enrich add. maths f4

8/13/2019 Enrich Add. Maths F4

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Enrichment Programme ForAdditional Mathematics

Form 4

l

Prepared By:Committee members of Additional MathematicsEducation Department of Kota Setar District


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Introductions

The Kota Setar Additional Mathematics Committee was formed inOctober 1998. The main objective of this committee is to helpstudents improve their performance in the subject. Variousactivities are organized to help teachers as well as students forthis purpose.

This compilation is one of the efforts by the committee tohelp increase the students' basic knowledge in the subject. It istargeted for the Form Four students who are not doing well inthe subject. The questions cover topics considered appropriatefor this group of students.

This enrichment programme is a joint effort by the committee andthe officer- in-charge. We hope that this effort will help enhancethe performance of the students in the SPM examination.


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Com mittee Members Of Additional MathematicsOf Kota Setar District:

Official In - Charge : En Saadon Bin Nayan(PPD Kota Setar)En Suhaimi B Haris(PPD Kota Setar)

Chairman

Vice Chairman

Secretary

Mr Goh Hock Loo(SMK Tunku Abd. Aziz)

Mr Koay Kheng Hooi(SMK Darul Aman)

Mr Ng Siak Lim(SMK Tunku Abd. Rahman)Mdm Yeoh Sock Leng(SM Teknik Alor Star)

Committee Members : En Rizuan B. Hj Husin(Kolej Sultan Abdul Hamid)Pn Norma Bt. Anis(SMK Sultanah Asma)Pn Sharifah Maznah Bt . Mohd Isa(SMK Bukit Payong)


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ContentsChapters Topics Pages

1 . Functions 1 -62. Quadratic Equations 7-123. Quadratic Functions 13-184. Simultaneous Equations 19-245. Indices And Logarithms 25-306. Coordinate Geometry 31-367. Statistics 37-428. Circular Measure 43-489. Differentiation 49-541 0. Solution Of Triangle 55-601 1 . Index Number 61-68


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(a) One to one

(b) Many to one

(c) One to many

(d) Many to many

1 A relation connects elements in set A (domain)to elements in set B codomain) according tothe definition of the relation.

2 A relation can be represented by:(a) Arrow diagrams (c) Graphs(b) Ordered pairs

3 The element in the domain is called the objectwhereas the element in the codomain which isrelated to the object is called the image. The setof all the images is the range.

4 A relation can be classified as follows:

5 A function is a relation in which every e lement(object) in the domain is connected to a unique(exactly one) e lement ( image) in the codomain.

6 If x is an object in set X which is mapped to animage y in set Yb y a function f, it can be writtenasf:xHyorf(x)=y.

7 Two funct ions f and g can be combined toproduce composite function , gf or fg such that:gf (x) = g [f(x)] orfg (x) =f[g(x)]

8 A one to one function has an inverse . For theo n e to o n e f u n c t i o n f : x H y, the inversefunction is denoted by y i , x ory =f(x) then x


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I";iii ichmcnt I'rohranrnrc For 1,tlditit ual N 1athcnraticsPe,labat I'endidikan 1)aeraIi Kota Setar

'T'opic : FunctionsProgressive Exercise1. Diagr,mr 8 shows the relation between set R and set

5 ' .

S c t . S 'Diagram Y

State(a) the range of' the relation,(b) the type of' the relation.

Answer .............2. Diagram 1 shows a relation between set P and set Q.

Q

Diagram 1(a) Rewrite the above relation in ordered pairs.(b) State the range of the relation.

Answer .............3. The relation of set P = (2, 4, 6) to set Q = (2, 3, 4)

is defined by the ordered pairs of ((2, 3), (2, 4), (4,2),(6, 4) ). Name(a) the images of 2(b) the objects of 4

4. he relation of set S= (2, 3, 4) to set T= (13, 26, 35 , 41, 5 9 )is given by the difference of the digits of'. Draw anarrow diagram to represent the relation.

Answer .............Diagram 6 shows the mapping off : x -> 30ax+b

x J ( x )

Diagram 6

F i n d t h e v a l u e o f a a n d b .

6 .Answer .............

Sketch the graph off : x - 2x - 3 1 for thedomain 0 5 x


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I,;nrichment 11 rohramnre I or Additional MathematicsPejahat I'endidilcan Uaerah Kota Setar

7. Function f is given by f : x -> ( 2x - 3 - 1.(a) Find the image of-1.(b) Iff (x) = x, find the possible values of x.

8 .Answer .............

The arrow diagram shows part of the functionf : x +b,x k. Find the values of a ) k ,(b) a and b,(c) p.

Answer .............9 Given function g : x ->12 x qpx q, p

If g (2) = -2 and g ( 10) = 6, find(a) the values ofp and q,( b) the values of x so that g (x) _ -x.

10. Given the func t ion fx) =4x- 5. Find the value of pif f(2p)=p.

Answer .............11. Given the function g : x ---> 5x + 3. Find the value of

pifg(3)=4p+2.

Answer .............12. Given function f : x -+ x 5 2 , find

(a) f (-4),( b) the value of x when f x) = x.


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i";nrichment I' 'ohramme For Additional MathematicsPejabat I'endi(likan 1)aerah Kota Setar

1 3 .

The diagram shows functions f and g givenbyf:x ->-4 x aancig :x -abx+c.Find(a) the values of a, b and c,(b) the value of k,(c) an expression for function ifin the same

form.

Answer .............14. Given the functionf(x) = x - 3 and gf(x) = 2x - 5.

Find the function of K.

Answer .............15. Given the function f(x) = 6 , where x # -1.x+l

Find an expression for the function of f 2(x).

16. Given function f : x -- 2x + 1 and g : x --> x2 -1,find the value Of X SO that(a) f g (x) = 15,(h) g'f (_r) = 3.

Answer.............17. Given the function f(x) = 2x + 3 and g(x) = 43 -x'where x # 3. Find the composite function of gf

18.Answer.............

Function is defined as f : x 1 , x # 0.XExpress the following in the same form:(a) fz(b) fa(c) f 2


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I :nrichmcnt l'rograi it e For Additional Mathematics1'cj:rbat Penrlirlikan 1)aera11 Kota Setar'

1 9 .

20 .

Given func t ions f:x-->px +2and g:x--^qx+3.(a) Iffg=gf, find a relation betweenp and

(I(b) Up = 4, find the values of x if

(ii) g'Z =g.

Answer .............Given the function h : x where x ^ -1.Find the function of /i-'(x). 'V + 1 '

Answer .............21. Given the function (x) = 6x + 5 , where x # 0.

Find the value of ,i '(17).

322. Given functions f : x --4 2x and g : x x - 2x x 2, find(a) f (-3(b) the value of x when g f (x) = 2.

Answer.............23. Given functions f : x ->x - 1. and gf : x --^

x'2 - 4x + 4, find(a) the function g,(b) the value off -' g (3).

24.Answer .............

Given functions g : x -> 2x + 1 and Ia : x x ,find(a) an expression for g-' h,(b) the values of x that satisfy the equation

g' (x)=x2+6.


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h:nricbnicnt I'rugranimc Fur Additional N1,1111cillaticsI'c,labat Ncndi(1iI a l Uaerah Kota Setar

Answers :( Function)

(a) {--2,2.3}(b) one to mane

(a) u=10,6-3,c=-2(b) A=613. (c) .x > -- , x 1010--x

2 .

3

4 .

5

6 .

7 .

(a) (2 I),(3,1).(b) { LL 3}( a ) { 3 , 4 }b) {2.6}

u=0.3, b=2

00 f0 )s0

,

a ) 4(b) x=0.5,4

15 .

16 .

>2+2 x

6(x+I)f (x) - x-7x+7(a) x = 2.828, -2.828(b) x =- 3 12 2

17. gf'(x) 2 , x:pt: 0x(a) f. (x) = x

18 . (b) .f (x) x t 0. x(c) f zo(x) = x

(a) 3p-2q = l(b) (1) x =19 . 3

(ii) x=-2


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Quadratic Equations

I5 .

II the sum of roots,a baIi

the product of roots, a/3 = ca

7. The nature of the roots of a quadratic equation,

E x p r e s s N o t e s1. A quadratic equation is an equation with an unknown

and the highest power of the unknown is 2.2. The general form of a quadratic equation is

ax 2 + bx+c = 0, where a, b and c are constants, a # 0and x is an unknown.3. The root of a quadratic equation is the value of an

unkno\Gn that satisfies the equation.4. The quadratic equation, ax2 + bx + c = 0, can be solved

by the following methods:(a) Factorisation(b) Completing the square(c) Quadratic formula

If x = p and x = q are the roots, then the quadraticequation iso r

(x-p)(x-q)=0xz- pq)x+pq0Product of roots)^Surn of root

6. Given the quadratic equation, axe + bx + c = 0, witla and /3 are the roots.Then,


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V ,nrichmcnt Programme For Additional WithenlaticsPe,labaI PendidILan 1)aerah Kota Setar

topic : Quadratic EquationsProgressive Exercise1. Express 2(x + 1)22 = 5x + 3 in the general form of aquadratic equation.

Answer.............2. By using the quadratic formula , solve the equation

22-5x-1=0.

Answer .............3. Solve the quadratic equation

3w(w -2) = (w + 6)(1-w). Give your answercorrect to four significant figures.

4. Find the roots of the equation J V2 + 5x = 12.

Answer .............5. Find the roots of the quadratic equation 2x2 = 5x + 8.

Give your answer correct to 3 decimal places.

Answer .............6. Form a quadratic equation which has the roots of

-2 and 3. Give your answer in the form ofa x 2 + bx + c = 0, where a, b and c are constants.


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Enrichment Programme For Additional M athematics1'ejabat Pendidikan I)aerah Kota Setar

7. Given that 1 and -2 are the roots of a quadraticequation, write the equation in the form ofn_x` + bx + c = 0, where [t, b and c are constants.

10. Given the roots of the quadratic equat ion AV' + )x + 4 = 0is 4 and 2 . Find the value of p.

Answer .............8. Given that a and 0 are the roots of the quadratic

equa t ion 2x2 + 7x - 15 = 0. Form an equation withthe roots of 2a and 20.

Answer .............9. If a and J3 are roots of the quadratic equation

3x22 - 4x - 6 = 0, form a quadratic equationwith roots U and 3j .

Answer .............11. Given that 4 is a root of the quadratic equation

2x2 - 5x +p = 0, find the value of p.

Answer .............12. Given that 3 and k are roots of the quadratic

equation x(x + 1) = 12. Find the value of k.


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Enrichment P rogramm e F or Additional MathematicsPejabat Pendidikan l)aerah Kota Setar

13. A root of the quadratic equation x2 + x + p = 0 is 3.Find the value of the other root.

16. Given that the equation kx' + 31,,x + h + p = 0,where k 0, has two equal real roots, find(a) p in terms of h,(b) the roots of the equation,

Answer .............14. Given that 3 and m are roots of equation

2(x+1)(x+2)=k(x-1),wherekis a constant.Find the values ofm and k.

Answer .............15. The quadratic equation 2x2 + mx +k = O hasroots -7 and 4. Find

the values of in and le ,

Answer .............17. Given the quadratic equation x2 + px + 9 = 0

has two equal roots. Find the value of p.

Answer .............18. Given that the equation 4x2 - hx + 25 = 0 hastwo equal roots. Find the value of h.


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Enrichment Programme For Additional MathematicsPejabat Pendidikan Daerah Kota Setar

19. The quadratic equation x2 - 2.v + 1 = k(-.v - 2)has two real and equal roots. Find the possible valuesof k.

22. Find the values of k so that equation(2-k)x'2-2(k+1)x+le +3=0 has equalroots. Hence, find the roots of the equationbased on the values of k obtained.

Answer .............20. Given that the equation x'2 - 4x + k + 1 = 0

has two different roots, find the largest integerof, 1 .

Answer ... ..........21. The quadratic equation (p + 5)x2 = 8x - 1 has two

distinct roots. Find the range of p.

Answer .............23. Given that in + 3 and it -1 are roots of equation

x2 + 6x = -5, find the possible values of inand n..

Answer.............24. Solve the quadratic equation (5x - 3)(x + 1) = x(2r - 5).

Give your answer correct to four significant figures.


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Program Peningkatan Prestasi Matematik TambahanPejabat Pendidikan Daerah Kota Setar

Answers:1 . 2x -x-1=0 1 3 . -42 . x2.686, - 0.186 1 4 . m=4,k=203 . w=1.356, -1.106 1 5 . m6,k=-564 . x- 3,4 16. (a)p= 4k,(b)- 2

5 . x3.609,-1.109 1 7 . p=66 . X -x-6 =0 1 8 . h=20

E.. 3x +5x-2=0 19 . k=0,128. X2 +7x-30=0 20. k=29. 9x -4x-2=0 21 . p


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CEIAPTER 3

Express Notes1 Quadratin Function (General/Standard Form):- f : x -- axe + bx + C Conditions:

or f(x) = axe + bx + C (i) a, b and c are constantsor y = axe + bx + c J (ii) x is the variable(Ili) a/02 It exhibits symmetry.3 It has a minimum value and a minimum point when a > 0.4 It has a maximum value and a maximum point when a < 0.5 Quadratic Function (after applying the Method of Completing the Square)

f(x) =a(x+p)2 + q(i) a, p and q are cons(ii) x is the variable(iii) a v-' 0

tants

6 When a > 0, the graph shape is 'U' AND When a < 0, the graph shape is AND(i) Axis of symmetry: x = -p (i) Axis of symmetry: x = -p(ii) Minimum value of fix: q (ii) Minimum value o f(x): q(iii) Minimum point: (-p, q) (iii) Mnmumpon: (-p, q)

7 When sketching graphs of f (x) = a(x + p)2 + q, there are 3 essential steps:Step 1 Determine its shape: n or nStep 2 Find the minimum or maximum point

(Note: The Axis or Line of Symmetry must pass through this point)Step 3 A Find Y and Y where Y is the point of intersection on the y-axis, and

Y' is the mirror image about the axis of symmetry ORB X and X which are the 2 points of intersection on the x-axis (when b2 - 4ac > 0).

8 The general Graph Shapes of a Quadratic Function are as follows:

Conditions : b2 -,4ac > 0',,: b2 _ 4ac = 0 4ac < 0

a > 0 4L-a < 0

x-- x

Any Intersection At x-axis? At 2 points At 1 point NIL9 Quadratic Inequation - is a Quadratic 'Function' with an ' inequation ' sign (E g. >, =


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F,nrichtncnt 1'roor:rntnie I or Additional i\1athcrnaticsI'cjabat Pcndidikan I)acrah Kota 'star

'I'onic : Quadratic FunctionsProgressive Exercise1 Diagram 5 shows the graph of Y = 2x2 + 4px + 1 8

which touches the x-axis. 4.Sketch the graph of f(x) = (x - 3) 2 - 6 for thedomain 0


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7. Given the graph y = p - 2(q - x)2 has a maximumpoint of (3, -4). Find the value of p and q.

10. Given that the maximum value of the quadraticfunction f (x) = 2k - 4 (x - 1)2 is 6, find thevalue of k.

Answer .............8 . Given that the maximum point of quadratic

function f(x) = -3 [(x - 1)2 - 6 ] is h, k),find the values of h and k .

Answer.............9- Express the fiinctionf(x)=2 [(x-7)2+(x+5)2]

in the form of a (x + p)2 + q. Hence, find theminimum value off (x) and the correspondingvalue ofx.

Answer .............11. Express quadratic function f (x) = - 2x2 + 4x - 3

in the form of a (x + p)2 + q. Hence, state themaximum or minimum value of the function.

Answer .............12. Given the curve ), = p - (x - q )2 intersects the

x-axis at (2, 0) and passes through (-2, -4).Find(a) the values ofp and q,(b) the maximum value ofy.


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E n r ic h m e n t P r o g r a m m e F o r Additional M at he mat icsPejabat Pendid ikan Uaerah Ko ta Setar

13. Find the smallest integer of k with the conditionthat the equation kx2 + (2k - 15)x + k = 0 doesnot have a real root.

1 6. Find the range of k such that 2x2 - 5x + k is alwayspositive.

Answer .............14. Given y = 2x + 3 intersects with the curve of

y = 3x2 + nix + 6 at two different points. Find therange of ni.

Answer .............15. Find the range of p if y = 2x + p does not intersect

at the curve of x2+y2=5.

Answer.............17. Find the range of values of p if

f(x) = 2x2 + 2px + 5p - 12 intersects the x-axisat two different points.

Answer .............18. Find the values of p if the graph of

g(,,C) = x2 + (p - 1) x + 1 touches the x-axis atonly one point.


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h;nrichment Programme. For Additional MathematicsPejabat Pentlidikan Uaerah Kota Setar

19. The quadratic equation x(x +I) = -hx - 9 hastwo distinct roots. Find the range ofvailues of'r ; .

22. Find the range of x if (2x + 1)(x - 3) < 0.

Answer .............20. Given x = 8 - y3 . Find the range of x if y > 10.

Answer .............21. Find the range of x if 2x >0.2x + 1

Answer.............23. Find the range of x for 3y + I = 4x and 2y> 1 +x.

Answer .............24. Given f(x) = x2 - 7x + 6. Find the range of x

that sat is f ies 0


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l':nrichnicnl Progr.unnle For Additional I\lathcnraticsI'clabat Pendidikan l)acrah Kota Setar

Answer-:( (quadratic Function)

f(x) = (x - 3) ' - 6(6,3)

(a) p-6.q 2(b )

5 .

6 .

7 .

y=(x-3)'-p -4, q = 3

2 58

17. p6

18 . p= - 1,31 9 . k< 7ak>5


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CEIAPTER 4

SIMULTANEOUS EQUATIONSExpress Notes

1 Simultaneous means at the same time .2 Simultaneous Equations represent the occurrence of 2 (or more equations) 'at the same time'.3 ' These equations can involve Linear and/or Non-Linear Equations.4 The Linear Equations to be studied involve 2 variables and are of degree 1 (E.g. 2x - 3y = 7)5 The Non-Linear Equations likewise involve 2 variables and are of degree 2 E.g. xy = 4)6 The Substitution Method (one of many methods available) will be emphasized in this Chapter.7 We solve the Simultaneous Equations by finding the actual values of the variables involved.8 The results obtained are referred to as Solution Sets.9 Examples include:

A Straight Line and a C urve)Y )Y )YG(x7, y,) and H(xe, ye) /(x9, y9) and J(x1o, yio)

10 Uses of Simultaneous Equations include the determining of:(i). Permeer (ii) Aea(iii) Volume (iv) Point of intersection(s) between a curve and a straight line, etc.

The steps in solving simultaneous equations are:Step IFrom the linear equation, an unknown (let it be y) isexpressed in terms of the other unknown (let it be x).Step 2


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7. Solve the following simultaneous equations:p + 3q = 1

2p2-q2+pq = 36

1 0 . So lve the s imul taneous equa t ionsx+2y=0andx2+y2+8x+10y-8=0.

Answer .............8. Solve the following simultaneous equations:y-x =-3

5x-y2 = 19

Answer .............9. Solve the following simultaneous equations:

2x--y =4x2-2y2-.Ay =-5

Answer .............11. Solve the following equations by giving the

answers correct to two decimal places.2x + 3y = 1x2+xy =9

Answer.............12. Solve the following simultaneous equations:5v-6x=2

4y2-3x2-4xy =0


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Enrichment Programm e For A ddit ional Mathema ticsPejabat Pendidikan Daer ah Kota Setar

13. Solve the s imu ltaneous eq uat ions2y=x-2and y +y=5.

16. Solve the equation 2x + 3y = 12x2 + Gx y = 1.

Answer .............14. Solve the following simultaneous equations:

x-2y = 2x+1'^^=5N x

Answer .............15. Solve the equation 2x - 5y = 12x2 - 5y' = 7.

Answer .............17. Solve the simultaneous equations

x + 2y + 3 = y + 2 = 5.

Answer .............18. Solve the simultaneous equations3x-2y=5 andy(x+y)+5 =x(x+y).


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Enrichment 1'rogranime For Additional MathematicsPejabat Pendidikan Daerah Kota Setar-

19. Solve the simultaneous equationsx +Y =1and (x+l)y=5x+2.2 4

xcm

22. Diagram 1 shows a right-angled triangle.

2x+3)cm

(x+y)cmDiagram I

The length of the hypotenuse is (2x + 3) cm and thelength of the other two sides are x cm and (x + y) cmrespectively. If the perimeter of the triangle is 30 cm ,find the value of x and y.

Answer .... .........20. Given that x = 8, y = -2 are the solutions of

the simultaneous equations x + by = k andx' - hy2 = 14k, find the values of h and k.Hence, find the other solution.

Answer.............23. The difference between two numbers, p and q,

is 3 where p > q. The product of these twonumbers is 88. Find the values ofp and q.

Answer .............21. The straight liney = 6 - 2x intersects the curve

.v2 + xy - 8 = 0 at points A and B. Find thecoordinates of points A and B.

Answer .............24. A 52 cm wire is cut into two parts. Each part is

bent into the shapes as shown in Diagram 2.

x cill y cnl


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Enrichment Programme For Additional MathematicsPejabat 1'endidikan llaerah Kota Setar

Answers : ( Simu ltaneous Equations) x=-4 v=0 1 3 1x=6

x = 0 , v = 42x=-^ 9y= 4 x=6 y = 22 =0

3 x = y = 12 ; _1 5 x= 1 y=-1x=4 =-1 x=-2 Y=-2

4 x=2 Y= -2 6 x=4 , y= 6x4 y=1 1x-_- 2y ^5 x3 y- 7 1 7

- 5 -Y- 5

x= -5 -Y=-5 x= l y =6 x=7.16 , y=0.56 ; 1 8 x=3 , y=2x=0.84 y =4.777 p= -565 , q=2 ;17 19 x=2 y=3P= 144 q=-1 4 x=-2 , y=88 x=4 , y=l ; 20 h=2 , k =4x=7 , =4 x=-16 , y 109 x = 3 , y=2 ; 2 1 (2 , 2) , (7 , -8).x=1 =-2

10 x=5 Y= -5 22 x=5 , y=7x=-4 =21 1 x=4.72 , y=-2.81 ; 23 p= ll q=8x=-5.72 , =4.15 = -8 = -111 2 x=3 2 . 24 x=8 , y=5

4 2-17 y=_17


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CHAPTER 5

INDICES AND LOGARITHMS

I

Zero index: a = 1, a # 0 Negative index: a- = Fractional indices: a _ a

a=V )n

iLaw of Indices

am x an = am + n aman= am- n /am\n = amn

If am=a , then m=n. If am=bm,then a=b.

Law of Logarithms logo xy = logo x + logo y log log x - log ), log"x"=mlog"x


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E n r ic h m e n t P r o g r a m m e F o r A d d i t io n a l M a t h e m a t ic sPejab at Pend id ikan D aerah Ko t a Se tar

Topic : Indices and Logar i thm sProgr ess ive Exerc ise1. Simplify each of the following: 4. Solve the equation 5x'z - 256 -2X = 0.(a) 31 2n x31x33 2 .(b) hn2X1) 3 -Y

: iPs

Answerr .............2. Show that 8 4 "+1X 93 , 1 -1 - 126 1, 1 _ 23

Answer .............3. Show that 3 + 3 '' + 3i-1 can be divided by13 for all positive integers n.

Answer .............5. Solve the following equations:

(a) 32x=82-1(b) 3> . 9v' 1 = 243y '

Answer .............6. Find the value ofy that satisfies the equation

(6>')2. 2 6,, = 36.


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7. Find the values of x and y that satisfy bothequations below:3(9x)=271,-x =48

10. Solve the following equations:(a) 2 log, (x - 1) = 1 + log, (x - 1)(b) 4log,,, 2 + log,,, (x - 1 ) - log,, 8x = 0

Answer .............8. Given 22x-1 = 32Y and 25x = 125.5'

(a) Find the values of x and y .(b) Hence, find the value of log, y.

Answer .............9. Solve the following equations:(a) log,. 128 - 5 log, 2 = 2

(b) 2 log , 3 + 3 log ,, , y = log,,, y

Answer.............11. Solve the following equations:

(a) loges16-log,, 2 = 6(b) log,,, 2x + 2 log 1, 3 - log,,, (3x + 1) = 0

Answer.............12. Solve the following equations:(a) 2, . 3, = 12

(b) 3Y - 8 (3-)) = 2


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I';nrichment Programme For Additional MathematicsPejabat Pendidikan 1)aerah Kota Setar

19, (a) Given x = log, 5, find the value of 9 . hence,find the value of 9' Ify = 1 + X. [5 mocks]

(b) Given logy, (i = 1.(331 .Without using the logarithm tables,(i) prove that log,; 216k = 5.893,(ii) solve the equation

f6 XI?' '=6. 15nuzrks]

Answer .............20. (a) Solve 5 b09: = 125. [3 marks]

(b) If 4 , = 8 (3 1r), prove thatx log,, 6 4 = log 8. [3 marks]

Answer .............21. Given log, 3 = 1.585 and logs 5 = 2.322.

Without using the logarithm tables, find(a) loge 30, [3 marks](b) log 2 5 . [3 marks]

22. (a) Given log,,, x = 3 and log,( )y = -2, showthat xy-10000y2=9. [3 marks]

(b) Solve the following equations:(i) 5,'I = 1.20 + 5'

(ii) log, x = log2r, (3x + 4) [7 marks]

Answer .............23. (a) Given log, 2=k. If 7f3i -' = 14, find n in

terns of k. [3 marks](b) Solve the equationlog,3 (5t - 3) - log, 3t = -1. 13 marks]

24. (a)Answer.............

Given 61og3 x - 12 logs y = 3. Express x interms of y. [5 marks](b) Solve the simultaneous equations

2 ' -s x 16 k.' = 32 and 7-2 ' x 2 4011 -3k = 1,where in and k are constants. [5 marks]


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Program Peningkatan Prestasi Ma tern at 1k 1am ha I) auPejabat Pendidikan Daerah Kota Setar

Answers: (indices & Logarithms)(b) p" 13. (a) 5 .492 (b). 0 .26 3 (c) 0 .5119

2 . (a) 3 (b) 2 1 4 . (a) 31) (b) - p (c) 23. 2 , - 6 15 . (a) y - x (b) 2+3(x-y)4 . - 2 1 6 . 3(a) log,, I (b) 455 . - 1 7 . (a) - (b) 9.6346 . - 1 8 . (a) 6 (b) 37. 3 4 19 . (a) x = 25 y= 225 (b) ii. 1.8155

2 38. 7 1

(b) - 1.2386a) , 20 . (a) 274 29. (a) 2 (b) 1 21. (a) 4.907 (b) 0.491331 0. (a) 3 (b) 2 22. (a) - b) i) 1 ii) 411. 1(a) 2 (b) 23. k +2 3(b)a) n =15 3 412. (a) 1.387 (b) 1.262 24 (a) x = 3y (b) m=8 , k = - 1


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I .nrichnient I>rograH inte For Additional Mathematics


36/72

I ejabat Pendidikan Daerah Kota SetarTopic : Coordinate G eo metryProgre ss ive Exercise1 . Given the distance betwee n points A(1 , 3) and B(7, k) 4, The e(luation of a circler- 4.v + y2- 8y = 5 intersects

is 10 units. Find the value of k. v-axis at points P and Q. Find the distance of PQ .

Answer .............2. Given points A (k, 5), B (0, 3) and C (5, 4),

find the possible values of k if the length ofAB is twice the length of BC.

Answer.............3. Points A, B and C are (6, -2), (7, -3) and

(0, 4) respectively. Given M is the midpointof AC, find the distance of BM.

Answer .............5. Find the area of a triangle with vertices A(-1, -2),B(1, 8) and C(5, -3).

Answer .............6. Given the area of a triangle PQR with vertices P(-3, 0),

Q(7, h) and R(4, -2) is 24 unit2 . Find the value of h.

Enrichment Programme For Additional Mathematics


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Pcjabat Pendidikan I)aerah Kota Setar

7. Find the equation of a straight line with gradient 23that passes through (1, 5 ). Ex press your answer in

intercept form.

Answer .............8. Diagram 1 shows a Cartesian plane.

yDiagram I

(-3,2)0 1

Find the equation of the straight line.

(4, 5 )

x

Answer .............9. Given a straight line 3y = nix + 1 is parallel to

x + y = 1. Find the value of m.3 5

10. Find the equation of the straight line thatpasses through A (-4, 2) and is parallel to thestraight line y = 3x + 1.

Answer .............11. Given points P(-2, 12), Q(2, k) and R(4, 3) are

collinear. Find the value of k.

Answer .............12. Find the equation of the straight line that is

perpendicular to the straight line y = - 2 x + 3and passes through point (3, -4).

F, nrichment Prohraninte For Additional Mathematics


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I ejahat l'endidikan l)aerah Kota Setar

13. Find the equation of the perpendicular bisector ofpoints P( 1, 6) and Q(3, 0).

16. Points P, Q and R are (6, 0), (-4, 2) and(-n, -3) respectively.(a) Prove that ZPQR - 90.(h) Find the coordinates of point S which lies

on the line P1? so that PS: SR=1:3.

Answer .............14.In Diagram 2, AB and BC are two perpendicular

straight lines.V

Diagram 2

AGiven that the equation of AB is 3x - 2y = 6, find theequation of BC.

Answer .... .........15. ABCD is a parallelogram. Given A (-2, 7),

B (4, -3) and C(8,-1.1),Find(a) point D,(b) the area of the parallelogram.

Answer .............17. PQR is a straight line such that PQ : QR = 1 : 2.

Given points P(-1, 3) and Q(2, 5). Find the coordinatesof R.

Answer .............18. The coordinates of A and B are (1, 5) and (5, 15)

respectively. If point M divides AB to the ratio of2 : 3, find the coordinates of point M.

l';nrichnient Programme For Additional Mathematics


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ejahat Pendidikan Uaerah Kota Setar

19, Given po uts A(-2, 0) and point B(2, 3). Point I movessuch that PA : PB = 3 : 2. Find the equation of thelocus of 1'.

22. A moving point P with its distance from two fixedpoints, A(-4, 0) and B(1, 3), are in the ratio of4PA = 3PB. Find the equation of locus P.

Answer .............20. Point P moves on the Cartesian plane such

that its distance from A (1, -3) is always 5units. Find the equation of the locus of P.

Answer .............21. Find the equation of a locus for a moving point P

such that it is equidistant from A(-3, 4) and B(2, -6).

Answer .............23. Points A, B and C are (k, 8), (-4, -2) and

(5, 10) respectively. Point D lies on the lineBC where BD : BC = 1: 3. Find(a) point D,(b) the values of k if AD is perpendicu'ar to

BC.

Answer .............24. A point P moves in such a way that its distance from

two fixed points Q (0, 1) and R (6, 2) is always in theratio PQ : PR = 1 : 2. Find(a) the equation of locus P,(b) the coordinates of the point where locus P meets

the x-axis.

I': nrirhntcnt Pro^grantntc For i\ddilioii,il Ma thematics


40/72

Ile, lahat Penclitlikan Dacrah Kola Saar

Answers: (Coordinate Geometry)7. k=-5ork=11 13 . x+-} 3 3

172. 1) 14-x+33 .565715) (2 ,-1) b) 8 un it2

1 34 6 unit 16 . b) (- --)' 4

5 . 3 1 unit2 17 . R (8 , 9 )6 . 68h4@7- 18 . 1 3( ,9)7 57 . +--1-1313 19x2 +5y2 - 52x-54y+101 0

2 -8 .

- --x 202+y2-2x+6y-1507 79 .m5212x-4y-3=01 0 . y=3x 14122x2 + 7y2+146x+54y+166=01 1 .k=623a) -1,2) b)-9

a) 3x2 + 3y2+12x-4y-36=01 2 .y=2x1024b) (2,0),(-6,0)


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C H A P T E R 7

IIII

Express Notes1. Measures of central tendency:

(a) Ungrouped data(i) The mode of a set of data is the value or

observation that occurs most frequently in theset of data.

(ii) The median for an odd number of data isthe middle observation when the data arearranged in ascending order.

The median for an even number of data isthe average of the two middle observationswhen the data are arranged in ascendingorder.

Mean x = sum of observations, Ix( ii i ) total number of observations, N(b) Grouped data

(i) The modal class of a set of data is the classwith the highest fr quency in the set of data.

(ii) Median , m= L+ 12 f I Cwhere L = lower boundary of median class

N = total frequencyC = s'ze of median classH = cumulative frequency before medianclass

= frequency of median classMean , - = sum of (frequency x class mark)(ii i

I f xIf

total frequency

Range = the highest value - the lowest valueInterquartile range = third quartile - first quartile=Q3_Q

rT2 = .. (x - X)t = xN2where x = value of a quantity

x = mean of the quantityN = total frequency of the quantity

(b) Grouped data(i) Range = the highest class mark - the lowestclass mark(ii) Interquartile range

1 N- F lwhere Q, =L,+(4ICQ+4NF3)C

(iii) variance , 02 = I fX2 - z2=, - ( f )2Y - fwhere x = class markf = frequency of class

1?nrichment Progranunc For ,k(i(litionaf N athem.itics


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Pejabat PentlIdikan I)acrah Kota SetarTopic : StatisticsProgressive Exercise

The mean of three numbers is 11. If another twonumbers p and y are added to the set of numbers,its new mean is 14. Find the mean of p and q.

Answer.............2. Numbers 8, 15, 13, 22, x,y have the same mode

and mean, that is 15. Determine the values ofx andy.

Answer .............3. Numbers 3, 9, x, 15, 17, 21 have been arrangedin ascending order. If the mean is the same as

the median, determine the value of x.

4. Determine the median and the interquartile for theset of data 18, 14, 11, 15, 20, 13, 21, 17.

Answer .............5. Table 1 shows the scores obtained after throwing the

dice 20 times.Score 1 2 3 4 5 6

Frequency 4 4 m n 3 2Table 1

If the mode score is 4, determine the minimumvalue of n. With the value of n, find the value of mand determine the median for the data.

Answer .............6. Table 2 shows the weight of 50 students.

Weight ( kg) 48 49 50 51 52Frequency 4 9 c 1 4 1 3 1 0

Table 2



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Pejahat Pendidikan llaerah Kota Setar

ScoreFrequency

07

11 0

2x

31 5

48

7 .

Table 3

10. Table 7 shows the number of story books read bya group of 20 students.Number of books 1 2 3 4 5

Number of students 2 4 5 6 3The table shows the score distributionobtained by a group of pupils in a quiz. If themedian is 2, find the smallest value of x.

Answer .............8. Table 4 shows the distribution of lengths of 50

wooden poles in a factory.Length 10-14 15-19 20-24 25-29 30-34(cm)

Frequen t y 8 1 0 9 1 3 1 0Table 4

Find the mean length of the wooden pole.

Answer .............9 Table 5 shows the weight of 40 students in a class.

Weight (kg) 46-50 51 -55 5 6 - 6 0 61 -65 6 6 - 7 0umber ofstudents 7 10 12 8 3

Table 5

Tab le 7Find the mean number of story books read by the students.

Answer .............11. Table 8 shows the number of medals won by a group

of students.Number of medals 2 3 4 5

Frequency 3 7 1 2 21 1 7Table 8

Find the interquartile range for medals won by the students.

Answer .............12. Find the mean and standard deviation of 8, 9, 7, 10

and 6.

h;nrichment Programme For Additional Mathematics


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Pejabat Pendidikan Daerah Kota Setar

13. The set of data 3, in, n, in + 6 and 15 is arranged inascending order. Given that the median is 6 and themean is 8, find the value of in and n.

16. A set of data has 7 numbers. The mean is 8 and thestandard deviation is 4. When a number k is addedto the data, the tnean is unchang ed. Find(a) the value of k,(b) the standard deviation of the 8 numbers.

Answer .............14. The mean of four integers is 12. When two new integers,

x and x + 2, is added to it, the new mean is 13. Findthe value of the new integers.

Answer .............15. The mean weight of a group of 20 male workers is

70 kg and the mean weight of 30 female workers is52 kg. Find the mean weight of all the workers.

Answer .............17. The mean and the standard deviation for the

numbers 1, 3, 5, 6 and 8 are 4.6 and 2.42respectively. Find in terms of k for each of thefollowing:(a) The mean for the numbers 1 + k, 3 + k,3+k,6+kand8+k.(b) The standard deviation for the numbers

k+4,3k+4 5k + 4, 6k + 4 and 8k + 4.

Answer.............18. The mean and standard deviation for the

numbers x ,, x2, ..., x are 6.5 and 1.5respectively. Find the(a) mean for the numbers: x, + 6, x2 + 6, ...,

x + 6.(b) standard deviation for the numbers: 4x1,

4x2, . , 4x,,.



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Pejabat Pendidikan 1)aeraIi Kota Setar

1 9 . Number of classes Number of pupils7 404 352 37

2 2 . The mean of the data 2, k, 3k, 11, 12 and 17which has been arranged in an ascendingorder, is in. If each element of the data isreduced by 2, the new median is 9 na. Find

The table shows the results of a survey of'the number of pupils in several classes ina school. Find(1) theImeall,(ii) the standard deviation,of the number of'pupils in each class.

Answer .............20. The list ofnumbers x - 1, x + 6, 2x - 1, 2x + 3,

x + 3 and x - 2 has a mean of 8. Find(a) the value of x,(b) the variance.

21.Answer .............

The following data shows the number ofpins knocked down by two players in apreliminary round of bowling competition.PlayerA: 6,8,9,6,6,7Player B: 6. 7, 5, 9, 9, 6

(a) the values Of' 1 1 ? and k ,(b) the variance of the new data.

Answer .............23. A set of data consists of 10 numbers. The sum

of the numbers is 150 and the sum of thesquares of the numbers is 2 760.

Find the mean and variance of the 10numbers.

Answer .............24. A set of examination marks x1, x2i x3, x4i x5, x6has a mean of 7 and a standard deviation of

1.4 .Find(i) the sum of the marks, Ex,


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V.nrichnient I'rohr: mn1c For ildtlitionai illathernatics


48/72

P& ihat Pcntlitiikan I)acrali Kota SetarTopic : Circular (MeasureProgressive Exercise1, PQ is an are of a circle with centre 0 and radius5 cm. if tlic angle subtended by I'Q at the centre. of

the circle is 1.5 radian, find the length, in cm, of arcPQ .

Answer .............2. FG is an arc of a circle with centre 0 and radius

6 cm. If the length of are FG is 15 cm, find the anglesubtended FOG, in radian.

4. Diagram 2 shows a simple pendulum which swingsfrom P to Q.

Diagram 2If LPOQ = 25 and the length of arc PQ is 22.4 cm,find the length of OQ.

Answer......... . ...5 Diagram 4 shows a sector POQ with centre 0 and

radius 8 cm.

0Diagram 4

Answer .............3.The length of arc JK of a circle with centre 0 is

12.5cm and angle subtended JOK is 1.84 radian.Find the radius of the circle.

If LPOQ = 130, calculate the area of the sectorOPQ.

Answer .............6. Diagram 5 shows a sector OVW with centre 0 andradius 8 cm.



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Pejabat Pendidikan Uacrah Kota Setar

7. Diagram 8 shows a circle with centre 0 and radius10 cm.

Diagram 8An angle 60 was subtended at centre 0 by theminor arc PQ . Calculate the perimeter of the shadedsegment.

Answer .............8. Diagram 10 shows a circle with centre 0 and radius

6.4 cm.

Diagram 10Given the distance of the chord FG from centre 0is 4.8 cm. Calculate the area of the shaded segment.

Answer .............9. Diagram 11 shows a sector OPQ with centre 0 andradius 12cm P

10. Diagram 14 shows a sector I_OM with centre 0.L

MDiagram 14The length of aic L M is 14 .25 cm and t h e p e r i m e t e r o f

the sector LOM is 34.25 cnm. Calculate the value of0, in radian .

Answer .............11. Diagram 15 shows a circle with centre 0.

Diagram 15Given that the length of the major arc is 48.24 cm,find the radius, in cm. (Use rt = 3.142)

12.0

Answer .............

In the diagram, AB and CD are arcs of two

Enrichment Prog nnie For Additional Mathematics


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I'cjabat Pendidikan Uacrah Kota Setar1

1 3 .

14.

15 .

8 in UThe diagram shows a semicircle with radius8 cm and centre O. Given the area of the shadedsector is 73.5 cm -, find(a) the angle ofAOl3,(b) the length of arc BC.

Answer .............

RUrn, IThe diagram shows a semicircle with centre0 ,nd radius 6 cm. Find the value of 0, into ms of 7t, for each of the cases below:(a) If the length of arc PQ is the same as the

perimeter of the shaded region.(b) If the area of sector OPQ is 3 times the

area of'sector OQR.

6 c n ,

Answer .............

1 6 .

17 .

18 .

SIn the diagram, PQ and RS are arcs of twoconcentric circles with centre O. Given OP= 8 cm and LPOQ = 3 radian. If the area ofthe shaded region is 6 cm z, find(a) the length of PS,(b) the perimeter of PQRS.

Answer .............1 3

The diagram shows two sectors of twoconcentric circles with centre O. OAB andODC arc straight lines. Find the length of arcAD.

Answer .............

UThe diagram shows a circle with centre 0 and

Enrichment Program m e.I'or Additional Mathem atics


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Pejabat Pendidikan I)aerah Kota Setar

1 9

RThe diagram shows two arcs, PS and QR, oftwo circles with centre 0 and with radii OSand OR respectively. Given the ratio OS : SR= 3: 1, find(a) the angle 0 in radians,(b) the area of the shaded region PQRS.

20 .Answer .............

SQ0The diagram shows two sectors OPQ andORS of two concentric circle with centre0. Given LPOQ = 0 rad., the length of arcPQ is twice the length of radius OP andthe length of radius OR = 8 cm, find(a) the value of 0,( b) the perimeter of the shaded region.

Answer .............21 T

2 2 .I

The diagram shows two sectors OA B and OCDof two concentric circles with centre 0, whereA OD and B OC are straight lines. Given OB =k cm , OD = h - 2) cm and perimeter of thefigure is 50.4 cm, find(a) the value of k,(b) the difference between the areas of sector

OA B and sector OCD.

Answer.............23. The diagram shows a sector POQ with centre

0 and radius 16 cm. Point R lies on OP suchthat OR: OP = 5: 8.

P

0Calculate(a) the value of x, in radians.(b) the area of the shaded region, in cm2.

24.Answer .............

1 SO7 R

E n r ic h m e n t P r o g r a m m e For Additional M a t h e m a t i c s


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Pejabat Pendidikan Daerah Kota Setar

Answers : (Circular Measure)

1 7.5 cm 1 3 a) 0.8447 rad.b ) 1 8.38 cm

2 2.5 rad. 1 4 a) 2g rad.b) rad

3 6.793 cm 1 5 a) 9 cmb 9.045 cm2

4 51 .34 cm 1 6 a) 2 cmb ) 10 cm5 72.61 cm2 1 7 3 cm

6 11435' 18 35 cm

7 20.47 cm 19 a) 0.875 rad.b ) 12.25 cm28 9.284 cm2 20 a) 2 rad.b ) 329 120 cm2 21 a) 2.485 cm

b 4.243 cm10 1.425 rad. 22 a) 10 cmb) 14.4 cm211 8.251 cm 23 a) 0.8957 rad.b) 52.20 cm'12 a) 5 cm 24 a) 0.9 rad.

b ) 6.4 cm h) 23.07 cm2

C E I A P T E R 9


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Express NotesA Concept of langents to a Curve and its Relation to

Differentiation1 The limit off (X) as xapproaches a is denoted by

lim f(x).X 4 d i2 If = lim & then `^ represents theA , I X

gradient of a tangent to a curve y = f(x) at acertain point.

B Differentiation of Algebraic Functions3 Ify= ax , then _y = at>_x -`.d x4 If y= uv, w h e r e u and v are functions in terms of

x , b y u s i ng th e p ro d u c t ru l e , th endy = u-X + v^ or by using the direct method,

then y =(copy( diff ) + ( copy( diffdx left right right left

5 If v=v, w h ere u a nd v a re f u nc t i o ns in t erm s o fx, by us ing the q uot ien t rule , thenu vvx xdxv

method, thenor by using the direct

copy cliff copy)( diffd bottom top) ( top )y bottomdx(botom

6 For a composite function y = k(ax' + bx + c)"= kn((u + bx+ c)" '(2 ax + b).

7 Ify= f(u) and u = g(x), then by using the chainrue _ Xdux u x

C Gradients of Tangents Equations of Tangents andNormals

1) Second - order Dif ferent iat ion , Maximum andMnimum Points

11 or f (x) me ans di f ferent iat ing `^ anotherdx dxt i m e .

1 2 At a max imum point, -- 0 and the sign oftix is negative.

13 At a minimum point , ^Z = 0 and the sign ofdxis p ositive.dx'

E Problems on Maxima and Minima14 Let the maximum/minimum value that has to be

determined be V. The steps of solution are:(a) Express Vin teems of one variable only, letit be r.(h) Find dV and solve the equation dV = 0 to

determine the value of r. z(c) Determine the sign of _rv . If the sign ofd2 Vd rmaximum and if the sign of -V ispositive, then the value of V is a minimum.

(d) Calculate the maximum / minimum value ofV .

F Connected Rates of Change15 The problems on rates of change can be solved

using the chain rule , i.e. A =di X dG Small Changes and Approximations



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Pejahat Penditlikan Dacrah Kota SetarTopic : DifferentiationProgressive Exercise1 Find the value of each of the followingexpressions.

(a) limit 14-(h) limit n2 25n5/ - 5

2 .Answer .............

Given y = 4x2 - 3, find lY using the firstprinciple.

4. Given f (x) (x xT 1)- find f (x).+

5 .Answer .............

Dif ferent ia te 3x + 1 with respect to x.

Answer Answer............6.x' (2x - 7)(' with respect to x. . Given f(x) = 2x4 + 3x2-x + 1. F ind f"(x).,.

Enrichment Programme For Additional iA1athcmaticsPejalrat Pendidikan DacraIi Kota Setar


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2 ,7. Giveny= 1-.v. Find dX(12 . 10. Given a curve with an equationf (x) =3 -xI,find the gradient of the curve at point (1, 2).

Answer ...... .......8. Given f (x) = (3x - 2)', find f (x).

Answer .............9. Given f(x) =x +2 5 , find f'(-3).

11.

12.

Answer .............Given y = x2 - 3x. Express y d2y +x dy + 2 in termsd x 2 dxof x in its simplest form. Hence, find the values of xthat satisfies the equation

yy +x + 2 = 0.CIX

Answer .............Given y = x2- 2x + 1. Find the value of x ifx2UX+(x+1) dy+y=6.rI r2dx

F,nrichment Programme For Additional Mathematics


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Pcjahat Pendidikan Daerah Kota Setar

13. Find the coordinates of the point on the curvey = (2x -3) ' where the gradient of the tangentis -4.

16. Find the equation of the tangent to the curvey=x:'-4atthepoint (-1,-5).

Answer .. ...........14. Find the coordinates of a point on the curve

y = 2r' - 4x -h 5 where the gradient of normal at thepoint is 1 .

2

Answer .............15. The gradient of the curve y = a + bx2 at thex

point (2,-1) is 3. Find the values ofa and b.

Answer.............17. Find the equation of the normal to the curve

y = -3x2 + 5 at the point (1, 2).

Answer.............18. The curve ' y = 2x3 - px2 + qx has a turning

point at (1, 5 ). Find(a) the values of p and q,(h) the other turning point.

Enrichment I'rohrai nme For Additional MathematicsPejabat Pendidilcan l)aerah Kota Setar


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19. Two positive quantities, i and varyaccording to ti'u = 21. Another quantity, s, isdefined as s = 14u 4 9u . Find the values of iiand u so that s is minimum.

22. Given y = x1 , find x ifx = 10. Hence, estimatethe value of (10.2)2

Answer .............20. Given that y = 8x (5 - x), calculate

(a) the value of x when y is a maximum,(b) the maximum value ofy.

Answer .............21, Given y = 2x' - 1. Find the rate of change ofy at

(1 , 4) when the rate of change of x is 2 units persecond.

Answer .............23. The radius of a spherical balloon increases at

a rate of 0 . 4 cm s-' . Find the rate of increaseof(a) the volume of the balloon,(b) the surface area of the balloon,if its radius, r = 5 cm.

Answer .............24. The side of a cube changes at the rate of

0.2 cm s-'. Find the rate of change of its volumewhen its surface area is 96 cml.

I;nrichn ent I'roor;u n e For Additional Niathen aticsPejabat 1'eudidikan 1)aerah Kota 5etar


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Answer:( Dfferentiation)

, (b) 1 0 1 3 (1 , 1 )

2. 8x 14. L22

3 . 1>.r2 (6.v - 7)(2.v 7) 1 5 . 1 0{

^ r 1 2=- - -4 . 1 x 1 6 . y 3x -2

-35 , (3x + 1 ) 2 1 7 . 6y=x+11

6 . 24x' +6 18. (a) p = 9, q = 12 , (b) (2, 4)

7 . 2 1 9 . a =3,v=7x 38. 180(3x- 2) 20. (a) x = 2 ,(b) y = 50

9. - 21214

21. 3 units per second

1 0. -2 22 . 0 .0768

1 1 . , 2 23 . 9 .6 ( a) 40^r crn3 s 1 ( b) 16 7r cmzs

1 2. 77, -1 24 . 9 .6

CHAPTER 1 0


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SO L UTIO N O F TRIANGL ESE x p r e s s N o t e s

a, bsinA sin B 'sin C

2 Sine Rule that Involves the Ambiguous CaseC 2

(a) Based on the above diagram, if a < c and theangle A is an acute non-included angle, thenthe ambiguous case will occur.

(b) Ambiguous case means that there are twotriangles that can be formed, i.e. AABC, andAABC,.

The cosine rule is given by:a2 = b2+c2-2bccosAb2 = a2+c2-2accosB

PO- orthogonal projectionQO- normal

- 2 acsin B5 Three-Dimensional Geometry

(a) Angle between a line and a planeThe angle between the line QPand theplane ABCD is LQPO, where

I nrichnient Programme For AAdtlitional MathematicsI'ejahat Pendidikan l)aerah Kota Setar


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Top ic : Solution of Triang lesProgress ive Exerc i se1 4

Diagram 4 show s a tr iangle ABC with BC= 10 cm,. Diagram 1 shows a triangle A B C . AB = 7 cm a n d L B =125.

A

Diagram I

Given AC = 13.2 cm, LB = 25 and LC: = 70.Find the length of A R .

Answer .............2 .The length of three sides of a triangle is 3.5 cm,4 cm and 6 cm respectively. Find the value of thebiggest angle in the triangle.

10 cmDiagram 4

Find the area of DABC.

Answer .............5. Diagram 6 shows a triangle BCD.

D4cm/ \5 cm

A

Answer .............3. Diagram 3 shows a triangle PQR.

P

7cmDiagram 6

The length of BC, BD and CD is 7 cm, 4 cm and5 cm respevtively. Find ZABD.

Answer .............6. Diagram 2 shows two triangles, ABC and ADC.B

I .nrichinent I'rograi nmc For Additional MathematicsPcjahat Pendidikan l)aerah Kota Setar


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7. In Diagram 4, PQS and SQR are two triangles.S

Diagram 4PQIK is a straight line. Calculate(a) the length of PQ,(b) the area of A S Q R .

Answer .............8.In triangle PQR, LQ = 42, PQ = 12.5 cm andPR = 8.5 cm.(a) Sketch APQR and show two possible sides of

PR.(b) Calculate LQRF.

1 0 . In Diagram 6 , sin L QRS = 4 such that LQRS is an5

Diagram 6Find(a) the length of Q S,(b) LPQS,( c) the area of the w hole diagrarn.

11.Answer.............

Diagram 7 shows a cuboid ABCDEFGH.D12 cm

Diagram 7Find the angle between the line AG and the planeEFGH.

Answer .............9. Diagram 5 shows a quadrilateral ABCD withAB = lO cm,AD= 7 cm, BC= 16 cm, LBAD= 60and ZBDC = 48.

Answer .............12. Diagram 8 shows a cuboid ABCDEFGH.

H GF cm

H iF- - -- - C

5 cm

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13. Diagram 9 shows two triangles, PT Q and PSR ,

4cm Q 6cmDiagram 9

Y TS an d PQ R are straight lines. Calculate(a) Z SPR,(b) the length of SR,(c) the area of equilateral TARS.

Answer .............14. Diagram 10 shows two triangles , HML and H L K .

H

KDiagram 10

Find(a) L KLH,(b) the length of HM, correct to 3 decimal places,( c) the area of the whole diagram.

16. Diagram 12 shows a cuboid.1 1

P Acm C)Diagram 12

M and K are m idpoints of TH and PS respectively.Find(a) the length of MR,(b) Z M R K ,( c ) t h e a r e a o f AMPR

17 .Answer .............

Diagram 13 shows a cuboid ABCDEFGH.1 1

Find(a) the length o f A G(b) the angle between line ,,G and plane ABFE,( c) the area of A H A G .

Answer..............15. Diagram 11 shows a pyramid OABC with base ABC,

a right-angled triangle.

Answer .............18. In triangle ABC, LA = 35, c = 12 cm and a = 9 cm

Calculate(a) the two possible values of LC,(b) the corresponding sides of b.

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n Dagram 14, ABCD is a q uadrilateral.19.C

2 2 . Diagram 3 shows a triangle ABC.C

Dingranr 31U cam

Diagram 14LDAB=90,LA13D=32 , LDBC=50, AB= 10cmand PC = 9 cm. Calculate(a) the length of CD, correct to 3 dec imal places ,(b) LBDC,(c) the area of triangle BDC. Answer .............

20. Diagram 15 shows a quadrilateral ABCD.B5 C11

D5cm Diagram 15 CGiven AB =5cm,BC =7cln,CD 5cm AD =6cm Diagram 7,and LA = 60. Find BCD is a straight line. Find(a) the length of B D , correct to 2 decimal places, (a)

(b)LADC,he length of CD ,

(b)(c) LBCD,the area of the whole diagram. c) the area of the whole diagram.

Answer .............21.In triangle ABC, LB = 35, c =10 cm andb = 6 cm. Find

(a) the possible value of LC,(b) the length of BC.

Find a) LA B,(b) the area of A A B C .

Answer.............23. Diagram 7 shows two triangles, ABC and ACD.

24.Answer.............

In triangle ABC, LB = 120, b = 16.8 cma n d a = 7.2 cm. Find the angle of C.

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Answers: ( S o l u t i o n s Of Triangles)

1 129 35 cm

5 ABD = 135 35'6 la) 14.42 cm B = 2647'

7 a) 5.354 cm b) 28 48 cm'8

0a) 79 42' o r 10 0 18'

9 a) 8.89 cm b ) 2423'10 ^a) 9. 9 3 cm b) 28 1 4 ' c) 3 5 . 7 7 cm211 a) 4518 b) 11.16 cm c) 39.49 cm2 i

a) JQ = 9.43cm; JL = 11.18cm1 QL = 12.8112 Icm I

13 a) 36'' 52 b) 6.08 cm c) 15 cm214 (;a)86' 25 b)5.276 cm c) 22.15

cm215a) 9 74 cm b)13 88 cm c) 55 15

16 a) 7 07cm b; 45 c) 18.68 cm`

17 a) 13 75 cm b) 21 19 c) 47.17 Xm218 a) 4954' or 1306' b) 15.62 cmor 4.03 cm19 Ia) 9 144 cm b) 4856' c) 40.56 cm220 a) 5 568 cm: b) 526' c) 26.80 cm2

21

23

2 4

a) 10559' b) 40.86 cm2

a)7256' or 1074'b) 9 .95 2 cm or 6 .430 cm

a) 45 48' b) 11.16 cm c) 39.49cm23813'

CEIAPTER 11


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uUUv u u v E x p r e s s N o t e s

Index number 1,x 100 where Qo = Quantity at base yearQQ=uantity at specific year

ase Specific QLyear year 100 1Qo Q Q x 100. 1= Qoo r Qo _ Qi100 1

Q0 1 = 100Q 1a E Lw.2. Composite index, , jII W + 12W2 + 13W3 + ... + INWN

w +w2+W3+..+WNwhere I,.= price index

w. = weightage

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Index Num berProg ressive Ex erciseI A tin of biscuits costs EM10 in the year 1995and RM1:1.50 in the year 2001. 1 sing the year

1995 as the base year calculate the price indexof a tin of biscuits in the year 2001.

3. Complete the fo l lowing table :

P r o d u c t P r i c e iny e a r 1 9 9 0P r i c e in

y e a r 1 9 9 5P r i c e i n d e x

( b a s e = 1 9 9 0 )P RM0.50 RMO.80 aQ RM1.20 b 1 1 5R c RM4,27 122

Answer ................2. The price index of a certain product in the year

1999, using the year 1997 as the base year, is125. If the price of the product in the year 1999is RM300, what is its price in the year 1997:%

4 .Answer ................

The price indices of a certain product in theyear 2000 and 2001 based on the year 1998are 115 and 120 respectively. Calculate theprice indices for the year 1998 and 2000, ifthe year 2001 is used as the base.

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5. The table below shows the price indices andweightages allocated for four products.Calculate the composite price index,

Product Price index WeightageA 1 2 0 3P 112 5C 1 2 5 3D 1 0 5 4

7. The table below shows the price indices andweightages of three products in the year 2001based on the year 1999.

Product Price i n d e x W e ig h t a g eA 1 2 0 51 3 140 2C 1 3 0 y

Given the composite price index in the year2001 is 127, find the value of y.

Answer .............6. The table below shows the price indices and

weightages of four products in the year 1996based on the year 1993.Pr o du c t Price index WeightageP 115 6Q 108 3R 135 2S 12 0 4

Calculate the composite price index in the year

Answer .............Product Index number Weightage

A 110 2B x wC 115 z

The table above shows the index number inthe year 1996, using the year 1995 as the baseyear.(a) If w = 2 and z = 6, the composite index in

the year 1996 is 112, using the year 1995as the base year. Find the value of x.


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13. Table I shows the mean price of four food items in19 95 and 2000. The monthly expenditure in 1995 isalso included.

1 5 . Table 2 shows the price index of expenditure of afew items in year 2000 based on the yea r 1995.

Price ( kg ) MonthlyFood i tems 1 9 9 5 2000 expenditure(1995)

Rice RM 1.50 RM 1.80 RM36Chicken RM4.00 RM5.00 RM24Vegetable RM1,20 RM1,80 RM12Milk powder RM 1 2.0 0 R M1 5 .0 0 RM48

Table 1Find the composite index in 2000 ba sed on 19 95 .

Answer ................14. By taking 1997 as the base year, the price index of

flour in 1998 and 1999 are 106 and 112 respectively.Find the price index of flour in 1997 and 1999 bytaking 1998 as the base year.

Item Price Index WeightageF o o d 1 1 5 2 0House instalment 120 10Electric andwater bill 105 8Others 1 0 8 1 2

Table 2Find the composite index of living in 2000.

Answer ................16. Table 3 shows the price index and the weightage of

three items in 2000 based on 1996.Item Price index WeightageP 12 5 5Q X 3R 1 1 0 2

Table 3The composite index in 2000 is 116. Find the valueof X.

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17. The pr ice inde x o f a k ilo g r a m o f r ice in 1 998 a nd1999 based on 1993 are 120 and 105 respect ive ly.Given that a kilogram of rice costs RM 2.80 in 1998,f ind the price of rice in 1999.

1 8 . Answer ................The composite index number of the cost ofbattery production for 2004 based on the 2000is 118.Calculate(i) the value of x.(ii) the price of a box of batteries in 2000 if the

corresponding price in 2004 is RM21.25.

1 9 .

20 .

Inde x nu m be r , X 105 98 110W e ig ht a ge , W 1 6-X x 5The composite index number of the data inthe above table is 106. Find the value of x.

Answer ................The price index of a certain item in the year1997 is 110 when 1995 is used as the baseyear and 132 when 1993 is used as the baseyear. Given the price of the item in the year1995 is RM480, calculate its.price in the year1993. [3 marks]

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21.

22.

Item Price index WeightageP X 6Q 115 yR 1 23 13- y

The table above shows the price indices ofitems 1', Q and ft with their respectiveweightages. Given the price of P in the year1998 is RM20 and decreases to RM19.60 inthe year 1999. By using 1998 as the base year,calculate the value of x. Hence, find the valueofy if the composite price index is 113.

Answer ................Price Of) Price index Namber

Item Year 2000 Year 2002 (Base year 2000: o f i te m sA 5 2 x 1 5 0 1 0 0 0B 6 7 5 1 2 5 yC 4 5 5 4 1 2 0 4 0 0

The table shows the prices, price indices andthe number of items.(a) Find the value of x.(b) If the composite price index of the threeitems in the year 2002 using year 2000 as

the base year is 136.5, find the value of y.

2 3 .

24 .

Item Price index W e ig h t ag eBag 1 20 nShirt 110 5T r o u s e r s 1 4 0 2S h o e s 1 0 0 3

The above table shows the price indices andweightages of four items in the year 1997based on the year 1992. Given the compositeprice index in the year 1997 is 115, calculate(a) the value ofit, [3marks](b) the price of a shirt in 1997 if its price in

1992 is RM50. [2 m arks]

Answer ................The table below shows the monthly expensesof Azman's family.

yearExpenses

1997 1999

Food 420 488Transportation 100 105R e n t a l 310 36 0Electricity and water 5 0 50Find the composite index in the year 1999 byusing the year 1997 as the base year. Hence,if Azman's monthly income in the year 1997is RM1 200. find the monthly income requiredin the y car 1999 so that the increase in hisincome is in line with the increase in his

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Answ ers: ( Index Num ber)

1 1352 RM2403 a=160 b = RM1 98 c = RM3 50

4 83 3 95 8

5 11436 117.67

8

[ 3

105 (b) 111

9 a) 115 (b) RM3706 56

10 137.5

11 a) 112 5, 125, 150, (b) x = 10 y = 70

1 2 6

1 4 94 34 105.66 105 6 6

1 5 112 72

1 6 1 0 5

1 7 2 45

18 IRM18 y = 18

19 ,X=2

20 RM400

21 a) 78 (b) 600

22 IX 98, y 5

23 a) n = 4 (b) RM 55

enrich add. maths f4

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