kertas 1 spm matematik tambahan 2004-2010

7/30/2019 Kertas 1 SPM Matematik Tambahan 2004-2010

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Disediakan oleh Mohamad Esmandi Bin Hapni

FUNCTIONS

1 Diagram shows the relation between set P and set Q.State

a) the range of the relation,b) the type of relation [2 marks]P1,2004

2 Given the function mxxh 4: and8

52:1 kxxh , where m and k are constants, find the value of

m and ofk. [3 marks]P1,2004

3 Given the function 0,6)( xx

xh and the composite function hg(x) = 3x, find

a) g(x)b) the value of x when gh(x) = 5 [4 marks]P1,2004

4 In Diagram, the function h maps x to y and the function g maps y to zDetermine

a. h-1(5)b. gh(2) [2 mark]P1,2005

5 The function w is define as 2,2

5)(

x

xxw

a) w-1(x)b) w-1(4) [3 marks]P1,2005

6 The following information refers to the functions h and gFind gh

-1(x) [3 marks]P1,2005

7 In diagram, Set B shows the images of certain elements of set Aa) State the type of relation between set A and set B.b) Using the function notation, write a relation between set A and set B

[2 marks]P1,2006

d

e

f

w

x

y

z

Set P Set Q

xh

yg

z

2

5

8

14:

32:

xxg

xxh

5

4

-4

-5

25

16

Set A Set B


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8 Diagram show the function 0,: xx

xmxh , where m is a constant.

Find the value ofm

[2 marks]P1,2006

9 Diagram shows the linear functions ha) State the value of mb) Using the function notation, express h in terms of x.

[2 marks]P1,2007

10 Given the function 3: xxf , find the values of x such that f(x) = 5 [2 marks]P1,200711 The following information is about the function h and the composite function h2.

Find the value of a and of b [3 marks]P1,2007

12 Diagram show the graph of the function 12)( xxf , for the domain 50 x State

a) the value of tb) the range of f(x) corresponding to the given domain

[3 marks]P1,2008

13 Given the functions 25: xxg and 34: 2 xxxh , finda) g

-1(6)

b) hg(x) [4 marks]P1,2008

14 Given the functionsf(x) =x1 and g(x) = kx + 2, finda) f(5)b) the value of k such that gf(5) = 14 [3 marks]P1,2008

15 Diagram shows the relation between setXand set Yin the graph form.Statea) the objects ofqb) the codomain of the relation

[2 marks]P1,2009

8

21

xx

xm h

0

1

m

5

1

2

4

6

x h(x)

baxxh : , where a and b are constants, and a > 03536:

2 xxh

O t 5x

y

1

y = f(x)

2 4 6SetX

p

qr

s

Set Y


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16 Given the function 32: xxg and xxh 4: , finda) hg(x)b) the value ofx if )(

2

1)( xgxhg [4 marks]P1,2009

17 Given the function 13: xxg , finda) g(2)b) the value ofp when 11)(1 pg [3 marks]P1,2009

18 Diagram shows the relation between setXand set Yin the graph form.

[3 marks]P1,2010

19 Given the function 12: xxg and 63: xxh , find

(a) g-1

(x)

(b) hg-1

(9)

[3 marks]P1,2010

20 Given the functions 8: xxg and32,

23:

x

xxxg , find the value ofhg(10)

[3 marks]P1,2010

QUADRATIC EQUATIONS

1 Form the quadratic equation which has the roots -3 and2

1. Give your answer in the form

ax2

+ bx + c = 0, where a, b and c are constants [2 marks]P1,2004

2

Solve the quadratic equation x(2x5) = 2x -1.Give your answer correct to three decimal places. [3 marks]P1,2005

3 A quadratic equationx2 + px + 9 = 2x has two equal roots. Find the possible value of p.[3 marks]P1,2006

4 (a) Solve the following quadratic equation :3x

2+ 5x2 = 0

(b)The quadratic equation hx2 + kx + 3 = 0, where h and k are constants, has two equal roots. Express hin terms of k

[4 marks]P1,2007

5 It is given that -1 is one of the roots of the quadratic equationx24xp = 0.Find the value of p.

[2 marks]P1,2008

SetX

Set Y

p

q

r

s

1 2 3 4

State

(a) the relation in the form of ordered pairs.(b) the type of the relation.(c)

the range of the relation.


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6 The quadratic equation 22 2 ppxxx , wherep is a constant, has two different root. Find the rangeof values ofp. [3 marks]P1,2009

7 The quadratic equation (1p)x26x + 10 = 0, wherep is a constant, has two different roots.Find the range of values ofp.

[3 marks]P1,10

QUADRATIC FUNCTIONS

1 Find the range of values ofx for which 12)4( xx . [3 marks]P1,04

2 Diagram shows the graph of the function 2)( 2 kxy , where kis a constant.

Find

(a) the value ofk(b)the equation of the axis of symmetry(c) the coordinates of the maximum point.

[3 marks]P1,04

3 The straight liney = 5x1 does not intersect the curvey = 2x2 +x +p. Find the range of values of p.[3 marks]P1,05

4 Diagram shows the graph of a quadratic functionsf(x) = 3(x +p)2 + 2, wherep is a constant.The curvey =f(x) has the minimum point (1,q), where q is aconstant. State

a) the value ofp,b) the value ofq,c) the equation of the axis of symmetry.

[3 marks]P1,05

5 Diagram shows the graph of a quadratic functiony =f(x). The straight liney = -4 is a tangent to thecurvey =f(x)

a) Write the equation of the axis of symmetry of the curve.b) Expressf(x) in the form (x + b)2 + c, where b and c areconstants.

[3 marks]P1,06

6 Find the range of the values ofx for (2x1)(x + 4) > 4 +x [2 marks]P1,067 Find the range of values ofx for which xx 12 2 [3 marks]P1,078 The quadratic functionf(x) =x2 + 2x4 can be expressed in the formf(x) = (x + m)2n, where m and n

are constant. Find the value ofm and ofn. [3 marks]P1,07

(2,-3)-3

O

y

x

(1,q)x

y

y =f(x)

O

y = -4

x

yy =f(x)

O 1 5


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9 The quadratic functionf(x) =p(x + q)2 + r, wherep, q and rare constants, has a minimum value of -4.The equation of the axis of symmetry isx = 3. State

a) the range of values ofpb) the value ofqc) the value ofr [3 marks]P1,08

10 Find the range of the values ofx for (x3)2 < 5x [3 marks]P1,0811 Diagram shows the graph of quadratic function qpxxf 2)( , wherep and q are constant.

State

a) the value ofpb) the equation of the axis of symmetry

[2 marks]P1,09

12 The quadratic function 22 4)( axxxf , has maximum value 8. Find the value ofa.[3 marks]P1,09

13 Diagram shows the graph of a quadratic functiony =f(x).

[3 marks]P1,10

14 The quadratic functionf(x) = -x2 + 4x3 can be expressed in the form off(x) = -(x2)2 + k, where kisa constant.

(a)Find the value ofk.(b)Sketch the graph of the functionf(x).

[4 marks]P1,10

INDICES AND LOGARITHM

1 Solve the equation 324x

= 48x + 6

[3 marks]P1,2004

2 Given that m2log5 and p7log5 , express 9.4log5 in terms ofm andp. [4 marks] P1,2004

(0, -9)

(-3, 0)

y

x

y =f(x)

O

y

x

y =f(x)

- 1 3

5

State

(a) the roots of the equationy =f(x)(b)the equation of axis of symmetry of the

curve.


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3 Solve the equation 2x + 4

- 2x + 3

= 1 [3 marks] P1,2005

4 Solve the equation 1)12(log4log 33 xx [3 marks] P1,2005

5 Given that pm 2log and rm 3log , express

4

27log

mm

in terms ofp and r. [4 marks] P1,2005

6 Solve the equation 232

4

1

8

xx

. [3 marks] P1,2006

7 Given that yxxy 222 loglog32log , express y in terms of x. [3 marks] P1,2006

8 Solve the equation xx 33 log)1(log2 [3 marks] P1,2006

9 Given that xb 2log and yc 2log , express

c

b8log4 , in terms ofx andy. [4 marks] P1,2007

10 Given that 9(3n1

) = 27n, find the value ofn. [3 marks] P1,2007

11 Solve the equation, 162x3 = 8

4x[3 marks] P1,2008

12 Given that 3loglog 24 x , find the value ofx. [3 marks] P1,2008

13 Given 243273 3 nn , find the value ofn. [3 marks]P1,200914 Given that 0loglog 28 qp , expressp in terms ofq. [3 marks]P1,200915 Solve the equation :

9

833 2 xx [3 marks]P1,2010

16 Given a3log2 and b5log2 , express 45log8 in terms ofa and b. [3 marks]P1,2010

COORDINATE GEOMETRY

1 Diagram shows a straight line PQ with the equation 132

yx. The point P lies on thex-axis and the

point Q lies on they-axis.

Find the equation of the straight line perpendicular to PQ and passingthrough the point Q.

[3 marks] P1,2004

2 The pointA is (-1, 3) and the pointB is (4, 6). The point P moves such that PA : PB = 2 : 3. Find theequation of the locus ofP. [3 marks] P1,2004

x

y

Q

PO


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3 The following information refers to the equations of two straight lines, JK and RT, which areperpendicular to each other.

Expressp in terms ofk.

[2 marks] P1,2005

4 Diagram shows the straight lineAB which is perpendicular to the straight line CB at the pointB.The equation of the straight line CB isy = 2x -1

Find the coordinates ofB.

[3 marks] P1,2006

5 The straight line 16

h

yxhas ay-intercept of 2 and is parallel to the straight liney + kx = 0. Determine

the value ofh and ofk. [3 marks] P1,2007

6 The vertices of a triangle areA(5, 2),B(4, 6) and C(p, -2). Given that the area of the triangle is 30 unit2,find the values ofp. [3 marks] P1,2007

7 Diagram shows a straight line passing through S(3, 0) and T(0, 4)a) Write down the equation of the straight line STin form

1b

y

a

x.

b) A point P(x,y) moves such that PS= PT. Find the equation ofthe locus ofP.

[4 marks] P1,2008

8 The point (0, 3), (2, t) and (-2, -1) are the vertices of a triangle. Given that the area of the triangle is 4unit

2, find the values oft.

[3 marks] P1,2008

9 Diagram shows a straight lineAC

The pointB lies onACsuch thatAB :BC= 3 : 1

Find the coordinates ofB

[3 marks]P1,2009

JK : y =px + k

RT : y = (k2)x +p

Wherep and kare constants.

C

B

A(0, 4)

Ox

y

y

x

T(0, 4)

S(3, 0)

O

O C(4, 0)x

B(h, k)

A(-2, 3)

y


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10 A straight line passes throughA(-2, -5) andB(6, 7).(a) Given C(h, 10) lies on the straight lineAB, find the value ofh.

(b) PointD divides the line segmentAB in the ratio 1 : 3. Find the coordinates ofD.

[4 marks]P1,2010

11 Point P moves such that its distance is always 5 units from Q(-3 , 4). Find the equation of the locusofP.

[3 marks]P1,2010

STATISTICS

1 The mean of four numbers is m . The sum of the squares of the numbers of the numbers is 100 and thestandard deviation is 3k. Express m in term ofk.

[3 marks]P1,2005

2 A set of positive integers consists of 2, 5 and m. The variance for this set of integers is 14. Find thevalue ofm. [3 marks]P1,2006

3 A set of data consists of five numbers. The sum of the numbers is 60 and the sum of the squares of thenumbers is 800.

Find, for the five numbers

a) the meanb) the standard deviation. [3 marks]P1,2007

4 A set of seven numbers has a mean of 9.a) Find xb) When a number kis added to this set, the new mean is 8.5. Find the value ofk.

[3 marks]P1,2008

5 A set of 12 numbersx1,x2, ,x12, has a variance of 40 and it is given that x2 = 1080Find

a) the mean, x b) the value ofx [3 marks]P1,2009

6 A set of data consists of 2, 3, 3, 4, 5, 7 and 9. Determine the interquartile range of the data.[3 marks]P1,2010

CIRCULAR MEASURE

1 Diagram shows a circle with centre O.Given that the length of the major arcAB is 45.51 cm, find the length, in

cm of the radius.

(use = 3.142)

[3 marks]P1,2004

A

B

O 0.354 rad


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2 Diagram shows a circle with centre OThe length of the minor arcAB is 16 cm and the angle of the major sector

AOB is 290o. Using = 3.142 find

a) the value of, in radians.(Give your answer correct to four significant figures.)

b) the length in cm of the radius of the circle.[3 marks]P1,2005

3 Diagram show sector OAB with centre O and sectorAXYwith centreA.

Given that OB = 10 cm,AY= 4 cm, XAY= 1.1 radians and the length of arcAB

= 7 cm, calculate

a) the value of, in radian,b) the area in cm2, of the shaded region.

[4 marks]P1,2006

4 Diagram shows a sectorBOCof a circle with centre O.

It is given thatAD = 8 cm and

BA =AO = OD =DC= 5 cm, Find

a) the length in cm of the arcBCb) the area in cm2 of the shaded region.

[4 marks]P1,2007

5 Diagram shows a circle with centre O and radius 10 cm.

Given that P, Q, andR are points such that OP = PQ and OPQ = 90o,

find

(Use = 3.142)

a) QOR, in radians,b) The area in cm2 of the shaded region.

[4 marks]P1,2008

A

B

O

O

Y

A

B

X

C

B

D

A

O

1.85 rad

Q

P

O

R

10 cm


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6 Diagram shows a sectorBOCof a circle with centre O.It is given that BOC= 1.42 radians,AD = 8 cm and OA =AB = OD =DC= 5 cm

Find

a) the length in cm of arcBCb) the area in cm2 of the shaded region

[4 marks]P1,2009

7 Diagram shows sector OPQ of a circle with centre O, and sectorNRSof a circle with centreN.

[3 marks]P1,2010

DIFFERENTIATION

1 Differentiate 42 523 xx with respect tox.[3 marks]P1,2004

2 Two variables,x andy, are related by the equationx

xy2

3 . Given thaty increases at a constants rate

of 4 units per second, find the rate of change ofx whenx = 2

[3 marks]P1,2004

3 Given that 253

1)(

xxh , evaluate )1(h .

[4 marks]P1,2005

O

A D

B C

8 cm

10 cm

O R Q

N

P

S

Given POQ = 1.5 radians and

RNS= 0.5 radian, find the area, in cm2

of the shaded region.


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4 The volume of water, Vcm3 in a container is given by hhV 83

1 3 , where h cm is the height of the

water in the container. Water is poured into the container at the rate of 10 cm3

s-1

. Find the rate of

change of the height of water, in cm s-1

, at the instant when its height is 2 cm.

[4 marks]P1,2005

5 The point P lies on the curve 25 xy . It is given that the gradient of the normal at P is 41 . Find thecoordinates ofP.

[3 marks]P1,2006

6 It is given that 73

2uy , where u = 3x5. Find

dx

dyin terms ofx.

[4 marks]P1,2006

7 Given that 43 2 xxy a) find the value of dxdy whenx = 1b) express the approximate change iny, in terms ofp, whenx changes from 1 to 1 +p, wherep is a

small value.

[4 marks]P1,2006

8 The curvey =f(x) is such thatdx

dy= 3kx + 5, where kis a constant. The gradient of the curve atx = 2 is

9. Find the value ofk.

[2 marks]P1,2007

9 The curve 64322 xxy has a minimum point atx =p, wherep is a constant. Find the value ofp.[3 marks]P1,2007

10 Two variables,x andy are related by the equation2

16

xy . Express in terms ofh, the approximate

change iny, whenx changes from 4 to 4 + h, where h is a small value.

[3 marks]P1,2008

11 The normal to the curve xxy 52 at point P is parallel to straight liney = -x + 12. Find the equationof the normal to the curve at point P.

[4 marks]P1,2008

12 A block of ice in the form of a cube with sidesx cm, melts at a rate of 9.72 cm3 per minute. Find the rateof change ofx at the instant whenx = 12 cm.

[3 marks]P1,2009

13 The gradient function of a curve is 6 kxdx

dy, where kis a constant. It is given that the curve has a

turning point at (2, 1). Find

a) the value ofkb) the equation of the curve.

[4 marks]P1,2009


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14 Giveny = 2x(x6), find(a)

dx

dy

(b)the value ofx wheny is minimum(c) the minimum value ofy. [3 marks]P1,2010

15

The volume of a sphere is increasing at a constant rate of 12.8

cm

3

s

-1

. Find the radius of the sphere atthe instant when the radius is increasing at a rate of 0.2 cm s-1

.

[Volume of sphere = 3

3

4rV ] [3 marks]P1,2010

PROGRESSIONS

1 Given a geometric progression ...,,4,2, py

y expressp in terms ofy.

[2 marks]P1,2004

2 Given an arithmetic progression -7, -3, 1,, state three consecutive terms in this progression which sumup to 75.

[3 marks]P1,2004

3 The volume of water in a tank is 450 litres on the first day. Subsequently, 10 litres of water is added tothe tank everyday. Calculate the volume, in litres, of water in the tank at the end of the 7

thday.

[2 marks]P1,2004

4 Express the recurring decimal 0.969696 as a fraction in its simplest form.[4 marks]P1,2004

5 The first three terms of a sequence are 2, x, 8. Find the value of x so that the sequence isi) an arithmetic progression,ii) a geometric progression.

[2 marks]P1,2005

6 The first three terms of an arithmetic progression are 5, 9, 13. Findi) The common difference of the progression,ii) The sum of the first 20 terms after the 3rd term.

[4 marks]P1,2005

7 The sum of the first nterms of the geometric progression 8, 24, 72, is 8744. Finda) the common ratio of the progression,b) the value ofn

[4 marks]P1,2005

8 The 9th term of an arithmetic progression is 4 + 5p and the sum of the first four terms of the progressionis 7p10, wherep is a constant. Given that the common difference of the progression is 5, find the

value ofp.

[3 marks]P1,2006

9 The third term of a geometric progression is 16. The sum of the third term and the fourth term is 8. Finda. the first term and the common ratio of the progression,b. the sum to infinity of the progression.

[4 marks]P1,2006


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10 a) Determine whether the following sequence is an arithmetic progression or a geometricprogression.

16x, 8x, 4x,

b) Give a reason for the answers in (a).

[2 marks]P1,2007

11 Three consecutive terms of an arithmetic progression are 5x, 8, 2x. Find the common difference of theprogression.

[3 marks]P1,2007

12 The first three terms of a geometric progression are 27, 18, 12. Find the sum to infinity of the geometricprogression.

[3 marks]P1,2007

13 It is given that the first four terms of a geometric progression are 3, -6, 12 andx. Find the value ofx.[2 marks]P1,2008

14 The first three terms of an arithmetic progression are 46, 43 and 40. The nth term of this progression isnegative. Find the least value ofn. [3 marks]P1,2008

15 In a geometric progression, the first term is 4 and the common ratio is r. Given that the sum to infinityof this progression is 16, find the value ofr.

[2 marks]P1,2008

16 Given the geometric progression ,...,9

20,

3

10,5 find the sum to infinity of the progression.

[3 marks]P1,2009

17 Diagram shows three square cards.

The perimeters of the cards form an arithmetic progression. The terms of the progression are in

ascending order.

a. Write down the first three terms of the progression.b. Find the common difference of the progression.

[2 marks]P1,2009

18 The first three terms of a geometric progression arex, 6, 12. Finda. the value ofxb. the sum from the fourth term to the ninth term.

[4 marks]P1,2009

19 The sum of the first n terms of an arithmetic progression is given by ]13[2

nn

Sn .

Find(a) The sum of the first 5 terms,(b) The 5th term

[4 marks]P1,2010

3 cm

5 cm

7cm


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20 It is given that 1,x2,x4,x6,. Is a geometric progression and its sum to infinity is 3.Find

(a) the common ratio in terms ofx,(b) the positive value ofx.

[3 marks]P1,2010

21 The first three terms of an arithmetic progression are 3h, k, h + 2.(a)Express kin terms ofh.(b)Find the 10th term of the progression in terms ofh.

[4 marks]P1,2010

LINEAR LAW

1 Diagram shows a straight line graph ofx

y againstx

Given thaty = 6xx2, calculate the value

ofkand ofh.

[3 marks]P1,2004

2 The variablesx andy are related by the equationy = kx4, where kis a constant.a) Convert the equationy = kx4 to linear form.b) Diagram shows the straight line obtained by plotting y10log against x10log .

Find the value of

i) k10log

ii) h

[4 marks]P1,2005

(2, k)

(h, 3)

xO

x

y

O

(0, 3)

(2, h)

y10log

x10log


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3 Diagrams shows the curvey = -3x2 + 5 and the straight line graph obtained wheny = -3x2 + 5 isexpressed in the linear form Y= 5X+ c.

ExpressXand Yin terms ofx and/ory

[3 marks]P1,2006

4 The variablesx andy are related by the equationy2 = 2x(10x). A straight line graph is obtained byplotting

x

y2

againstx, as shown in diagram.

Find the value ofp and ofq.

[3 marks]P1,2007

5 The variablesx andy are related by the equationx

ky

5 , where kis a constant.

a) Express the equationx

ky

5

in its linear form used to

obtain the straight line graph shown in the diagram.

b) Find the value ofk.

[4 marks]P1,2008

(p, 0)

(3, q)

x

y2

xO

(0, -2)

y10log

xO

y

xO

Y

XO

-3

y = - 3x +5


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6 The variablesx andy are related by the equation hy = kx2 + hk. A straight line graph is obtained byplottingy againstx

2as shown in the diagram below.

[3 marks]P1,2010

VECTORS

1 Given that O(-3, 4) andB(2, 16), find in terms of the unit vectors~

i and~

j

a) AB

b) the unit vector in the direction of AB [4 marks]P1,2004

2 Given thatA(-2, 6),B(4, 2) and C(m,p), fin the value ofm and ofp such that~~

12102 jiBCAB .

[4 marks]P1,2004

3 Diagram show vector OA drawn on a Cartesian plane

a) Express OA in the form

y

x

b) Find the unit vector in the direction ofOA

[2 marks]P1,2005

4 Diagram shows a parallelogram, OPQR, drawn on a Cartesian planeIt is given that

~~46 jiOP and

~~54 jiPQ .

Find PR

[3 marks]P1,2005

y

x O

(0 , 6)

Given the gradient of the straight line is 3,

find the value ofh and ofk.

Ox

y

R

Q

P

O 2 4 6 8 10 12

2

4

6

y

x

A


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5 Diagram shows two vectors OA and AB

Express

a) OA in the form

y

x

b) AB in the form~~jyix

[2 marks]P1,2006

6 The points P, Q andR are collinear. It is given that~~

24 baPQ and ~~

13 bkaQR , where kis a

constant. Find

a) the value ofkb) the ratio ofPQ : QR

[4 marks]P1,2006

7 The following information refers to the vectors~a and

~b .

8

2

~a ,

4

1

~b

Find

a) the vector~~

2 ba

b) the unit vector in the direction of~~

2 ba [4 marks]P1,2007

8 The vector ~a and ~b are non-zero and non-parallel. It is given that ~~ 53 bkah , where h and kareconstants. Find the value of

a) hb) k [2 marks]P1,2008

9 Diagram shows a triangle PQR.The point Tlies on QR such that QT: TR = 3 : 1

Express in terms of~

a and~

b :

a) QR b) PT

[4 marks]P1,2008

10 Diagram shows a triangle PQR.

Given~

3aPQ ,~

6bPR and point Slies on QR such that

QR : SR = 2 : 1,

Express in terms of~a and

~b

a) QR

b) SP [4 marks]P1,2009

O

-5

y

x

A(4, 3)

B

P

Q

T

R4

~

a

4~

b

P

R

S

Q


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11 Given that jia 13 and jkib 7 , finda) ba in form jyix

b) the value ofkif 10ba [4 marks]P1,2009

12 Diagram shows the vector OR .

[3 marks]P1,2010

13 Diagram shows a triangle OAB andMis a point onAB.

[4 marks]P1,2010

TRIGONOMETRIC FUNCTIONS

1 Solve the equation xxx sinsincos 22 for 0ox 360o[4 marks]P1,2004

2 Solve the equation 5sin82cos3 xx for 0ox 360o[4 marks]P1,2005

3 Solve the equation oxx 30sin4sinsin15 2 0ox 360o[4 marks]P1,2006

4 Solve the equation 0cos2cot xx for 0ox 180o.[4 marks]P1,2007

5 Given that sin=p, wherep is a constant and 90ox 180o. Find in terms ofpa) cosecb) sin2

[3 marks]P1,2008

y

O x

R(3 , 4)Express in the form jyix :

(a) OR (b)the unit vector in the direction ofOR .

O

A

M

B

Given aOA 5 , bOB 4 , and 2AM= 3MB,

find

(a) AB (b)OM


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6 Solve the equation 0cossin3 coxxx , 0ox 360o.[3 marks]P1,2009

7 It is given that13

5sin A and

5

4cos B , whereA is an obtuse angle andB is an acute angle. Find

a) tanAb)

cos (AB) [3marks]P1,2009

8 Given cos =p, find tan2. [2 marks]P1,2010

PROBABILITY

1 A box contains 6 white marbles and kblack marbles. If a marble is picked randomly from the box, theprobability of getting a black marble is 5

3

. Find the value ofk.[3 marks]P1,2004

2 Table shows the number of coloured cards in a box.Two cards are drawn at random from the box.

Find the probability that both cards are of the same colour.

[3 marks]P1,2005

3 The probability that Hamid qualifies for the final of a track event is5

2while the probability that Mohan

qualifies is3

1. Find the probability that

a) both of them qualify for the finalb) only one of them qualifies for the final.

[3 marks]P1,2006

4 The probability that each shot fired by Ramli hits a targets is 31

.

a) If Ramli fires 10 shots, find the probability that exactly 2 shots hit the target.b) If Ramli fires n shots, the probability that all the n shots hit the targets is

243

1. Find the value ofn.

[4 marks]P12007

5 The probability of Sarah being chosen as a school prefect is5

3while the probability of Aini being

chosen is12

7. Find the probability that

a) neither of them is chosen as a school prefect,b) only one of them is chosen as a school prefect.

[4 marks]P1,2008

Colour Number of Cards

Black 5

Blue 4

Yellow 3


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6 The probability that a student is a librarian is 0.2. Three students are chosen at random. Find theprobability that

a) all three are librarians,b) only one of them is librarian.

[4 marks]P1,2009

7 In a selection of a class monitor, the probability that studentXis chosen is3

1, while the probability that

either studentXod student Yis chosen is5

2.

Find the probability that

(a)student Yis chosen,(b)studentXor student Yis not chosen.

[3 marks]P1,2010

PROBABILITY DISTRIBUTIONS

1 Xis a random variable of a normal distribution with a mean of 5.2 and variance of 1.44. Finda) theZscore ifX= 6.7b) P(5.2 X 6.7)

[4 marks]P1,2004

2 The masses of students in a school has a normal distribution with mean of 54 kg and a standarddeviation of 12 kg. Find

a) the mass of the students which gives a standard score of 0.5,b) the percentage of students with mass greater than 48 kg.

[4 marks]P1,2005

3 Diagram shows a standard normal distribution graph.The probability represented by the area of the shaded

region is 0.3485.

(a)Find the value ofk.(b)Xis a continuous random variable which is normally

distributed with a mean of 79 and a standard deviation

of 3. Find the value ofXwhen thez-score is k.[4 marks]P1,2006

4 Xis a continuous random variable of a normal distribution with a mean of 52 and a standard deviationof 10. Find

a) thez-score whenX= 67.2b) the value ofkwhen P(z < k) = 0.8849

[4 marks]P1,2007

5 The masses of a group of students in a school have a normal distribution with a mean of 40 kg and astandard deviation of 5 kg. Calculate the probability that a student chosen at random from this group has

a mass ofa) more than 45 kgb) between 35 kg and 47.8 kg

[4 marks]P1,2008

O k z

f(x)

0.3485


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6 The masses of apples in a stall have a normal distribution with a mean of 200 g and a standard deviationof 30 g.

a) Find the mass, in g, of an apple whosez-score is 0.5b) If an apple is chosen at random, find the probability that the apple has a mass of at least 194 g.

[4 marks]P1,2009

7 The discrete random variableXhas a binomial probability distribution with n = 4, where n is the numberof trials. Diagram shows the probability distribution ofX.

[4 marks]P1,201016

1

4

1

k

P(X=x)

x0 1 2 3 4

Find

(a) the value ofk(b)P(X 3)


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FUNCTIONS

1 a) {x , y}

1 b) many to one

2) m =

2

5 , k =

8

1

3 a) 0,2

xx

3b) x = 15

4a) 2

4b) 8

5a)x

x 52

5b)4

3

6) 2x + 57a) many to one

7b) 2: xxf

8) m = 4

9a) m = 3

9b) h(x) = x + 1

10) x =8 , x = -2

11) a = 6 , b = -5

12a) t =2

1

12b) 0 f(x) 9

13 a)5

4

13 b) 25x2

-1

14 a) 4

14 b) k = 3

15 a) 2 and 6

15 b) {p, q, r, s}

16 a) 8x12

16 b) x =2

3

17 a) 517 b) p = 32

18 a) {(1,p), (2,r), (3,5), (4,p)}

18 b) many to one

18 c) {p, r, s}

19 a)2

1x

19 b) 18

20 )2

1

QUADRATIC EQUATIONS

1) 2x2

+ 5x -3 = 0

2) x = 0.149 , x = 3.351

3) p = -4 , p = 8

4 a) x =3

1, x =-2

4 b)12

2k

h

5) p = 5

6) p 10

1

QUADRATIC FUNCTIONS

1) -2 x 6

2a) k =1

2b) x = 1

2c) (1, -2)3) p > 1

4a) p = -1

4b) q = 2

4c) x = 1

5a) x = 3

5b) f(x) = (x3)24

6) x < -4 , x > 1

7)2

1 x 1

8) m = 1, n = 5

9a) p >09b) q = -3

9c) r = -4

10) 1 < x < 4

11a) p = 3

11b) x = -3

12) a = -2 , 2

13a) -1 , 3

13b) x = 1

14a) k = 1

14b)

f(x

)

x

1

3

(2,1)

1-3


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INDICES AND LOGARITHM

1) x = 3

2) 2pm -1

3) x = -3

4) x =

2

3

5) 3r2p +1

6) x = 1

7) y = 2x

8) x =8

9

9)2

3 yx

10) n =2

1

11) x = -312) x = 9

13) n =2

14) p = q3

15) x = -2

16)3

2 ba

COORDINATE GEOMETRY

1) y = 3

2

x + 32) 5x

2+ 5y

2+ 50x6y118 = 0

3) p =k2

1

4) B(2,3)

5) h = 2 , k =3

1

6) p = -9 , p = 21

7a)3

x+4

y= 1

7b) 6x8y + 7 = 08) t = 3 , t = 11

9) B(4

3,

2

5)

10a) h = 8

10b) D(0,-2)

11) x2

+ y2

+ 6x - 8y = 0

STATISTICS

1) m = 25 - 9k2

2) m = - 4 , 11

3a) 12

3b) 4

4a) 63

4b) k = 5

5a) 7.071

5b) 84.85

6) 4

CIRCULAR MEASURE

1) 7.675 cm

2a) 1.222

2b) 13.09 cm

3a) 0.7

3b) 26.2 cm2

4a) 18.5

4b) 80.48

5a) 1.047 rad

5b) 30.70 cm2

6a) 14.26b) 58.64

7) 59 cm2

DIFFERENTIATION

1) 6x(6x5)(2x5)2

2)5

8

dt

dx

3)8

27

4)6

5

dt

dh

5) P(7,4)

6) 14(3x5)6

7a) 7

7b) y = 7p

8) k =3

2

9) p = 16

10) y = 2

1

h11) y = -x3

12)dt

dx= - 0.0225

13a) k = 3

13b) y = 762

3 2 xx

14a) 4x12

14b) x = 3

14c)18

15) r = 4

PROGRESSIONS


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1) p =2

8

y

2) T8 = 21, T9 = 25 , T10 = 29

3) T7 = 510

4)33

32

5i) x = 55ii) x = 4

6i) 4

6ii) 1100

7a) 3

7b) 7

8) p = 8

9a) T1 = 64 , r =2

1

9b)

3

242

10a) G.P

10b) Common ration, r =2

1

11) d = 14

12) 81

13)24

14) n = 17

15)4

3

16)3

17a) 12, 20, 28

17b) d = 8

18a) x = 3

18b) 1512

19a) 40

19b) 14

20a) x2

20b) 0.8165

21a) k = 2h + 1

21b) 96h

LINEAR LAW

1) k = 4 , h = 3

2a) kxy 101010 loglog4log

2bi) 3

2bii) 11

3) X =2

1

x, Y =

2x

y

4) p = 10 , q = 14

5a) kxy 101010 log)5log(log

5b) k =100

1

6) h = 2 , k = 6

VECTORS

1a) 5 i + 12 j

1b)13

5i +

13

12j

2) m = 6 , p = -2

3a) 12i + 5j3b)

13

12i +

13

5j

4) -10i + j

5a)

3

4

5b)4i8j

6a)2

5

6b) 4 : 3

7a)

12

5

7b)13

125 ji

8a) h = -3

8b) k = 5

9a) 4a6b

9b) 3a +2

3b

10a) -3a + 6b

10b) - a - 4b11a) 6i + (1 +k)j

11b) k = -9 , 7

12a) 3i + 4j

12b)5

3i +

5

4j

13a) -5a + 4b

13b) 2a +5

12b

TRIGONOMETRIC FUNCTIONS

1) x = 30o, 150

o, 270

o

2) x = 41o49, 138

o11

3) x = 23o35, 156

o25, 199

o28, 340

o32

4) x = 90o, 210

o, 270

o, 330

o

5a)p

1

5b) 212 pp

6) x = 19o28, 90o, 160o32, 270o

7a)12

5


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7b)65

33

8)2

21

p

p

PROBABILITY

1) k = 9

2)66

19

3a)15

2

3b)15

7

4a) 0.1951

4b) n = 5

5a)16

1

5b)60

29

6a) 0.008

6b) 0.384

7a)15

1

7b)5

3

PROBABILITY DISTRIBUTIONS

1a) 1.25

1b) 0.3944

2a) 60 kg

2b) 69.15%

3a) k = 1.03

3b) 82.09

4a) 1.52

4b) 1.25a) 0.1587

5b) 0.7819

6a) 215 g

6b) 0.5793

7a) k =8

3

7b)16

5

kertas 1 spm matematik tambahan 2004-2010

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