past year collection f5 with answers

7/29/2019 Past Year Collection F5 With Answers

1/16

1 Prepare by Tan Sze Haun

Progression (SPM 2003 2007, Paper 2)

1.

2004P2Q6

Diagram shows the arrangement of the first three of an infinite series of similar triangles. The firsttriangle has a base of x cm and a height of y cm. The measurements of the base and height ofeach subsequent triangle are half of the measurements of its previous one.a) Show that the areas of the triangles form a geometric progression and state the common ratio.b) Given that x = 80 cm and y = 40 cm,

i) determine which triangle has an area of 164

cm2.

ii) Find the sum to infinity of the areas, in cm2, of the triangles.

2.

2005P2Q3

Diagram shows part of an arrangement of bricks of equal size. The number of bricks in the lowestrow is 100. For each of the other rows, the number of bricks is 2 less than in the row below. Theheight of each brick is 6 cm. Ali builds a wall by arranging bricks in this way. The number of bricksin the highest row is 4. Calculatea) the height, in cm, of the wall,b) the total price of the bricks used if the price of one bricks is 40 sen.

3.

2006P2Q3

Two companies, Delta and Omega, start to sell cars at the same time.a) Delta sells kcars in the first month and its sales increase constantly by m cars every

subsequent month. It sells 240 cars in the 8th month and the total sales for first 10 month are1900 cars. Find the value ofkand ofm.

b) Omega sells 80 cars in the first month and its sales increase constantly by 22 cars everysubsequent month. If both companies sell the same number of cars in the nth month, find thevalue ofn.

4.

2007P2Q6

Diagram shows the side elevation of part of stairs built of cement blocks. The thickness of eachblock is 15 cm. The length of the first block is 985 cm, the length of each subsequent block is 30cm less than the preceding block as shown in diagram.a) If the height of the stairs to be built is 3m, calculate

i) the length of the top most block,ii) the total length of the blocks.

b) Calculate the maximum height of the stairs.

Answers:1 a) 1

4r =

b) (i) The fifth triangle

(ii) 13

2133

2 (a) 294, (b) 1019.20

3 a) k = 100, m = 20b) n = 11

4 a) (i) 415, (ii) 14000

b) 495


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Progression (SPM 2003 2007, Paper 1)

1. 2003P1Q7

The first three terms of an arithmetic progression are k 3, k + 3, 2k + 2. Finda) the value of k,b) the sum of first 9 terms of the progression.

K = 7, 252

2. 2003P1Q8

In a geometric progression, the first term is 64 and the fourth term is 27. Calculatea) the common ratio,b) the sum to infinity of the geometric progression.

34 , 256

3. 2004P1Q9

Given a geometric progression y, 2,4

y, p, , express p in terms of y.

2

8p

y=

4. 2004P1Q10

Given an arithmetic progression 7, 3, 1, , state three consecutive terms in thisprogression which sum up to 75.

29, 25, 21

5. 2004P1

Q11

The volume of water in a tank is 450 litres on the first day. Subsequently, 10 litresof water is added to the tank everyday. Calculate the volume, in litres, of water in

the tank at the end of the 7th

day.

510

6. 2004P1Q12

Express the recurring decimal 0.969696 as a fraction in its simplest form. 3233

7. 2005P1Q10

The first three terms of a sequence are 2, x, 8. Find the positive value of x so thatthe sequence is(a) an arithmetic progression, (b) a geometric progression.

5, 4

8. 2005P1Q11

The first three terms of an arithmetic progression are 5, 9, 13. Finda) the common difference of the progression,b) the sum of the first 20 terms after the 3rd term.

4, 1100

9. 2005P1Q12

The sum of the first n terms of the geometric progression 8, 24, 72, is 8744.Finda) the common ratio of the progression,b) the value of n.

3, n = 7

10. 2006P1Q9

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

terms of the progression is 7p 10, where p is a constant. Given that the commondifference of the progression is 5, find the value of p.

8

11. 2006P1Q10

The third term of a geometric progression is 16. The sum of the third term and thefourth term is 8. Findthe first term and the common ratio of the progression,the sum to infinity of the progression.

64, 12

,

23

42

12. 2007P1Q9

a) Determine whether the following sequence is an arithmetic progression or ageometric progression. 16x, 8x, 4x,

b) Given a reason for the answer in (a).

GP, hascommon ratio

of 12

13. 2007P1Q10

Three consecutive terms of an arithmetic progression are 5 x, 8, 2x. Find thecommon difference of the progression.

14

14. 2007P1Q11

The first three terms of a geometric progression are 27, 18, 12. Find the sum toinfinity of the geometric progression.

81


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Linear Law (Paper2, SPM 2003 - 2007)

1. 2003P2Q7

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

experiment. It is known that x and y are related by the equation2xy pk= ,

where p and k are constants.

x 1.5 2.0 2.5 3.0 3.5 4.0y 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.(b) Use the graph in (a) to find the value of

(i) p, (ii) k.

p = 1.259k = 1.115

2. 2004P2Q7


experiment. It is known that x and y are related by the equationxy pk= , 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 2cm to 0.2 unit on the log10 y-axis. Hence, draw the line of best fit.

(b) Use the graph in (a) to find the value of(i) p, (ii) k.

p = 1.820k = 1.309

3. 2005P2Q7


experiment. It is known that x and y are related by the equationr

y pxpx

= + ,

where p and r are constants.

x 1.0 2.0 3.0 4.0 5.0 5.5

y 5.5 4.7 5.0 6.5 7.7 8.4

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

(b) Use the graph in (a) to find the value ofi) p, (ii)r.

p = 1.375r = 5.5

4. 2006P2Q7


experiment. Variables x and y are related by the equation x 1y pk += , 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 log10 y against (x + 1) by using a scale of 2 cm to 1 units on the (x + 1)-axis and 2 cm to 0.2 unit on the log y-axis. Hence, draw the line of best fit.

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

p, (ii) k.

p = 1.778k = 1.486

5. 2007P2Q7


experiment. Variables x and y are related by the equation2 py 2kx x

k= + ,

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 yx

against x, using a scale of 2 cm to 1 units on both axes. Hence, draw

the line of best fit.(b) Use your in (a) to find the value of

(i) p, (ii) k. (iii) y when x = 12.

p = 0.75k = 0.25y = 4.32


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Linear Law (Paper1, SPM 2003 - 2007)

1. 2003P1Q10

x and y are related by the equation 2y px qx= + ,

where p and q are constants. A straight line is

obtained by plottingy

x

against x, as shown in the

diagram. Calculate the values of p and q.

p = 2q = 13

2. 2004P1Q13

Diagram shows a straight line graph ofy

xagainst x.

Given that y = 6x x2, calculate the value of k andof h.

k = 4h = 3

3. 2005P1Q13

The variables x and y are related by the

equation4y kx= , where k is constants.

(a) Convert the equation4

y kx= to linear form.(b) Diagram shows the straight line obtained by

plotting log10 y against log10 x. Find the value of(i) log10 k, (ii) h.

10 10 10log 4log logy x k= +

(b) (i) 3(ii) 11

Diagram (a) shows the curve y = 3x2 + 5.Diagram (b) shows the straight line graph obtained when y = 3x2 + 5 is expressedin the linear form is expressed in the linear form Y = mX + c.Express X and Y in terms of x and/or y.

4. 2006P1Q11

(a) (b)

2

2

1X

x

yY

x

=

=

5. 2007P1Q12

The variables x and y are related by the equation2y 2x(10 x)= . A straight line graph is obtained

by plotting2y

xagainst x, as shown in the diagram.

Calculate the value of p and of q.

q = 14h = 10

Integration (Paper 1, SPM 2003 - 2007)

1. 2003P1Q17

Given that = + ++

n

4

5dx k(1 x) c

(1 x), find the values of k and n. k =

5

3 ,

n = 3

2. 2003P1Q18

Diagram shows the curve y = 3x2 and the straightline x = k. If the area of the shaded region is 64unit2, find the value of k.

k = 4


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3. 2004P1Q22

Given that

=k

1(2x 3)dx 6 , where k > 1, find the value of k.

k = 5

4. 2005P1Q21

Given that =6

2f(x) dx 7 and =

6

2(2f(x) kx)dx 10 , find the value of k. k =

1

4

5. 2006P1Q20

Diagram shows the curve y=f(x) cutting the x-axis at x = a and x = b. Given that the area ofthe shaded region is 5 unit2, find the value of

b

a2f(x)dx .

10

6. 2006P1Q21

Given that =5

1g(x) dx 8 , find

(a) the value of 15 g(x)dx ,(b) the value of k if =51 [kx g(x)]dx 10 .

(a) 8

(b) k =3

2

7. 2007P1Q21

Given that7

2h(x) dx 3= , find

(a) 27h(x)dx ,

(b) 72[5 h(x)]dx .

(a) 3(b) 22

Integration (Paper 2, SPM 2003 - 2007)

1. 2003P2Q3

(a) Given that = +dy

2x 2dx

and y = 6 when x = 1, find y in terms of x.

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

2

2

d y dyx (x 1) y 8

dxdx.

a)2

2 7y x x= + +

b)3

, 15

x =

(a) Diagram shows a conical container ofdiameter 0.6 m and height 0.5 m. Wateris poured into the container at a rate of0.2 m3s1. Calculate the rate of changeof the height of the water level at theinstant when the height of the water

level is 0.4 m. (use = 3.142; volume of

a cone = 21

r h3

)

1.1052. 2003P2Q9

(b) Diagram shows a curve x=y2 1 whichintersects the straight line 3y = 2x atpoint A. Calculate the volume generatedwhen the shaded region is revolved 360about the y-axis.

52

15

3. 2004P2

Q5

The gradient function of a curve which passes through A(1, 12) is 3x26x. Find

(a) the equation of the curve,(b) the coordinate of the turning points of the curve and determine

whether each of the turning points is a maximum or a minimum.

a)3 2

3 10y x x=

b) (0, 10) maximum,(2, 10) minimum


6/16


4. 2004P2Q10

Diagram shows part of the curve =

2

3y

(2x 1)

which passes through A(1, 3).(a) Find the equation of the tangent to the

curve t the point A.(b)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 360about the x-axis. Find the volume

generated, in terms of.

a) y= 12x+15

b) i)1

5

ii)49

1125

5. 2005P2Q2

A curve has a gradient function px2 4x, where p is a constant. Thetangent to the curve at the point (1, 3) is parallel to the straight line y

+ x 5 = 0. Find(a) the value of p,(a) the equation of the curve.

a) p = 3b) f(x) = x3 2x2 + 4

6. 2005P2Q8

In the diagram, the straight line PQ is normal to

the curve = +2

xy 12

at A(2, 3). The straight line

AR is parallel to the y-axis. Find(a) the value of k,(b) the area of the shaded region,

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

a) k = 8

b) 1123

c) 4

7. 2006P2

Q8

Diagram shows the straight line y = x +4 intersecting the curve y = (x 2)2 at

the points A and B. Find(a) the value of k,(b) the area of the shaded region P,(c) the volume generated, in terms of

, when the shaded region Q isrevolved 360 about the x-axis.

a) k = 5

b)5

206

c)2

65

8. 2007P2Q4

A curve with gradient function2

22x

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

(a) Find the value of k,(b) Determine whether the turning point is a maximum or minimum point.(c) Find the equation of the curve.

a) k = 1b) Minimum

c)2 2

5y xx

= + +

9. 2007P2Q10

Diagram shows part of the curve y = k(x 1)3, where k is a constant. The curveintersects the straight line x = 3 at point A.

At point A,dy

24dx

= .

(a) Find the value of k.(b) Hence, calculate

(i) the area of the shaded region P,(ii) the volume generated, in terms of

, when the region R which isbounded by the curve, the x-axis

and the y-axis, is revolved through360 about the x-aixs.

a) k = 2b) i) 8

ii)4

7


7/16


Vector (Paper 1)

1. 2003P1Q12

Diagram shows two vectors, OP

and QO

. Express

(a) OP

in the formx

y

.

(b) OQ

in the form x i y j+

.

2. 2003P1Q13

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

3. 2003P1Q14

Diagram shows a parallelogram ABCD with BED as a

straight line. Given that AB 6p=

, AD 4q=

and DE =

2EB, express in terms of p

and q

:

(a) BD

(b) EC

4. 2004P1Q16

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

and j

,

(a) AB

,

(b) the unit vector in the direction of AB

.

5. 2004P1Q17

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

AB 2BC 10i 12j+ =

.

6. 2005P1Q15

Diagram shows vector OA drawn on a Cartesian

plane.

(a) Express OA in the form xy

.

(b) Find the unit vector in the direction of OA .

7. 2005P1

Q16

Diagram shows a parallelogram, OPQR, Drawn on a

Cartesian plane. It is given that OP 6i 4j= +

and

PQ 4 i 5j= +

. Find PR .

8. 2006P1Q13

Diagram shows two vectors, OA

and AB

. Express

(a) OA

in the formx

y

,

(b) AB

in the form x i yj+

.


8/16


9. 2006P1Q14

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

and QR 3a (1 k)b= + +

, where

k is a constant. Find(a) the value of k,(b) the ratio of PQ : QR.

10. 2007P1

Q15

Diagram shows a rectangle OABC and the point Dlies on the straight line OB. It is given that OD =

3DB. Express OD

in terms of x

and y.

11. 2007P1Q15

The following information refers to vectors a

andb

2 1a , b

8 4

= =

Find(a) the vector 2a b

,

(b) the unit vector in the direction of 2a b

Answers:1.

(a)5

3

(b) 8 4i j +

2. h = 2, k= 133. (a) 6 4p q +

(b)8

32p q+

4.(a)

5

12

(b)51

1213

5. m = 6, p = 26.

(a)12

5

(b)121

513

7. 10i j + 8. 5

2, 4 : 3k =

9.(a)

4

5

(b) 4 8i j

10. ( )34 9 5x y+ 11.

(a)5

12

(b)51

1213

Vector (Paper 2)

1. 2003P2Q6

Given that5

AB

7

=

,2

OB

3

=

andk

CD

5

=

, find

(a) the coordinates of A,(b) the unit vector in the direction of OA ,(c) the value of k, if CD is parallel to AB .

2. 2004P2Q8

Diagram shows triangle OAB. The straight line AP intersects the straight line OQ at R. It is given

that1

OP OB3

= ,1

AQ AB4

= , OP 6x=

and OA 2y=

.

(a) Express in terms of x

and/or y

:

(i) AP

(ii) OQ

(b) (i) Given that AR h AP= , state AR in terms of h, x

and y

.

(ii) Given that RQ k OQ=

, state RQ

in terms of k, x

and y

.

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


9/16


3. 2005P2Q6

ABCD is a quadrilateral. AED and EFC are straight lines. It is

given that AB 20x=

, AE 8y=

, DC 25x 24y=

,

1AE AD

4= and

3EF EC

5= .


and/or and y

.

(i) BD

(ii) EC

(b) Show that the points B, F and D are collinear.(c) If x 2=

and y 3=

, find BD

4. 2006P2Q5

Diagram shows a trapezium ABCD. It is given that AB 2y=

,

AD 6x=

,

2AE AD

3=

and5

BC AD6

=

.

(a) Express AC , in terms of x and y.(b) Point F lies inside the trapezium ABCD such that

2EF 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 thevalue of m.

5. 2007P2Q8

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

point Q lies on AB. The straight line BP intersects thestraight line OQ at the point S. It is given that OA : OP = 4 :

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


and y

.

(i) BP

. (ii) OQ

.

(b) Using OS hOQ= and BS kBP= , where h and k areconstants, find the value of h and of k.

(c) Given that x 2 unit, y 3 unit= =

and AOB 90 = ,

find AB

.

Answers:1.

a) ( 3, 4) b)

35

45

c) 257

k =

a)

9 32 2

) 6 2

)

i x y

ii x y

+

2.

b)

9 32 2

) 6 2

)

+

i hx h y

ii k x k y

c) 1 12 3

,h k= =

a ) 32 20

) 25

i y x

ii x

3.

c) 104 unit

a) 5 2AC x y= +

4.b) 4AF x m y= +

8

5m=

a) ) 6 2

) 4 3

i x y

ii x y

+

+

b) 2 45 5

,h k= =

5.

c) 24.08 unit


10/16


Trigonometric Functions (Paper 1)

1. 2003 P1 Q20 Given that tan = t, 0 < < 90, express, in terms of t:(a) cot , (b) sin (90 ) (a)

1

t(b)

2

1

1t +

2. 2003 P1 Q21 Solve the equation 6 sec2 A 13 tan A = 0, 0 A 360. 33.69, 56.31,213.69

, 236.31

3. 2004 P1 Q18 Solve the equation cos2 x sin2 x = sin x for 0 x 360. 30, 150, 2704. 2005 P1 Q17 Solve the equation 3 cos 2x = 8 sin x 5 for 0 x 360. 41.81, 138.195. 2006 P1 Q15 Solve the equation 15 sin2 x = sin x + 4 sin 30 for 0 x 360. 2335, 15625,

19928, 34032

6. 2007 P1 Q17 Solve the equation cot x + 2 cos x = 0 for 0 x 360. 90, 210, 270,330

Trigonometric Functions (Paper 2)

1. 2003P2Q8

(a) Prove that tan + cot = 2 cosec 2.

(b) (i) Sketch the graph of3

y 2 cos x2

= for 0 x 2.

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

3 3cos x x 1

2 4=

. Hence, using the same axes, sketch the straight line

and state the number of solutions to the equation3 3

cos x x 12 4

=

for

0 x 2.

32,

2

3 solutions

=

xy

2. 2004P2Q3

(a) Sketch the graph of y cos 2x= for 0 x 180.

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

of solutions satisfying the equation 2x

2sin x 2180

= for 0 x 180.

1,180

2 solutions

= x

y

3. 2005P2Q5

(a) Prove that cosec2 x sin2 x cot2 x = cos 2x.(b) (i) Sketch the graph of y cos 2x= for 0 x 2.

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

number of solutions to the equation 2 2 2x

3(cos ec x 2sin x cot x) 1 =

for 0 x 2. State the number of solutions.

1,

3 3

4 solutions

=

xy

4. 2006P2Q4

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

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

solutions to the equation 2 cos x 0x

+ = for 0 x 2. State the number of

solutions.

,

2 solutions

=y

x

5. 2007P2Q3

(a) Sketch the graph of y 3cos2x= for 0 x 2.

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

of solutions for the equationx

2 3cos 2x

2 =

for 0 x 2. State the

number of solutions.

2 ,2

8 solutions

=

xy


11/16


Permutations and combinations

1. 2003P1Q22 Diagram shows 5 letters and 3 digits. A code is to be formed using those letters and

digits. The code consists of 3 letters followed by 2 digits. How many codes can beformed if no letters or digit is repeated in each code?

360

2. 2003P1Q23

A badminton team consists of 7 students. The team will be chosen from a group of 8boys and 5 girls. Find the number of teams that can be formed such that each teamconsists of(a) 4 boys, (b) not more than 2 girls.

a) 700b) 708

3. 2004P1Q23

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

side.

a) 120b) 48

4. 2005P1Q22

A debating team consists of 5 students. These 5 students are chosen from 4 monitors,2 assistant monitors and 6 prefects. Calculate the number of different ways the teamcan be formed if(a) there is no restriction,(b) the team contains only 1 monitor and exactly 3 prefects.

a) 792b) 160

5. 2006P1Q22

Diagram shows seven letter cards. A four-letter code is to be formed using four ofthese cards. Find

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

a) 840b) 480

6. 2007P1Q23

A coach wants to choose 5 players consisting of 2 boys and 3 girls to form abadminton team. These 5 players are chosen from a group of 4 boys and 5 girls. Find(a) the number of ways the team can be formed,(b) the number of ways the team can be arranged in a row for a group photograph, if

the three girls sit next to each other.

a) 60b) 18

Probability

1. 2004P1Q24

A box contains 6 marbles and kblack marbles. If a marbles is picked randomly from

the box, the probability of getting a black marbles is 3

5. Find the value ofk.

k= 9

Colour Number of Cards

Black 5

Blue 4

Yellow 3

2. 2005P1Q24

Table 1 shows the number of coloured cards in abox. Two cards are at random from the box. Find theprobability that both cards are of the same colour.

1966

3. 2006P1Q23

The probability that hamid qualifies for the final of a track event is 2

5while the

probability that Mohan qualifies is1

3. Find the probability that

(a) both of them qualify for the final,(b) only one of them qualifies for the final.

(a) 215

(b) 7

15


12/16


Probability Distribution (Paper 1)

1. 2003P1Q24

Diagram shows a standard normal distributiongraph. If P(0 < z < k) = 0.3128, find P(z > k).

0.1872

2. 2003P1Q25

In an examination, 70% of the students passed. If a sample of 8 students israndomly selected, find the probability that 6 students from the sample passed theexamination.

0.2965

3. 2004P1Q25

X is a random variable of a normal distribution with a mean of 52 and a variance of

144. Find(a) the Z score if X = 67, (b) P(52 X 67).

a) 1.25b) 0.3944

4. 2005P1Q25

Diagram shows a standard normaldistribution graph. The probabilityrepresented by the area of the shaded

region is 03485.(a) Find the value of k.(b) X is a continuous random variable

which is normally distributed with amean of 79 and a standard deviation of3. Find the value of X when the z-scoreis k.

a) 1.03b) 82.09

5. 2006P1Q25

The mass of students in a school has a normal distribution with a mean of 54 kg

and a standard deviation of 12 kg. Find(a) the mass of the students which gives a standard score of 0 5.(b) The percentage of students with mass greater than 48 kg.

a) 60

b) 69.146

6. 2007P1Q24

The probability that each shot fired by Ramli hits a targets is1

3.

(a) If Ramli fires 10 shots, find the probability that exactly 2 shots hit the target.(b) If Ramli fires n shots, the probability that all the n shots hit the target is 1

243.

Find the value of n.

a) 0.1951b) 5

7. 2007P1Q25

X is a continuous random variable of a normal distribution with a mean of 52 and a

standard deviation of 10. Find(a) the z-score when X = 67.2,(b) the value of k when p(z < k) = 0.8849.

a) 1.52

b) 1.2

Probability Distribution (Paper 2)

(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?

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

a variance of 4225 kg2. 200 of the workers in the factory weigh between 50 kg and 70kg. Find the total number of workers in the factory.

1. 2003P2Q10


13/16


(a) A club organizes a practice session for trainees on scoring goals from penalty kicks. Eachtrainee takes 8 penalty kicks. The probability that a trainee scores a goal from a penaltykick is p. After the session, it is found that the mean number of goals for a trainee is 48.(i) Find the value of p,(ii) If a trainee is chosen at random, find the probability that he scores at least one goals

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

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

2. 2004P2Q11

(a) The result of a study shows that 20% of the pupils in a city cycle to school. If 8 pupilsfrom 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,

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

mean of 3.2 kg and a standard deviation of 05 kg. Find(i) the probability that a water-melon chosen randomly from the orchard has a mass of

not more than 40 kg.(ii) the value of m if 60% of the water-melons from the orchard have a mass of more

than m kg.

3. 2005P2Q11

4. 2006P2Q11

An orchard produces lemons. Only lemons with diameter, x greater than k cm are graded andmarketed. Table below shows 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 anda standard deviation of 15 cm.

(a) If one lemon is picked at random, calculate the probability that it is of grade A.(b) In a basket of 500 lemons, estimated the number of grade B lemons.(c) If 857% of the lemons is marketed, find the value of k.

2007P2Q11

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.

5.

b) A group of workers are given medical check up. The blood pressure of a worker has anormal distribution with a mean of 130 mmHg and a standard deviation of 16 mmHg.Blood pressure that is more than 150 mmHg is classified 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 114 mmHg and 150 mmHg.

(ii) It is found that 132 workers have high blood pressure. Find the total number ofworkers in the group.

Answers (Paper 2):a) i) 0.4233

ii) 8001.

b) 319

a) i) 0.6ii) 0.9993446

2.b) i) 0.2743

ii) 67.625

a) i) 0.2936ii) 0.79691

3.b) i) 0.9452

ii) 3.0735

a) 0.2119b) 0.4912

4.c) 4.1965

a) i) 0.2090ii) 0.8936

5.b) i) 0.73569

ii) 1250


14/16


Prepare by Tan Sze Haun

Motion along a straight line

1. 2003P2Q12

A particle moves in a straight line and passes through a fixed point O, with a

velocity of 24 ms1. Its acceleration, a ms2, t s after passing through O is given bya = 10 2t. The particle stops after k s.(a) Find

(i) the maximum velocity of the particle,(ii) the value of k.

(b) Sketch a velocity-time graph for 0 t k. Hence, or otherwise, calculate thetotal distance traveled during that period.

a) i) 49

ii) 12

b) 432

2. 2004P2Q15

A particle moves along a straight line from a fixed point P, Its velocity, V ms1, isgiven by V 2t(6 t)= , where t is the time, in seconds, after leaving the point P.

(Assume motion to the right is positive.) Find(a) the maximum velocity of the particle,(b) the distance traveled during the third second,(c) the value of t when the particle passes the point P again.(d) the time between leaving P and when the particle reverses its direction of

motion.

a) 18

b) 13

17

c) 9

d) 6

3. 2005P2Q15

Diagram shows the positions and directions of motion of two objects, P and Q,moving in a straight line passing two fixed points, A and B, respectively. Object Ppasses the fixed point A and object Q passes the fixed point B simultaneously. Thedistance AB is 28 m.

The velocity of P, vp ms

1, is given by 2pv 6 4t 2t= + , where t is the time, inseconds, after it passes A while Q travels with a constant velocity of 2 ms1.Object P stops instantaneously at the point C. (Assume that the positive directionof motion is towards the right.) Find(a) the maximum velocity, in ms1, of P,(b) the distance, in m, of C from A,(c) the distance, in m, between P and Q when P is at the point C.

a) 8

b) 18

c) 4

4. 2006P2Q12

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

velocity, v ms1, is given by 2v t 6t 5= + , where t is the time, in seconds,

after leaving O. (Assume motion to the right is positive.)(a) Find

(i) the initial velocity of the particle,(ii) the time interval during which the particle moves towards the left,(iii) the time interval during which the acceleration of the particle is

positive.(b) Sketch the velocity-time graph of the motion of the particle for 0 t 5.(c) Calculate the total distance traveled during the first 5 seconds after

leaving O.

a) i) 5

ii) 1 < t < 5

iii) t > 3

b)

c) 13

5. 2007P2Q12

A particle moves in a straight line and passes through a fixed point O. Its velocity,

v ms1, is given by 2v t 6t 8= + , where t is the time, in seconds, after passing

through O. (Assume motion to the right is positive.) Find(a) the initial velocity, in ms 1,(b) the minimum velocity, in ms 1,(b) the range of values of t during which the particle moves to the left.(c) the total distance, in m, traveled by the particle in the first 4 seconds.

a) 8

b) 1

c) 2 < t < 4d) 8


15/16



Linear Programming

1. 2003P2Q14

Yahya has an allocation of RM225 to buy x kg of prawns and y kg of fish. The total mass of thecommodities is not less than 15 kg. The mass of prawns is at most three times that of fish. Theprice of 1 kg of prawns is 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 of the aboveconditions.

(b) Hence, using a scale of 2 cm to 5 kg for both axes, construct and shade the region R thatsatisfies all the above conditions.

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

2. 2004P2Q14

A district education office intends to organize a course on the teaching of Mathematics andScience in English. The course will be attended by x Mathematics participants and y Scienceparticipants. 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 RM120 and for a Science participant is RM80.

Write down three inequalities, , other than x 0 and y 0, which satisfy the above constraints.Hence, by using a scale of 2 cm to 10 participants for both axes, construct and shade the region R

which satisfies all the above constraintsUsing your graph from (b), find

(i) the maximum and minimum number of Mathematics participants when the number ofScience participants is 10,

(ii) the minimum cost to run the course.

3. 2005P2Q14

An institution offers two computer courses, P and Q. The number of participants for course P is xand for course Q is y. The enrolment of participants is based on the following constraints:I : The total number of participants is not more that 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 participant 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 the aboveconstraints.

(b) Hence, by using a scale of 2 cm to 10 participants for both axes, construct and shade theregion R that satisfies all the above constraints(c) Using your graph from (b), find

(i) the range of the number of participants for course Q if the number of participants forcourse P is 30,

(ii) the maximum total fees per month that can be collected if the fees per month for coursesP and Q are RM50 and RM60 respectively.


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4. 2006P2Q14

A workshop produces two types of rack, P and Q. The production of each type of rack involvestwo processes, making and painting. Table below shows the time taken to make and paint a rackof type P and a rack of type Q.

Time taken (minutes)Rack

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 racksper 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 down three inequalities, other than x 0 and y 0, which satisfy the above constraints.(b) Hence, by using a scale of 2 cm to 2 racks for both axes, construct and shade the region R

that satisfies all the above constraints(c) Using your graph from (b), find

(i) the maximum 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 RM 24 and from

one rack of type Q is RM 32.

5. 2007P2Q14

A factory produces two components, P and Q. In a particular day, the factory produced x piecesof component P and y pieces of component Q. The profit from the sales of a piece of componentP is R 15 and a piece of component Q is R 12. The production of the components per day is basedon the following constraints:I : The total number of component is at most 500.II : The number of component P is not more than three times the number of component Q.III : The minimum total profit for both components is R 4200.

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

Hence, by using a scale of 2 cm to 50 components for both axes, construct and shade theregion R that satisfies all the above constraints

(d) Use your graph in (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.

Answers:I x + y 15

II x 3y

a)

III 9x + 5y 225

1.

c) RM 130

I x + y 40II y 2x

a)

III 3x + 2y 180i) 30, 53

2.

c)

ii) RM 3760

I x + y 100

II y 4x

a)

III y x + 5i) 35 y 70

3.

c)

ii) RM 5800

I 3x + y 36II 3x + 4y 36

a)

III y 3x

4.

c) i) 4 RM ii)720

I x + y 500II 3y x

a)

III 5x + 4y 1400i) 225

5.

c)

ii) RM 7125

past year collection f5 with answers

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