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8/6/2019 STPM Pahang Maths S P2 09 Q n A
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CONFIDENTIAL*
PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN
PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNPPEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN
PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP
PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAANPERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP
PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN
PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP
PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAANPERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP
PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAANPERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP
PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN
PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNPPEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN
PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP
PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAANPERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP
PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN
PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAANJPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP
PEPERIKSAAN PERCUBAAN
SIJIL TINGGI PERSEKOLAHAN MALAYSIA
NEGERI PAHANG DARUL MAKMUR 2009
Instructions to candidates:
Answerall questions. Answers may be written in either English or Malay.
All necessary working should be shown clearly.
Non-exact numerical answers may be given correct to three significant figures, or one
decimal place in the case of angles in degrees, unless a different level of accuracy is
specified in the question.
Mathematical tables, a list of mathematical formulae and graph paper are provided.
This question paper consists of 7 printed pages.
950/2 STPM 2009
Three hours
MATHEMATICS S
PAPER 2
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2
Mathematical Formulae for Paper 2 Mathematics S :
Logarithms :
a
xx
b
b
alog
loglog =
Series :
)1(2
1
1
+==
nnrn
r
)12)(1(6
1
1
2++=
=
nnnrn
r
22
1
3)1(
4
1+=
=
nnrn
r
Integration :
= dxdxdu
vuvdxdx
dv
u
cxfdxxf
xf+= )(ln)(
)('
ca
x
adx
xa+
=
+
1
22tan
11
c
a
xdx
xa
+
=
1
22sin
1
Series:
Nnwhere ++
++
+
+=+
,
21)( 221 nrrnnnnn bba
r
nba
nba
naba
Coordinate Geometry :
The coordinates of the point which divides the line joining (x1 ,y1) and (x2 ,y2) in
the ratiom :n is
+
+
+
+
nm
myny
nm
mxnx 2121 ,
The distance from ),( 11 yx to 0=++ cbyax is
22
11
ba
cbyax
+
++
Maclaurin expansions
1,!
)1()1(!2
)1(1)1( 2
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Mathematical Formulae for Paper 2 Mathematics S :
Numerical Methods :
Newton-Raphson iteration for 0)( =xf :
)('
)(1
n
n
nnxf
xfxx =+
Trapezium rule :
+++++ b
ann yyyyyhdxxf ])(2[
2
1)( 1210
n
abhrhafyr
=+= andwhere )(
Correlation and regression :
Pearson correlation coefficient:
) )( ) ( )
=
22
yyxx
yyxxr
ii
ii
Regression line ofy onx :
y =a +bx
where(((( ))))(((( ))))
(((( ))))
====
2
i
ii
xx
yyxxb
xbya =
Trigonometry
BAAABA sincoscossin)sin( = BABABA sinsincoscos)cos( =
BA
BABA
tantan1
tantan)tan(
=
AAAAA2222 sin211cos2sincos2cos ===
AAA3sin4sin33sin =
AAA cos3cos43cos3
=
++++====++++
2
BAcos
2
BAsin2BsinAsin
++++====
2
BAsin
2
BAcos2BsinAsin
++++====++++
2
BAcos
2
BAcos2BcosAcos
++++====
2
BAsin
2
BAsin2BcosAcos
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1. An auditor of a credit card company knows that, on average, the monthly balance of any
given customer is and standard deviation of RM56.00. In a random sample of 49 customers
it was found out that the probability that the monthly balance is between RMaand
RM156.00 is 0.6826. Find the sample mean anda. [5]
2. A six-sided die is biased such that there is an equal chance of scoring each of the numbersfrom 1 to 5 but a score of 6 is three times more likely than each of the other numbers.
(a) Write down the probability distribution for the random variable,X, the score on a single
throw of the die. [4]
(b) Show that E(X) = 833 . [3]
3. A school is preparing to participate in a sports carnival in the district. It selects students
from the inter house games for the various sports. The table below shows the various
activities involved in the preparation for the carnival.
Activity Description Predecessor(s) Duration
(days)
A House selection and training I 19B House selection and training II 13
C Football A 8
D Hockey A 7
E Tennis B 9
F Squash B 10
G Table tennis B 7
H Rugby C, D 8
I Hand ball E, F, G 9
J School selection and training III H 20
K School selection and training IV I 20
L Centralised training J, K 30
(a) Construct an activity network for the project. [3](b) Determine the minimum time required to complete the project, and the corresponding critical
path. [4]
(c) How many days will the project be extended if the tennis game is delayed by 4 days and the
squash game is also delayed by 4 days? [3]
4. The price per kg of a type of fish is monitored in February 2009.
Price / RM per kg
Week \ Day Monday Tuesday Wednesday Thursday Friday1st 9.10 8.40 6.90 7.10 7.60
2nd 8.60 7.70 7.60 6.90 6.80
3rd 8.30 7.30 7.40 7.10 8.10
4th 8.20 7.50 7.30 6.90 7.60
(a) Plot the above data as a time series. [2]
(b) Find the 5point moving averages for the data given [3]
(c) Compute the seasonal index using the multiplicative model. [3]
(d) Deseasonalise the data. [3]
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5. The following table show the mean and standard deviation of the marks of the male and
female students who sat for a semester test.
Student Number of students Mean Standard deviation
Male 80 52. 9 5. 3
Female 100 61. 4 4. 1
Calculate the mean and standard deviation of the marks of all the students. [6]
6. A music teacher recorded the number of hours her students spent in practising and the
scores obtained in the ARSM piano examination. She believes that the more time her students
spent in practising the higher the score obtained.
Time / x
hours
80 130 190 60 180 150
Score /y 118 135 146 125 135 60
(a) Calculate the correlation coefficient for the score obtained and the time spent in practice.
[5]
(b) Comment on the result. [1]
7. There are one or two flowers on the faces of 50 cents stamps. 90% of all these 50 cents
stamps have two flowers while the rest of the stamps have single flower.
From the stamps which have single flower, 95% of these stamps have a flower at the centre
of the stamps while the rest have a flower on the left side of the stamps.
By using a suitable approximation, determine the probability that between 5 and 15 stamps
inclusive have one flower , out of a random sample of 100 pieces of 50 cents stamps. [5]
8. When using the simplex method to solve a particular linear programming problem
involving two variablesx andy, the initial tableau was:
P x y s t u v
1 -3 -5 0 0 0 0
0 1 1 1 0 0 10
0 1 2 0 1 0 14
0 3 2 0 0 1 18
a) State the three non-trivial inequalities inx andy and state the objective function. [4]
b) Apply one iteration of the simplex method by increasing y. [2]
What point does the new tableau represent? [1]
Explain how you know that the optimum point has not been reached. [1]
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9. The following stemplot shows the masses, in gm of mangoes harvested in a particular
day.
9
60
97
4211
974222
9880
75432
Leaf
9
8
7
6
5
4
3
Stem
(a) Find the mean and standard deviation of the masses of the mangoes . [3]
(b) Find the median and semi-interquartile range of the masses of the mangoes . [3]
(c) Draw a boxplot to represent the data and identify possible outliers .Comment on the shape of the distribution and give a reason for your answer. [5]
10. A survey is done to estimate the proportion of people who supports the national
service project for youths. In a random sample of 100 people, 92 of them supported
the project.
(a) Find the 90% symmetric confidence interval for the proportion of people who
support the national service project. [4]
(b) Without calculation, will the confidence interval be bigger or smaller if the
confidence level is increased to 98%? Explain your reason. [2]
(c) What is the minimum sample size to ensure that the estimated proportion is within0.1 of the actual proportion with probability 0.90? [4]
11. Pencilshop have stores selling stationary in each of 6 towns. The population, P, in tens of
thousands and the monthly turnover, T, in thousands of RM for each of the shops are as
recorded below.
Town Population,P(0
000s)
monthly turnover, T
(RM 000)
Arau 3.2 11.1
Bember 7.6 12.4
Camelon 5.2 13.3Dinding 9.0 19.3
Ehsan 8.1 17.9
Fama 4.8 11.8
(a)Represent these data on a scatter diagram with Ton the vertical axis. [3]
(b)Which towns shop might appear to be underachieving given the populations of
the towns? [1]
You may assume that P = 37.9, T= 85.8, P2
= 264.69, T2
= 1286, PT= 574.25.
(c)Find the equation of the regression line ofTon P. [6]
(d)Estimate the monthly turnover that might be expected if a shop were opened in Gersang, atown with a population of 68 000. [2]
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(e)Why might the management ofPencilshop be reluctant to use the regression line to
estimate the monthly turnover they could expect if a shop were opened in Hanlet, a
town with a population of 172 000? [1]
12. Chef Wah sells pre packed food in his shop and has recorded the following information
where value is (price) (quantity). The base year is taken to be 2002.
2002 2003
Price Value Price Value
Nasi lemak 2.00 780 2.50 1200
Fried mee 1.20 696 1.50 1110
Fried rice 1.30 598 1.80 936
Egg sandwich 1.60 928 2.10 840
(a) Find the Paasche price index and comment on the change in the sales of packed food. [4]
(b) Find the Laspeyres quantity index and comment on the change in the sales of packed
food. [4]
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12. (a) A machine is used to fill up bottles with a mean volume of 550 ml. Suppose that the
volume of water delivered by the machine follows a normal distribution with mean , ml and
standard deviation 4 ml. Find the range of values of mean , , if it is required that not more
than 1% of the bottles contain less than 550 ml. [4]
(b) The mass of a box of chocolate cereal is distributed normally with mean 100 g and
standard deviation 2 g.
(i) Calculate the probability that three boxes of chocolate cereal chosen at random,
each has a mass less than 98 g. [2]
(ii) Calculate the probability that three boxes of chocolate cereal chosen at random,
have a total mass exceeding 305 g. [2]
(iii) Calculate the probability that out of three boxes of chocolate cereal chosen at
random, exactly two have a mass greater than 98 g while the mass of the remaining box
is greater than 105 g. [3]
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MARKING SCHEME PEPERIKSAAN PERCUBAAN STPM SABS Mathematics S
Paper 2 ( 950 / 2 )
Question Scheme Marks
1 Standard error of the sample mean, 8=49
56=
n
=s
P( s
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Question Scheme Marks
4. (a)
4. (b)
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Question Scheme Marks
5
Standard deviation =
22
F
2
M
180
10372
180
xx
++++
=
2
180
10372
180
604797
= 6.2978 = 6.298 or 6.30
M1
A1
6(a)
B1
B1
6(a)
B1
M1
A1
(b) The correlation coefficient is rather small. So there is no linearrelationship between for the score obtained and the time spent inpractising the piano.
A1
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Question Scheme Marks
7 X represents the number of stamps with one flower
X B ( 100 , 0.1 )
Normal Approximation : mean=10 , variance = 9
(((( ))))15X5P =
3
105.15Z
3
105.4P
= (((( ))))833.1Z833.1P = (((( ))))0334.021 = 0. 9332
B1
B1M1
Standardiz
e
M1
Continuity
correction
A1
8(a) x +y 10
x + 2y 14
3x + 2y 18
B1
B1
B1
The objective function: P = 3x + 5y B1
(b) M1
A1
The tableau represents the point (0, 7). A1
The optimum point has not been reached because there is a negativeentry in the rowthat corresponds to the objective function.
A1
9. (a)Mean =
24
1351= 56.29
B1
Standard deviation =
2
24
1351
24
83379
=17.475
B1 for
83379
M1
A1
9.(b)Median = 53
2
5452====
++++
Q 1=2
xx 76 ++++ Q 1 = 442
4840====
++++
Q3 =2
xx 1918 ++++ Q3 = 632
6462====
++++
Semi-interquartile Range = [[[[ ]]]] 5.944632
1====
B1
M1 A1
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Question Scheme Marks
9.(c) Lower boundary = 44 1.5 ( 63 44 ) =15. 5
Upper boundary = 63 + 1.5 ( 63 44 ) =91. 5
Outlier is 99.
M1
l.b.&u.b.
A1 outlier
D1 Box
D1
Whiskers
9(c) 10QQ 23 ==== , 9QQ 12 ====
Since , 23 QQ > 12 QQ the distribution is skewed to the right.B1
10 (a)
(b)
B1 ( 0.92 )
B1 (1.645)
M1
A1
M1
A1
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Question Scheme Marks
10 (c)
11(a) T ( Monthly turnover, RM000 )
D1 (scale &
label)
D1(plot
allow one
mistake)
D1(all
correct)
11(b) Bember A1
11(c) SPT= 574.25 (37.9x85.8)/6 = 32.28SPP= 264.69 (37.9)
2/ 6= 25.288b= 32.28/ 25.288 = 1.2765
a= 85.8/6- 1.2765(37.9/6) = 6.2369T= 6.24 + 1.28P
M1M1
M1 A1
M1A1A1
P ( Population 0000 )
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(d) P = 6.8 giving T= 14.917 hence 14 900 M1 A1
(e) P = 17.2 which lies outside the set of values used to obtain theequation
B1
12 (a) value = (price)x(quantity), quantity =value/ price
Question Scheme Marks
12
B1 (4086 or
3164)
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B1(3164, or
3002)
M1 A1
A1