2 f00(1) =¡8:

Useato determine the coefficients off(t):

34 Let A=

·

3 0 0 4

¸

, B=

·

1 ¡2

¡1 1

¸

, C=

·

2 ¡2

3 1

¸

: a Find the inverse matrices A¡1 and B¡1.

b Hencefind the matrixXwhich satisfies AXB=C.

35 Because of overhunting, the population of Sei whales in the Indian Ocean has decreased rapidly this century. The biologists modelling this decrease chose a time unitt such thatt = 1 corresponds to the year1950,t= 2 to the year1960, t= 3 to1970, and so on. The estimate of the size of the population

P(t), in1000s of whales, is shown below for these times.

t 1 2 3

P(t) 23 21 17

a quadratic P(t) =a+bt+ct2 wherea,bandcare constants.

a Explain why the data and relationship above produce the equations: a+b+c = 23

a+ 2b+ 4c = 21

a+ 3b+ 9c = 17

b Solve the system inafora,b, andc.

c In 1980 (that is, when t = 4) the population was 3000 whales. Determine whether the biologists’ model would overestimate or underestimate the population, and by how much.

36 Let A =

·

x 0 2

1 ¡1 0

¸

and B=

"

_{0 1}

2 x

0 ¡1

#

Findxsuch that det(AB)= det(BA) .

37 Let A=

"

_{1 0 ¡1}

3 ¡2 0

¡1 0 1

#

.

FindA3and show that A3= 4A. Hence, or otherwise, find A7.

38 Evaluate the determinant

¯¯

¯¯

¯

a ¡a a

¡a a ¡a

a ¡a a

¯¯

¯¯

¯

:

39 a Let A=

"

_{3 ¡2 10}

0 1 ¡3

¡1 1 ¡4

#

. i Find A2.

ii Verify that A3=I and deduce that A2=A¡1: b Write the equations 3x¡2y+ 10z = 1

y¡3z = ¡6

¡x+y¡4z = ¡5

in matrix form and use the inverse ofAto solve the system.

40 The general equation of a circle in the plane has the form

x2+y2+ax+by+c= 0:

The three points (1,3), (1,¡7) and (¡2, 2) lie on a circle. By substituting their coordinates into the equation, obtain a system of three equations ina,b, andc. Solve the system, and hence write down the equation of this circle.

41 A nut wholesaler prepares packs of mixed nuts.

They come in three types of1kg packs: Premium, Deluxe and Special.

By weight, the Premium packs consist of40%hazelnuts,30% cashews and30%peanuts.

The Deluxe packs consist of30%hazelnuts,20%cashews and 50%peanuts.

The Special packs consist of30%hazelnuts,10%cashews and 60%peanuts.

The company has an order to produce3000packets of Premium mix,4000of Deluxe mix and7000of Special mix. Determine the number of kilograms of each nut variety needed to fill the order.

42 Findxandygiven that

£

¡2 3 x

¤

"

_{¡1 4}

3 2

5 6

#

=

£

31 y

¤

43 IfA,BandCare invertible square matrices and ABC= 2I, find:

a A¡1in terms ofBandC b B¡1in terms ofAandC c C¡1in terms ofAandB

TOPIC 5 VECTORS

VECTORS IN 2-D

v=

·

v1

v2

¸

=v1i+v2j

whereiandjare unit vectors parallel to thexandy-axes in the positive directions.

LENGTH

The length of v=

·

v₁ v2

¸

is jvj=

p

v₁2+v₂2.

THE SUM OF TWO VECTORS

For

v+w starts at the beginning ofvandwis added onto the end ofv.

v+w ends at the end ofw.

THE DIFFERENCE BETWEEN TWO VECTORS v¡w=v+(¡w)

Note: v+w and v¡w are represented by the diagonals of a parallelogram with defining vectorsvandw.

VECTOR EQUATIONS

a¡b¡c+d=0

or

a+d=b+c UNIT VECTORS

Aunit vectoris any vector which has a length of one unit. The twospecial unit vectorsare i=

·

1 0

¸

and j=

·

0 1

¸

.

For a=

·

a1

a2

¸

,

² 1

jaj

·

a1

a₂

¸

is the unit vector in the direction ofa

² 1

jaj

·

a₁ a2

¸

and ¡1

jaj

·

a₁ a2

¸

are unit vectors parallel toa

² 1

jaj

·

¡a2

a1

¸

and 1

jaj

·

a2

¡a1

¸

are unit vectors perpendicular toa. v or _~v

vz

vx

v _w

v

w

v w+

-w v v-w

v

w

v-wv+w

a

VECTORS BETWEEN TWO POINTS ¡!

AB=

·

xB¡xA yB¡yA

¸

and

¡!

BA=

·

xA¡xB yA¡yB

¸

PARALLEL VECTORS

Ifaandbare parallel there exists a scalar k such that

a=kb.

Ifaandbhave the same direction, k >0. Ifaandbhave opposite direction, k <0.

THE SCALAR PRODUCT OF TWO VECTORS

If v=

·

v1

v2

¸

and w=

·

w1

w2

¸

then v²w =jvj jwjcosµ

=v₁w₁+v₂w₂

VECTOR EQUATION OF A 2-D LINE

If an object is initially at A and is moving a direction parallel to vectorb, then aftertunits of time

r = ¡!OA+¡!AR

So, r = a+tb (vector equation)

tis called aparameter. For example,

thevector equationof the line passing through (¡1,4) in the direction

·

5 2

¸

is

·

x y

¸

=

·

¡1 4

¸

+t

·

5 2

¸

:

Theparametric equationsof this line are

x=¡1 + 5t and y= 4 + 2t, t2R

Thecartesian equationof this line is obtained by equatingts, i.e., x+ 1

5 =

y¡4

2 which simplifies to 2x¡5y=¡22.

APPLICATION TO MOTION PROBLEMS r=a+tb where t represents time

a represents the initial position

b represents velocity

jbj represents speed.

VECTORS IN 3-DIMENSIONAL SPACE COORDINATES AND POSITION VECTORS

Theposition vectorof P is

¡!

OP=

"

_x

y z

#

.

)

,

(

B x_B y_B

)

,

(

A xA yA

b a

q v

w

a r

b A

R

O

Z

X

Y

x z

y

P( , , )x y z Any point P, in space can be

specified by an of

numbers ( , , ) where , and are the in the , and directions from the origin O, to P.

ordered triple steps

x y z x y z

X Y Z

Note:

DISTANCE AND MIDPOINTS

If A(x1,y1,z1) and B(x2,y2,z2) are two points in space then

² AB=

p

(x2¡x1)2+ (y2¡y1)2+ (z2¡z1)2

² the midpoint of AB=

³

_x

1+x2

2 ,

y1+y2 2 ,

z1+z2 2

´

.

3-D VECTORS

² If P is (x1,y1,z1) then ¡!OP=

"

_x

1

y1

z₁

#

² If A(x₁,y₁,z₁) and B(x₂,y₂,z₂) are two points in space then:

¡!

AB=

"

_x

2¡x1

y2¡y1

z2¡z1

#

_x-step

y-step

z-step

² ¡!AB is called ‘vector AB’ or

‘theposition vector of B relative to A(or from A)’.

THE LENGTH OF A VECTOR

Themagnitude (length)is represented by

j¡!OAj or OA or jaj or j

e

aj.

If a=

"

_a

1

a2

a₃

#

then jaj=

p

a₁2+a₂2+a₃2:

VECTOR EQUALITY

Two vectors areequalif they have the same magnitude and direction. So, if arrows are used to represent

vectors, then equal vectors are par-allelandequal in length.

If a=

"

_a

1

a2

a3

#

and b=

"

_b

1

b2

b3

#

, then

a=b , a1=b1, a2=b2, a3=b3.

a=b implies that vectorais parallel to vectorb.

Consequently,aandbare opposite sides of a parallelogram, and certainly lie in the same plane.

Z

X

Y

x z

y

P( , , )x y z To help us visualise the -D

position of a point on our -D paper it is useful to complete a rectangular prism (or box) with the origin O as one vertex, the axes as sides adjacent to it, and P at the vertex opposite O.

3 2

Cartesian axes can be rotated and so viewed from different directions. For example:

Z

Y

X

Z

X Y

a

a

a

a

b

This means that equal vector arrows are translations of one another, but in space.

O

)

, ,

(a1 a2 a3

1 For p=

·

¡4 1

¸

and q=

·

2 3

¸

show how to

geometrically construct: a p+q b p¡q ZERO VECTORS

Thezero vectoris written as0and for any vectora,

a+(¡a)=(¡a)+a=0.

SOME PROPERTIES OF VECTORS ² a+b=b+a

(a+b)+c =a+(b+c)

a+0 =0+a=a a+(¡a) =(¡a)+a=0

² jkaj =jkj jaj where ka is parallel toa

length ofka

modulus ofk

length ofa

Notice that if ¡!OA=a and ¡!OB=b

where O is the origin

then ¡!AB=b¡a and ¡!BA=a¡b.

PARALLELISM

If two vectors areparallel, then one is a scalar multiple of the other and vice versa.

Note:

² Ifais parallel tob, then there exists a scalar,ksay, such that a=kb.

² If a=kb for some scalark, then I ais parallel tob, and

I jaj=jkj jbj: UNIT VECTORS

Aunit vectoris any vector which has a length of one unit. For example,

²

"

₁ 0 0

#

is a unit vector as its length is p12+ 02+ 02= 1

²

2

6

4

1

p

2 0

¡_p1 2

3

7

5

is a unit vector as its length is

r³

1

p

2

´

2 + 02+

³

¡_p1 2

´

2 = 1

1

jaj

"

_a

1

a₂ a3

#

is the unit vector in the direction of a=

"

_a

1

a₂ a3

#

i=

"

₁

0 0

#

, j=

"

₀

1 0

#

and k=

"

₀

0 1

#

are special unit vectors in the direction of theX,Y and

Z-axes respectively. Notice that a=

"

_a

1

a2

a₃

#

, a=a1i+a2j+a3k.

component form unit vector form

COLLINEAR POINTS

Three or more points are said to becollinearif they lie on the same straight line.

Notice that,

A, B and C are collinear if ¡!AB=k¡!BC for some scalark.

a b

A

B

O

a _b

C B A

THE SCALAR PRODUCT OF 3-DIMENSIONAL VECTORS

If a=

"

_a

1

a2

a3

#

and b=

"

_b

1

b2

b3

#

,

the scalar productofaand b(also known as the dot product) is defined as a²b=a1b1+a2b2+a3b3.

Properties of scalar product

I a²b= b²a

I a²a= jaj2

I a²(b+c) = a²b+a²c and

(a+b)²(c+d) = a²c+a²d+b²c+b²d

I Ifµ is the angle between vectorsaandbthen:

a²b= jaj jbjcosµ

I For non-zero vectorsaandb:

a²b= 0 , aandbare perpendicular.

r=a+tb ²

"

_x

y z

#

=

"

_a

1

a2

a₃

#

+t

"

_b

1

b2

b₃

#

is thevector equationof the line.

² x =a1+b1t y =a2+b2t

z =a₃+b₃t are theparametric equationsof the line. LINE CLASSIFICATION

2-dimensions

3-dimensions

) (

R x,y,z

3 2

1 )

(

Aa,a,a

ú ú ú

û ù

ê ê ê

ë é =

3 2 1

b b b b

O LINES IN 3-DIMENSIONAL SPACE

may intersect may be parallel may coincide

may intersect may be parallel may coincide

{ {

ormay beskew(neither parallel nor intersecting)

2 Given a=

·

3

¡1

¸

and b=

·

¡1 2

¸

show how to

geometrically construct 2a¡3b.

3 Find a vector equation for:

a b

4 For v=

·

¡4 3

¸

and w=

·

2 6

¸

find:

a jvj b jv+wj c 3w¡2v

d v²w e w²w

5 A is (2,10), B(¡2,2) and C(16,6). Find:

a ¡!AC b ¡!AM where M is the midpoint of BC

c the size of angle BAC.

6

Use vector methods to find the coordinates of D for parallelo-gram ABCD.

7 Find scalarsrandssuch that r

·

3

¡6

¸

+s

·

7 2

¸

=

·

5 22

¸

.

8 For p= 4i¡9j and q=¡12i+ 5j find:

a jqj b p²q

c the acute angle betweenpandq d a unit vector perpendicular top.

9 Findtif

·

t

1

¸

and

·

t¡4

¸

are perpendicular vectors.

10 For points P(6, ¡4), Q(4,11) and R(2,1), find the size of angle PQR.

11 Find the size of the acute angle between the lines with equations

y= 2x+ 5 and 3x+ 4y= 20.

12 Find the vector equation of:

a the line through (1,6) and parallel to

·

4 2

¸

b the points (¡3,2) and (5,¡2).

13 A line passes through the point (¡5,¡2) and has a direction parallel to

·

2 3

¸

. Find:

a the vector equation of the line

b the parametric equations of the line

c the Cartesian equation of the line.

14 Vectorsvandware given by v= 5i¡2j and w=i+ 3j. Find scalarsrandssuch that r(v¡w) = (r+s)i¡20j.

15 Find the vector equation of the line through (6,¡2) and (¡3,5), giving your answer in the form r=a+tb, t2R.

16 Find the vector equation of the line through (0,4) and parallel to the vector 3i+j.

17 Find, to the nearest degree, the angle between a= 3i¡3j

and b= 5i¡j.

18 Triangle ABC is defined by the following information:

¡!

OA=

·

¡2 4

¸

, ¡!AB²¡!OA= 0, ¡!CA=

·

¡6 2

¸

and

¡!

CB is parallel to

·

0 1

¸

.

a On grid paper, draw an accurate diagram of¢ABC.

b Write down the vector ¡!OB.

19 The vector equations of two linesl₁andl₂are:

l1 is r1 =

·

4

¡2

¸

+t

·

3 2

¸

l2 is r2 =

·

¡1

¡1

¸

+¸

·

5

¡1

¸

Use vector methods to find the coordinates of the point where

l1meetsl2.

20 Quadrilateral ABCD has vertices A(2,10), B(12,8), C(11,5) and D(¡1,2).

Find the acute angle between the diagonals¡!AC and¡!BD.

21 A line passes through the point (¡2,5) in a direction perpen-dicular to the vector

·

3 4

¸

.

Find the Cartesian equation of the line.

22 A boat moves in a straight line defined by the parametric equa-tions x= 3 +t and y= 4t¡3, t>0 wheretis the time in seconds andxandyare measured in metres. What is the:

a initial position of the boat b velocity vector of the boat

c speed of the boat?

23 Suppose

·

1 0

¸

represents a1km displacement due East and

·

0 1

¸

represents a1km displacement due North. A lighthouse

is at the point (0, 10). A ship is moving in a straight line according to the parametric equations x= 3¡2t, y= 3t+1,

t>0 wheretis the number of hours after8:30am.

a What was the initial position of the ship?

b Find the ship’s i velocity ii speed.

c Find the distance between the ship and the lighthouse at 10:30am.

d Find the time when the ship is directly West of the light-house.

e At what time (nearest minute) will the ship be closest to the lighthouse?

24 Write vector equations (form r= a+tb) to define the position at timetfor these remote controlled toy racing cars. (Distance in metres, time in seconds.)

a Car A which is initially at (10, 6) and is travelling at 13ms¡1in the direction 5i+ 12j.

b CarBwhich is initially at (¡5,3) and is travelling parallel to

·

4

¡3

¸

at15ms¡1.

c Car C which is initially at (¡1, 4) and is travelling at a constant velocity for3seconds to (5,7).

25 Triangle PQR is formed by these lines:

(PQ) :

·

x y

¸

=

·

11 13

¸

+r

·

4

¡6

¸

(QR) :

·

x y

¸

=

·

9 3

¸

+t

·

3 2

¸

(PR) :

·

x y

¸

=

·

2 7

¸

+¸

·

5 12

¸

a

b d

c

p

q

r

s t

A ,(1 ¡7)

B(-1 ¡2, )

C ,(8¡4)

D

5

where r,tand¸ are scalars.

a Accurately draw the three lines on a grid and hence find the coordinates of P, Q and R (extending lines if necessary).

b Prove that¢PQR is right angled.

c Find the area of¢PQR.

26 For these points in space:

A(2,0,5), B(¡3,1,7) and C(4,¡2,9) find:

a the distance AB b the distance BC

c ¡!AC d ¡!AB e the size of]BAC

27 For p=

"

₃

1 3

#

, q=

"

_¡4

2 4

#

and r=

"

_¡4

¡3 2

#

, find:

a p+ 2q¡r b jr+qj

c a unit vector parallel toq

d p²q e (p+q)²q

28 For m= 3i+j+ 2k and n=¡2i+ 5j¡4k find:

a 3m¡n b m²n

c jm+nj d m²(2i+k)

29 If a=

"

₄

¡2 1

#

and b=

"

₃

2

¡4

#

find x if a¡4x= 2b.

30 Find scalarsaandbgiven that

"

₄

¡1 5

#

and

"

_a

b ¡15

#

are parallel.

31 a Find a unit vector that is parallel to 3i+j¡k.

b Find a vector of length4units which is parallel to 3i+j¡k.

32 Findtgiven that

"

_2t

¡4 7

#

and

"

₃

t ¡8

#

are perpendicular vectors.

33 A(2,¡5,3), B(7,¡3,¡1), C(1,3,0) and D(¡4,1,4) are vertices of a quadrilateral.

a Prove that ABCD is a parallelogram.

b

34 Find the angle between theZ-axis and the vector

"

₄

¡1 3

#

.

35 Given the points K(4,¡2,7), L(6,1,¡1) and M(3,¡2,5), find the measure of angle KLM.

36 Given ¡!AB=

"

₄

¡6 2

#

and ¡!AC=

"

_¡2

7

¡3

#

find:

a ¡!BC b the size of angle BAC.

37 Given P(4,¡2,11), Q(13,¡8,14) and R(19,¡12,16) :

a prove that P, Q and R are collinear.

b Hence, find the ratio in which Q divides (PR).

38 Find thevector equationof these lines:

a parallel to

"

₁

2 3

#

and passing through the point (5,0,¡2)

b parallel to 2i ¡3j + 4k and passing through the point (¡2,5,4)

c perpendicular to theXOZ plane and passing through the point (2,¡4,1).

39 Find theparametric equationsof these lines:

a passing through (5,¡1,2) and parallel to the vector

"

₇

2

¡3

#

b passing through (0,2,¡6) and parallel to 2i¡3j+k c passing through (2,5,1) and (8,6,2)

40 At what point does the line

"

_x

y z

#

=

"

_¡1 4 3

#

+t

"

₃ 2

¡1

#

meet theY OZ plane?

41 Find the shortest distance from the point (1,4, 6) to the line with parametric equations

x= 3 +t, y=¡1 + 2t, z= 2 +t, t2R.

42 Lines l₁,l₂andl₃are defined by:

l₁:

"

_x

y z

#

=

"

₂ 1

¡1

#

+t

"

₃

¡6

¡3

#

l2: x= 5¡¸, y=¡4 + 2¸, z= 1 +¸ l3: x= 5¡3s, y= 5 + 4s, z= 1 + 2s a Show thatl1is parallel tol2.

b Show thatl1 and l3 intersect and find the angle between them.

POPULATIONS AND SAMPLES

Apopulationis a collection of items under consideration. Asampleis a part of the population.

A population generally consists of a large number of items. Because of the expense and time factors it is often easier to select a sample rather than use the whole population.

Arandom sampleis a sample where every item has the same chance of being selected.

MEASURES OF A SAMPLE’S CENTRE The median

When a set of data is written in order, themedianis the middle value of the set.

The mean

Themeanis x= k

P

i=1

f_ix_i

n where n=

k

P

i=1

f_i.

MEASURES OF SPREAD OF A SAMPLE The range

Therange=the maximum data value¡the minimum data value

The interquartile range

TheIQR=Q3¡Q1 fthe upper quartile¡the lower quartileg

Percentiles

Apercentileis the score, below which a certain percentage of the data lies.

For example, Q3=the upper quartile=the75th percentile. So,75%of the data lies below (or equal) Q3.

TOPIC 6 STATISTICS AND PROBABILITY

KEY FACTS AND CONCEPTS