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4/22/12 SOALAN RAMALAN SPMSOALAN RAMALAN SPM 2010CIKGU VEN, DITAIP OLEH

11

SPM MEGA WORKSHOP2011ADDITIONAL

MATHEMATICS

REPRESENTED BYMR VENKATES

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4/22/12 SOALAN RAMALAN SPM

Solve the following simultaneous equations :

2x + y = 1y - x + xy = 11

Solve the following simultaneous equations, giveyour answer correct to three decimal places.

2 + 1 = x + 3y = 5x y

SOALAN RAMALAN SPM 22

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Solve the following simultaneous equations andgive your answer correct to three decimal places:

2x + y = 3xy 2y + 5 = 0

Solve the following simultaneous equations2m n + 1 = 0 and m + 2mn 4n = 0, giveyour answer correct to three decimal places.


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Solve the following simultaneous equations andgive your answer correct to three decimal places:

p + 2q = 1p 3p + 2q = 3


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a. Sketch the graph

y = tanx for 0 x 2

b. Hence, on the same axes, sketch a suitablestraight line to find the number of solutionsto the equation:

tanx -- x = 1 for 0 x 22

State the number of solutions :


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a. Sketch the graph

y = sin 2x + 2 for 0 x

b. Hence, using the same axes, sketch a suitable

straight line to find the number of solutionsfor the equation:

1 sin2x = x for 0 x

2 2


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a. Sketch the graph

y = 3 + sin 2x for 0 x 2

b. Hence, on the same axes, sketch a suitable

straight line graph to find the number ofsolutions for the equation:

x = sin2x for 0 x 2

2

State the number of solutions :


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a. Sketch the graph

y = sin 2x for 0 x 2

b. Hence, using the same axes, draw a suitable

straight line to find the number of solutionsto the equation:

2

sin 2x =x for 0 x 2

State the number of equations


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a. Prove that cosecx cos x cosecx = sinx.

b. Sketch the graph of y = sin 2 x for 0 x 2

3

c. Hence, using the same axes, sketch a suitablestraight line to find the number of solutions forthe equation

cosecx cos x cosecx 3x = 0 for 0 x 25



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a. Sketch the graph

y = -2 sin 2x for 0 x 2

b. Hence, using the same axes, sketch a suitable

straight line to find the number of solutionsfor the equation:

x + 4

sin 2x = 0for 0 x 2



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a. Prove that 2 = sin 2x

tanx + cotx

b. Sketch the graph of y = sin 2x for 0 x

c. Hence, using the same axes, sketch a suitablestraight line to find the number of solutionsfor the equation .

4 + x = 0 for 0 x tanx + cotx 2


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a. Sketch the graph of

y = 2 - sin 2x for 0 x 2

c. Hence, using the same axes, sketch a suitable

straight line to find the number of solutionsfor the equation

x = sin 2x for 0 x 2

2

State the number of solutions

SOALAN RAMALAN SPM2011 CIKGU VEN,DITAIP OLEH SHAH

1212

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SLOT 3

GRADIENT FUNCTIONS


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Given that a curve has a gradient function

hx - 9, such that h is a constant. The tangentto the curve at the point (1, 2) is parallel thestraight line2y 6x = 1

Find :

a.) the value ofh

b.) the equation of the curve.

SOALAN RAMALAN SPM 2011 CIKGU VEN,1414

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A curve has a gradient function ax - 2x, where

a is a constant. The tangent to the curve at thepoint (1,3) is perpendicular to the straight line2y = -x+ 3.

Find :

a.) the value ofa

b.) the equation of the curve.


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a.) Given that dy = 2x + 2 and y = 5 whenx = 2, dx

Findy in term ofx

b.) Hence, find the value ofx if

x dy + (x 1) dy + y + 2 = 0dx dx


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A curve has a gradient function hx 4, whereh is a constant. The equation of normal to the

curve at the point (1, -3) is2y = -x 5.

Find :

a.) the value ofh.

b.) the equation of the curve


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a.) Giveny = x - 3x + 5.Expressy dy + dy + 6in terms inx

dx dx


y dy + dy + 6 = 0dx dx


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A curve has a gradient function 3px + 5x, where h is aconstant. The tangent to the curve at the point ( 2, 1 ) is

parallel to the straight line 2y +x 5 = 0.

Find :

a.) the value ofp.

b.) the equation of the curve

SOALAN RAMALAN SPM

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a.) Given that dy/dx = 2x + 2 andy = 6 whenx = -1,Findy in terms ofx.


x dy + (x 1 ) dy + y = 8dx dx


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SLOT 4

STATISTISOALAN RAMALAN SPM 2121

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Given that setp = (x1 ,x 2,x3,x4,x5,x6,x7). The sum of thenumbers is 140 and the sum of the squares of the numbers is 3080.

a. Find the mean and the variance for the 7 numbers.

b. When kis added to setp, the mean increased by 2.

Find :

i. The value ofk

i. The standard deviation of the set of 8 numbers


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Table 1 shows the frequency distribution of the masses ofa group of students.

Mass (kg) frequency

41 45 18

46 50 20

51 55 40

56 60 26

61 - 65 12

66 70 10

a.) Calculate the first quartile mass ofdistribution.

b.) use the graph paper to answer thisquestion.

Using a scale 0f 2 cm to 5 kg on thehorizontal axis and 2cm to 5 students on the

vertical axis, draw a histogram to representthe frequency distribution of the masses.

Find the mode score.

SOALAN RAMALAN SPM 2011 CIKGU VEN,2323

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Table 5 shows the frequency distribution of the age of agroup of tourists who visited a National Museum.

Age frequency

5 - 9 3

10 - 14 6

15 - 19 8

20 - 24 15

25 - 29 m

30 - 34 1

a.) It is given that the first quartile ageof the distribution is 15.125. calculatethe value ofm

b.) By using a scale 0f 1 cm to 5 unitson thex-axis and 1 cm to 2 units on the

y-axis, draw the histogram, Hence, find

the modal age


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A set of fifty numbers, x1 ,x 2,x3 x50, has a mean of 11and a standard deviation 0f 8,

a. Find

i.) x

ii.) x

b. If each of the numbers is multiplied by 1, 8 and

then increased by 5, find the new value for the.

i.) Mean ii.) Variance


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SLOT 5

SECTION B

INTEGRATIONSOALAN RAMALAN 2626

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Diagram 1 shows the curvey = x + 2 and the straight liney = -x + 8.

Find :a.) the value ofk

b.) the are of the shaded region

c.) the volume generated in terms of , when the

region bounded by the curve. They-axis andy = 8is revolved 360 about they - axis


Diagram 2 shows the curve y = x + 2 the straight line

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Diagram 2 shows the curvey = x + 2. the straight line2

QR is the normal to the curve at ( 4, 10 )

Find :a.) the equation ofQR


c.) the volume of revolution, in terms of , when theregion bounded by the curve, the straight linex = 4and they-axis is rotated through 360 about the x -axis

SOALAN RAMALAN SPM2011 CIKGU VEN,

2828

Diagram 3 shows the curve y = x + 3 intersects the straight

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Diagram 3 shows the curvey = x + 3. intersects the straightlineACat pointB.

It is given that the equation of straight lineACisy +x = 9and the gradient of the curve at pointB is 4

Find :

a.) the value of k,

b.) the are of the shaded regionP

about the x - axisc.) the volume of revolution, in terms of , when theshaded region Q is rotated through 360 x-axis

SOALAN RAMALAN SPM2929

Di 4 th t i ht li PQ i t t t th

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Diagram 4, the straight linePQ is tangent to the curvey =x + 3 at T ( 3, 12 ).

Find :a.) the value ofk


c.) the volume generated in terms of , when theregion bounded by the curve. They-axis and thestraight liney = 12 is revolved 360 about they -axis

SOALAN RAMALAN SPM2011 CIKGU VEN,

3030

Diagram 5 shows part of the curve y = 4 x which passes

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Diagram 5 shows part of the curvey = 4 xwhich passesthrough pointA (1, 3).

Find :

a.) find the equation of normal to the curve at the pointA

b.) find the are of the shaded region

c.) the shaded region is revolved 360 about they - axis


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SLOT 6

VECTORSSOALAN RAMALAN SPM 3232

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Diagram 1 shows the vectorsAB = 5x , AD = 4y andDC = x

a.) Express in terms ofx andy :I. AC

II. BD

NEXT QUESTION WILL BE IN NEXT S


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Diagram 1 shows the vectorsAB = 5x , AD = 4y andDC = x

b.) GivenAP = h ACandBP = k BD, where h and kare constants.ExpressAP,

i.) In terms ofh, x andy ,

ii.) In terms ofk,x andy,

c.) Find the value ofh and ofk,


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Diagram 3 shows a triangle OAB. The straight line OCintersects thestraight lineAD atE.

a.) It is given that OA = a, OB = b, AC = 2CB and

OD = OB

Express in terms ofa and b :

i.) OC

ii.) AD

SOALAN RAMALAN SPM

Di 3 h t i l OAB Th t i ht li OC i t t th

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Diagram 3 shows a triangle OAB. The straight line OCintersects thestraight lineAD atE

b.) It is given thatEC = m OCand AE = n AD

i.) Express ECin terms ofm, a and b

ii.) Express AEin terms ofn, a and b

c.) Using ECand AEfrom (b), find the value ofm and ofn


Di 4 h t ll l li AB d DC

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Diagram 4 shows two parallel linesAB andDC.

It is given thatAD = 4x, CD = 9y, AB = 1/3 DC and BQDIs a straight line such that BQ : BD = 1 : 4

a.) Expressin terms ofx, and/ory

i. AC

i. BD

b.) show that the pointsA, Q, and Care collinear.


In diagram 5, ABCD is a quadrilateral. BFC and DEF are straight line.

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In diagram 5,ABCD is a quadrilateral. BFCand DEFare straight line.

Given thatBA = 24x, BF = 10y, CD = 30 x 30 y, BF = 1/4 BCandDE =2/5DF,

a.) Express in terms ofx and/ory :

i.) OC ii.) AD

b.) show that the pointsA, Eand Care collinear.


Diagram 6 shows a triangle OPQ. The point R lies on OP and the point

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Diagram 6 shows a triangle OPQ. The pointR lies on OPand the pointSlies on the PQ. The straight line QR intersects the straight line OSat

point T

Given OP: OR = 4 : 3,PQ : PS = 2 : 1, OP = 12xand OQ = 9y


i. QR

i. OS

b.) ifOT = h OS and QT = k QR, where h and k

are constants, find the values ofh and k.

c.) Given that x = 3 units, y = 5 units and POQ = 90, find PQ ,


Diagram 7 shows a triangle OAB. The straight line OQ intersects

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g g g QThe straight lineAB at the pointP.

It is given OA : 2 x , OB = 3 y and

OB : AQ = 1 : 2


i. AB

i. OQ

b.) UsingAP = k AB and OP = h OQ,where h and kare constants, find the values of

h and k.

c.) Using the value ofh and kin (b),find the k AB + h OQ.


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SLOT 7

COORDINATEGEOMETRY


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Solution to this question by scale drawing will not be accepted.

In diagram 1,J,KandL lie on the straight line

x 2y + 10 = 0.

a.) Find

I. The coordinates ofJII. The equation of a straight line which passes throughKand

perpendicular toJL.


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b.) Given that the coordinates ofL is ( 14, 12) and the ratioJK : KL = m : n,Find the value of m and n .

c.) A pointPmoves such that its distance from the origin, O , isalways twice its distance from pointJ, find the equation of the locusofP.


Solution by scale drawing is not accepted.

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Diagram 2 shows a straight line y = 2x + 10 that isperpendicular to the straight lineAB and divides it in the ratioAB : BC = 1 :3.

a.) Find the equation of the straight lineThe coordinates ofABC.b.) calculate the coordinates of point B.

c.) calculate the coordinates of pointA.

d.)A point Wmoves such that its distance from pointA isalways 3 units. Find the equation of the locus ofW



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Diagram 3 shows a triangle RST, the equation of the straight lineRQT isy 3x + 4 = 0.

a.) Find :

i. the coordinates ofR

ii. The coordinates ofQ ifRQ : QT = 3 : 2b.) Given that the area of QST is 14 units,find the positive value ofk.

c.) Find the equation of the straight line RS

d.)A point P ( x, y ) moves such that PS = 2PQ. Find the equation ofthe locus for P


Solution by scale drawing is not accepted. SOALAN RAMALAN SPM2011 CIKGU VEN

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Diagram 4 shows a triangleofABC and O is the origin.

a.) Calculate the midpoint ofAC.Hence, find the equation of the perpendicular Bisector ofAC

b.) Find the equation of the straight lineAB

c.) Find the intersection point of straight lineAB and the perpendicular bisector ofAC

d.) A point P moves such that its distance from point C is always 4units.

2011 CIKGU VEN,DITAIP OLEH SHAH

4646


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y g p

In the diagram 5, the straight lineAB has an equationy/9+ x/3 = 1.AB intersects thex-axis and they-axis at pointA

and B respectively.

Point P lies onAB such thatAP : AB = 2 : 3

Find

a.) The coordinates ofP

b.) The equation of theperpendicular bisector ofAP.


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SLOT 8

SECTION C

SOLUTION OFTRIANGLE


Diagram 1 shows a quadrilateral ABCD

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Diagram 1 shows a quadrilateralABCD.

The area of triangle QRS is 11.4 cm and QRS isacute.

Calculate

a.) QRS.

b.) the length, in cm, ofQS,

c.) PQS, SOALAN RAMALAN SPM 4949

Diagram 2 shows a quadrilateral PQRS

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Diagram 2 shows a quadrilateralPQRS.

Calculate

a.) The length ofQS.

b.) QPS,

c.) the area of quadrilateral PQRS,

d.) the perpendicular distance from Q to PS.SOALAN RAMALAN SPM 5050

Diagram 3 shows a quadrilateral ABCD

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Diagram 3 shows a quadrilateralABCD.

Calculate

a.) The length, in cm, ofAC.

b.) ACB,

c.) ADC,

d.) the area, in cm, of the quadrilateralABCD.SOALAN RAMALAN SPM 5151

Diagram 4 shows two triangle ABC and ACD. ABC is obtuse

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Diagram 4 shows two triangleABCandACD. ABCis obtuseand sin ABC = 3/5. Given thatAB = 7 cm,

BC = 5 cm and CD = 9 cm

Calculate

a.) The length, in cm, ofAC.

b.) ADC,

c.) the area, in cm, of theABCD.


Diagram 5 shows a triangle PQR. The point T lies on the straight

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Diagram 5 shows a trianglePQR. The point Tlies on the straightlinePR. The area of triangle QTR is 17.5 cm and QTR is obtuse.

Calculate

a.) QTR.

b.) the length, in cm, ofQR,

c.) QPT

d.) the area, in cm, of triangle PQR.SOALAN RAMALAN SPM 5353

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SLOT 9

INDEX

NUMBERSOALAN RAMALAN SPM 5454

Table 1 shows the prices and the price indices for the four ingredients J

SOALAN RAMALAN SPM2010 CIKGU VEN

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Table 1 shows the prices and the price indices for the four ingredients,J,K, L and M, used in making particular kind of cakes. Diagram 1 is barchart which represents the relative amount of the ingredientsJ, K, L andM, used in making these cakes.

Ingredients Prices per kg (RM) for

the years

2003

2007

Prices index for the year

2007 based on the year2004

J 0.400.60

150

K a

0.40

80

L 2.00

b

140

M 0.80

1.00

c


5555

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a.) find the values ofa, ofb and ofc.

b.) i. Calculate the composite index for the cost of

making these cakes in the year 2007 based on the year2003.

ii. Hence, calculate the corresponding cost of baking

these cakes in the year 2003 if the cost in the year 2007was RM 3125

c.) the cost of baking these cakes is expected to increaseby 40% from the year 2007 to the tear 2008. find the

expected composite index for the year 2008 based on theyear 2003

SOALAN RAMALAN SPM2010 CIKGU VEN,DITAIP OLEH SHAH5656

Table 2 shows the prices and the price indices for the four ingredients A

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Table 2 shows the prices and the price indices for the four ingredients,A,B, C and D, used in baking particular kind of cakes. Diagram 2 is barchart which represents the relative amount of the ingredientsA, B, C andD, used in baking these cakes.

Ingredients Prices per kg (RM) for

the years

2005

2007

Prices index for the

year 2007 based on theyear 2004

A X

8.00

160

B 2.00

2.50

125

C 6.00

y

115

D 7.00

8.40

z


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a.) find the values ofx, ofyand ofz.

b.) i. Calculate the composite index for the cost of

baking these cakes in the year 2007 based on the year2005.

ii. Hence, calculate the corresponding cost of making

these cakes in the year 2005 if the cost in the year 2007was RM 4600

c.) the cost of baking these cakes is expected to increaseby 40% from the year 2005 to the tear 2008. find the

expected composite index for the year 2008 based onthe year 2007


Table 3 shows the prices and the price indices for the four ingredients P

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Table 3 shows the prices and the price indices for the four ingredients, P,Q, R and S. Diagram 3 shows a bar chart of the sales for these items forthe year 2005.

ITEM Prices (RM)

2005

2007


year 2007 based on

the year 2005

P 1.25

x

120

Q 3.204.40

y

R 2.60

3.25

125

S z

2.70

135


a ) find the values of x of y and of z

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b.) Calculate the composite index for these item

for the year 2007 based on the year 2005.

c.) the sales of these item for the year 2005 isRM 400. find the correspond sales for the year

2007.

d.) calculate the price index of item Q for the year2009 based on the year 2005, if its price increase

at the same rate as it increase from the year2005 to the year 2007.SOALAN RAMALAN SPM


6060

Table 4 shows the prices and the price indices for the four ingredients A

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Table 4 shows the prices and the price indices for the four ingredients,A,B, C and D. used in making a type of biscuits.Diagram 4 shows a piechart which represents the relative amount of the ingredientsA, B, C andD used in making these biscuits

INGREDIENTS Prices per kg (RM)

2002

2007


year 2007 based onthe year 2002

P 1.80

2.25

125

Q 3.00

4.20

x

R 0.80

y

150

S z

0.60

80


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Table 6 shows a bar chart representing the quantities of five types of

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Table 6 shows a bar chart representing the quantities of five types ofmagazines sold in bookstores. Table 6 shows the prices and price indicesfor the five types of magazines.

TYPES OF MAGAZINES Prices (RM) for the

year

2005

2007


year 2007 based on

the year 2005

HOME x

24.00

120

CAR 30.00

34.80

Y

COOKERY 12.00

18.00

125

SPORTS 9.50

z

135

FASHION 10.0012.50 125


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b.) Calculate the composite index for the price ofthe magazines in the year 2007 Based on theyear 2005.

c.) The total sales of the magazines in the year2005 isRM45 000.

d.) The price of the magazines increase by 20%from the year 2007 to the year 2008. calculatethe composite index in the year 2008 based onthe year 2005


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HORMATI IBUBAPA &

JANGAN

TINGGAL

SEMBAHYANGSOALAN RAMALAN SPM 6565

i k l h

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Hormati guru sekolah

kerajaan

MR VENKATES

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Add Math 01