Perfect Score Sbp Add Mth 2007[1]

SEKTOR SEKOLAH BERASRAMA PENUH BAHAGIAN SEKOLAH KEMENTERIAN PELAJARAN MALAYSIA ADDITIONAL MATHEMATICS

NAME : SCHOOL :

PERFECT SCORE PROGRAM SBP 2007

MODULE 1 ADD MTH ( 3472 / 1 )

3472/1

5

Answer all questions 1 It is given that set P = { 4 , 6 , 9 , 25 } and set Q = { 2 , 3 , 5 }. If the relation between set P and set Q is ‘ the multiple of ’, state a) b)

the domain the image of 9 [2 marks]

Answer : (a)…………………………… (b)…………………………… _________________________________________________________________________ 2 Diagram 1 shows the relation between set A and set B. set B



20 15







10 5

 2

 4

6

8

set A

DIAGRAM 1

State (a) the objects of 20 (b) the range of this relation. [ 2 marks] Answer: (a)…………………………. (b)………………………….

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6

3

3x  2 Given that h 1 ( x)  , find 3

(a) h(5), (b) the value of m if h 1 (3m)  2m  3 . [4 marks]

Answer : (a) ……………………… (b) ………………………

Given the function f: x  2x + 5 , g : x 

4

x2 mx  n and fg: x  , 5 5

where m and n are constants , find (a) the value of m and of n, (b) the value of gf(2). [4 marks ]

Answer: (a) m =…………..n =……………. (b)…………………..……………..

5 The roots of the quadratic equation x 2  10 x  3  3k  0 are in the ratio of 2 : 3. (a) find the roots (b) hence, find the value of k. [4 marks]

Answer : (a)……………………….. (b)..………………………

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7

6

Diagram 2 shows the graph of the function f(x) = ax2+bx + c y

O

– 7

x

 3,25 

DIAGRAM 2

The point  3,25 is a minimum point of a curve. Find the equation of the curve. [ 3 marks]

Answer :..........................................

7

Find the range of values of p if f ( x)  px( x  1)  px  2 is always positive. [3 marks]

Answer : …………………………

8

5n   If the minimum value of the function f ( x)  3  ( x  2) 2   is 3, 2   find the value of n.

[2 marks]

Answer : …………………

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8

9 Find the range of values of x for which ( x  1)( x  2)  12 [3 marks]

Answer : ………………………… _________________________________________________________________________ 10 Solve the equation 2 3 x  4  2 3 x  2  24 . [3 marks]

Answer : …………………… 11 Solve the equation log 3 2  log 9 3x  2 . [ 3 marks ]

Answer : …………………………

12 Given that log m 27  y and log

n

3  x , express log 9 m 4 n 3 in terms of x and y. [4 marks]

Answer : …………………………

13 Given that x = 5k and y = 5h , express log 5

x3 in terms of k and h. 125 y 2 [ 4 marks ]

Answer : …………………………

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14 The sum of the first n terms of an arithmetic progression is given by S n  3n 2  13n. Find (a) the ninth term, (b) the sum of the next 20 terms after the 9th terms. [4 marks]

Answer : (a) ……………………… (b) ……………………… _________________________________________________________________________ 15 The first term of a geometric progression is a , and the common ratio, r , is positive. 10a Given that the sum of the second and the third term is and the sum of the first 9 four terms is 65. Find (a) the common ratio, (b) the first term. [ 4 marks ]

Answer : (a) ……………………… (b) ……………………… 16 Diagram 3 shows a straight line y  3 x  6 which is perpendicular to the straight line that joins points A(2, 3) and B(m,n). y y=3x+6 B(m,n)

A(2 , 3) x O DIAGRAM 3

Express m in terms of n.

[3 marks]

Answer…………………………

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17 Diagram 4 shows a semicircle KLMN, of diameter KLM , with centre L. y

N (x,y)



K

L 0

x

M DIAGRAM 4

x y   1 and point N( x , y ) lies on 4 3 the circumference of a circle KLMN , find the equation of the locus of the moving point N. [ 3 marks ]

Given that the equation of the straight line KLM is

Answer………………………… 18 Given that x and y are related by the equation y = Ax4k, where A and k are constants. A straight line is obtained by plotting log 8 y against log 8 x, as shown in diagram 5. log 8 y ( 143 , 10)

( 53 , 4) log 8 x

0 DIAGRAM 5

Calculate the value of A and of k.

[4 marks]

Answer : k = …………………… A = …………………… 3472/1

11

x2 y2 19 Given that x and y are related by the equation 2  2  1 , where p and q are p q positive constants. When the graph of y 2 against x 2 is plotted, a straight line with 1 9 gradient and passes through the point (0, ) is obtained. 4 4 Find the values of p and q. [ 4 marks]

Jawapan : p = …………..………… .

q = ………………………. 20 Find the equation of the tangent to the curve y 

5 at the point (3, 4). ( x  5) 3

[2 marks]

Answer : …… …………………… ________________________________________________________________________ d2y  6 x and gradient of the curve is –12 when x = 2. If P (2,–4) dx 2 lies on the curve, find

21 Given that

(a) the equation of the normal at P, (b) the equation of the curve. [ 4 marks ]

Answer : (a) ……………………… (b) ………………………

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22

Given that y  m 2  1 and x 

1 dy  m. , find in terms of m. m dx

[3 marks]

Answer : …… ……………………

23 A

T

B O

DIAGRAM 6

Diagram 6 shows a circle, with center O and radius 10 cm. Tangent to the circle at A meet the line OB at T. Given the area of the triangle OAT = 60 cm², find the area of sector OAB. [ use  = 3.142] [4 marks]

Answer : …………………………

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24

Diagram 7 shows a semicircle ABC with center O.

C

 A

B

O DIAGRAM 7

The length of arc BC is 20 cm and the area of sector BOC is 105.68 cm2, find the value of  in radian. Give your answer correct to four significant figures. [ 4 marks ]

Answer : …………………………

25 Given that A(-1, 4), B(2, -3) and O is origin. 

(a) express AB in term of xi  y j , 

(b) find AB . [3 marks] Answer:(a)……………………… (b)…..…………………… 26 The information below shows the vectors AB , CB and AC 

AB  2i  3 j 

CB  3i  k j 

AC  4hi  2 j,

h, k are constant

Find the value of h and of k. [3 marks] Answer : h = ………………………… k = ………………………… 3472/1

14

27 Given that (4a  44) p = (b + 5) q , where p and q are not parallel. Find the value of a and of b. [2 marks]

Answer : a = ……………………… b = ………….……………

28

Diagram 8 shows a parallelogram ABCD such that AEC is a straight line.

D

C E

A

B DIAGRAM 8

Given AD = 4a + 2b, AC = 6a + 3b and EC =

1 AC . Express BE in terms of a and b. 3

[3 marks]

Answer: …………………………

29

 2n 2 Find the value of lim n  3  n 2 

  .  [ 2 marks ]

Answer : …………………………

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30

The height of a cone is 10 cm. If its radius is increasing at the rate of 0.5 cm s -1, find the rate of increase of its volume at the instant its radius is 5 cm. [ 3 marks]

Answer : …………………………

31

Given that y  10 x  (a) the value of

27 , calculate x2

dy when x = 3, dx

(b) the approximate value of y, in terms of p, when x  3  p, where p is small. [ 4 marks ]

Answer : (a)……………………… (b)…..…………………… 32 The equation of a curve is y  x 

4 . Find the coordinate of the turning point of the x2

curve.

[ 3 marks ]

Answer : (a)…………………… (b)…..……………… ..

33 Given that y =

x2  3 dy and 5h( x)  , find the value of 3 dx x

2

 [h( x)  4]dx

.

1

[ 4 marks ]

3472/1

Answer : .................................... 16

x2 34 Diagram 9 shows the shaded region bounded by the curve y  k  . 2 y A B 2

yk

x 2

C O

DIAGRAM 9

2

x

Given that the volume generated when the shaded region OABC is revolved by 360o about y  axis is 28, find the value of k. [ 4 marks ]

Answer : .……………………...

35 Diagram 10 shows the curve y 

2 , the straight lines x = 1 and x = k x2

y

y O

1 X

k

2 x2

x

DIAGRAM 10

Find the value of k if the area of shaded region is

8 unit 2 . 5

[4 marks] Answer:.……………………... 36 Solve the equation sin(60 0  x)  (sin 60 0  x) 

3 for 0  x  360. 2

[ 3 marks ]

Answer : .…………………………. 3472/1

17

37 Find all the values of x, between 0 0 and 360 0 , which satisfy the equation 2 sin 2 x  cos x(1  sin x) . [4 marks]

Answer : ………………………… 38 Solve the equation 6 sec 2 x  tan x  8 for 0  x  360. [4 marks]

Answer : (a)……………………… (b) ……………………..

39 Diagram 11 shows graph for the function y = a sin bx y 3

O

180 0

360

0

x

-3 DIAGRAM 11

Find the value of a and b. [ 2 marks ]

Answer : a =……………………… b= ……………………..

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40 A chess club has 10 members of whom 6 are men and 4 are women. A team of 4 members is selected to play in a match. Find the number of different ways of selecting the team if (a) all the players are to be of the same gender, (b) there must be an equal number of men and women. [3 marks]

Answer : (a)……………………… (b)……………………... 41 It is given that six digits numbers 1, 2, 3, 4, 5, and 6. Calculate the different ways of odd numbers which are less than 200 000 can be formed with out repetitions. [ 3 marks ]

Answer : ………………………….

42 Five letters from the word ‘INTEGRAL’ are to be arranged . Calculate the number of possible arrangements if they must begin and end with a vowel. .

[2 marks]

Answer : ………………………….

43 Diagram 12 shows 6 letters and 4 digits .

A B C D E F 2 3 4 5 DIAGRAM 12

A code is to be formed using the letters and digits. Each code must consist of 4 letters followed by 2 digits. Find the different codes that can be formed if repetitions are not allowed. [ 3 marks ] Answer : …………………………. 3472/1

19

44

Diagram 13 shows a set of data with a mean of 4.

1 , 1 , 7 , 2 , 1 , 3 , 7, m , n DIAGRAM 13

Given that m + n = 14 and standard deviation Find the values of m and n if m  n.

76 . 3

[4 marks]

Answer : …………………………. 45

Table 1 shows the frequency distribution of ages of workers. Age ( years ) Number of workers

28-32 16

33-37 38

38-42 26

43-47 11

48-52 9

TABLE 1

75  F  5 . Find the values of  G 

Given the third quartile of ages of workers is K  L   K , L , G and F.

[ 4 marks ]

Answer : K =…………………………. L = ........................................ G =………………………… F=……….......………………

46

There were 12 girls and 3 boys in a group of children. One child was chosen at random from the group. Another child was chosen at random from the remaining children. Calculate the probability that a child of each gender was chosen. [ 3 marks ]

Answer : ………………………….

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47

Hanif , Zaki and Fauzi will be taking a driving test. The probabilities that Hanif , 1 1 1 Zaki and Fauzi will pass the test are , and respectively. Calculate the 2 3 4 probability that (a) only Hanif will pass the test (b) at least one of them will pass the test. [ 3 marks ]

Answer : (a)……………………… (b) ….....………………..

48

In a lucky draw, the probability to obtain a prize is p . (a)

(b)

Find the number of draws required and the value of p such that the mean is 15 3 6 and the standard deviation is . 2 If 8 draws are carried out, find the probability that at least one draw will win the prize. [ 4 marks ]

Answer : (a)……………………… (b) ……………………..

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49

Diagram 14 shows the graph that represents the binomial probability distribution. P(X=x)

0.4 0.3 0.2 0.1 0

1

2

3

x

DIAGRAM 14

Calculate (a) P ( X = 1) (b) P ( X < 2 ) [ 2 marks ]

Answer : (a)…………………………. (b)………………………….

50 Diagram 15 shows a standard normal distribution graph. f(z)

-k

k

z

DIAGRAM 15

Given that the area of shaded region in the diagram is 0.7828 , calculate the value of k. [ 2 marks ]

Answer :....................................... END OF QUESTION PAPER 3472/1

22

PERFECT SCORE PROGRAM ADDITIONAL MATHEMATICS 2007

Module 1(3472/1) 1.

{4 , 6 , 9 , 25 } 2 and 3 4 dan 6 {5,15,20} 17/3 4/5 a = 2, b = 29 11/5

26

h=-¼,k=1

27

k =10/3

28

2a + b

29

2 3

5 6

k=–7

30 31

7 8

0
2 3 4

(a) (b) (a) (b) (a) (b) (a) (b)

f(x) = 2x

2

12 x  7

2 5 2 x 5 n

9 10 11

x=1/3

6

3 4

(a) (b)

32 33

1/8 8 33+ 8p (2,3) k = 26

34 35 36

k=5 3 3/8

240 0 ,300 0

12

6x  3y xy

37

19 0 28' ,90 0 ,160 0 32' ,270 0

13

3k-2h-3

38

33.7 0 ,153.4 ,213.7 0 ,333.4 0

64 2540 r = 23 a = 27 , 81 m = 11 –3n

39

a=3,b=2

40

14 553

x  y  4x  3y  0

41 42

48 720

k=½,A=4 p = 3 , q = 3/2 15x+16y-109=0

43 44 45

4320 m=5,n=9 K = 41.5385 , L = 37.5 G = 26 , F = 54

12y = x – 50

46

12

9/35 5/6

14 15

(a) (b) (a) (b)

16 17

2

18 19 20 21

(a) (b)

2

y  x  4 3

22

2m 3 m2 1

47

23

j = 6.875 cm

48

24 25

(a) (b)

 = 1.2491 rad

49

3i – 7j 7.6158

50

35

(a) (b)

p = 1 , n = 150

(a) (b)

05695 0.35 0.55 k = 1.234

10

PERFECT SCORE SBP 2007

MODULE 2 ADD MTH ( 3472/2 ) Part A

23

x y 7   1 . y x xy [5 marks]

1

Solve the simultaneous equations 2 x  y  7 and

2

A straight line 2 x  3 y  1  0 intersects a curve x 2  xy  2 at two points. Find the coordinate of the points. [5 marks]

3.

Diagram 1 shows the curve has a gradient function k - 2x , where k is a constant. The straight line x+ y = 4 is tangent to the curve at ( 1,3 ).

y

A x+ y = 4.

B x

0 DIAGRAM 1 Find a) the value of k , b) the equation of the curve, c) the area of the shaded region.

[2 marks] [4 marks] [2 marks]

24

4

Diagram 2 shows a circle with centre O and a radius of 5 cm. Radius OA is perpendicular to the radius OB. T is the mid point of OB. D

O

A

T C B DIAGRAM 2 Find (a)

5

AOC,

[2 marks]

(b) the length of the major arc of ADC,

[2 marks]

(c)

[4 marks]

the area of the shaded region.

A study has been carried out in a village to determine the age of a male got married. A sample of 150 males had been studied and the table below shows the results. Age (years) 16-20 21-25 26-30 31-35 36-40

Number of males 9 63 49 20 9

Calculate (a) the mean (b) the variance

[ 3 marks ] [ 3 marks ]

25

6

Table 1 shows the points in a competition for 40 students. Point Number of students

20-29

30-39

40-49

50-59

60-69

70-79

3

5

9

12

7

4

TABLE 1 Find (a) the mean, (b) the standard deviation of frequency of distribution.

7

[ 5 marks ] [ 2 marks ]

Diagram 3 shows a series of cones. The base radius of each one is fixed at 4 cm. The height of the first cone is h cm. The height of the second cone is ( h + 1 ) cm and the height of the third cone is ( h + 2 ) cm. The height of each cone is increase by 1 cm compared to the previous cone.

DIAGRAM 3

a) Determine whether the volumes, in cm 3 , of these cones are in an arithmetic or geometric progression. Hence, state the common difference or common ration. [ 4 marks ] b) If h = 3 , find the sum of the volumes of the first 13 cones, in term of  . 1 [ Volume of cone = r 2 h ] 3 [ 3 marks ]

26

8

Diagram 4 shows a sector OABC , with centre O and a radius of 4 cm. Given that the AOC = 135o and BOC = 900 . B A

O O

4 cm

C

DIAGRAM 4 Find

9

(a)

the perimeter of the shaded region,

[4 marks]

(b)

the area of the shaded region.

[4 marks]

Solutions to this question by scale drawing will not accepted. Diagram 5 shows a quadrilateral ABCD whose vertices A (1,5), B (-6,2), C (0, h) and D( k,2). Given that the straight line AD is perpendicular to the straight line CD and the equation of the straight line BC is 2x + 3y +6 =0. y y A (1,5)

M

B (-6,2)

D (k,2) x

C (0,h) DIAGRAM 5

27

a)

Find (i)

b)

10

the value of h and of k,

[2 marks ]

(ii) the equation of the straight line AC ,

[2 marks ]

(iii) the area of quadrilateral ABCD.

[2 marks ]

If M is the intersection point between AC and BD, find the ratio of AM : MC. [2 marks ]

A vessel is filled with water after t second. The depth of the water, x cm in the vessel increases at the rate of 1.44 cm s-1.Given that the vessel is empty when t = 0. Find (a)

the value of t when x = 18.

(b)

small change in x when t increases from 4.0 to 4.1 [ 5 marks ]

11

  4 1 h Given that PQ =   , QR =   and RS =   ,  7   2  20  (a) express as a column vector of PQ + 3 QR , (b) find the value of h if RS is parallel to PQ , (c) the unit vector in the direction of PQ . [7 marks]

12

The minimum value of f ( x)  x 2  2mx  4n is  4 when x = 6. (a)

Without using differentiation method, find the values of m and of n.

( b)

Hence, sketch the graph of f ( x)  x 2  2mx  4n

for the domain

2  x  9.

(c)

Find the range of f ( x)  x 2  2mx  4n for the domain 2  x  9 . [7 marks]

28

MODULE 3 ( 3472/2 ) Part B

29

Part B 13

The variables x and y are known to be related by the equation y2 = ax + bx2 , where a and b are constants. Table 2 shows the corresponding values of x and y obtained from an experiment : x y

2 2.82

4 4.91

6 7.01

8 9.02

10 11.2

TABLE 2 y (a)

By using a scale of 2 cm to represent 1 unit on both axes, plot

against x x [5 marks]

Hence, draw the line of best fit. (b)

Use the graph in (a) to find (i) (ii)

14

2

the values of a and b, the values of x satisfies (a5)x + bx2 = 0.

[2 marks] [3 marks]

The values of two variables, x and y , obtained from an experiment are as shown in the Table 3. x y

3 4.5

4 8.9

5 18.1

6 36.0

TABLE 3 It is known that the variables x and y are related by the equation y = pq x 3 where p and q are constants. a)

Plot a graph of log 10 y against (x3) and draw a line of best fit. [5 marks]

b)

Use your graph from (a) to find i) the value of y when x = 2 ii) the values of p and q

[2 marks] [3 marks]

30

15

Diagram 6 shows a curve y 2  x  9 and a straight line y = x – 3. y C A

y2  x  9

y = x–3 0

x

B

DIAGRAM 6

(a)

Find the coordinates of A , B and C. Hence, calculate the area of shaded region. [5 marks]

(b)

A region which bounded by the curve y 2  x  9 and the y-axis is revolved through 3600 about the y-axis. Calculate the volume generated in term of  . [5 marks]

31

16 ( a) A circular cylinder, open at one ends, has radius r cm and external surface area 27  cm2 . Given that the volume of the cylinder , V cm3 , is given by  V  27 r  r 3  . Find the stationary value of V, hence, determine whether this 2 value is a maximum or a minimum. [ 5 marks ] y

(b)

y = x2 + 1 A

B y =5 PP QQ

O

C

x

DIAGRAM 7 Diagram 7 shows a rectangle enclosed by the y-axis , the x-axis, the straight line y = 5 and the straight line BC. The curve y = x2 + 1 divides the rectangle OABC into two sections , P and Q . Find the ratio of the area P : area Q. [ 5 marks ]

17

  In Diagram 8, OM  2 p and ON  5q .   A

B

M L 2p  O

5q 

N

DIAGRAM 8

32

(a)

Given that OM  MA and AB  2ON . Express each of the following in terms of p and/or q .    (i) MN  (ii) OB [3 marks]

  (b) Given that LN  h MN

  and LB  k OB , show that

  OL  2h p  5(1  h) q and OL  (8  8k ) p  (10  10k ) q .     Hence , find the value of h and of k. [5 marks]

18

In Diagram 9, ABCD is a quadrilateral such that the line DB intersects the line EC at F. A

E

B

F

C

D DIAGRAM 9

Given that DE 

DF = m DB and a) b) c)

2 1 DA , AB  EC , DA  10 x , DC  10 y , 5 2 DF  DE  n EC.

Find the value of m and of n. [4 marks ] Hence, find DF:FB. [3 marks ] If the area of triangle DEF is 4 unit2, evaluate the area of triangle DAB. [3 marks ] 33

19

a) Prove (cos 2x + 1) tan x = sin 2x. [4 marks] b) i) Sketch the graph of y  1  3cos x for 0  x  2 ii) Find the equation of a suitable line for solving the equation   3 cos x  2 x . Hence, using the same axes, sketch the straight line and state the number of solutions to the equation   3 cos x  2 x for 0  x  2 . [ 7 marks]

20 (a) Given that sin 3 A  3sin A  4sin 3 A and kos3 A  4kos 3 A  3kosA Prove that

sin 3 A  sin A  tan A kosA  kos3 A

(b) Solve the equation 2 sin 2 ( x  30 0 )  3 sin( x  30 0 )  1  0 for 0 o  x  360 o .

  2 x) for 0  x  2 . 2 Hence , find the number of solutions to the equation x cos 2 x   1 for 0  x  2 

[3 marks]

[3 marks]

(c) Sketch the graph of y = 3 sin (

[4 marks]

21 (a) A study shows that 40 % of the students in a school entered university after the SPM. A sample of 10 students was chosen at random. Calculate i) the probability of at least 9 students entering university. ii) the number of SPM students to be taken in order that the probability of at least one student who enters university is more than 0.85 [ 5 marks] b) The marks for 500 candidates in an Additional Mathematics examination in normally distributed with a mean of 45 and a standard deviation of 5 marks. i) If a candidate is chosen at random, calculate the probability of his marks between 47 and 52. ii) Given that 5 % of the students obtained excellent grades, find the minimum mark for a candidate to obtain an excellent grade. [ 5 marks]

34

22 (a) A study on post graduate students, revealed that 70% out of them obtained jobs six months after graduating. (i)

If 15 post graduates were chosen at random, find the probability of not more than 2 students not getting jobs after six months.

(ii)

It is expected that 280 students will succeed in obtaining jobs after six months. Find the total number of students involved in the study. [5 marks]

(b)

The mass of 5000 students in a college is normally distributed with a mean of 58kg and variance of 25kg2. Find (i)

the number of students with the mass of more than 90kg.

(ii) the value of w if 10% of the students in the colleges are less than w kg. [5 marks]

3 23 (a) The volume , V cm3 , of the sphere of radius r cm , is given by the formula V   r3 4 . A pump puts air into a spherical balloon at the rate of 250 cm3 s-1. Calculate

(i)

the rate of surface area of the balloon when the radius is 10 cm,

(ii)

approximate change in volume as the radius decreases from 10 cm to 9.95 cm. ( give your answer in terms of  ) [ 6 marks ]

(b) Given that g ( x) 

1 1 , evaluate g ''   . 2 8( 4 x  5) 2

[ 4 marks ]

35

24 Diagram 10 shows the straight lines PQS and QRT. Q is the midpoint of PS. y

S Q R (0 ,1)

8x + 3y = 12 P

x

0 T DIAGRAM 10

a) Find i) the coordinate of point Q, ii) the area of the quadrilateral OPQR. [ 3 marks ] b) Given that QR : RT = 1: 3, find the coordinates of point T. [ 2 marks ] c) A point W moves in such a way that its distance from point T is twice its distance from point S. i) Find the equation of the locus of the locus of point W, ii) Hence, determine whether the locus will intersect the x-axis or not. [ 5 marks ]

36

25

Diagram 11 shows two arcs, AD and BC , of two concentric circles, with the same centre O. B

Q

P

C

A

D

O

DIAGRAM 11

BD is perpendicular to OC. It is given that OA = OD = 5 cm, OC = 14 cm and AOD = 1.2056 radians. Using  = 3.142, calculate (a) the area of region P, in cm² [4 marks] (b) the perimeter of region Q , in cm

[3 marks]

(c) the total perimeter , in cm, of regions P and Q

[3 marks]

37

MODULE 4 ( 3472/2 ) Part C

38

Part C 26

A particle moves in a straight line so that its distance, s metres, from a fixed point A on the line is given by s  2t 2  4t  9, for t  3 , where t is the time in seconds after passing through a point B on the line. Find

(a) the distance AB,

[1 mark]

(b) the distance from A of the particle when it is instantaneously at rest, [2 marks] (c) the total distance traveled by the particle in the period t = 0 to t = 3, [3 marks] (d) If t =3 , the acceleration of the particle is changed to ( t – 8 ) ms-2 , the instantaneous velocity remaining unchanged. Hence, find the next value of t at which the particle comes to instantaneous rest. [4 marks]

27

A car moves along a straight horizontal road so that, t seconds after passing a fixed point A with a speed of 5 ms 1 , its acceleration , a ms-2, is given by a  8  2t. On reaching its greatest speed, the brakes are applied and the car decelarates at a constant rate of 3 ms 2 , coming to rest at point B . For the journey from A to B,

(a)

sketch the velocity – time graph

[3 marks]

(b)

find the time taken

[3 marks]

(c)

find the distance traveled

[4 marks]

39

28

29

A summer gala is being held in a village to raise funds for the school and one lady offers to make cushions and table cloths. One cushion requires 50 minutes of preparation time and 75 minutes of machine time. One table cloth requires 60 minutes of preparation time and 45 minutes of machine time. The lady makes x cushions and y 1 table cloths. Given that at least 12 hours is spent on preparation and that the machine 2 is available for a maximum of 15 hours. Given also that the total preparation time is less than or equal to the total machine time. (a)

Write three inequalities , other than x  0 and y  0 , which satisfy all the conditions described above.

(b)

Using a scale of 2 cm to 2 hours on both axes , construct and shade the region R in which every point satisfies all the conditions.

(c)

Based on the graph obtained in (b) , find the maximum profit made by the lady if the profit on each cushion is RM4 and the profit on each table cloth is RM2. [10 marks]

Pak Abu plans to plant x papaya trees and y rambutan trees on a plot of land of area 1000 m2 . He has allocated RM2000 to buy some seedlings. A papaya seedling costs RM2 and requires a land area of 1.5 m2. A rambutan seedling costs RM10 and requires a land area of 2 m2 . The number of papaya trees Pak Abu intends to plant is more than that of rambutan trees by at least 100 trees. (a)

Write three inequalities , other than x  0 and y  0 , which satisfy all the conditions described above.

(b)

Using a scale of 2 cm to 100 trees on both x–axis , and 2 cm to 50 trees on the y–axis, construct and shade the region R in which every point satisfies all the conditions.

(c)

Based on the graph obtained in (b) , answer each of the following questions. (i) If the cost for buying the seedlings is a maximum, find the land area required to plant the most number of both types of trees. (ii) During a certain period, a papaya tree yields RM120 of profit whereas a rambutan tree yields RM 400 of profit, find the maximum total profit that Pak Abu can acquire during the period. [10 marks]

40

30 Diagram 12 shows BDC is a straight line. Given that  ADB = 115° 10', AD = 7.2 cm and DC = 8.1cm. A

48

0

8'

B D

C

DIAGRAM 12 Calculate (a) the length of AC

[2 marks]

(b) the length of AB

[2 marks]

(c) the area of ABC

[4 marks]

(d) the length of the perpendiculer line from A to BC.

[2 marks]

41

31 Digram 13 shows triangle PQR . Given that PB = 20 cm, BR = 6 cm, RC = 8 cm , 24 CQ = 7 cm and sin  QRP = 25

DIAGRAM 13

B Calculate

32

(a) the length of PQ,

[3marks]

(b) sin  QPR,

[3 marks]

(c) the length of AP if the area of triangle PAD and RBC are equal.

[4 marks]

Diagram 14 shows PSR is a straight line. Given that PQ = 9 cm, QR = 6 cm, and  QPR = 30°. S is a point on PR such that QS = 6 cm. P

30 0 S 9 cm

6 cm

Q

DIAGRAM 14

R

Calculate (a) the length of PS,

[3 marks]

(b) the length of SR,

[4 mars ]

(c) the area of the triangle PQR .

[3 marks] 42

33

Diagram 15 shows, PQ is parallel to TS . Given that PQ = 12 cm, PT = 7 cm, RT = 5 cm and  PST = 20°.

T

P

DIAGRAM 15 Given that the area of PQT is 35 cm2, calculate

34

(a)

 QPT,

[2 marks]

(b)

the length of RS

[5 marks]

(c)

the area of the triangle PRT,

[3 marks]

(a) Table 4 shows , prices indices and weightages for four items in the year 2003 based on the year 2000. Item A B C D

Price index 120 130 145 110

weightage 7 4 m n

TABLE 4 The composite index in the year 2003 based on the year 2000 is 128 and the total of weightage is 20 (a) Calculate the price of item A in the year 2003 if the price in the year 2000 is RM42.50. (b) Find the value of m and n (c) The price index of item A increase by 20%, the price index of item B decrease by 15%,the cost of item C and D are not changing from the year 2003 to 2005. Find the composite index in the year 2005 based on the year 2003. [10 marks]

43

35

Table 5 shows the price indices of four raw materials, K, L, M and N, needed to produce a type of weed killer. The pie chart below the table shows the relative amount of the materials K, L, M and N used in producing the weed killer. Material K L M N

Unit price (RM) Year 2003 Year 2005 1.40 1.75 4.00 6.00 2.00 q r 2.40

I 2005 (based on the year 2003) p 150 140 80

TABLE 5

L

K 155 75

M

N

a) Find the value of p, q and r. [3 marks] b) i) Calculate the composite index of the cost for producing the weed killer for the year 2005 based on the year 2003. ii) Hence, calculate the corresponding selling price of a bottle of weed killer in the year 2003 if its selling price in the year 2005 was RM38.00 [5 marks] c) From the year 2005 to the year 2007, the cost of producing the weed killer is expected to increase by the same margin as from the year 2003 to the year 2005. Calculate the expected composite index for the year 2007 based on the year 2003. [2 marks]

END OF QUESTION PAPER

44

PERFECT SCORE PROGRAM ADDITIONAL MATHEMATICS 2007 ANSWER PAPER N0

MODULE 2 (3472/2) PART A

3

x  2, y  3; x  3, y  1 6 7 (1,1), ( , ) 5 15 a) k = 1 b) y = x – x2 + 3

4

a)

5

(a) 26.567

6

a)

1 2

7

2.2134 rad

a) 16.798 cm

9

a) i) h = -2 , k = 4 unit2 ii) y = 7x - 2 a) t=5

11

12

(b) 23.602 b) 13.67 (b) 144 

4  3

8

10

b) 20.3485 cm

51.25

(a) AP,

5 6 c) 17.673

c)

b) 14.283 cm2 iii) 23

b)

3 : 4

b) x  0.576

  1 a)    13 

, b)

a) m = -6 , n = 8

12

 80 7

c)

65

0  f ( x)  12

5 0

 4i  7 j

2

4

8 9

N0

MODULE 3 (3472/2) PART B

13 14

b) i) a = 1.80, b = 1.075 b) i) 2.239

15

a) A (0,3), B (0,-3), 21.667 C(7,4) (a) 84.4 cm3 , Maximum value

16

17

i) 5q – 2p

18

a) m = 2/5 , n = 1/5 graph

19

ii) x = 3.0 ii) p = 4.47 , q = 2.01 b) 583.2  b) 8 : 7

( ii) 8p + 10q b)

b) h = 2/3 , k = 5/6 c) 25 unit2

5 : 2

ii) y 

2x , 

No. of solutions = 2 20

b) x = 600, , 1200 , 1800

3 1  -

21

22

Bil. Peny. = 4

a) i) 0.001677 ii) 4

b) i) 0.2638

(a) (i) 0.1268 (ii)400 a) (i) 50

24

25

(ii) 53.225

(b) (i) 82or 83

(ii) - 20 

(b)

23 3  a) i)  , 2  4  7 ii) 1 8 a) 85.46

2

b)  1   2 , 2   4  b) 28.11

(ii) 38.77 or 38.79

64 27

c)(i) 48 x 2  48 y 2  72 x  576 y  879  0 ii) No c) 40.91

MODULE 4 26

(a)

9

(b)

(3472/2) PART C (c) 10

7

v

(d) 5

(b) 11

(c) 136

27 21

4

28

29

5 x  6 y  75 5 x  3 y  60 5 y x 3 y  x  100 x  5 y  1000 3 x  4 y  2000

t (c) RM44

(c) (i) 999.5 m2 (ii) RM 101 800

30

a) 8.237

b) 8.751

c) 35.44

31

a) 26.13

b) 0.5511

c) 8.353 cm

32

a) 3.826 cm

b) 7.937 cm

c) 26.47 cm2

33

a) 56o 26’

b) 8.641

c) 17.48 cm2

34

a) RM 51.00

b) m = 6

d) 6.516

c) 132.5

n=3 35

a) p = 125 q = 2.80 r = 3.00

b) i) 119.65 ii) RM 31.76

c) 143.16

1 6