STPM Trial 2009 MathS2 Q&A (Pahang)

8/14/2019 STPM Trial 2009 MathS2 Q&A (Pahang)

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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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3

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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4. (a)

4. (b)

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4 (c)

(d)

5 Mean =FM

FM

nn

xx

++++

++++ , Mean =

(((( )))) (((( ))))

10080

4.611009.5280

++++

++++

Mean =180

61404232 ++++, Mean =

180

10372= 57.622

M1

A1

Male : (((( )))) [[[[ ]]]]222M nx ++++==== (((( )))) [[[[ ]]]]222M 9.523.580x ++++==== = 226 120

Female : (((( )))) [[[[ ]]]]222

F 4.611.4100x ++++==== = 378 677B1

B1

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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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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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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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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


12

B1 (4086 or

3164)

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B1(3164, or

3002)

M1 A1

A1

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STPM Trial 2009 MathS2 Q&A (Pahang)

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