# JABATAN PELAJARAN NEGERI SABAH SIJIL PELAJARAN MALAYSIA TAHUN 2008

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```					                                                                NAMA        : _____________________
SULIT                                                           KELAS       : _____________________
NO K.P      : _____________________
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MATHEMATICS
PAPER 2
AUGUST 2008
2 ½HOURS

JABATAN PELAJARAN NEGERI SABAH
SIJIL PELAJARAN MALAYSIA TAHUN 2008
EXCEL 2
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PAPER 2 (KERTAS 2)
TWO HOURS THIRTY MINUTES (DUA JAM TIGA PULUH MINIT)

___________________________________________________________________________

JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU

1.    This question paper consists of three sections: Section A, Section B and Section C.
2.    Answer all questions in Section A, four questions from Section B and two questions
from Section C.
3.    Give only one answer / solution for each question.
5.    The diagrams in the questions provided are not drawn to scale unless stated.
6.    The marks allocated for each question and sub-part of a question are shown in
brackets.
7.    A list of formulae is provided on pages 2 to 4.
8.    A booklet of four-figure mathematical tables is provided.
9.    You may use a non-programmable scientific calculator.
___________________________________________________________________________
This question paper consists of 13 printed pages.
(Kertas soalan ini terdiri daripada 13 halaman bercetak.)
[Turn over (Lihat sebelah)
The following formulae may be helpful in answering the questions. The symbols given are the
ones commonly used.

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ALGEBRA

b  b 2  4ac                                          log c b
1.       x                                      8.    log a b 
2a                                                 log c a

2.       a m  a n  a m n                      9.    Tn  a  (n  1)d

3.       a m  a n  a mn                                 n
10.   Sn  [2a  (n  1)d ]
2
4.       (a m ) n  a mn
11.   Tn  ar n1
5.       loga mn  loga m  loga n
a(r n 1) a(1  r n )
12.   Sn                          ,r 1
6.       log a
m
 log a m  log a n                           r 1      1 r
n
a
7.       log a m  n loga m
n                             13.   S          , r 1
1 r

CALCULUS

dy   dv  du               4.    Area under a curve
1.       y  uv,          u v                            b
dx   dx  dx
=  y dx or
a

b

u dy
v
du
u
dv                            =    x dy
2.       y ,    dx 2 dx                                  a

v dx       v
5.    Volume generated

b

dy dy du                                      =   y 2 dx or
3.                                                       a
dx du dx
b
=   x 2 dy
a

STATISTICS

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1.        x
x                                                                    W I      i i
N                                               7.         I 

 fx                                                                        W     i

2.        x
f
n!
8.          n
P       
r
 n  r !
3.               (x  x )     2


x        2

x   2

N                  N
n!
9.          n
C   n  r  !r !
r

4.  
 f (x  x )    2


 fx   2

 x2
f                       f                      10.        P  A  B   P  A  P  B   P  A  B 

11.        P  X  r   nCr p r q nr , p  q  1

1      
 2N F                                         12.        Mean, μ = np
5.        m  L       c
 fm                                            13.          npq
       

x
6.        I
Q1
100                                           14.        Z
Qo                                                                        

GEOMETRY

1.       Distance                                                4.         Area of triangle =

 x1  x2    y1  y2                                1
2                  2
=                                                               ( x1 y2  x2 y3  x3 y1 )  ( x2 y1  x3 y2  x1 y3 )
2
2.       Midpoint
5.          r  x2  y 2

x1  x2 y1  y2 
 x, y   
        ,                                                          xi  yj
 2         2 
6.         r
ˆ              
            x2  y 2
3.       A point dividing a segment of a
line
nx  mx2 ny1  my2 
 x, y    1
         ,          
 mn        mn 

TRIGONOMETRY

1.        Arc length, s  r                                          8.         sin ( A  B)  sin A cos B  cos A sin B
9.         cos ( A  B)  cos Acos B  sin A sin B

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1 2                            tan A  tan B
2.   Area of sector, A         r   10.   tan ( A  B) 
2                             1  tan A tan B

3.   sin 2 A  cos2 A  1                                2 tan A
11.   tan 2 A 
4.   sec2 A  1  tan 2 A                              1  tan 2 A

5.   cosec2 A  1  cot 2 A                  a     b     c
12.             
6.   sin 2 A  2sin A cos A                sin A sin B sin C
7.   cos 2 A  cos2 A  sin 2 A      13.   a 2  b 2  c 2  2bc cos A
 2 cos 2 A  1
 1  2 sin 2 A                                      1
14.   Area of triangle       ab sin C
2

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Section A
[40 marks]

1        Solve the simultaneous equations 4 x  y  x 2  x  y  3 .                   [5 marks]

2        Diagram 1 shows a straight line CD which meets a straight line AB at point D. The
point C lies on the y-axis.

Diagram 1

(a)    State the equation of AB in the intercept form.                          [1 mark]
(b)    Given that 2AD = DB, find the coordinates of D.                          [3 marks]
(c)    Given that CD is perpendicular to AB, find the y-intercept of CD. [3 marks]

3        (a)    Sketch the graph of y  3sin 2 x for 0  x  2 .                       [4 marks]
(b)    Hence, using the same axes, sketch a suitable straight line to find the number
x
of solutions for the equation 3sin 2 x        =1 for 0  x  2 . State the number

of solutions.                                                            [3 marks]

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4        Given that the gradient of the tangent to the curve y  2 x3  6 x 2  9 x  1 at point P is
3, find
(a)       the coordinates of P,                                                  [2 marks]
(b)       the equation of the tangent and normal to the curve at P.              [4 marks]

5        Table 1 shows the distribution of the ages of 100 teachers in a secondary school.
Age
<30        <35       <40       <45       <50           <55    <60
(years)
Number of
8           22      42        68        88           98     100
teachers
Table 1
(a)       Based on Table 1, copy and complete Table 2.
Age
25 - 29
(years)
Frequency
Table 2
[2 marks]
(b)       Without drawing an ogive, calculate the interquartile range of the distribution.
[5 marks]

6        The first three terms of a geometric progression are also the first, ninth and eleventh
terms, respectively of an arithmetic progression.
(a)       Given that all the term of the geometric progressions are different, find the
common ratio.                                                          [4 marks]
(b)       If the sum to infinity of the geometric progression is 8, find
(i)    the first term,
(ii)   the common difference of the arithmetic progression.            [4 marks]

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Section B

[40 marks]

7        Use graph paper to answer this question.
Table 3 shows the values of two variables, x and y, obtained from an experiment.
Variables x and y are related by the equation y  ab  x , where a and b are constants.
x            1             2             3          4        5        6
y           41.7       34.7             28.9       27.5     20.1     16.7
Table 3
(a)       Plot log10 y against x by using a scale of 2 cm to 1 unit on the x-axis and 2 cm

to 0.2 unit on the log10 y -axis.

Hence, draw the line of best fit.                                     [4 marks]
(b)       Use your graph from (a) to find
(i)     the value of y which was wrongly recorded, and estimate a more
accurate value of it,
(ii)    the value of a and of b,
(iii)   the value of y when x = 3.5.                                  [6 marks]

    

8        Diagram 2 shows a trapezium PQRS. U is the midpoint of PQ and PU  2SV . PV and
TU are two straight lines intersecting at W where TW : WU = 1 : 3 and PW = WV.

S           V   R

T
W

P                 U              Q

Diagram 2


                  
It is given that PQ  12a, PS  18b and QR  18b  5a .
                     
(a)      Express in terms of a and/or b ,
       

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
(i)     SR ,

(ii)    PV ,

(iii)   PW .                                                       [5 marks]



(b)    Using PT : TS = h : 1, where h is a constant, express PW in terms of h,
a and/or b and find the value of h.                                [5 marks]
        

9        Diagram 3 shows a circle with centre C and of radius r cm inscribed in a sector OAB
of a circle with centre O and of radius 42 cm. [Use  = 3.142]

Diagram 3

Given that AOB           rad , find
3
(a)    the value of r,                                                    [2 marks]
(b)    the perimeter, in cm, of the shaded region,                        [4 marks]
(c)    the area, in cm2, of the shaded region.                            [4 marks]

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10       Diagram 4 shows part of the curve y  x 1 .
y

A
y  x 1
y=k

R             S

x
O

Diagram 4
The curve intersects the straight line y = k at point A, where k is a constant. The
1
gradient of the curve at the point A is     .
4
(a)    Find the value of k.                                                   [3 marks]
(b)    Hence, calculate
(i)     area of the shaded region R : area of the shaded region S.
(ii)    the volume generated, in terms of π, when the region R which is
bounded by the curve, the x-axis and the y-axis, is revolved through
360o about the y-axis.                                         [7 marks]

11       (a)    A committee of three people is to be chosen from four married couples. Find
how many ways this committee can be chosen
(i)     if the committee must consist of one woman and two men,
(ii)    if all are equally eligible except that a husband and wife cannot both
serve on the committee.                                        [5 marks]
(b)    The mass of mango fruits from a farm is normally distributed with a mean of
820 g and standard deviation of 100 g.
(i)     Find the probability that a mango fruit chosen randomly has a
minimum mass of 700 g.
(ii)    Find the expected number of mango fruits from a basket containing
200 fruits that have a mass of less than 700 g.                [5 marks]

Section C

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[20 marks]

12       A particle moves along a straight line and passes through a fixed point O. Its velocity,
v m s–1, is given by v  pt 2  qt  16 , where t is the time, in seconds, after passing
through O, p and q are constants. The particle stops momentarily at a point 64 m to
the left of O when t = 4.
[Assume motion to the right is positive.]
Find
(a)    the initial velocity of the particle,                                [1 mark]
(b)    the value of p and of q,                                             [4 marks]
(c)    the acceleration of the particle when it stops momentarily,          [2 marks]
(d)    the total distance traveled in the third second.                     [3 marks]

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13       Table 4 shows the prices of four types of book in a bookstore for three successive
years.

Price in year (RM)       Price index in     Price index in
Book                                   2001               2002            Weightage
2000 2001 2002           based on 2000     based on 2000

P      w      20     30            150                225                6

Q     50       x     65            115                130                5

R     40      50     56            125                140                3

S     80       z     150                y              y                 2
Table 4

(a)       Find the values of w, x, y and z.                                 [4 marks]
(b)       Calculate the composite index for the year 2002 based on the year 2001.
[4 marks]
(c)       A school spent RM4, 865 to buy books for the library in the year 2002. Find
the expected total expenditure of the books in the year 2003 if the composite
index for the year 2003 based on the year 2002 is the same as for the year
2002 based on the year 2001.
[2 marks]

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14       Use graph paper to answer this question.
A farmer wants to plant x-acres of vegetables and y-acres of tapioca on his farm.
Table 5 shows the cost of planting one acre and the number of days needed to plant
one acre of vegetable and one acre of tapioca.

Vegetables                Tapioca
Cost of planting
RM100                   RM 90
per acre
Number of days
4                      2
needed per acre

Table 5
The planting of the vegetables and tapioca is based on the following constraints:
I      The farmer has a capital of RM1800.
II     The total number of days available for planting is 60.
III    The area of his farm is 20 acres.
(a)    Write down three inequalities, other than x  0 and y  0 , which satisfy all the
above constraints.                                                   [3 marks]
(b)    By using a scale of 2 cm to 4 acres on both axes, construct and shade the
region R that satisfies all the above constraints.                   [3 marks]
(c)    By using your graph from (b), find
(i)     the maximum area of tapioca planted if the area of vegetables planted
is 10 acres,
(ii)    the maximum profit that the farmer can get if the profit for one acre of
vegetables and one acre of tapioca planted are RM60 and RM20
respectively.                                                [4 marks]

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15       Diagram 5 shows a quadrilateral ABCD such that ABC is acute.

Diagram 5
(a)   Calculate
(i)     ABC ,
(iii)   the area, in cm2, of quadrilateral ABCD.                   [8 marks]
(b)   A triangle AB’C has the same measurement as triangle ABC, that is, AC = 15
cm, CB’ = 9 cm and B ' AC  30 , but is different in shape to triangle ABC.
(i)     Sketch the triangle ABC .
(ii)    State the size of AB 'C .                                 [2 marks]

END OF QUESTION PAPER

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