XPY Paper 2

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1 Prepared by Tan Sze Haun

Quadratics Equations

2003 P2 Q1 Solve the equations = +23 2( 1) 7x x 5, 1

3

2004 P2 Q7Solve the equations

+=

+

22 52

1

m m

m

1, 2

2


=

22 53

3

kk

1,5

2


= +

3 ( 1)6

2

x xx

4,3

3

2007 P2 Q3 Solve the equations = 24 15 17x x 3, 5

4

Extra exercise1 Solve the equations = +25 6( 3) 26x x 4 ,2

5

2Solve the equations

+=

+

23 42

4

p p

p

4, 2

3

SIMULTANEOUS LINEAR EQUATIONS

2003P2 Q2

Calculate the values of kand wthat satisfy the following simultaneous equations:2k 3w= 10 and 4k+ w= 1

1k ,w 32= =

2004P2 Q5

Calculate the values ofpand qthat satisfy the following simultaneous equations:

=1

2 132p q and 3p+ 4q= 2

p 6,q 5= =

2005P2 Q2

Calculate the values ofpand qthat satisfy the following simultaneous equations:2p 3q= 13 and 4p+ q= 5

p 2,q 3= =

2006P2 Q4

Calculate the values of xand ythat satisfy the following simultaneous equations:

x+ 2y= 6 and = 3

72

x y

x 2,y 4= =

2007P2 Q2

Calculate the values of gand hthat satisfy the following simultaneous equations:g+ 2h= 1 and 4g3h= 18

g 3,h 2= =

Extra Exercise1 Calculate the values ofpand qthat satisfy the following simultaneous equations:

+ = 1

13p q andp2q= 17

p 9,q 4= =

2 Calculate the values ofpand qthat satisfy the following simultaneous equations:3p q= 10 andp3q= 10

5 5p ,q2 2

= =

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Mathematical Reasoning

2003P2 Q8

(a) Is the following a statement or a non-statement.

4 is a prime number

(b)Write down two implications based on the following statement:

PRif and only if PR

The number of subsets in a set with 2 elements is 22.The number of subsets in a set with 3 elements is 23.

The number of subsets in a set with 4 elements is 24.

(c) Based on the information above, make a general conclusion by induction regarding thenumber of subsets in a set with kelements.

2004P2 Q4

(a) State whether the following statement is true or false.

8 > 7 or 3

2

= 6

(b)Write down two implications based on the following statement:

m3 = 1000 if and only if m= 10

(c) Write down premise 2 to complete the following argument:

Premise 1: All hexagons have six sides.Premise 2: ______________________________________.Conclusion: PQRSTU has six sides.

2005

P2 Q8

(a) State whether each of the following statement is true or false.

(i) 8 2 = 4 and 82= 16.(ii) The elements of set A = {12, 15 18} are divisible by 3 or the elements of set B =

{4, 6, 8} are multiple of 4.

(b)Write down premise 2 to complete the following argument:

Premise 1: If xis greater than zero, then xis a positive number.Premise 2: ______________________________________.Conclusion: 6 is a positive number.

(c) Write down two implications based on the following statement:

3m > 15 if and only if m> 5

2006P2 Q6

(a) Complete each of the following statements using the quantifier all and some so that it willbecome a true statement.

(i) _____________ of the prime numbers are odd numbers.(ii)_____________ pentagons have five sides.

(b) State the converse of the following statement and hense determine whether its converse istrue or false:

If x> 9, then x> 5

(c) Complete the premise in the following argument:

Premise 1: If set K is a subset of set L, then K L = L.Premise 2: ______________________________________.Conclusion: Set K is not a subset of set L.

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2007P2 Q7

(a) Complete the following statement using the quantifier all and some. To make it a truestatement.

_____________quadratic equations have two equal roots.

(b)Write down premise 2 to complete the following argument:

Premise 1: If M is a multiple of 6, then M is a multiple of 3.

Premise 2: ______________________________________.Conclusion: 23 is not a multiple of 6.

(c) Make a general conclusion by induction for the sequence of numbers 7, 14, 27, whichfollows the following pattern.

7 = 3(2)1+ 114 = 3(2)2+ 227 = 3(2)3+ 3 = .

(d)Write down two implications based on the following statement:

p q> 0 if and only ifp> q

Extra Exercise1 (a) Write two implications from the sentence below.

1=+b

y

a

xif and only if abaybx =+ .

(b) Complete the conclusion and premise in each of the arguments below.

Premise 1 : All rectangles have two diagonals of the same length.Premise 2 : A square is a rectangle.Conclusion:

2 (a) Complete the conclusion in the argument below.

Premise 1: If a number is a factor of 12, then the number is a factor of 48.Premise 2: 7 is not a factor of 48.Conclusion:

(b) Make a general conclusion by induction for the sequence of numbers 10, 35, 70, 115, accordingto the pattern below:

10 = 5(2)2 10

35 = 5(3)2 1070 = 5(4)2 10115 = 5(5)2 10

(c) Determine whether each of the following statements is true or false.

(i) 72= 14 or 34

0.75=

(ii) 4 x 5 = 20 and 4 < 5

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Inverse Matrices

2003P2Q11

M is a 2 2 matrix where3 -2 1 0

M5 -4 0 1

=

(a) Find the matrix M.(b) Write the following simultaneous linear equations as a matrix equation.

3 2 75 4 9x yx y

=

=

Hence, calculate the values of x and yusing matrices.

4 21(a)

5 32

(b) x 5,y 4

= =

2004P2 Q8 (a) The inverse matrix of

3 4 6is

5 6 5 3m

. Find the value of mandp.

(b) Using matrices, calculate the value of xand of ythat satisfy the followingsimultaneous linear equations.

3 4 1

5 6 2

x y

x y

=

=

1(a)m ,p 4

2

11(b) x 7,y

2

= =

= =

2005P2Q11

It is given that matrix2 5

1 3P

=

and matrix

3

1 2

hQ k =

such that

1 0

0 1PQ

=

.

(a) Find the value of kand of h.(b) Using matrices, calculate the value of xand of ythat satisfy the following

simultaneous linear equations.2 5 17

3 8

x y

x y

=

+ =

1a) k ,h 5

11

b) x 1,y 3

= =

= =

2006P2Q11

(a) It is given that

1 2

1

2n

is the inverse matrix of 3 41 2

. Find the value

of n.(b) Write the following simultaneous linear equations as a matrix equation.

3 4 5

2 2

u v

u v

=

+ =

Hence, using matrices, calculate the values of uand v.

3a) n 2

1b) u 1,v

2

=

= =

2003P2Q11

(a) Given4 2 2 1 01

5 3 5 4 0 1

n

m

=

, find the value of mand of n.

(b) Using matrices, calculate the value of xand of ythat satisfy the matricesequations:

4 2 1

5 3 2

x

y

=

a) m 2,n 3

1 3b) x ,y

2 2

= =

= =

Extra Exercise

M is a 2 x 2 matrix where4 3 1 0

M2 1 0 1

=

.

(a) Find the matrix M.

(b) Write the following simultaneous linear equations as a matrix equation.4x 3y= 262x+ y= 7

Hence, calculate the values of x and y using matrices.

1 31a)

2 410

1b) x ,y 8

2

= =

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Straight Line

2003P2Q5

The graph shows that PQ, QRand RS are straight lines. P is on

the yaxis, OP is parallel to QRand PQ is parallel to RS. Theequation of PQ is 2 5x y+ = .

(a) State the equation of thestraight line QR.

(b) Find the equation of thestraight line RS and hence,state its y-intercept.

5a) x

2

b) y 2x 9,

y int ercept 9

=

= +

=

2004P2Q6

OPQR is a parallelogram and O isthe origin.(a) Find the equation of the

straight line PQ,(b) Find the y-intercept of the

straight line QR.

a) y 3x 15

b) 20

=

2005P2Q5

Point R lies in the x-axis and pintP lies on the y-axis. Straight linePU is parallel to the x-axis andstraight line PR is parallel tostraight line ST. The equation ofstraight line PR is 2 14x y+ = .

(a) State the equation of the

straight line PU.(b) Find the equation of thestraight line ST and hence,state its x-intercept.

a) y 7

1b) y x 4,

2

x int ercept 8

=

=

=

2006P2Q10

The straight line PQ is parallel tostraight line RS.(a) Find the equation of the

straight line PQ,(b) Find the x-intercept of the

straight line PQ.

a) y = 2x + 11

b)11

2

2007P2Q5

Straight line KL is parallel to thestraight line MN. The equation ofstraight line KL is 2x+ y= 4.The points L and N lie on the y-axis.(a) Find the equation of the

straight line MN,(b) Find the x-intercept of the

straight line MN.

a) y = 2x + 1

b)1

2

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Solid Geometry

2003P2Q6

Diagram 3 shows a solid formed bycombining a right pyramid with a halfcylinder on the rectangular planeDEFG. DE= 7 cm, EF= 10 cm andthe height of the pyramid is 9 cm.

Calculate the volume, in cm3, of the

solid. [Use22

7 = ]

1402

2cm3

2004P2Q2

Diagram below shows a solid formedby joining a cone and a cylinder. Thediameter of the cylinder and thediameter of the base of the cone areboth 7 cm. The volume of the volume

is 231 cm3

. By using22

7 = ,

calculate the height, in cm, of thecone.

6 cm

2005P2Q6

Diagram below shows a solid conewith radius 9 cm and height 14 cm. Acylinder with radius 3 cm and height 7cm is taken out of the solid. Calculate

the volume, in cm3, of the remaining

solid. (Use =722 )

990 cm3

2006P2Q5

Diagram below shows a combinedsolid consists of a right prism and aright pyramid which are joined at theplane EFGH. Vis vertically above the

base EFGH. TrapeziumABGF is theuniform cross section of the prism.The height of the pyramid is 8 cm andFG= 14 cm.

(a) Calculate the volume, in cm3, of

the right pyramid.(b) It is given the volume of the

combined solid is 584 cm3.Calculate the length, in cm, of

AF.

a) 224 cm3b) 5 cm

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2007P2Q11

Diagram shows a solid, formed byjoining a cylinder to a right prism.Trapezium AFGB is the uniform cross-section of the prism. AB = BC = 9 cm.The height of the cylinder is 6 cm andits diameter is 7 cm. Calculate the

volume, in cm3, of the solid. [Use

22

7 = ]

987 cm3

Gradient and area under the graph

SPM2003P2Q10

Diagram 6 shows the speed-time graph of aparticle for a period of 17s.(a) State the length of time, in s, that

particle moves with uniform speed.(b) Calculate the rate of change of speed, in

ms-2, in the last 4 s.(c) Calculate the value of u, if the total

distance travelled in the first 13s is195m.

a) 7

b)1

42

c) 5

SPM

2004P2Q11

Diagram 5 shows the distance time graph

for the journeys of a car and a bus. Thegraph JKLM represents the journey of thecar and the graph JLN represents the

journey of the bus. Both vehicles departfrom town T at the same time and travelalong the same road.(a) State the length of the time, in hours,

during which the car is stationary.(b) Calculate the average speed, in km h-1

of the car over the 2 hour period.(c)At the certain time during the journey,

both vehicles are at the same location.

(i) Find the distance, in km,

between that location and townT.(ii) State the time taken by the bus

to reach that location fromtown T.

a) 0.3

b) 85c) (i) 48(ii) 0.8

SPM2005P2Q10

Diagram 5 shows the speedtime graph of aparticle for a period oftseconds.(a) State the length of time, in s, that the

particle moves with uniform speed.(b) Calculate the rate of change of speed, in

ms-2, in the first 5 seconds.(c) Calculate the value of t, if the total

distance travelled for the period of tseconds is 148 metres.

a) 7b) 1.6c) 16

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SPM2006P2Q9

Diagram 4 shows the speed time graph forthe movement of a particle for a period of20 seconds.a) State the uniform speed, in ms-1, of the

particle.b) The distance traveled by the particle

with uniform speed is 84 m. Calculate

(i) the value of t.(ii) the average speed, in ms-1, of the

particle for the period of 20seconds.

a) 14b) (i) 18

(ii) 16

SPM2007P2Q10

Diagram shows the distance-time graph ofthe journey of a bus and taxi. The graphPQRS represents the journey of the busfrom town A to town B. The graph JKrepresents the journey of the taxi from townB to town A. The bus leaves town A and thetaxi leaves town B at the same time and

they travel along the same road.(a) State the length of time, in minutes,during which the bus is stationary.

(b) (i) If the journey starts at 9.00 a.m., atwhat time do the vehicles meet?

(ii) Find the distance, in km, from townB when the vehicles meet.

(c) Calculate the average speed, in

km h1, of the bus for the wholejourney.

a) 25b) (i) 9.4

(ii) 60c) 36

Perimeters and area of a circles

2003P2Q7

Diagram below shows two sectors OMN andOPQ with the same centre O and a quadrantQTO with centre Q. OM = 14 cm and QT = 7

cm. Using22

7 = , calculate,

(a) the perimeter of the whole diagrams,(b) the area of the shaded region.

a)2

53 3

b)1

1152

2004P2

Q9

In the below diagram, PQ and PS are arcs oftwo different circles with centre O. RQ = ST = 7

cm and PO = 14 cm. Using22

7 = , calculate

(a) the area, in cm2, of the shaded region,(b) the perimeter, in cm, of the whole

diagram.

a)1

2482

b) 89.67

2005P2Q7

Diagram below shows two sectors ORST andOUV with the same centre O. RWO is asemicircle with diameter RO and RO = 2OV .ROV and OUT are straight lines. OV = 7cm

and = UOV 60 . Using 227

= , calculate

(a) the perimeter , in cm, of the whole diagram.(b) the area, in cm2, of the shaded region.

a)2

643

b) 154

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2006P2Q8

In the following diagram, OMRN is a quadrant ofa circle with centre O and PQ is an arc ofanother OMP and ORQ are straight lines. OM =

MP = 7 cm and 60POQ = . Using22

7 = ,

calculatea) the perimeter , in cm , of the whole

diagram.b) the area , in cm2, of the shaded region.

a) 46.34b) 89.83

2007P2Q6

The diagram below shows a quadrant OST andsemicircle PQR, both with centre O. OS = 21 cm

and OP = 14 cm. Using22

7 = , calculate

a) the area , in cm2, of the shaded region.b) the perimeter , in cm , of the whole

diagram.

a) 243.83b) 104.33

Probability

2003P2Q8

Diagram shows the route of a vehicle which carries a group of volunteers. The groupconsists of 7 male and 5 females who are dropped off at random to sell flags at variouspoints along the routes.(a) If two volunteers are dropped off at Taman Aman, calculate the probanility that both

are males.(b)Two volunteers of different gender are dropped off at Taman Aman. If two other

volunteers are then dropped off at Taman Sentosa, calculate the probability that atleast one of them is female.

7a)

22

2b)

3

2004P2Q8

Table shows the number of coupons in two boxes, A and B. The coupons are of variousvalues, RM1, RM2 and RM5. Students are given coupons as an incentive for sellingbookmarks.

Number of Coupons

Box RM1 RM2 RM5

A 1 6 8

B 2 5 3

Students selling more than 100 bookmarks are given the chance to draw at random acoupon from box A. Students selling less than 100 bookmarks are given the chance todraw at random a coupon from box B. Ali sells 120 bookmarks. Lim sells 52 bookmarks.Find the probability that(a) both of them draw a RM5 coupon,(b) the total value of the two coupons drawn by them is less than RM4.

4a)

25

19b)

150

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2005P2Q9

A group of 5 boys and 4 girls take part in a study on the type of plants found in areserved forest area. Each day, two pupils are chosen at random to write a report.(a) Calculate the probability that both pupils chosen to write report on the first day are

boys.(b) Two boys do write the report on the first day. They are then exempted from writing

the report on the second day. Calculate the probability that both pupils chosen towrite the report on the second day are of the same gender.

5a)

18

3b)

7

2006P2Q7

In a quiz contest, there are three categories of questions of 5 questions on sport, 3questions on entertainment and 7 questions on general knowledge. Each question isplaced inside an envelope. All of the envelopes are similar and put inside a box. All theparticipants of the quiz contest are requested to pick at random two envelopes from thebox.(a) the first envelope with a sport question and the second envelope with an

entertainment question,(b) two envelopes with questions of the same category.

1a)

14

34b)

105

2007P2

Q8

Diagram shows ten labeled cards in two boxes.

A card is picked at random from each of the boxes. By listing the outcomes, find theprobability that(a) both cards are labeled with a number,(b) one card labeled with a number and the other card is labeled with a letter.

1a)

12

5b)12

Lines and Planes in 3 dimensions

2003P2Q4

Diagram shows a prim witha horizontal square baseHJKL. Trapezium EFLK is theuniform cross-section of theprism. The rectangularsurface DEKJ is vertical whilethe rectangular surfaceGFLH is inclined. Calculatethe angle between the planeDLH and the base HJKL.

36.87 @

36 52 '

2003P2Q4

The base JKLM is ahorizontal rectangle. Q is themidpoint of JM. The apex Vis 8 cm vertically above thepoint Q. Calculate the anglebetween the line KV and thebase JKLM.

31.61 @

31 36 '

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2003P2Q4

Right angled triangle PQR isthe uniform cross-section ofthe prism. Calculate theangle between the planeRTU and the plane PQTU.

33.69 @

33 41'

2006P2

Q2

The base is a horizontalrectangle. The right angled

triangle UPQ is the uniformcross section of the prism.Identify and calculate theangle between the line RUand the base PQRS.

PRU@

URP

34.70 @

34 42 '

2007

P2Q4

The base PQRS is a

horizontal rectangle. Rightangled triangle QRU is theuniform cross-section of theprism. V is the midpoint ofPS. Identify and calculatethe angle between the lineUV and the plane RSTU.

SUV@

VUS31.61 @

31 36 '

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Graph of functions

2003P2Q12

(a) Complete the following table for the equation = 4

yx

.

Answer:

x4

2.5

1

0.5 0.5 1 2 3.2 4y 1 1.6 8 8 4 1.25 1

(b) By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 2 units on the y-

axis, draw the graph of = 4

yx

for 4 x 4.

(c) From your graph, Find(i) the value of y when x = 1.8,(ii) the value of x when y = 34.

Answer:(i) y = __________ (ii) x = __________

(d) Draw a suitable straight line on your graph to find all the values of x which

satisfy the equation = +4

2x 3x

for 4 x 4. State these values of x.

Answer:x = ___________, x = ___________

(a) 4, 2

(c) (i) 2.2

(ii) 1.2

(d) y 2x 3=

x = 2.35, 0.85

2004P2Q12

(a) Table below shows values of x and y which satisfy the equation

= 2y 2x 4x 3 .

x 2 1 0 1 2 3 4 4.5 5

y k 3 3 5 3 m 13 19.5 27

Calculate the value of k and of m.


axis, draw the graph of = 2y 2x 4x 3 for 2 x 5.

(c) From your graph, Find(i) the value of y when x = 15,(ii) the value of x when y = 0.

(d) Draw a suitable straight line on your graph to find a value of x which satisfies

the equation + =22x x 23 0 for 2 x 5. State this value of x.

Answer:

(a) k = ________ m = ________

(c) (i) y = ___________ (ii) x = ___________

(d) x = __________

(a) k = 13,

m = 2

(c) (i) 7

(ii) 2.6, 0.5

(d) y 5x 20= +

x = 3.1, 3.2

2005P2Q12

(a) Complete the following table for the equation = 2y 2x x 3 .

Answer:

x 2 1 0.5 1 2 3 4 4.5 5

y 7 2 2 3 12 33 42


axis, draw the graph of = 2

y 2x x 3 for 2 x 5.


(a) 0, 25

(c) (i) 19(ii) 4.7

(d) y = 2x + 7

x = 1.6, 3.1

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Answer:(i) y = _______ (ii) x = ________

(d)Draw a suitable straight line on your graph to find all the values of x which

satisfy the equation =22x 3x 10 for 2 x 5. State these values of x.

Answer:

x = ___________ , ____________

2006P2Q13

(a) Complete table in the answer space for the equation =24

yx

by writing down

the values of y when x = 3 and x = 1.5.


axis, draw the graph of =24

yx

for 4 x 4.

(c) From your graph, Find(i) the value of y when x = 2.9,

(ii) the value of x when y = 13.

(d)Draw a suitable straight line on your graph to find a value of x which satisfy

the equation + =22x 5x 24 for 4 x 4. State this value of x.

Answer:

(a)

x 4 3 2 1 1 1.5 2 3 4

y 6 12 24 24 12 8 4

(c) (i) y = __________

(ii) x = __________

(d) x = _____________

(a) 8, 16

(c) (i) 8(ii) 1.85

(d) y = 2x + 5x = 2.45

2007P2Q12

(a) Complete table in the answer space for the equation 3y 6 x= by writing

down the values of y when x = 2 and x = 2.


axis, draw the graph of 3y 6 x= for 3 x 2.5.


(d)Draw a suitable straight line on your graph to find the values of x which

satisfy the equation 3x 8x 6 0 = for 3 x 2.5. State these values of x.

Answer:

(a)

x 3 2.5 2 1 0 1 2 2.5

y 33 21.63 14 6 5 9.63

(c) (i) y = _____________

(ii) x = _____________

(d) x = _______________

(a) 7, 2

(c) (i) 2.5(ii) 1.6

(d) y = 8x

x = 0.8, 2.4

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Statistics

2003P2Q14

The data in diagram below shows the donations, in RM, of 40 families to their children's schoolwelfare fund.

40 23 27 30 20 24 28 35 34 32 35 39 27 1417 33 45 31 29 30 33 21 37 26 38 19 28 35

22 39 38 40 32 26 34 22 32 22 28 44(a) Using the data in the diagram, and a class interval of RM5, complete the following table.

Answer:

Donation (RM) Frequency Cumulative frequency

11 15

16 20

(b) By using a scale of 2 cm to RM 5 on the x-axis and 2 cm to 5 families on the y-axis, draw anogive based on the data.

(c) From your ogive in (b),(i) find the third quartile,(ii) hence, explain briefly the meaning of the third quartile.

Answer:

(i) Third quartile = _____________________

(ii)__________________________________

2004

P2Q14

The data in diagram below shows the masses, in kg, of suitcases for a group of tourists. Each

tourist has one suitcase.

27 29 25 22 28 25 16 21 29 26 2729 10 19 16 13 21 23 24 24 25 3127 18 22 33 19 20 14 24 26 27

(a) Based on the data in the diagram and by using a class interval of 3 , complete the tableprovided in the answer space.

(b) Based on the table in (a), Calculate the estimated mean mass of the suitcases.(c) By using a scale of 2 cm to 3 kg on the x-axis and 2 cm to 1 suitcase on the y-axis, draw a

histogram for the data.(d) State one in formation obtained based on the histogram in (c).

Answer:

(a) Class Interval Frequency Midpoint

10 12

13 15

(b)

(d)

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2005P2Q14

The data in diagram below shows the marks for an English Language monthly test for 42 pupils.

51 25 46 33 42 25 47 34 35 38 32 20 45 3738 39 21 26 31 41 37 29 35 48 31 26 42 2843 44 34 28 31 23 36 40 30 38 39 23 22 54

(a) Using the data in the diagram, and a class interval of 5 marks, complete the following table.

Answer:

Marks Midpoint Frequency

20 24 22

25 29

(b) Based on your table in (a),(i) state the modal class,

(ii) calculate the mean marks for the English Language monthly test and give your answercorrect to 2 decimal places.

Answer

(i)

(ii)

(c) For this part of the question, use the graph paper provided.By using a scale of 2 cm to 5 marks on the horizontal axis and 2 cm to 1 pupils on the verticalaxis, draw a histogram for the data.

2006P2Q14

The data in diagram below shows the donations, in RM, collected by 40 pupils.

49 22 27 34 26 30 24 34 37 40 38 33 3 2548 39 39 40 34 25 41 45 43 46 47 45 43 4023 30 45 39 38 35 29 43 31 35 37 28

(a) Based on the data in the diagram and by using a class interval of 5, complete the table in theanswer space.

(b) Based on the table in (a), Calculate the estimated mean of the donation collected by a pupil.(c) For this part of the question, use the graph paper provided.

By using a scale of 2 cm to RM 5 on the horizontal axis and 2 cm to 1 pupil on the vertical axis,draw a frequency polygon for the data.

(d) Based on the frequency polygon in (c), state one piece of information about the donations.

Answer:

(a) Class Interval Midpoint Frequency

21 25 23 5

26 30

(b)

(d)

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2007P2Q16

Table shows the frequency distribution of the mass, in kg, of a group of 80 students.

Mass(kg) Frequency

30 34 5

35 39 8

40 44 11

45 49 21

50 54 22

55 59 1060 64 3

(a) (i) State the modal class(ii) Calculate the estimated mean of the mass of the group of students.

(b) Based on table above, complete table in answer space to show the cumulative frequencydistribution of the masses.

(c) For this part of the question, use the graph paper provided.By using a scale of 2 cm to 5 kg on the horizontal axis and 2 cm to 10 students on the verticalaxis, draw an ogive based on the data.

(d) 25% of all the students in the group have a mass of less thanpkg. These students will besupplied with nutritional food. Using the ogive you have drawn in (c), find the value ofp.

Answer:(a) (i)

(ii)

(b)

Upper Boundary Cumulative Frequency

(d)

Answers:2003 (a) Frequency = 1, 3, 6, 10, 11, 7, 2

(c) (i) RM 35(ii) 75% donation less than RM 35

2004 (a) Frequency = 1, 2, 3, 5, 6, 9, 4, 2

(b) 23.19(d) model class = 25 27

2005 (a) 5, 7, 8, 10, 6, 4, 2(b) (i) 35 39

(ii) 34.98

2006 (a) Frequency = 5, 6, 8, 10, 7, 4(b) RM 35.50(d) Modal class = 36 40

2007 (a) (i) 50 - 54

(ii) 47.56(d) p = 43

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Plan and Elevations

(a) Diagram beside shows a solid rightprism with rectangular base FGPNon a horizontal table. EFGHJK is theuniform cross-section of the prism.Rectangle EKLM is an inclined plane.Rectangle JHQR is a horizontalplane. EF, KJ and HG are verticaledges. Draw a full scale plan of thesolid.

2003P2Q15

(b) A solid cuboid is joined o the prismin the diagram in (a) at the verticalplane PQRLMN. The combined soldis as shown in Diagram 1(ii). Thesquare base FGSW is horizontal.Draw full scale(i) the elevation of the combined

solid on a vertical line parallelto FG as viewed from C.

(ii) the elevation of the combinedsolid on a vertical line parallelto GPS as viewed from D.

2004P2Q15

(a) Diagram shows a solid consisting of a cuboid and a half-cylinder which are joined at the planeHJMN. The base GDEF is on a horizontal plane and HJ = 3 cm. Draw to full scale, theelevation of the solid on a vertical plane parallel to DG as viewed from X.

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(b) A solid right prism is joined to the solid in the diagram in (a) at the vertical plane ELMW. Thecombined solid is a shown in diagram 2 below. Trapezium PMWU is the uniform cross-sectionof the prism and PQRM is an inclined plane. The base DEUTSWFG is on a horizontal plane andEU = 2 cm. Draw to full scale,(i) the plan of the combined solid.(ii) The elevation of the combined solid on a vertical plane parallel to UT as viewed from Y.

(a) Diagram beside shows a solid rightprism with rectangular base PEFN ona horizontal table. The surface EFGHJis the uniform cross-section of theprism. Rectangle KJHL is an inclinedplane and rectangle LHGM is ahorizontal plane. JE and GF arevertical edges. Draw full scale, the

elevation of the solid on a verticalplane parallel to EF as viewed form X.

2005P2Q15

(b) A solid right prism with prism cross-section UVWX is removed from thesolid in the diagram in (a). Theremaining solid is as shown in thediagram beside. Rectangle VSTW is ahorizontal plane. UV and XW are

vertical edges. VS = 3 cm and SE = 2cm. Draw full scale,(i) the plane of he remaining solid.(ii) the elevation of the remaining

solid on a vertical plane parallelto PE as viewed from Y.

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(a) The diagram shows a combinedsolid of a prism and a cuboid

joined together at the verticalplane EHIN and lying on ahorizontal table. ABCD is theuniform cross section of theprism. BCFG is a horizontal

plane and the base ADKL is arectangle. Draw a full scale planof the solid.

2006P2Q15

(b) A half-cylinder is joined to thesolid in (a) at the horizontalplane IJMN. The combined solidis as shown in the diagrambelow. IJ is the diameter of thehalf-cylinder. Draw to full scale(i) the elevation as viewed

from X(ii) the elevation as viewed

from Y.

(a) Diagram beside shows asolid right prism withrectangular base ABCD on ahorizontal plane. Thesurface ABJHGF is theuniform cross-section of theprism. AF, HG and BJ arevertical edges. RectangleJKLH is a horizontal planeand rectangle GMEF is aninclined plane. Draw fullscale, the plan of the solid.

2007P2Q13

(b) A half-cylinder solid of

diameter 6 cm is joined tothe prism in the diagram in(a) at the plane SKLT. Thelength of SK is 4 cm. Thecombined solid is shown infollowing diagram. Draw fullscale,(i) the elevation of the

combined solid on avertical line parallel to

AB as viewed from P.(ii) the elevation of the

combined solid on avertical line parallel toBC as viewed from Q.

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Transformations

2003P2Q13

Diagram below shows trapeziumABCD,ABEF, GHJK, and LMNPon a Cartesian plane.

(a) Transformation R is a reflection about the line y = 3. Transformation T is the translation

2

4

. State the coordinates of the image of point Hunder the following transformations:

(i) RT (ii) TR

(b) ABEFis the image ofABCDunder transformation V and GHJKis the image ofABEFundertransformation W. Describe in full(i) transformation V(ii) a single transformation which is equivalent to transformation WV.

(c) LMNPis the image ofABCDunder an enlargement.(i) State the coordinates of the centre of the enlargement.

(ii) Given that the area of LMNPis 325.8 unit2, calculate the area ofABCD.

2005P2Q13

(a) Diagram below shows two points, M and N, on a Cartesian plane.

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Transformation T is the translation3

1

. Transformation R is an anticlockwise rotation of 90

about the centre (0, 2).(i) State the coordinates of the image of point M under transformation R.(ii) State the coordinates of the image of point N under the following transformation:

(a) T2 (b) TR

(b) Diagram below shows three quadrilaterals, ABCD, EFGH and PQRS on a Cartesianplane. EFGH is the image of ABCD under transformation V. PQRS is the image of EFGHunder transformation W.

(i) Describe in full the transformation:(a) V, (b) W.

(ii) Given that quadrilateral PQRS represents a region of area 45.6 cm2, calculate the

area, in cm2, of the region represented by the shaded region.

2004P2Q13

Diagram below shows quadrilaterals, ABCD, PQRS and KLRM, drawn on a Cartesian plane.

(a) Transformation T is a translation4

2

. Transformation V is a reflection in the line y = 1.

State the coordinates of the image of point A under each of the following transformations:(i) Translation T,(ii) Combined transformations VT.

(b) (i) KLRM is the image of ABCD under the combined transformations WU. Describe in full, thetransformation U and the transformation W.

(ii) Given that the shaded region KLQPSM represents a region of area 120 m2, calculate thearea, in m2, of the region represented by PQRS.

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2006P2Q13

(a) Transformation T is a translation3

2

and transformation P is an anticlockwise rotation of

90 about the centre (1, 0). State the coordinates of the image of point (5, 1) under each offollowing transformations:(i) Rotation P(ii) Translation T

(iii) Combined transformations T

2

.(b) Diagram below shows three quadrilaterals, ABCD, EFGH and JKLM, drawn on a Cartesianplane.

(i) JKLM is the image of ABCD under the combined transformations VU. Describe in full thetransformation:

(a) U, (b) V.(ii) It is given that quadrilateral ABCD represents a region of area 18 m2. Calculate the

area, in m2, of the region represented by the shaded region.

2007P2Q115

The following diagram shows the quadrilaterals ABCD, EFGH and JKLM drawn on a Cartesian Plane

a) Transformations R is a rotation of 90anticlockwise about the centre (0, 2). Transformation Pis a reflection in the straight line x= 2. State the coordinates of the image of point A undereach of the following transformations:(i) TR, (ii) T2.

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(b) EFGH is the image of ABCD under the combined transformation MN. Describe in full,

(i) the transformation M,(ii) the transformation N.

(c) JKLM is the image of EFGH under an enlargement at centre (3, 0).(i) State the scale factor of the enlargement.

(ii) Given that EFGH represents a region of area 112 m2, calculate the area, in m2, of theregion represented by JKLM.

Answers:2003 (a) (i) (7, 0) (ii) (7, 8)

(b) (i) V = reflection in the line AB.(ii) rotation through 90anticlockwise about the centre (6, 5).

(c) (i) (6, 2) (ii) 36.2

2005 (a) (i) (3, 4) (ii) (a) (2, 1) (b) (2, 3)

(b) (i) V = reflection in the line x = 3.W = enlargement with scale factor 2 about the centre (1, 2).(ii) 34.2

2004 (a) (i) (2, 4) (ii) (2, 2)

(b) (i) U = rotation through 90clockwise about the centre (0, 1).W = enlargement with scale factor 3 about the centre R.

(ii) 15

2006 (a) (i) (0, 4) (ii) (2, 3) (iii) (1, 5)

(b) (i) U = reflection in the line x = 1.V = enlargement with scale factor 3 about the centre (2, 4).

(ii) 144

2007 (a) (i) (3, 5) (ii) (3, 3)(b) (i) M = enlargement with scale factor 2 about the centre (3, 0).

(ii) rotation through 180about the point (2, 4).

(c) (i)1

2 (ii) 28

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XPY Paper 2