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Additional Mathematics SPM

FORM 4

TOPIC: FUNCTION

PAPER 1

10}8,6,4,2,{

3}2,1,{

=

=

Q

P

YEAR 2003

1. Based on the above information, the relation betweenPand Q is defined by the set of ordered

pairs {(1, 2), (1,4), (2, 6), (2, 8)}.

State

(a) the image of 1,

(b) the object of 2.

[2 marks]

2. Given that 15: + xxg and 32: 2 + xxxh , find

(a) )3(1g ,

(b) )(xhg .

[4

marks]

YEAR 2004

3. Diagram 1 shows the relation between setPand set Q.

SetP Set Q

Diagram 1

State

(a) the range of the relation,

(b) the type of the relation.[2 marks]

4. Given the functions8

52:and4:

1++

kxxhmxxh , where m and kare constants, find

the values ofm and k.

[3 marks]

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d

e

f

w

x

y

z


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5. Given the function 0,6

)( = xx

xh and the composite function xxhg 3)( = , find

(a) )(xg

(b) the value ofx when 5)( =xgh .

[4

marks]

YEAR 2005

6. In Diagram 2, the function h mapsx toy and the functiongmapsy toz.

h g

x y z

Diagram 2

Determine

(a) )5(1h ,

(b) )2(gh .

[2 marks]

7. The function w is defined as 2,25)(

= xx

xw .

(a) )(1 xw ,

(b) )4(1w .

[3 marks]

8. The following information refers to the functions h andg.

14:

32:

xxg

xxh

Find )(1 xgh .

[3 marks]

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8

2

5


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YEAR 2006

9. In Diagram 3, setB shows the images of certain elements of setA.

SetA SetB

Diagram 3

(a) State the type of relation between setA and setB.

(b) Using the function notation, write a relation between setA and setB.

[2 marks]

10. Diagram 4 shows the function ,0,:

xx

xmxh where m is a constant.

xx

xm

8

2

1

Diagram 4

Find the value ofm.

[2 marks]

PAPER 2

YEAR 2006

1. Given that 23: + xxf and 15

: +x

xg , find

(a) )(1 xf , [1 marks]

(b) )(1 xgf , [2

marks]

(c) )(xh such that 62)( += xxgh . [3

marks]

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5

4

-4

-5

2

5

1

6

h


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ANSWERS (FUNCTION)

PAPER 1

1.

(a) 2 or 4(b) 1

2.

1 2( ) (3)

5a g =

(a) )(xhg = 25x2 + 2

3.

(a) range = {x,y}

(b) many to one relation.

4.

1 5

8 2

k m= =

5.

2(a) ( ) , 0g x x

x=

(b) 15x =.

6.

(a) 2)5(1 =h

(b) (2) 8gh =

7.

1 2 5(a) ( ) , x 0x

w xx

=

1 3(b) (4) 4w

=

8. 1( ) 2 5gh x x = +

9.

(a) Many to one relation

(b) 2: xxf

10. 4m =

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PAPER 2

1.

1 2(a) ( )

3

xf x

=

1 5

(b) ( ) 15

x

f g x

=(c) ( ) 10 4h x x=

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TOPIC: QUADARTIC EQUATION

PAPER 1

YEAR 2003

1. Solve the quadratic equation 2x(x 4) = (1 - x)(x + 2). Give your answer correct to foursignificant figures.

[3 marks]

2. The quadratic equation x(x + 1) = px 4 has two distinct roots. Find the range of values of p.

[3 marks]

YEAR 2004

3. From the quadratic equation which has the roots -3 and 2

1

. Give your answer in the form

ax2 + bx + c = 0, where a, b and c are constants. [2 marks]

YEAR 2005

4. Solve the quadratic equationx(2x 5) = 2x 1. Give your answer correct to three decimal

places.

[3 marks]

YEAR 2006

5. A quadratic equationx2 +px + 9 = 2x has two equal roots. Find the possible values ofp.

[3 marks]

ANSWERS (QUADRATIC EQUATION)

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PAPER 1

1. x = 2.591 or -0.2573

2. p < -3 or p > 5

3. 2x2 + 5x 3 = 0

4. x = 8.153 or 0.1495. p = 8 or -4

TOPIC: QUADARTIC FUNCTION

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PAPER 1

YEAR 2003

1. The quadratic equation 4)1( =+ pxxx has two distinct roots. Find the range of values ofp .

[3

marks]

YEAR 2004

2. Find the range of values ofx for which 12)4( xx . [3marks]

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

Find

(a) the value ofk,

(b) the equation of axis of symmetry,

(c) the coordinates of the maximum point. [3 marks]

YEAR 2005

4. Diagram below shows the graph of a quadratic functions 2)(3)( 2 ++= pxxf , wherep is a

constant.

The curve )(xfy = has a minimum point (1,q) where q is a constant. State

(a) the value of p ,

(b) the value of q ,

(c) the equation of the axis of symmetry.

[3 marks]

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0

(1,q)

y

x

y=f(x)

-3 (2,-3)

y

x0


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YEAR 2006

5. Diagram below shows the graph of a quadratic function )(xfy = . The straight line 4=y

is a tangent to the curve )(xfy = .

(a) Write the equation of the axis of symmetry of the curve.

(b) Expressf(x) in the form of cbx ++ 2)( , where b and c are constants.

[3 marks]

6. Find the range of the values of xxx +>+ 4)4)(12( . [2marks]

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y = -4

x

y

y = f(x)

O 1 5


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ANSWERS (QUADRATIC FUNCTIONS)

PAPER 1

1. 53 >< porp

2. 62 x

3. (a) 1=k(b) 1=x

(c) Maximum point (1,-2)

4. (a) 1=p

(b) 2=q

(c) 1=x

5. (a) 3=x(b) 4)3()( 2 = xxf

6. 14 >< xorx

TOPIC: SIMULTANEOUS EQUATION

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PAPER 2

YEAR 2003

1. Solve the simultaneous equations 4x +y = -8 andx2 +x y = 2. [5

marks]

YEAR 2004

2. Solve the simultaneous equationsp - m = 2 andp2 + 2m = 8.

Give your answers correct to three decimal places. [5

marks]

YEAR 2005

3. Solve the simultaneous equationsx +2

1y = 1 andy2 - 10 = 2x. [5

marks]

YEAR 2006

4. Solve the simultaneous equations 2x +y = 1 and 2x2 +y2 +xy = 5.

Give your answers correct to three decimal places. [5marks]

YEAR 2007

5. Solve the simultaneous equations 022 = yx and 09102 2 =++ yxx . [5

marks]

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ANSWERS (SIMULTANEOUS EQUATIONS)

PAPER 1

1. x = -2 or -3

y = 0 or 4

2. m = 0.606 or -6.606

p = 2.606 or -4.606

3. x = 3 or -2

1

y = - 4 or 3

4. x = 1.443 or -0.693

y = -1.886 or 2.386

5. x = 2.707 ,y = 3.404

x = 1.293 ,y = 5858

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TOPIC: INDICES & LOGARITHM

PAPER 1

YEAR 2003

1. Given that 3loglog 42 = VT , express T in terms of V. [4marks]

2. Solve the equation xx 74 12 = . [4marks]

YEAR 2004

3. Solve the equation 684 432 += xx . [3marks]

4. Given that m=2log5 and p=7log5 , express 9.4log5 in terms ofm andp. [4marks]

YEAR 2005

5. Solve the equation 122 34 = ++ xx . [3marks]

6. Solve the equation 1)12(log4log 33 = xx . [3marks]

7. Given that pm =2log and rm =3log , express

4

27log

mm in terms ofp and r. [4

marks]

YEAR 2006

8. Solve the equation2

32

4

18

+

=x

x. [3 marks]

9. Given that 222 loglog32log += xxy , express y in terms ofx . [4 marks]

10. Solve the equation .log)1(log2 33 xx =+ [3 marks]

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ANSWERS (INDICES & LOGARITHMS)

PAPER 1

1. VT 8=

2. 677.1=x3. 3=x4. 12 mp

5. 3=x

6.2

3=x

7. 123 + pr

8. 1=x

9. xy 4=

10.8

11=x

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TOPIC: COORDINATE GEOMETRY

PAPER 1

YEAR 2003

1. The pointsA(2h, h),B(p, t) and C(2p, 3t)are on a straight line.B dividesACinternally in the

ratio 2 : 3. Expressp in terms oft.

[3 marks]

2. The equation of two straight lines are 2435and135

+==+ xyxy

. Determine whether the lines

are perpendicular to each other.

[3 marks]

YEAR 2004

3. Diagram 3 shows a straight linePQ with the equation 132=+

yx.

The pointPlies on thex-axis and the point Q lies on they-axis.

y

Q

O P x

Diagram 3

Find the equation of the straight line perpendicular toPQ and passing through the point Q.

[3 marks]

4. The pointA is (-1, 3) and the pointB is (4, 6). The pointPmoves such thatPA :PB = 2 : 3.

Find the equation of the locus ofP.

[3 marks]

YEAR 2005

5. The following information refers to the equations of two straight lines,JKandRT, which are

perpendicular to each other.

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constants.areandwhere

)2(::

qp

pxkyRTkpxyJK

+=

+=

Expressp in terms ofk.

[2 marks]

YEAR 2006

6. Diagram 6 shows the straight lineAB which is perpendicular to the straight line CB at the point

B.

y

A(0, 4)

B

O x

C

Diagram 6

The equation of the straight line CB isy = 2x 1.

Find the coordinates ofB.

[3 marks]

PAPER 2

YEAR 2003

1. Solutions to this question by scale drawing will not be accepted.

A pointPmoves along the arc of a circle with centreA(2, 3). The arc passes through Q(-2, 0)

andR(5, k).

(a) Find

(i) the equation of the locus of the pointP,

(ii) the value ofk.

[6 marks]

(b) The tangent to the circle at point Q intersects they-axis at point T.

Find the area of triangle OQT.

[4 marks]

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YEAR 2004

2. Diagram 7 shows a straight line CD which meet straight lineAB at the pointD. The pointD

lies on they-axis.

y

C

O B(9, 0) x

D

A(0, 6)

Diagram 7

(a) Write down the equation ofAB in the form of intercepts.

[1 marks]

(b) Given that 2AD =DB, find the coordinates ofD.

[2 marks]

(c) Given that CD is perpendicular toAB, find they-intercept ofCD.

[3 marks]

YEAR 2005


yA(4, 9 )

B

O x

2y +x + 6 = 0

C

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Diagram 8

(a) Find

(i) the equation of the straight lineAB.

(ii) the coordinates ofB.

[5 marks]

(b) The straight lineAB is extended to a pointD such thatAB :BD = 2 : 3.Find the coordinates ofD.

[2 marks]

(c) A pointPmoves such that its distance from pointA is always 5 units.

Find the equation of the locus ofP.

[3 marks]

YEAR 2006


Diagram 9 shows the triangleAOB where O is the origin.

PointPlies on the straight lineAB.

y

A(3, 4 )

C

O x

B(6, 2)

Diagram 3

(a) Calculate the area, in unit2, of triangleAOB.

[2 marks]

(b) Given thatAC: CB = 3 : 2, find the coordinates ofC.

[2 marks]

(c) A pointPmoves such that its distance from pointA is always twice its distance from

pointB.

(i) Find the equation of the locus ofP.(ii) Hence, determine whether or not this locus intercepts they-axis.

[6 marks]

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ANSWERS (COORDINATE GEOMETRY)

PAPER 1

1. p = -2 t

2.

1

2

1 2

1 , 5 3 245 3

5 3 24,

3 5 53

5

5 3

3 51

the lines are perpendicular to each other

y xy x

m y x

m

m m

+ = = +

= = +

=

=

=

.

3.

23

3y x= +

4.2 25 5 50 6 118 0x y x y+ + =

5.

1 1or2 2

p pk k= =

6. (2, 3 )B

PAPER 2

1.

(a)

2 2(i) 4 6 12 0

(ii) 1 or 7

x y x y

k

+ =

=

(b)28Area of unit

3OQT =

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2.

(a) 169=+

yx

(b)

(3, 4)D =

1(c) - intercept

2y = .

3.

(a)(i) Equation of line , 2 17

(ii) ( 8, 1)

AB y x

B

= +

(b) ( 14, 11)D

2 2(c) 8 18 72 0x y x y+ + + =

4. 2(a)area 9 unit .=

(b)

12 2Coordinates of ,5 5

C =

2 2

2

2

2

2

(c)

(i) locus of : 3 54 3 24 135 0

(ii) when 0, 3 24 135 0

8 45 01, 8, 45

4 64 180244

4 0,locus of intercepts the -axis

P x x y y

x y y

y ya b c

b ac

b acP y

+ + =

= =

=

= = =

= +

=

>

Q

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TOPIC: STATISTICS

PAPER 1

YEAR 2005

1. The mean of four numbers is m . The sum of the squares of the numbers is 100 and the

standard deviation is 3k. Express m in terms ofk.

[3 marks]

PAPER 2

YEAR 2003

1. A set of examination marks x1,x2,x3,x4,x5,x6 has a mean of 5 and a standard deviation of

1.5.

(a) Find(i) the sum of the marks, x ,

(ii) the sum of the squares of the marks, .2x [3

marks]

(b) Each mark is multiplied by 2 and then 3 is added to it.

Find, for the new set of marks,

(i) the mean,

(ii) the variance. [4 marks]

YEAR 2004

2. A set of data consists of 10 numbers. The sum of the numbers is 150 and the sum of the

squares of the numbers is 2472.

a. Find the mean and variance of the 10 numbers,

[3 marks]

b. Another number is added to the set of data and the mean is increased by 1.

Find

(i) the value of this number,

(ii) the standard deviation of the set of 11 numbers.[4 marks]

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YEAR 2005

3. Diagram below shows a histogram which represents the distribution of the marks obtained by

40 pupils in a test.

14

9

7

6

4

0.5 10.5 20.5 30.5 40.5 50.5

a. Without using an ogive, calculate the median mark. [3 marks]

b. Calculate the standard deviation of the distribution. [4 marks]

YEAR 2006

4. Table below shows the frequency distribution of the scores of a group of pupils in a game.

Score Number of pupils

10 19 1

20 29 2

30 39 8

40 49 12

50 59 k

60 69 1

(a) It is given that the median score of the distribution is 42.

Calculate the value of k.

[3 marks]

(b) Use the graph paper to answer this question

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Using a scale of 2 cm to 10 scores on the horizontal axis and 2 cm to 2 pupils on the vertical

axis, draw a histogram to represent the frequency distribution of the scores, find the mode score.

[4 marks]

(c) What is the mode score if the score of each pupil is increased by 5

[1 marks]

ANSWER (STATISTICS)

PAPER 1

1. m = 25 9k2

PAPER 2

1. (a) x = 30 , .2x =163.5(b) mean = 13, variance = 9

2. (a) mean = 15, variance = 22.2(b) k = 26, standard deviation = 5.494

3. (a) median = 24.07 (b) standard deviation= 11.74

4. (a) k = 4 (b) mode = 43 (c) mode score = 48

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TOPIC: CIRCULAR MEASURE

PAPER 1

YEAR 2003

1. Diagram 1 shows a sector ROS with centre O .

The length of the arc RS is 7.24 cm and the perimeter of the sector ROS is 25 cm. Find thevalue of , in radian. [ 3 marks

]

YEAR 2004

2. Diagram 2 shows a circle with centre O .

Given that the length of the major arc AB is 45.51 cm , find the length , in cm , of the

radius.

( Use = 3.142 ) [ 3marks ]

YEAR 2005

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O

R

S

DIAGRAM 1

O

A

B

0.354 rad

DIAGRAM 2


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3. Diagram 3 shows a circle with centre O .

The length of the minor arc is 16 cm and the angle of the major sector AOB is 290 o .

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 ]

YEAR 2006

4. Diagram 3 shows sector OAB with centre O and sector AXY with centre A .

Given that OB = 10 cm ,AY= 4 cm , XAY= 1.1 radians and the length of arcAB = 7cm ,calculate

( a) the value of , in radian ,

( b) the area, in cm2 , of the shaded region . [ 4 marks ]

PAPER 2

YEAR 2003

1. Diagram 1 shows the sectors POQ, centre O with radius 10 cm. The point R on OP is such

that OR : OP = 3 : 5 .

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O

A

B

DIAGRAM 3

A

Y

BO

DIAGRAM 4

X

O

R

Q

DIAGRAM 1

P


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Calculate

(a) the value of , in radian . [ 3 marks ]

(b) the area of the shaded region , in cm2 . [ 4 marks ]

YEAR 2004

2. Diagram 2 shows a circle PQRT , centre O and radius 5 cm. JQK is a tangent to the circle

at Q . The straight lines , JO and KO , intersect the circle at P andR respectively. OPQR is a

rhombus . JLK is an arc of a circle , centre O .

Calculate

(a) the angle , in terms of , [ 2 marks ]

(b) the length , in cm , of the arc JLK , [ 4 marks ]

(c) the area , in cm2 , of the shaded region. [ 4 marks ]

YEAR 2005

3. Diagram 3 shows a sector POQ of a circle , centre O. The pointA lies on OP, the pointB

lies on OQ andAB is perpendicular to OQ.

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O

rad

Q

RP

J

L

K

T

DIAGRAM 2

O

A

P

QB

rad6

DIAGRAM 3

8 cm


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It is given that OA: OP= 4 : 7 .

( Using = 3.142 )

Calculate

(a) the length , in cm , of AP, [ 1 mark

]

(b) the perimeter , in cm , of the shaded region , [ 5

marks ]

( c) the area , in cm2 , of the shaded region . [ 4 marks

]

YEAR 2006

4. Diagram 4 shows the plan of a garden.PCQ is a semicircle with centre O and has a radius

of 8 m.RAQ is a sector of a circle with centreA and has a radius of 14 m .

Sector COQ is a lawn . The shaded region is a flower bed and has to be fenced . It is giventhat AC = 8 m and COQ = 1.956 radians . [ use = 3.142 ]

Calculate

(a) the area , in m2 , of the lawn . [ 2 marks ]

(b) the length , in m , of the fence required for fencing the flower bed , [ 4 marks ]

(c ) the area , in m2 , of the flower bed . [ 4 marks ]

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P A O Q

R

C

DIAGRAM 4


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ANSWERS (CIRCULAR MEASURE)

PAPER 1

1. = 0.8153 rad.

2. 675.7=r

3. (a) = 1.222 rad

(b) 09.13=r

4. (a) 7.010

7 ==

(b)A = 26.2

PAPER 2

1. (a) 9273.0=(b) Area of the shaded region = 22.37

2. (a) POR = 3

2

(b) The length of arc JLK = 20.94

(c) Area of the shaded region = 61.40

3. (a) AP = 6

(b) Perimeter of ehe shaded region = 24.40


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4. (a) Area of COQ = 62.59

(b) The perimeter = 38.25


TOPIC: DIFFERENTIATION

PAPER 1

YEAR 2003

1. Given that )5(14 xxy = , calculate(a) the value ofx wheny is maximum,

(b) the maximum value ofy. [3 marks]

2. Given that xxy 52 += , use differentiation to find the small change in y whenx increases from 3

to 3.01. [3 marks]

YEAR 2004

3. Differentiate42

)52(3 xx with respect tox. [3 marks]

4. Two variables, x and y are related by the equation .2

3x

xy += Given that y increases at a

constant rate of 4 units per second, find the rate of change ofx whenx= 2.[3 marks]

YEAR 2005

5. Given that 2)53(

1)(

=

xxh , evaluate h(1). [4 marks]

6. The volume of water, V cm3, in a container is given by hhV 83

1 3 += , when h cm is the height

of the water in the container. Water is poured into the container at the rate of 10 cm3 s1. Find

the rate of change of the height of water, in cm s1, at the instant when its height is 2 cm.

[3 marks]

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YEAR 2006

7. The point P lies on the curve 2)5( = xy . It is given that the gradient of the normal at P is

4

1 . Find the coordinates ofP. [3 marks]

8. It is given that

7

3

2

uy = , when .53 = xu Find dxdy

in terms ofx. [4 marks]

9. Given that ,43 2 += xxy

(a) find the value ofdx

dywhenx= 1,

(b) express the approximate change in y, in terms ofp, when x changes from 1 to 1 + p,

where p is a small value. [4 marks]

PAPER 2

YEAR 2007

11. The curve )(xfy = is such that 53 += kxdx

dy, where kis a constant. The gradient of the curve

at 2=x is 9 .

Find the value ofk. [ 2 marks ]

12. The curve 64322 += xxy has a minimum point atx =p , where p is a constant.

Find the value ofp . [ 3 marks ]

YEAR 2003

1. (a) Given that 22 += xdx

dyandy= 6 whenx=1, findy in terms ofx.

[3 marks]

(b) Hence, find the value ofx if .8)1(2

22

=++ ydx

dyx

dx

ydx [4 marks]

2. (a) Diagram 2 shows a conical container of diameter 0.6 m and height 0.5 m. Water is poured

into the container at a constant rate of 0.2 m 3 s1.

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0.6 m

0.5 m water


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Calculate the rate of change of the height of the water level at the instant when the height of

the water level is 0.4 m.

(Use

=3.142; Volume of a cone = hr

2

3

1 ) [4 marks]

YEAR 2004

3. The gradient function of a curve which passes through A(1, 12) is .63 2 xx Find

(a) the equation of the curve, [3 marks]

(b) the coordinates of the turning points of the curve and determine whether each of the turning

points is a maximum or a minimum. [5 marks]

4. Diagram 5 shows part of the curve 2)12(

3

= xy which passes through A(1, 3).

(a) Find the equation of the tangent to the curve at the point A. [4 marks]

YEAR 2007

5. A curve with the gradient function2

22

xx has a turning point at ( k, 8 ) .

(a) Find the value of k. [ 3 marks ]

(b) Determine whether the turning point is a maximum or a minimum point .

[ 2 marks ]

(c) Find the equation of the curve . [ 3 marks ]

LTS 2007 31

Diagram 2

2)12(

3

=

xy

y

xO

A(1, 3)


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ANSWERS (DIFFERENTIATION)

PAPER 1

1. (a)2

5=x

(b)2

175=y

2. x= 0.11

3. 3)52)(56(6 xxx

4.5

8unit second1

5.8

27

6. 0.8333 cm s1

7. (7, 4)

8. 6)53(14 x

9. (a) 7 (b) 7p

PAPER 2

1. (a) 722 ++= xxy

(b)5

3=x orx=1

2. (a) 1063 2 = xxy

(b) (2, 10)

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3. (a) p= 3(b) 42)( 23 += xxxf

4. 1512 += xy

TOPIC: SOLUTION OF TRIANGLE

PAPER 2

YEAR 2003

1. Diagram 5 shows a tent VABC in the shape of a pyramid with triangle ABC as the horizontal

base. V is the vertex of the tent and the angle between the inclined plane VBC and the base is

50.

Diagram 5

Given that VB = VC = 2.2 m and AB = AC = 2.6 m, calculate

(a) the length of BC if the area of the base is 3 m2, [3 marks]

(b) the length of AV if the angle between AV and the base is 25, [3 marks](c) the area of triangle VAB. [4 marks]

YEAR 2004

2. Diagram 6 shows a quadrilateralABCD such that ABCis acute.

5.2 cm

9.8 cm

9.5 cm

Diagram 6

LTS 2007 33

V

C

B

A

A

C

D

B

40.5

12.3 cm


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(a) Calculate

(i) ABC,(ii)ADC,(iii) the area, in cm2, of quadrilateralABCD. [8 marks]

(b) A triangle ABC has the same measurements as those given for triangle ABC, that is,

AC= 12.3 cm, CB= 9.5 cm and BAC= 40.5, but which is different in shape totriangleABC.

(i) Sketch the triangleABC,

(ii) State the size ofABC. [2 marks]

YEAR 2005

3. Diagram 7 shows triangleABC.

Diagram 7

(a) Calculate the length , in cm, ofAC. [2 marks]

(b) A quadrilateralABCD is now formed so thatACis a diagonal, ACD = 40 andAD = 16cm. Calculate the two possible values ofADC. [2 marks]

(c) By using the acute ADCfrom (b), calculate(i) the length, in cm, ofCD,

(ii) the area, in cm2, of the quadrilateralABCD. [6 marks]

YEAR 2006

4. Diagram 5 shows a quadrilateral ABCD.

Diagram 5

LTS 2007 34

B

A

C

20 cm

15 cm

CD

B

A

5 cm

6 cm

9 cm

40


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The area of triangleBCD is 13 cm2 and BCD is acute.Calculate

(a) BCD, [2 marks](b) the length, in cm, ofBD, [2 marks]

(c) ABD, [3 marks](d) the area, in cm2, quadrilateralABCD. [3 marks]

ANSWERS (SOLUTION OF TRIANGLE)

PAPER 2

1. (a) 2.70 cm

(b) 3.149 cm

(c) 2.829 cm2

2. (a) (i) 57.23(ii) 106.07(iii) 80.96 cm2

(b) (i)

(ii) 122.77

3. (a) 19.27 cm(b) AD1C= 129.27, AD2C= 50.73(c) (i) 24.89 cm

(ii) 290.1 cm2

4. (a) 60.07 or 60 4(b) 5.573 cm

(c) 116.55 or 116 33(d) 35.43 cm2

LTS 2007 35

AB B

C


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TOPIC: INDEX NUMBER

PAPER 2

YEAR 2003

1. Diagram 1 is a bar chart indicating the weekly cost of the itemsP, Q ,R , Sand Tfor the

year 1990 . Table 1 shows the prices and the price indices for the items.

Items Price in 1900 Price in 1995

Price Index in 1995 based

on 1990

P x RM 0.70 175

Q RM 2.00 RM 2.50 125

R RM 4.00 RM 5.50 y

S RM 6.00 RM 9.00 150

T RM 2.50 z 120

(a) Find the value of

(i) x

LTS 2007 36

P Q R S T Items

P

12

15

24

P

30

33

DIAGRAM 1

Weekly cost ( RM )

0

TABLE 1


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(ii) y

(iii) z [ 3 marks ]

(b) Calculate the composite index for items in the year 1995 based on the year 1990 .

[ 2 marks ]

(c) The total monthly cost of the items in the year 1990 is RM 456 . Calculate thecorresponding total monthly cost for the year 1995 .

[ 2 marks ]

(d) The cost of the items increases by 20 % from the year 1995 to the year 2000 .

Find the composite index for the year 2000 based on the year 1990.

[ 3 marks ]

YEAR 2004

2. Table 2 shows the price indices and percentage of usage of four items ,P, Q ,R and S,which are the main ingredients in the production of a type of biscuit.

Item

Price index for the year 1995

based on the year 1993

Percentage of usage

(%)

P 135 40

Q x 30

R 105 10

S 130 20

(a) Calculate

(i) the price of Sin the year 1993 if its price in the year 1995 is RM 37.70 ,

(ii) the price index of Pin the year 1995 based on the year 1991 if its price index

in the year 1993 based on the year 1991 is 120.

[ 5 marks ]

(b) The composite index number of the cost of biscuit production for the year 1995

based on the year 1993 is 128.

Calculate

(i) the value of x ,

(ii) the price of a box of biscuit in the year 1993 if the corresponding price in the

year 1995 is RM 32 .

[ 5 marks ]

YEAR 2005

3. Table 3 shows the prices and the price indices for the four ingredients ,P, Q ,R and S,

used in making biscuits of a particular kind . Diagram 2 is a pie chart which represents therelative amount of the ingredientsP, Q ,R and S, used in making biscuits .

LTS 2007 37

TABLE 2


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Ingredients

Price per kg

( RM ) Price index for the

year 2004 based on

the year 2001Year

2001

Year

2004

P 0.80 1.00 x

Q 2.00 y 140

R 0.40 0.60 150S z 0.40 80

(a) Find the value of x ,y andz. [ 3 marks ]

(b) (i) Calculate the composite index for cost of making these biscuits in the year

2004 based on the year 2001 .

(ii) Hence , calculate the corresponding cost of making these biscuits in the year

2001 if the cost in the year 2004 was RM 2985 .

[ 5 marks ]

(c) The cost of making these biscuits is expected to increase by 50 % from the year 2004

to the year 2007 .

Find the expected composite index for the year 2007 based on the year 2001.

[ 2 marks ]

YEAR 2006

4. A particular kind of cake is made by using four ingredients ,P , Q , R and S. Table 4 shows

the prices of the ingredients .

Ingredient

Price per kilogram ( RM )

Year 2004 Year 2005

P 5.00 w

Q 2.50 4.00

R x y

S 4.00 4.40

LTS 2007 38

TABLE 3

Q

P

S

R

60o

100o

120o

DIAGRAM 2

TABLE 4


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(a) The index number of ingredientPin the year 2005 based on the year 2004 is 120 .

Calculate the value of w. [ 2 marks ]

(b) The index number of ingredientR in the year 2005 based on the year 2004 is 125 .

The price per kilogram of ingredientR in the year 2005 is RM 2.00 more than itscorresponding price in the year 2004 .

Calculate the value ofx and ofy . [ 3 marks ]

(c ) The composite index for the cost of making the cake in the year 2005 based on the

year 2004 is 127.5 .

YEAR 2007

5. Table 4 shows the prices and the price indices of five components ,P, Q ,R , S and T,used to produce a kind of toy .

Diagram 6 shows a pie chart which represents the relative quantity of components used.

Component

Price ( RM ) for the

year Price index for the

year 2006 based on

the year 2004Year

2004

Year

2006

P 1.20 1.50 125

Q x 2.20 110R 4.00 6.00 150

S 3.00 2.70 y

T 2.00 2.80 1.40

(a) Find the value of x andy . [ 3 marks ]

(b) (i) Calculate the composite index for the production cost of the toys in the year

2006 based 2004 .

[ 3 marks ]

(c) The price of each component increase by 20 % from the year 2006 to the year

LTS 2007 39

TABLE 4

S R

Q

P

36o

144o

72o

DIAGRAM 6

T

90o


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2008 .

Given that the production cost of one toy in the year 2004 is RM 55 , calculate the

corresponding cost in the year 2008.

[ 4 marks ]

ANSWERS (INDEX NUMBER)

PAPER 2

1. a) i) x = 0.40

ii) 137.5y =iii) z= 3.00

b) I = 140.9

c) 642.5RM

d) 169.10

2. a) i) 00.2993 RMP =

ii) I = 162

b) i) x = 125

ii) 2593 RMP =

3. a)125x

= , y = 2.80, z= 0.50

b) i) 129.4I=ii) =01P 2306.80

c) Expected composite index = 194.1

4. a) w = 6.00

b) x = 8.00

y = 10.00

c) i) 04 24.00P =ii) m = 4

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FORM 5

TOPIC: PROGRESSIONS

PAPER 1

YEAR 2003

1. The first three terms of an arithmetic progression are k 3, k + 3, 2k + 2.

Find

(a) the value ofk,

(b) the sum of the first 9 terms of the progression.

[3 marks]

2. In a geometric progression, the first term is 64 and the fourth term is 27. Calculate(a) the common ratio,

(b) the sum to infinity of the geometric progression.

[4 marks]

YEAR 2004

3. Given a geometric progression ,,4

,2, py

y expressp in terms ofy.

[2 marks]

4. Given an arithmetic progression 7, 3, 1, , state three consecutive terms in this

progression which sum up to 75.

[3 marks]

5. The volume of water in a tank is 450 litres on the first day. Subsequently, 10 litres of water is

added to the tank everyday.

Calculate the volume, in litres, of water in the tank at the end of the 7 th day.

[2 marks]

6. Express the recurring decimal 0.969696 as a fraction in its simplest form.

[4 marks]

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YEAR 2005

7. The first three terms of a sequence are 2,x, 8.

Find the positive value ofx so that the sequence is

(a) an arithmetic progression,

(b) a geometric progression.[2 marks]

8. The first three terms of an arithmetic progression are 5, 9 13.

Find

(a) the common difference of the progression,

(b) the sum of the first 20 terms after the third term.

[3 marks]

9. The sum of the first n terms of the geometric progression 8, 24, 72, is 8744.

Find(a) the common ratio of the progression,

(b) the value ofn.

[4 marks]

YEAR 2006

10. The 9th term of an arithmetic progression is 4 + 5p and the sum of the first four terms of the

progression is 7p 10, wherep is a constant.

Given that the common difference of the progression is 5, find the value ofp.

[3 marks]

11. The third term of a geometric progression is 16. The sum of the third term and the fourth term

is 8.

Find

(a) the first term and the common ratio of the progression.

(b) the sum of infinity of the progression.

[4 marks]

PAPER 2

YEAR 2006

1. Two companies, Delta and Omega, start to sell cars at the same time.

(a) Delta sells k cars in the first month and its sales increase constantly by m cars every

subsequent month. It sells 240 cars in the 8 th month and the total sales for the first 10

months are 1900 cars.

Find the value of k and of m.

[5 marks]

(b) Omega sells 80 cars in the first month and its sales increase constantly by 22 cars every

subsequent month.If both companies sell the same number of cars in the n th month, find the value of n.

[2 marks]

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ANSWERS (PROGRESSIONS)

PAPER 1

1. (a) 7k=

9(b) 252S = .2.

3(a)

4r=

(b) 256nS = .

3. 28

p

y

= .

4. 21, 25, 29

5. 7 510T =

6.

32

33

7. (a) 5x =(b) 4x = .

8.

(a) 459 ==d

20(b) 1100S = .

9.

(a) 38

24==r

(b) n = 7

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10. p = 8

11.

1(a)

2

64

2(b) 42

3

r

a

S

=

=

=

PAPER 2

1.(a) 20

100mk

==

(b) 11n =

TOPIC: LINEAR LAW

PAPER 1

YEAR 2003

1. x andy are related by the equationy =px2 + qx, wherep and q are constants. A straight line is

obtained by plottingx

yagainstx, as shown in Diagram 1.

x

y

(2 , 9)

(6 , 1)

0 x

Diagram 1

Calculate the values ofp and q. (4

marks)

YEAR 2004

2. Diagram 3 shows a straight line graph ofx

yagainstx.

x

y

(2 , k)

(h , 3)

0 x

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Diagram 3

Given thaty = 6x x2, calculate the value ofkand ofh. (3

marks)

YEAR 2005

3. The variablesx andy are related by the equationy = kx4, where kis a constant.

(a) Convert the equationy = kx4 to linear form.

(b) Diagram 3 shows the straight line obtained by plotting log10y against log10x.

log10y

(2 , h)

(0, 3)

0 log10x

Diagram 3

Find the value of

(i) log10 k,

(ii) h.

YEAR 20064. Diagram 4(a) shows the curvey = -3x2 + 5. Diagram 4(b) shows the straight line graph

obtained wheny = -3x2 + 5 is expressed in the linear form Y = 5X + c.

y Y

y = -3x2 + 5

0 x 0 X -3

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DIAGRAM 4(a) DIAGRAM 4(b)

Express X and Y in terms ofx and/ory.

YEAR 20075. The variablesx andy are related by the equation )10(22 xxy = . A straight line graph is

obtained by plottingx

y2

againstx , as shown in Diagram 2 .

x

y 2

(3 , q)

0 x

Diagram 2

Find the value ofp and ofq. [ 3 marks ]

PAPER 2

YEAR 2003

1. Use graph paper to answer this question.

Table 1 shows the values of two variables,x andy, obtained from an experiment. It is known

that x andy are related by the equationy =pkx2 , wherep and kare constants.

x 1.5 2.0 2.5 3.0 3.5 4.0

y 1.59 1.86 2.40 3.17 4.36 6.76

Table 1

(a) Plot logy against x2.

Hence, draw the line of best fit. [5

marks](b) Use the graph in (a) to find the value of

(i) p,

LTS 2007 46

(p , 0 )


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(ii) k. [5

marks]

YEAR 2004

2. Use graph paper to answer this question.

Table 1 shows the values of two variables,x andy, obtained from an experiment. Variablesxandy are related by the equationy =pkx , wherep and kare constants.

x 2 4 6 8 10 12

y 3.16 5.50 9.12 16.22 28.84 46.77

Table 1

(a) Plot log10y against x by using a scale of 2 cm to 2 units on thex-axis and 2 cm to 0.2 unit

on the log10 y-axis. Hence, draw the line of best fit. [4 marks]

(b) Use your graph from (a) to find the value of

(i) p,

(ii) k. [6 marks]

YEAR 2005

3. Table 1 shows the values of two variables,x andy, obtained from an experiment. Variablesx

andy are related by the equationy =px +px

r, wherep and rare constants.

x 1.0 2.0 3.0 4.0 5.0 5.5

y 5.5 4.7 5.0 6.5 7.7 8.4

Table 1

(a) Plotxy againstx2 by using a scale of 2 cm to 5 units on both axes. Hence, draw the line of best

fit. [5 marks]

(b) Use the graph from (a) to find the value of

(i) p,

(ii) r. [5 marks]

YEAR 2006

4. Use graph paper provided by the invigilator to answer this question.

Table 2 shows the values of two variables,x andy, obtained from an experiment. Variablesxandy are related by the equationy =pkx+1 , wherep and kare constants.

x 1 2 3 4 5 6

y 4.0 5.7 8.7 13.2 20.0 28.8

Table 1

(a) Plot logy against (x + 1) by using a scale of 2 cm to 1 units on the (x + 1)-axis and 2 cm to 0.2

unit on the log y-axis. Hence, draw the line of best fit. [5

marks]

(b) Use your graph from 7(a) to find the value of

(i) p,(ii) k. [5

marks]

LTS 2007 47


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YEAR 2007

4. Table 3 shows the values of two variables,x andy, obtained from an experiment. Variablesx

andy are related by the equationy = 2kx2+kx

p, wherep and kare constants.

x 2 3 4 5 6 7y 8 13.2 20 27.5 36.6 45.5

Table 1

(a) Plotx

yagainstx , using a scale of 2 cm to 1 units on both axes.

Hence, draw the line of best fit. [4 marks]

(b) Use your graph in 7(a) to find the value of

(i) p,

(ii) k.

(iii)y whenx = 1.2 . [5

marks]

ANSWERS (LINEAR LAW)

PAPER 1

1. p = - 2, q = 13

2. h = 3, k= 4

3. (a) log10y = 4 log10x + log10 k(a) (i) log10k= 1000

(ii) h = 11

4. X =2

1

x

Y =2

x

y

PAPER 2

1. (a)

x2 2.25 4.0 6.25 9.0 12.25 16.0

Log10y 0.20 0.27 0.38 0.50 0.64 0.83

(b) (i) p = 1.259

(ii) k= 1.109

2. (a)

x 2 4 6 8 10 12

Log10y 0.50 0.74 0.96 1.21 1.46 1.67

(b) (i) p = 1.820

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(ii) k= 1.309

3. (a)

x2 1 4 9 16 25 30.25

xy 5.5 9.4 15.0 26.0 38.5 46.2

(b) (i) p = 1.37

(ii) r= 5.48

4. (a)

x + 1 2 3 4 5 6 7

Logy 0.60 0.76 0.94 1.12 1.30 1.46

(b) (i) p = 1.738

(ii) k= 1.495

TOPIC: INTEGRATION

PAPER 1

YEAR 2003

1. Given that ( )

++=

+cxkdx

x

n1

)1(

54 , find the values of k and n [3marks]

2. Diagram below shows the curve y = 3x2 and the straight line x = k.

y y = 3x2

O x = k x

If the area of the shaded region is 64 unit2, find the value of k. [3marks]

YEAR 2004

3. Given that ( )

=k

dxx1

,632 where k> -1 , find the value of k. [4marks]

YEAR 2005

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4. Given that =6

2

7)( dxxf and ( ) =6

2

10)(2 dxkxxf , find the value ofk. [4marks]

YEAR 2006

5. Diagram below shows the curve y = f(x) cutting the x axis at x = q and x = b

y = f(x)

O a b

Given that the area of the shaded region is 5 unit2, find the value of b

a

dxxf )(2

[2marks]

6. Given that =5

1

8)( dxxg , find

(a) the value of 1

5

)( dxxg

(b) the value of k if =5

1

10)]([ dxxgkx

[4marks]

YEAR 2007

7. Given that =7

2

3)( dxxh , find

(a) =2

7

3)( dxxh

(b) 7

2

)(5[ dxxh [ 4 marks]

PAPER 2

YEAR 2003

1. Diagram below shows a curve x =y2 1 which intersects the straight line 3y =2x at point A.

3y =2x

LTS 2007 50

1

A

y


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x =y2 1

-1 O x

Calculate the volume generated when the shaded region is revolved 360o about

the y-axis.

[6marks]

YEAR 2004

2. Diagram below shows part of the curve( ) 212

3

=

xy which passes through

A(1,3).

A(1,3)

( ) 212

3

=

xy

(a) Find the equation of the tangent to the curve at the point A. [4marks]

(b) A region is bounded by the curve, the x-axis and the straight lines x = 2 and

x = 3.

(i) Find the area of the region

(ii) The region is revolved through 360o about the x axis. Find the volume

generated, in terms of [6marks]

3. The gradient function of a curve which passes through A(1, -12) is 3x2 6x.

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Find

(a) the equation of the curve [3marks]

YEAR 2005

4. A curve has a gradient function px2 4x, where p is a constant. The tangent to

the curve at the point (1,3 ) is parallel to the straight line y +x 5 = 0.Find

(b) the value ofp, [3marks]

(c) the equation of the curve. [3marks]

5. In Diagram below, the straight line PQ is normal to the curve 12

1 2 += xy at

A(2,3). The straight line AR is parallel to the y axis.

P

A(2,3)

O R Q(k,0)

Find

(a) the value of k, [3marks](b) the area of the shaded region, [4marks]

(c) the volume generated, in terms of , when the region bounded by the curve, the y axisand the straight line y = 3 is revolved through 360o about the y-axis.

[3marks]

YEAR 2006

6. Diagram below shows the straight line y = x + 4 intersecting the curve

y = (x 2 )2 at the points A and B.

y

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12

1 2 += xyy

2)2( = xy 4+=xy


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P

O k x

Find,

(a) the value of k [2marks]

(b) the are of the shaded region P [5marks]

(c) the volume generated, in terms of , when the shaded region Q is revolved 360o about thex axis. [3marks]

ANSWERS (INTEGRATION)

PAPER 1

1. k = - 5/3 , n = -3

2. k = 4

3. k = 44. k =

5. -10

6. (a) - 8 (b) k = 3/2

PAPER 2

1. volume =15

52

2. (a) y = -12x + 15 (b) area = 1/5 , volume =1125

49

3. y = 3x2 6x 104. p = 3 , y = x3 2x2 = 4

5. (a) k = 8 (b) area = 123

1(c) Volume = 4

6. (a) k = 5 (b) area = 20.83 (c) volume =5

32

LTS 2007 53

Q

B

AP

Q


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TOPIC: VECTORS

PAPER 1

YEAR 2003

1. Diagram below shows two vectors, OP and QO

Q(-8,4)

P(5,3)

Express

(a) OP in the form ,y

x

(b) QO in the form xi +yj [2marks]

LTS 2007 54

p = 2a + 3b

q = 4a b

r = ha + ( h k) b, where h and kare constants


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2. Use the above information to find the values of h and k when r = 3p 2q.

[3marks]

3. Diagram below shows a parallelogram ABCD with BED as a straight line.

D C

E

A B

Given that AB = 6p , AD = 4q and DE = 2EB, express, in terms of p and q

(a) BD(b) EC

[4marks]

YEAR 2004

4. Given that O(0,0), A(-3,4) and B(2, 16), find in terms of the unit vectors, i and j,

(a) AB

(b) the unit vector in the direction of AB[4marks]

5. Given that A(-2, 6), B(4, 2) and C(m,p), find the value of m and ofp such that

AB + 2 BC = 10i 12j. [4marks]

YEAR 2005

6. Diagram below shows vector OA drawn on a Cartesian plane.

yA

0 2 4 6 8 10 12 x

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6

4

2


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(a) Express OA in the form

y

x

(b) Find the unit vector in the direction of OA [2marks]

7. Diagram below shows a parallelogram, OPQR, drawn on a Cartesian plane.

y

Q

R P

O x

It is given that OP = 6i + 4j and PQ = - 4i + 5j. Find PR .

YEAR 2006

8. Diagram below shows two vectors, OA and AB .

y

A(4,3)

O x

-5

Express

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(a) OA in the form

y

x

(b) AB in the form xi + yj [2marks]

9. The points P, Q and R are collinear. It is given that PQ = 4a 2a and

bkaQR )1(3 ++= , where k is a constant. Find

(a) the value of k

(b) the ratio of PQ : QR [4marks]

PAPER 2

YEAR 2003

1. Give that

=

=3

2,

7

5OBAB and

=5

kCD , find

(a) the coordinates of A, [2marks]

(b) the unit vector in the direction of OA , [2marks]

(c) the value of k, if CD is parallel to AB [2marks]

YEAR 2004

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2. Diagram below shows triangle OAB. The straight line AP intersects the straight line OQ at

R. It is given that OP = 1/3 OB, AQ = AB, xOP 6= and .2yOA =

A

Q

R

O P B

(a) Express in terms ofx and/ory:

(i) AP(ii) OQ [4marks]

(b) (i) Given that ,APhAR = state AR in terms of h, x andy.

(ii) Given that ,OQkRQ= state RQ in terms of k, x andy.

[2marks]

(c) Using AR and RQ from (b), find the value of h and of k.

[4marks]

YEAR 2005

3. In diagram below, ABCD is a quadrilateral. AED and EFC are straight lines.

D

E F C

A B

It is given that =AB 20x, =AE 8y, DC= 25x 24y, AE = AD

and EF =5

3EC.

(a) Express in terms ofx and/ory:

(i) BD

(ii) EC [3marks]

(b) Show that the points B, F and D are collinear. [3marks]

(c) If |x| = 2 and |y | = 3, find | BD |. [2marks]

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YEAR 2006

4. Diagram below shows a trapezium ABCD.

B C

F

A E D

It is given that AB =2y, AD = 6x, AE =3

2AD and BC=

6

5AD

(a) Express AC in terms of x and y [2marks]

(b) Point F lies inside the trapezium ABCD such that 2 EF= m AB , and m is a

constant.

(i) Express AF in terms of m , x and y

(j) Hence, if the points A, F and C are collinear, find the value of m.

[5marks]

ANSWERS (VECTORS)

PAPER 1

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1. (a) ,

3

5

(b) -8 i + 4j

2. h = -2 , k = - 13

3. (a) = - 6p + 4q (b) 2p + 8/3 q

4. (a) AB = ,

12

5

(b) vector in direction AB =13

1 ,

12

5

5. m = 6, p = -2

6. (a) OA =

5

12(b) vector in direction OA=

13

1 ,5

12

7. PR = - 10 i + j

8. (a) OA =

3

4(b) AB = -4i 8j

9. (a) k = - 5/2 (b) 4 : 3

PAPER 2

1. (a) A( -3, -4 ) (b) OA =5

1

4

3(c) k =

7

25

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2. (a) AP= - 2y + 6x OQ =2

3y +

2

9x

(b) ),26( yxhAR =

+= yxkRQ

2

3

2

9

(d) k = 1/3 , h =

3. (a) BD = -20x + 32y , EC= 25x

(b) BF = - 5x + 8y BD = 4 ( - 5 x + 8y )

(c) | BD | = 104

4. (a) AC = 5x + 2y (b) AF = 4x+ my , m =5

8

TOPIC: TRIGONOMETRIC FUNCTION

PAPER 1

YEAR 2003

1. Given that tan = t, 0o < < 90o , express , in terms of t:

(a) cot (b) sin ( 90 - ) [ 3 marks ]

2. Solve the equation 6 sec2 A 13 tan A = 0 , 0o A 360o. [ 4 marks ]

YEAR 20043. Solve the equation cos2x sin2x = sinx for 0ox 360o . [ 4 marks ]

YEAR 2005

4. Solve the equation 3cos 2x = 8 sinx 5 for 0ox 360o . [ 4 marks ]

YEAR 2006

5. Solve the equation 15 sin2x = sinx + 4 sin 30o for 0ox 360o . [ 4 marks ]

PAPER 2

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YEAR 2003

1. (a) Prove that tan + cot = 2 cosec 2 . [ 4 marks ]

(b) (i) Sketch the graph y = 2 cos2

3x for 0ox 2 .

(ii) Find the equation of a suitable straight line for solving the equation

cos2

3x = 1

4

3x

.

Hence , using the same axes , sketch the straight line and state the number of

solutions to the equation cos2

3x = 1

4

3x

for 0ox 2.

[ 6 marks ]

YEAR 2004

2. (a) Sketch the graph of y = cos 2x for 0ox 180o. [ 3marks ]

(b) Hence , by drawing a suitable straight line on the same axes , find the number of

solutions satisfying the equation 2 sin2x = 2 -180

xfor 0ox 180o.

[ 3 marks ]

YEAR 2005

3. (a) Prove that cosec2x 2 sin2x cot2x = cos 2x. [ 2 marks ]

(b) (i) Sketch the graph of y = cos 2x for 0 x 2 .

(ii) Hence , using the same axes , draw a suitable straight line to find the number

of solutions to the equation 3(cosec2x 2 sin2x cot2x ) =

x- 1 for

0 x 2 . State the number of solutions . [ 6marks ]

YEAR 2006

4. (a) Sketch the graph of y = - 2 cosx for 0 x 2 . [ 4marks ]

(b) Hence , using the same axis , sketch a suitable graph to find the number of solutions

to the equationx

+ 2 cos x = 0 for 0 x 2 . State the number of solutions.

[ 3 marks ]

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ANSWERS (TRIGONOMETRIC FUNCTION)

PAPER 1

1. a)

tan

1cot =

=t

1

b) kos= )90(sin

=1

1

2 +t

2. A = 33.69 , 213.69 or 56.31 , 236.31

3. x = 30o , 50o , 270o

4. x = 41.81o , 138.19o

5. x = 23.58o , 156.42o , 199.47o , 340.53o

PAPER 2

1. a) kot+tan =

sin

sin kos

kos+

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1

t1

2 +t


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=

kos

kos

sin

sin 22 +

=kossin2

2

=2sin

2

= 2cosec .

b) (i) & (ii)

Number of solution = 3

2. a) & b)


3. a) cosec2x 2 sin2x cot2x = ( ) xxx 222 cotsin2cot1 += x2sin21= cos 2x.

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-2

-2

O

1

2

/3 5/42 2

22

3= xy

xy2

3cos2=

O

- 1

1

0.5 0.750.25

y = cos 2x

1180

=x

y


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b)


4.

a) & b)


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O

21.50.5

1

2

3

1

3=

xy

xy 2cos=

O

- 2

2

21.50.5

xy

=

y = cosx


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TOPIC: PERMUTATIONS & COMBINATIONS

PAPER 1

YEAR 2003

1. Diagram 6 shows 5 letters and 3 digits.

Diagram 6

A code is to be formed using those letters and digits. The code must consists of 3 letters

followed by 2 digits. How many codes can be formed if no letter or digit is repeated in each

code?

[3 marks]

2. A badminton team consists of 7 students. The team will be chosen from a group of 8 boys and 5

girls. Find the number of teams that can be formed such that each team consists of

(a) 4 boys,

(b) not more than 2 girls. [4 marks]

YEAR 2004

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A B C D E 6 7 8


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3. Diagram 6 shows five cards of different letters.

Diagram 6

(a) Find the number of possible arrangements, in a row, of all the cards.(b) Find the number of these arrangements in which the letters E and A are side by side.

[4 marks]

YEAR 2005

4. A debating team consists of 5 students. These 5 students are chosen from 4 monitors, 2 assistant

monitors and 6 prefects.

Calculate the number of different ways the team can be formed if

(a) there is no restriction,

(b) the team contains only 1 monitor and exactly 3 prefects. [4 marks]

YEAR 2006

5. Diagram 9 shows seven letters cards.

Diagram 9

A four-letter code is to be formed using four of these cards. Find

(a) the number of different four-letter codes that can be formed,

(b) the number of different four-letter codes which end with a consonant.

[4 marks]

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H E B A T

U N I F O R M


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ANSWERS (PERMUTATIONS & COMBINATIONS)

PAPER 1

1. 360

2. (a) 700 (b) 708

3. (a) 120 (b) 48

4. (a) 792 (b) 160

5. (a) 840 (b) 480

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TOPIC: PROBABILITY

PAPER 1

YEAR 2004

1. A box contains 6 white marbles and kblack marbles. If a marble is picked randomly from the

box, the probability of getting a black marble is5

3.

Find the value ofk. [3 marks]

YEAR 2005

2. The following table shows the number of coloured cards in a box

Colour Number of Cards

Black 5Blue 4

Yellow 3

Two cards are drawn at random from the box.

Find the probability that both cards are of the same colour. [3 marks]

YEAR 2006

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

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(a) both of them qualify for the final,

(b) only one of them qualifies for the final. [ 3 marks]

ANSWERS (PROBABILITY)

PAPER 1

1. k= 9

2.66

19

3. (a)15

2(b)

15

7

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TOPIC: PROBABILITY DISTRIBUTION

PAPER 1

YEAR 20031. The following diagram shows a standard normal distribution graph.

IfP(0 k). [2 marks]

2. In an examination, 70% of the students passed. If a sample of 8 students is randomly selected,

find the probability that 6 students from the sample passed the examination. [3 marks]

YEAR 2004

3. Xis a random variable of a normal distribution with a mean of 5.2 and a variance of 1.44.

Find

(a) theZscore ifX=6.7

(b) P(5.2X6.7) [4 marks]

YEAR 2005

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f(z)

0 k z

71


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4. The mass of students in a school has a normal distribution with a mean of 54 kg and a standard

deviation of 12 kg. Find

(a) the mass of the students which give a standard score of 0.5,

(b) the percentage of students with mass greater than 48 kg. [4 marks]

YEAR 20065. The diagram below 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 ofXwhenz-score is k. [4 marks]

PAPER 2

YEAR 2003

1. (a) Senior citizens make up 20% of the population of a settlement.(i) If 7 people are randomly selected from the settlement, find the probability that at least two

of them are senior citizens.

(ii) If the variance of the senior citizens is 128, what is the population of the settlement?

[5 marks]

(b) The mass of the workers in a factory is normally distributed with a mean of 67.86 kg and a

variance of 42.25 kg2. 200 of the workers in the factory weigh between 50 kg and 70 kg.

Find the total number of workers in the factory. [5 marks]

YEAR 2004

2. (a) A club organises a practice session for trainees on scoring goals from penalty kicks. Eachtrainee takes 8 penalty kicks. The probability that a trainee scores a goal from a penalty kick

isp. After the session, it is found that the mean number of goals for a trainee is 4.8

(i) Find the value ofp.

(ii) If a trainee is chosen at random, find the probability that he scores at least one goal.

[5 marks]

(b) A survey on body-mass is done on a group of students. The mass of a student has a normal

distribution with a mean of 50 kg and a standard deviation of 15 kg.

(i) If a student is chosen at random, calculate the probability that his mass is less than 41 kg.

(ii) Given that 12% of the students have a mass of more than m kg, find the value ofm.

[5 marks]

YEAR 2005

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f(z)

0 k z

0.3485

72


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3.For this question, give your answer correct to three significant figures.

(a) The result of a study shows that 20% of pupils in a city cycle to school.

If 8 pupils from the city are chosen at random, calculate the probability that

(i) exactly 2 of them cycle to school,

(ii) less than 3 of them cycle to school. [4 marks]

(b) The mass of water-melons produced from an orchard follows a normal distribution with amean of 3.2 kg and a standard deviation of 0.5 kg. Find

(i) the probability that a water-melon chosen randomly from the orchard has a mass of not

more than 4.0 kg,

(ii) the value ofm if 60% of the water-melons from the orchard has a mass of more than

m kg. [6 marks]

YEAR 2006

4. An orchard produces lemons.

Only lemons with diameter,x greater than kcm are graded and marketed.

Table below shows the grades of the lemons based on their diameters.

Grade A B CDiameter,x (cm) x > 7 7 x > 5 5 x >k

It is given that the diameter of lemons has a normal distribution with a mean of 5.8 cm and a

standard deviation of 1.5 cm.

(a) If one lemon is picked at random, calculate the probability that it is of gradeA. [2 marks]

(b) In a basket of 500 lemons, estimate the number of gradeB lemons. [4 marks]

(c) If 85.7% of the lemons is marketed, find the value ofk. [4 marks]

ANSWERS (PROBABILITY DISTRIBUTION)

PAPER 1

1. 0.1872

2. 0.2965

3. (a) 1.25 (b) 0.3944

4. (a) X = 60 (b) 69.146%

5. (a) 1.03 (b) 82.09

PAPER 2

1. (a) (i) 0.4232832 (ii) 800

(b) 319

2. (a) (i)p = 0.6 (ii) 0.9993

(b) (i) 0.2743 (ii) m = 67.625 kg

3. (a) (i) 0.2936 (ii) 0.79691

(b) (i) 0.9452 (ii) m = 3.0735

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4. (a) 0.2119 (b) 245 (c) k= 4.1965

TOPIC: MOTION ALONG A STRAIGHT LINE

PAPER 2

YEAR 20031. A particle moves in a straight line and passes through a fixed point O, with a velocity of

24 m s 1 . Its acceleration, a m s 2 , t s after passing through O is given by .210 ta = Theparticle stops afterk s.

(a) Find

(i) the maximum velocity of the particle,

(ii) the value of k.

(b) Sketch a velocity-time graph for kt 0 . [6 marks]

Hence, or otherwise, calculate the total distance traveled during that period.

[4 marks]

YEAR 20042. A particle moves along a straight line from a fixed pointP. Its velocity, Vm s 1 , is given by

)6(2 ttV = , where t is the time, in seconds, after leaving the point P.

(Assume motion to the right is positive)

Find

(a) the maximum velocity of the particle, [3 marks]

(b) the distance traveled during the third second, [3 marks]

(c) the value oftwhen the particle passes the pointsPagain, [2 marks]

(d) the time between leavingPand when the particle reverses its direction of motion.

[2 marks]

YEAR 2005

3. Diagram 9 shows the positions and directions of motion of two objects,Pand Q, moving in a

straight line passing two fixed points,A andB, respectively. ObjectPpasses the fixed pointA

and object Q passes the fixed pointB simultaneously. The distanceAB is 28 m.

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The velocity ofP, pv m s 1 , is given2246 ttvp += , where tis the time, in secondsA,

after

it passesA while Q travels with a constant velocity of -2 m s 1 . Object P stops

instantaneously

at point C.

(Assume that the positive direction of motion is towards the right.)

Find(a) the maximum velocity , in, m s 1 , ofP, [3 marks]

(b) the distance, in m, ofCfromA, [4 marks]

(c) the distance, in m, betweenPand Q whenPis at the points C. [3 marks]

YEAR 2006

4. A particle moves in a straight line and passes through a fixed point O.

Its velocity, v ms 1

, is given by 562 += ttv , where tis the time, in seconds, after leaving O.

[Assume motion to the right is positive.]

(a) Find

(i) the initial velocity of the particle,

(ii) the time interval during which the particle moves towards the left,

(iii) the time interval during which the acceleration of the particle is positive.

[5 marks]

(b) Sketch the velocity-time graph of the motion of the particle for 50 t .

[2 marks]

(c) Calculate the total distance traveled during the first 5 seconds after leaving O.

[3 marks]

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P Q

A C B

28 m

Diagram 9


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ANSWERS(MOTION ALONG A STRAIGHT LINE)

PAPER 1

1. (a) (i) 49 (ii) k = 12

(b)

432 m

2. (a) 18

(b) 1731 m

(c) t = 9

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5 120

24

49

y

x

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(d) t = 6

3. (a) 8

(b) 18

(c) 4

4. (a) (i) v= 5(ii) 1 < t < 5

(iii) t > 3

(b)

(c) 13 m

TOPIC: LINEAR PROGRAMMING

PAPER 2

1. Yahya has an allocation of RM 225 to buyx kg of prawns andy kg of fish. The total mass of the

commodities is not less than 15 kg. The mass of prawns is at most three times that of fish. Theprice of 1 kg of prawns is RM 9 and price of 1 kg of fish is RM 5.

(a) Write down three inequalities, other thanx0 andy0, that satisfy all of the aboveconditions. [3 marks]

(b) Hence, using a scale of 2 cm to 5 kg for axes, construct and shade the region R that satisfies

all the above conditions. [4 marks]

(c) If Yahya buys 10 kg of fish, what is the maximum amount of money that could remain from

his allocation? [3 marks]

2. A district education office intends to organise a course on the teaching of Mathematics and

Science in English.

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t

vy

1 5

5

0

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The course will be attended byx Mathematics participants andy Science participants.

The selection of participants is based on the following constraints:

I : The total number of participants is at least 40.

II : The number of Science participant is at most twice that of Mathematics.

III : The maximum allocation for the course is RM7200. The expenditure for a

Mathematics participant is RM120 and for Science participant is RM80.

(a) Write down three inequalities, other thanx0 andy0, that satisfy all of the aboveconstraints. [3 marks]

(b) Hence, using a scale of 2 cm to 10 participants on axes, construct and shade the region R

which satisfies all the above constraints. [3 marks]

(c) Using your graph from (b), find

(i) the maximum and minimum number of Mathematics participants when the number of

Science participant is 10,

(ii) the minimum cost to run the course. [4 marks]

3. An institution offers two computer courses,Pand Q. The number of participants for coursePis

x and for course Q isy.

The enrolment of the participants is based on the following constraints:

I : The total number of participants is not more than 100.

II : The number of participants for course Q is not more than four times the number of

participants for courseP.

III : The number of participants for course Q must exceed the number of participants for

coursePby at least 5.

(a) Write down three inequalities, other thanx0 andy0, which satisfy all of the aboveconstraints. [3 marks]

(b) By using a scale of 2 cm to 10 participants for axes, construct and shade the region R that

satisfies all the above constraints. [3 marks]

(c) By using your graph from (b), find

(i) the range of the number of participants for course Q if the number of participants for

coursePis 30, [3 marks]

(ii) the maximum total fees per month that can be collected if the fees per month for course P

and Q are RM50 and RM60 respectively. [4 marks]

4. A workshop produces two types of rack,Pand Q.

The production of each type of rack involves two processes, making and painting.

Table below shows the time taken to make and paint a rack of type Pand a rack of type Q.

RackTime taken (minutes)

Making Painting

P 60 30

Q 20 40

The workshop produces x racks of type P and y racks of type Q per day.

The production of the racks per day is based on the following constraints:

I: The maximum total time for making both racks is 720 minutes.

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II: The total time for painting both racks is at least 360 minutes.

III: The ratio of number of racks of type P and type Q is at least 1:3.

(a) Write down three inequalities, other thanx0 andy0, which satisfy all of the aboveconstraints. [3 marks]

(b) Using a scale of 2 cm to 2 racks on axes, construct and shade the region R which satisfies all

the above constraints. [3 marks](c) By using your graph from (b), find

(i) the minimum number of racks of type Q if 7 racks of type P are produced per day,

(ii) the maximum total profit per day if the profit from one rack of type P is RM24 and from

one rack of type Q is RM32. [4 marks]

ANSWERS (LINEAR PROGRAMMING)

Paper 2

1. (a)x +y 15 x 3y 9x + 5y 225

(b)

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R

55

50

45

40

35

30

25

20

15

10

5

10 20 3 40 50 60 70 80 90 100

(c)y =10 x =19 RM 130

2. (a) I:x +y 40 II:y 3x III: 3x + 2y 180

(b)

(c) (i) xminimum= 30 xmaximum= 53

(ii) RM 3760

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90

80

70

60

50

40

30

20

10

20 40 60 80 100 120 140 160

R


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3. (a) I:x +y 100 II:y 4x III: yx +5

(b)

(c) (i) 35 y 70 (ii) Maximum total fees = RM 5800

4. (a) I: 3x +y 36 II: 3x + 4y 36 III: 3xy

(b)

90

80

70

60

50

40

30

20

10

20 4 60 80 100 120 140 160

R

0

18

16

14

1

10

8

6

4

R

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