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KOLEKSI SOALAN SPM

KERTAS 2

NAMA

...........................................................................................

Koleksi Soalan Peperiksaan Sebenar SPM (Matematik Tambahan Kertas 2) Koleksi Soalan Peperiksaan Sebenar SPM (Matematik Tambahan Kertas 2)

http://cikgujep.tumblr.com

TOPIC: QUADRATIC FUNCTIONS

1.SPM 2003 P2 Q2

The function f(x) = x2 -4kx+5k2 +1 has a minimum value of r2 +2k, where k are constants.

(a) By using the method of completing the square, show that r =k -1 [4marks] (b) Hence, or otherwise, find the values of k and r if the graph of the function is symmetrical about

x= r2 -1 [4marks]

[k =3,r = -1]

2. SPM 2008

Diagram below shows the curve of a quadratic function f(x) = -x2 +kx-5. The curve has a maximum point

at B (2,p) and intersect the f(x)- axis at point A.

(a) State the coordinates of A [1 mark]

(b) By using the method of completing the square, find the value of k and of p. [4marks]

(c) Determine the range of values of x, if f(x) 5 [2marks]



TOPIC: SIMULTANEOUS EQUATION

1. SPM 2003 P2 Q1

Solve the simultaneous equations 4x+y = -8 and 2 2x x y [5 marks]

( ans :x= -2, -3 ; y= 0, 4 )

2. SPM 2004 P2 Q1

Solve the simultaneous equations p-m = 2 and 2 2 8p m [5 marks]

(ans: m= 0.606, -6.606 ; p=2.606, -4.606)

3. SPM 2005 P2 Q1

Solve the simultaneous equations 211, 10 2

2x y and y x [5 marks]

(ans: y= -4, 3 ; x =3, -1/2)

4. SPM 2006 P2 Q1

Solve the simultaneous equations 2x+y = 1 and 2 22 5x y xy [5 marks]

(ans: x=1.618, -0.618 ; y =-2.236, 2.236 )

5. SPM 2007 P2 Q1 Solve the following simultaneous equations:

2x-y-3 =0 , 2x2 -10x+y +9 =0 [5marks]

(ans : x= 1, 3 y= -1,3)

6. SPM 2008 P2 Q1 Solve the following simultaneous equations :

x-3y +4 =0 , x2 +xy-40 =0 [5marks]



TOPIC: CIRCULAR MEASURES

1. SPM 2003 P2 Q4

2. SPM 2004 P2 Q9

3.SPM 2005 P2 Q10

Diagram shows the sector POQ, centre O with radius 10

cm. The point R on OP is such that OR:OP=3:5. Calculate

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

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

(ans:0.9274, 22.37]

Diagram 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 and R 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]

.(ans:2/3 ,20.94, 61.40]

Diagram shows a sector POQ of a circle, centre O. The point A

lies on OP, the point B lies on OQ and AB is perpendicular to OQ.

The length of OA= 8 cm and 6

POQ rad

.

It is given that OA:OP= 4:7. (Use =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]

(ans:6, 24.403, 37.46]



4.SPM 2006 P2Q10

5. SPM 2007 P2Q9

Diagram 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 centre A

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 given that AC= 8 cm and

1.956COQ radians .(Use =3.142). Calculate

(a) the area of the lawn [2 marks]

(b) the length of the fence required for fencing the flower bed.

[4 marks]

(c ) the area of the flower bed [4 marks]

(ans:62.592, 38.252, 31.363]

Diagram shows a circle, centre O and radius 10 cm inscribed in a

sector APB of a circle, centre P. The straight lines, AP and BP,

are tangents to the circle at point Q and R, respectively.

[use =3.142]

Calculate

(a) the length, in cm, of the arc AB [5 marks]

(b) the area, in cm2, of the shaded region [5marks]



6.SPM 2008 P2Q9

Diagram below shows two circles. The larger circle has centre X and radius 12cm. The smaller circle has

centre Y and radius 8 cm. The circles touch at point R. The straight line PQ is a common tangent to the

circles at point P and point Q.

[use =3.142]

Given that PXR = radians,

(a) show that =1.37 (to two decimal places) [2marks]

(b) calculate the length, in cm, of the minor arc QR [3marks]

(c) calculate the area, in cm2, of the coloured region. [5marks]



TOPIC: STATISTICS

1. SPM 2004 P2 Q4

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 [15, 22.2] [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, [26]

(ii) the standard deviation of the set of 11 numbers. [5.494] [4marks ]

2.SPM2005 P2 Q4

3.SPM 2006 P2Q 6

Score Number of pupils

10-19 1

20-29 2

30-39 8

40-49 12

50-59 K

60-69 1

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

(a) It is given that the median score of the distribution is 42. Calculate the value of k. (3marks) [4] (b) 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. (4marks)

(c) What is the mode score if the score of each pupil is increased by 5? ( 1mark) [48]

(a) Without using an ogive, calculate the median mark ( 3 marks) [24.07]

(b) Calculate the standard deviation of the distribution. (4marks)

[11.74]



4. SPM 2007P2Q5

Table below shows the cumulative frequency distribution for the scores of 32 students in a competition.

Score < 10 <20 <30 <40 <50

Number of

students

4 10 20 28 32

(a) Based on the table, copy and complete the table below: Score 0-9 10-19 20-29 30-39 40-49

Number of

students

[1 mark]

(b) Without drawing an ogive, find the interquartile range of the distribution. [5marks]

Answers : 18.33

5. SPM 2008 P2Q5

Table below shows the marks obtained by 40 candidates in a test.

Marks Number of candidates

10-19 4

20-29 x

30-39 y

40-49 10

50-59 8

Given that the median mark is 35.5, find the value of a and of y. Hence, state the modal class.

[ 6 marks ]



TOPIC: DIFFERENTIATION

1.SPM 2003 P2 Q3

(a) Given that 2 2dy

xdx

and y=6 when x=-1, find y in terms of x [3marks]

(b) Hence, find the value of x if 2

2

2( 1) 8

d y dyx x y

dx dx [4 marks]

[ y = x2 +2x+7, 3/5, -1]

2.SPM 2003 P2 Q9(a)

Diagram below shows a conical container of diameter 0.6 m and height 0.5m. water is poured into the

container at a constant rate of 0.2 m3s-1.

[ 1.105]

3.SPM 2007 P2Q4

A curve with gradient function 2

22x

x has a turning point at (k,8)

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

(b) Determine whether the turning point is a maximum or a minimum point [2marks]

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

[ 1, (1,8) min point, y = x2 =2/x +5 ]

Calculate the rate of change of the height of the water

level at the instant when the height of gthe water level is

0.4 m

(Use 0.3142 ; Volume of a cone = 21

3r h ]

[4 marks]



TOPIC: SOLUTION OF TRIANGLES

1.SPM 2003 P2Q 15

Diagram below 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

[ANSWERS; 2.700, 3.149, 2.829 ]

2. SPM 2004 P2 Q13

Diagram below shows a quadrilateral ABCD such that ABC is acute.

(b) A triangle A’B’C’ has the same measurements as those given for triangle ABC, that is, A’C’=12.3 cm,

C’B’=9.5cm and ' ' ' 40.5B A C , but which is different in shape to triangle ABC.

(i) etch the triangle A’B’C’

(ii) State the size of A'B'C' [2 marks]

[ANSWERS ; 57.23, 106.07, 80.96, 122.77]

(a) Calculate

(i) ABC

(ii) ADC

(iii) the area, in cm2, of quadrilateral ABCD

[ 8 marks]

Given that VB =VC =2.2 m and AB =AC =2.6m, 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 250.

[3 marks]

(c) the area of triangle VAB [4 marks]



3.SPM 2005 P2Q12

Diagram below shows triangle ABC

[ANSWERS; 19.27, 50.73, 24.89, 290.1 ]

4.SPM 2006 P2 Q 13

Diagram below shows a quadrilateral ABCD.

[ANSWERS ; 60.07, 5.573, 116.55 35.43 ]

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

(b) A quadrilateral ABCD is now formed so that AC is a diagonal,

40ACD and AD =16 cm. Calculate the two possible values of

ADC [2 marks]

( c) By using the acute ADC from (b) , calculate

(i) the length , in cm, of CD (ii) (ii) the area, in cm2, of the quadrilateral ABCD

[6 marks]

The area of triangle BCD is 13 cm2 and BCD is

acute. Calculate

(a) BCD , [2 marks]

(b) the length, in cm, of BD, [ 2 marks]

(c) ,ABD [3 marks]

(d) the area, in cm2, quadrilateral ABCD [3 marks]



5.SPM 2007 P2Q15

[ 13.36, 23.880 , 13.8 ]

6. SPM 2008 P2 Q14

In the diagram below, ABC is a triangle. ADFB, AEC and BGC are straight lines. The straight line FG is

perpendicular to BC.

It is given that BD =19cm, DA =16cm, 080DAE and 045FBG .

(a) Calculate the length , in cm, of (i) DE (ii) EC [5marks]

(b) The area of triangle DAE is twice the area of triangle FBG. Calculate the length , in cm, of BG. [4 marks]

(c) Sketch triangle A’B’C’ which has a different shape from triangle ABC such that A’B’=AB,A’C’=AC and A’B’C’ =ABC. [1mark]

[ANSWERS: 19.344, 16.213, 10.502]

Diagram shows quadrilateral ABCD.

(a) Calculate

(i) the length, in cm, of AC

(ii) ACB [4 marks]

(b) Point A’ lies on AC such that A’B =AB.

(i) Sketch 'A BC

(ii) Calculate the area , in cm2 , of 'A BC

[6marks]



TOPIC: INDEX NUMBER

1. SPM 2003 P2 Q13

Diagram below is a bar chart indicating the weekly cost of the items P,Q,R ,S and T for the year 1990.

Table below shows the prices indices for the items.

Items Price in 1990 Price in 1995 Price index in 1995 based on

1990

P x RM 0.70 175

Q RM2.00 RM2.50 125

R RM4.00 RM5.50 y

S RM6.00 RM9.00 150

T RM2.50 Z 120

(a) Find the value of (i) x (ii) y (iii) z [3 marks]

(b) Calculate the composite index for the 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 RM456. Calculate the corresponding 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. [3marks]

[answer:RM0.40, 137.5,RM3.00, 140.9 RM642.5 ]



2. SPM 2004 P2Q12

Table below shows the price indices and the percentage of usage of four items, P,Q, R and S, which are

the main ingredients in the production of a type of biscuit.

Items 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 S in the year 1993 if its price in the year 1995 is RM 37.70 (ii) the price index of P in 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 lost 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 RM32. [5 marks] [answer: RM29, 162, 125, RM25]

3.SPM 2005 P2Q13

Table below shows the prices and the price indices for the four ingredients, P,Q, R and S, used in making

biscuits of a particular kind.

Ingredients

Price per kg (RM)

Price index for the year 2004 based

on the year 2001

Year 2001 Year 2004

P 0.80 1.00 x

Q 2.00 y 140

R 0.40 0.60 150

S Z 0.40 80

Diagram below show a pie chart which represents the relative amount of the ingredients P,Q,R

and S, used in making these biscuits.



(b) Find the value of x, y and z. (b) (i) Calculate the composite index for the 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 RM2985.

[Answer (a) 125, 2.80, 0.50 (b) 129.44, RM2306.09]

4.SPM 2006 P2 Q 15

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

(a) The index number of ingredient P in the year 2005 based on the year 2004 is 120. Calculate

the value of w. [ 2 marks]

(b) The index number of ingredient R in the year 2005 based on the year 2004 is 125. The price per kilogram of ingredient R in the year 2005 is RM2.00 more than its corresponding price in the year 2004. Calculate the value of x and of y. [3marks]

(c) The composite index for the cost of making the cake in the year 2005 based on the year 2004 is 127.5

Calculate

(i) the price of a cake in the year 2004 if its corresponding price in the year 2005 is RM 30.60

(ii) the value of m if the quantities of ingredients P,Q,R and S used in the ration of 7: 3: m : 2 [5 marks]

[ANSWERS: 6, 8, 10, RM24, 4 ]



5.SPM 2007 P2Q13

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

kind of toy.

component Price (RM) for the year Price index for the

year 2006 based on

the year 2004 2004 2006

P 1.20 1.50 125

Q x 2.20 110

R 4.00 6.00 150

S 3.00 2.70 y

T 2.00 2.80 140

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

(a) Find the value of x and of y [3marks]

(b) Calculate the composite index for the production cost

of the toys in the year 2006 based on the year 2004

[3marks]

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

the year 2008. Given that production cost of one toy in the year 2004 is

RM55, calculate the corresponding cost in the year 2008

[4marks]



6. SPM 2008 P2 Q13

Table below shows the prices and the prices indices of four ingredients, fish, flour, salt and sugar, used

to make a type of fish cracker.

Ingredients

Price (RM) per kg for the year Price index for the

year 2005 based on

the year 2004 2004 2005

Fish 3.00 4.50 150

Flour 1.50 1.80 h

Salt k 0.90 112.5

Sugar 1.40 1.47 105

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

(b) Calculate the composite index for the cost of making

these crackers in the year 2005 based on the year 2004

[3marks]

The composite index for the cost of making these crackers

increases by 50% from the year 2005 to the year 2009.

Calculate

(i) the composite index for the cost of making these crackers in the year 2009 based on the year 2004

(ii) (ii) the price of a box of these crackers in the year 2009 if its corresponding price in the year 2004 is RM25 [4marks]



TOPIC: PROGRESSION

1.SPM 2004 P2 Q6

Diagram below shows the arrangement of the first three of an infinite series of similar triangles. The first

triangle has a base of x cm and a height of y cm. the measurements of the base and height of each

subsequent triangle are half of the measurements of its previous one.

2. SPM 2005 P2 Q 3

The diagram above shows part of an arrangement of bricks of equal size.

The number of bricks in the lowest row is 100. For each of the other rows, the number of bricks is 2 less

than in the row below. The height of each brick is 6 cm.

Ali builds a wall by arranging bricks in this way. The number of bricks in the highest row is 4 , calculate

(a) the height, in cm, of the wall. [3marks]

(b) the total price of the bricks used if the price of one brick is 40 sen. [3marks]

[ 49, 294, 2548, RM1019.20 ]

(a) Show that the area of the triangles form a

geometric progression and state the common

ratio . [3 marks]

(b) Given that x=80 cm and y=40 cm,

(i) determine which triangle has an area of

216

4cm .

(ii) find the sum to infinity of the area, in cm2, of

the triangles. [5 marks]

[ans: ¼, n=5, 2133 1/3 ]



3.SPM 2007 P2 Q 6

Answer : 415, 14000, 495

4. SPM 2008 P2Q3

Muthu started working for a company on 1 January 2002 with an initial annual salary of RM18,000.

Every January, the company increased his salary by 5% of the previous year’s salary.

Calculate

(a) his annual salary, to the nearest RM, for the year 2007 [3marks]

(b) the minimum value of n such that his annual salary in the nth year will exceed RM36,000

[2marks]

(c) the total salary, to the nearest RM, paid to him by the company, for the years 2002 to 2007.

[2marks]

Diagram shows the side elevation of

part of stairs built of cement blocks.

The thickness of each block is 15 cm.

The length of the first block is 985 cm.

The length of each subsequent block is

30 cm less than the preceding block as

shown in the diagram.

(a) If the height of the stairs to be built is 3 m, calculate (i) the length of the top most block (ii) the total length of the blocks. [5marks]

(b) calculate the maximum height of the stairs [3marks]



TOPIC: LINEAR LAW

1. SPM2006 P2Q7.

Table below shows the values of two variables , x and y obtained from an experiment. Variables x and y

are related by the equation y=pkx+1, where p and k are constants.

x 1 2 3 4 5 6

y 4.0 5.7 8.7 13.2 20.0 28.8

(a) Plot lg y against (x+1) , using a scale of 2 cm to 1 unit on the (x+1) axis and 2 cm to 0.2 unit on the log

y-axis. Hence, draw the line of best fit. [5marks]

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

(i) p and k { p=1.778, k=1.483} [5marks]

2. SPM 2005 P2 Q7


are related by the equationr

y pxpx

, where p and r are constants.

x 1 2 3 4 5 5.5

y 5.5 4.7 5.0 6.5 7.7 8.4

(a) Plot xy against x2, by using a scale of 2 cxm to 5 units on both axes. Hence, draw the line of best fit.

[5marks]

(b) Use the graph from (a) to find the value of p and r {ans:p=1.37,r=5.48} [5marks]

3.SPM 2004 P2 Q7


are related by the equation y=pkX, where p and k are constants.

x 2 4 6 8 10 12

y 3.16 5.50 9.12 16.22 28.84 46.77

(a) Plot log10 y against x by using a scale of 2 cm to 2 units on the x-axis and 2 cm to 0.2 unit on the log10 y axis.

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

(b) Use your graph from part(a) to find the value of (i) p (ii) k {ans:p=1.820,k=1.309} [6marks]



4.SPM 2003 P2Q7


are related by the equation 2xy pk , where p and k are 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

(a) Plot log10 y against x2 Hence draw the line of best fit. [5marks]

(b) Use the graph in (a) to find the value of (i) p (ii)k { ans:p=1.259, k=1.109} [5marks]

5.SPM 2007 P2Q7

Use Graph paper to answer this question.

Table below shows the values of two variables, x and y, obtained from an experiment.

Variables x and y are related by the equation 22p

y kx xk

, where p and k are constants.

x 2 3 4 5 6 7

y 8 13.2 20 27.5 36.6 45.5

(a) Plot y

xagainst x, using a scale of 2cm to 1 unit on both axes. Hence, draw the line of best

fit. [4marks] (b) Use your graph in part (a) to find the value of

(i) p, (ii) k, (iii) y when x=1.2 [6marks]

Answers : 0.754, 0.26, 4.2



6.SPM 2008 P2 Q8

Table below shows the values of two variables, x and y , obtained from an experiment. Variables x and y

are related by the equation y = hk2x , where h and k are constant

x 1.5 3.0 4.5 6.0 7.5 9.0

y 2.51 3.24 4.37 5.75 7.76 10.00

(a) Based on the table, construct a table for the values of log 10y. [1 mark]

(b) plot log10 y against x, using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 0.1 unit on the log10 y

axis.

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

(a) Use the graph in part (b) to find the value of (i) x when y = 4.8 (ii) h. (iii) k [5 marks]

Answers : 4.95, 1.905, 1.096



TOPIC: INTEGRATION

1.SPM 2003 P2Q9b

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

2.SPM 2004 P2 Q10b

Diagram below shows part of the curve 2

3

(2 1)y

x

.

Calculate the volume generated when the shaded

region is revolved 360 about the y-axis

[6 marks]

[ 38pi/15]

A region is bounded by the curve, the x-axis and the straight line x=2

and x=3,

(i) Find the area of the region

(ii) The region is revolved through 360 about the x-axis, find the

volume generated, in terms of .

[6 marks]

[ 1/5]



3.SPM 2005 P2Q8

In the diagram below, the straight line PQ, is normal to the curve y=2

12

x at A(2,3). The straight line

AR is parallel to the y-axis.

4.SPM 2004 P2Q5

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

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

(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]

[ y = 3x2 -6x-10, ( 2, -10) max point]

Find

(a) the value of k [3 marks]

(b) the area of the shaded region.

[4marks]

(c) the volume generated, in terms of , when the region

bounded by the curve , the y-axis and the straight line y=3

is revolved through 360 about the y-axis.

[3 marks]

[ 8, 12 1/3, 4pi ]



5.SPM 2007 P2 Q10

[ 2, 8, 4pi/49]

6. SPM 2008 P2 Q 7

Diagram below shows the curve y = x2 +5 and the tangent to the curve at the point A(1,6)

Calculate

(a) the equation of the tangent at A. [3marks]

(b) the area of the coloured region, [4marks]

(c) the volume of revolution, in terms of , when the region bounded by the curve and the straight line

y= 7 is rotated through 3600 about the y-axis. [3marks]

Diagram shows part of the curve y=k(x-1)3, where k is a constant.

The curve intersects the straight line x=3 at point A. At point A,dy

dx

=24.

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

(b) Hence, calculate

(i) the area of the shaded region P

(ii) the volume generated, in terms of , when the region R is

bounded by the curve, the x-axis and the y-axis, is revolved through

360 about the x-axis. [7marks]



TOPIC: VECTOR

1 SPM 2003 P2 Q 6

Given that 5

7AB

,2

3Ob

and 5

kCD

, find

(a) the coordinate of A [2 marks]

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

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

(ans:( -3,-4), 31

45

, 25

7)

2 SPM 2004 P2 Q 8

(ans : -2y+6x, 3y/2 +9x/2, h(6x-2y), k(9x/2 +3y/2, k=1/3, h=1/2)

3. SPM 2005 P2 Q 6

(ans : -20x+32y, 25x, 104 )

Diagram shows triangle OAB. The straight line AP

intersects the straight line OQ at R. It is given that OP=

1 1, , 6

3 4OB AQ AB OP x and 2OA y

(a) Express in terms of x and y

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

(b) (i) Given that ,AR hAP state AR in terms of h, x and y

(ii) Given that ,RQ k OQ state RQ in terms of k, x and y [2marks]

(c) Using AR and RQ from (b) , find the value of h and of k. [4marks]

Diagram shows a quadrilateral ABCD. AED and EFC are straight lines.

It is given that

120 , 8 , 25 24 ,

4AB x AE y DC x y AE AD and

3

5EF EC

(a) Express in terms of x and y

(i) ( )BD ii EC [3marks]

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

(C) if 2 3x and y , find BD [2 marks]



4. SPM 2006 P2 Q 5

5. SPM 2007 P2Q8

[(a)(i) yxBP 62 (ii) yxOQ 34 (b) 54

52 , kh (c) 24.08 unit]

Diagram shows a trapezium ABCD.

It is given that 2

2 , 6 ,3

AB y AD x AE AD and

5.

6BC AD

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

(b) Point F lies inside the trapezium ABCD such that 2 ,EF mAB and m is a constant.

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

(ii) Hence, if the points A,F and C are collinear, find the value of m [5marks]

(ans : 5x+2y, 4x+my, m=8/5]

Diagram shows triangle AOB. The point P lies on OA and the point Q

lies on AB. The straight line BP intersects the straight line OQ at the

point S.

It is given that

OA: OP = 4:1, AB : AQ = 2 : 1, 8 , 6OA x OB y

(a) Express in terms of x and y :

(i) BP

(ii)OQ [3marks]

(b) Using OS hOQ and BS k BP , where h and k are constants, find the value of h and of k.

[5marks]

(c ) Given that 2 ,x units y =3 units and AOB =90 , find AB [2marks]



6. SPM 2008 P2 Q6

Diagram below shows a quadrilateral ABCD. The diagonals BD and AC intersect at point R. Point P lies on

AD.

It is given that 1 1

, ,3 3

AP AD BR BD AB x and AP y

(a) Express in terms of x and y :

(i) DB (ii) AR [3 marks]

(b) Given that DC kx y and AR hAC , where h and k are constants, find the value of h and of k.

[4 marks]

[Answer : (a)(i) yxDB 3 (ii) yxAR 32 (b)

34

21 , kh ]



TOPIC: TRIGONOMETRIC FUNCTIONS

1.SPM 2004 P2Q3

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

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

the equation 2 sin2 x = 2-180

xfor 0 180x (ans:2) [3 marks]

2 SPM 2005 P2 Q5

(a) Prove that 2 2 2( sec 2sin cot ) cos2co x x x x [2 marks]

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

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

equation 2 2 23( sec 2sin cot ) 1x

co x x x

for 0 2x . State the number of solutions.

(ans:4) [6 marks]

3 SPM 2006 P2 Q6

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

(b) hence, using the same axes, sketch a suitable graph to find the number of solutions to the equation

2cos 0xx

for 0 2x . State the number of solutions. [3 marks]

(ans:2)

4 SPM 2007 P2 Q3

(a) Sketch the graph of y = 3cos2 0 2x for x . [4marks]

(b) Hence, using the same axes, sketch a suitable straight line to find the number of solutions for the

equation 2 3cos 2 0 22

xx for x

. State the number of solutions. [3marks]

5 SPM 2008 P2 Q4b

(a) (i)Sketch the graph of y = -tan 2x for o x

(ii) hence, using the same axes, sketch a suitable straight line to find the number of solutions for the

equation 2

3 2 tan

2 sec

x x

x

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



TOPIC: PROBABILITY DISTRIBUTION

1.SPM 2003 P2 Q10

(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] The mass of the workers in a factory is normally distributed with a mean of 67.86 kg and a variance of 42.25kg2. 200 of the workers in the factory weigh between 50 kg and 70 kg. Find the total number of worker in the factory. [0.4233,800,0.62591, 319]

2.SPM 2004 P2 Q11

(a) A club organizes a practice sessions for trainees on scoring goals from penalty kicks. Each trainee takes 8 penalty kicks. The probability that a trainee scores a goal from a penalty kick is p. After the session, it is found that the mean number of goals for a trainee is 4.8.

(i) find the value of p (ii) If the trainee is chosen at random, find the probability that he scores at least one

goal. [5 marks] (b) A survey on the 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 of m.

[5 marks] [ 0.9993, 0.2743,67.625]

3. SPM 2005 P2 Q11

(a) The result of a study shows that 20% of the 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 [4marks]

(b) The mass of water melons produced from an orchard follows a normal distribution with a mean of

3.2kg and a standard deviation of 0.5 kg. find

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

4.0 kg

(ii) the value of m if 60% of the water melons from the orchard have a mass more than m kg [6 m]

[ 0.2936, 0.79691, 0.9452,3.0735,]



4 SPM 2006 P2 Q 11

An orchard produces lemons. Only lemons with diameter, x greater than k cm are graded and marketed.

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

Grade A B C

Diameter, x(cm) X > 7 7 > x > 5 5 > x > k

It is given that the diameter of the 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 grade A

[2marks]

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

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

[ 0.2119, 0.4912, 4.1965]

5. SPM 2007 P2Q11

(a) In a survey carried out in a school, it is found that 2 out of 5 students have handphones. If 8 students

from that school are chosen at random, calculate the probability that

(i) exactly 2 students have handphones

(ii) more than 2 students have handphones [5marks]

(b) A group of workers are given medical check up. The blood pressure of a worker has a normal

distribution with a mean of 130 mmHg and a standard deviation of 16mmHg. Blood pressure that is

more than 150 mmHg is classify as “ high blood pressure”

(i) a worker is chosen at random from the group. Find the probability that the worker has a blood

pressure between 114mm Hg and 150mmHg

(ii) It is found that 132 workers have “ high blood pressure”. Find the total number of workers in the

group. [5marks]

[0.6846, 0.7357, 0.1056,1250]



6.SPM 2008 P2 Q11

The masses of mangoes from an orchard has a normal distribution with a mean of 300 g and a standard

deviation of 80 g.

(a) Find the probability that a mango chosen randomly from this orchard has a mass of more than 168 g

[3marks]

(b) A random sample of 500 mangoes is chosen.

(i) calculate the number of mangoes from this sample that have a mass of more than 168 g

(ii) Given that 435 mangoes from this sample have a mass of more than m g, find the value of m

[7marks]

[0.95053, 475/476, 209.84]



TOPIC: LINEAR PROGRAMMING

1.SPM 2003 P2 Q 14

Use graph paper to answer this question.

Yahya has an allocation of RM 225 to buy x kg of prawns and y 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. The price of 1

kg of prawns is RM9 and the price of 1 kg of fish is RM5.

(a) Write down three inequalities, other than x 0 and y 0, that satisfy all the above conditions.

[3 marks]

(b) Hence, using a scale of 2 cm to 5 kg for both 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]

Answers: 15, 3 ,9 5 225, 130x y x y x y RM

2.SPM 2004 P2 Q14

Use graph paper to answer this question.

A district education office intends to organize a course on the teaching of Mathematics and Science in

English.The course will be attended by x Mathematics participants and y 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 participants is at most twice that of Mathematics.

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

Mathematics participant is RM 120, and for a science participant is RM80.

(a) Write down three inequalities, other than x 0 and y 0, which satisfy the above constraints. [3marks]

(b) Hence, by using a scale of 2 cm to 10 participants on both axes, construct and shade

the region R which satisfies all the above constraints. [3marks]

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



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

number of Science participants is 10,

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

Answer : 40, 2 ,120 80 7200,3 2 180x y y x x y x y RM3760

3. SPM 2005 P2 Q 14

An institution offers two computer courses, P and Q. The number of participants for the course P is x

and for course Q is y.

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 4 times the number of

participants for course P

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

Participants for course P by at least 5

(a) Write down three inequalities, other than x 0 and y 0, which satisfy all the above constraints. [3 marks]

(b) By using a scale of 2 cm to 10 participants on both axes, construct the shade the region R that satisfy all the above constraints. [3marks]

(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 course P is 30. (ii) The maximum total fees per month that can be collected if the fees per month

for courses P and Q are RM50 and RM 60 respectively. [4marks]

Answers 100, 4 , 5, 5800x y y x y x RM



4.SPM 2006 P2 Q14

A workshop produces two types of rack, P and 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 P

and a rack of type Q.

Rack Time 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 racks per day

is based on the following constraints:

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

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

III : The ratio of the number of racks of type P to the number of racks of type Q is at

least 1:3

(a) Write three inequalities, other than x 0 and y 0, which satisfy all the above constraints.

(b) Using a scale of 2 cm to 2 racks on both axes, construct and shaded the region R are satisfies all the above constraints.

(c) By using your graph from part (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 RM32.00

Answers

60 20 720,3 36,30 40 360,3 4 36

, 720

x y x y x y x y

xy RM

y



5.SPM 2007 P2Q14

Use graph paper to answer this question

A factory produces two components, P and Q. In a particular day, the factory produced x pieces of

component P and y pieces of component Q. The profit from the sales of a piece of component P is RM15

and a piece of component Q is RM12.

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

I : The total number of components produced is at most 500.

II : The number of component P produced is not more than three times the number

of component Q

III : The minimum total profit for both components is RM4200

(a) Write three inequalities, other than x 0 and y 0, which satisfy all the constraints. [3marks] (b) Using a scale of 2 cm to 50 components on both axes, construct and shade the region

R which satisfy all the above constraints. [3marks]

(c ) Use your graph in part (b) to find

(i) the minimum number of pieces of component Q if the number of pieces of

component P produced on a particular day is 100

(ii) the maximum total profit per day

[4marks]

[ ]7125,225,42001215,3,500 RMyxyxyx



6. SPM 2008 P2 Q15

Use graph paper for this question

The members of a youth association plan to organize a picnic. They agree to rent x buses and y vans. The

rental of a bus is RM 800 and the rental of a van is RM300. The rental of the vehicles for the picnics is

based on the following constraints:

I : The total number of vehicles to be rented is not more than 8

II: the number of buses is at most twice the number of vans

III: the maximum allocation for the rental of the vehicles is RM4000

(a) Write three inequalities, other than x ≥0 and y ≥ 0, which satisfy all the above constraints

[ 3 marks]

(b) Using a scale of 2 cm to 1 vehicle on both axes, construct and shade the region R which satisfy all the

above constraints. [3 marks]

(c) Use the graph constructed in part (a) , to find

(i) the minimum number of vans rented if 3 buses are rented

(ii) the maximum number of members that can be accommodated into the rented vehicles if a bus can

accommodate 48 passengers and a van can accommodate 12 passengers.

[4 marks]

Answers: ,4000300800,2,8 yxyxyx 2, (4,2) 216]

koleksi-soalan-spm-paper-2

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