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Koleksisoalanspmadditionalmathemat
icsadditionalkoleksisoalanspmadditio
nalmathematicsadditionalmathematic
sadditioadditionalmathematicsadditio
nalmathematicsadditionalmathematic
sadditioadditionalmathematicsadditio
nalmathematicsadditionalmathematic
sadditioadditionalKoleksisoalanspmad
ditionalmathematicsadditionalkoleksi
soalanspmadditionalmathematicsadd
Koleksisoalanspmadditionalmathemat
Koleksisoalanspmadditionalmathemat
icsadditionalkoleksisoalanspmadditio
nalmathematicsKoleksisoalanspmaddi
tionalmathematicsadditionalkoleksiso
alanspmadditionalmathematicsadditio
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KOLEKSI SOALAN SPM
KERTAS 2
NAMA
...........................................................................................
Koleksi Soalan Peperiksaan Sebenar SPM (Matematik Tambahan Kertas 2)
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]
Koleksi Soalan Peperiksaan Sebenar SPM (Matematik Tambahan Kertas 2)
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]
Koleksi Soalan Peperiksaan Sebenar SPM (Matematik Tambahan Kertas 2)
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]
Koleksi Soalan Peperiksaan Sebenar SPM (Matematik Tambahan Kertas 2)
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]
Koleksi Soalan Peperiksaan Sebenar SPM (Matematik Tambahan Kertas 2)
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]
Koleksi Soalan Peperiksaan Sebenar SPM (Matematik Tambahan Kertas 2)
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]
Koleksi Soalan Peperiksaan Sebenar SPM (Matematik Tambahan Kertas 2)
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 ]
Koleksi Soalan Peperiksaan Sebenar SPM (Matematik Tambahan Kertas 2)
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]
Koleksi Soalan Peperiksaan Sebenar SPM (Matematik Tambahan Kertas 2)
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]
Koleksi Soalan Peperiksaan Sebenar SPM (Matematik Tambahan Kertas 2)
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]
Koleksi Soalan Peperiksaan Sebenar SPM (Matematik Tambahan Kertas 2)
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]
Koleksi Soalan Peperiksaan Sebenar SPM (Matematik Tambahan Kertas 2)
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 ]
Koleksi Soalan Peperiksaan Sebenar SPM (Matematik Tambahan Kertas 2)
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.
Koleksi Soalan Peperiksaan Sebenar SPM (Matematik Tambahan Kertas 2)
(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 ]
Koleksi Soalan Peperiksaan Sebenar SPM (Matematik Tambahan Kertas 2)
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]
Koleksi Soalan Peperiksaan Sebenar SPM (Matematik Tambahan Kertas 2)
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]