Stpm Trial 2009 Maths2 Q&a (pahang)

papercollection

CONFIDENTIAL* 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 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 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 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 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 PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP Three hours

STPM 2009

950/2

MATHEMATICS S PAPER 2

PEPERIKSAAN PERCUBAAN SIJIL TINGGI PERSEKOLAHAN MALAYSIA NEGERI PAHANG DARUL MAKMUR 2009

Instructions to candidates: Answer all 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.

papercollection

2

CONFIDENTIAL*

Mathematical Formulae for Paper 2 Mathematics S : Integration :

Logarithms :

log a x =

log b x log b a

dv

Series :

∫

n

1 r = n( n + 1) ∑ 2 r =1 n

1 r = n( n + 1)(2n + 1) ∑ 6 r =1 2

n

∑r3 = r =1

du

∫ u dx dx = uv − ∫ v dx dx

1 2 n ( n + 1) 2 4

f ' ( x) dx = ln f ( x) + c f ( x) 1 1  x dx = tan −1   + c 2 a +x a

∫a

2

∫

 x dx = sin −1   + c a a2 − x2 1

Series: n n n (a + b) n = a n +  a n−1b +  a n− 2 b 2 + ⋯ +  a n − r b r + ⋯ + b n , where n ∈ N 1 2 r Coordinate Geometry : The coordinates of the point which divides the line joining (x1 , y1) and (x2 , y2) in the ratio m : n is  nx1 + mx 2 ny1 + my 2  ,   m+n   m+n The distance from ( x1 , y1 ) to ax + by + c = 0 is ax1 + by1 + c

a2 + b2 Maclaurin expansions (1 + x) n = 1 + nx +

n( n − 1) 2 n( n − 1) ⋯ ( n − r + 1) r x +⋯+ x + ⋯ , where x < 1 2! r!

x2 xr e = 1+ x + + ... + + ... 2! r! x

ln (1 + x ) = x −

x2 x3 (− 1) x r + ...,−1 < x ≤ 1 + − ... + 2 3 r r +1

2

papercollection

3

CONFIDENTIAL*

Mathematical Formulae for Paper 2 Mathematics S : Numerical Methods :

Correlation and regression :

Newton-Raphson iteration for f ( x ) = 0 : f ( xn ) x n +1 = x n − f ' ( xn )

Pearson correlation coefficient: ∑ xi − x y i − y r= 2 2 ∑ xi − x ∑ y i − y

(

(

)(

)

) (

Trapezium rule : 1 h[ y 0 + 2( y1 + y 2 + ⋯ + y n −1 ) + y n ] ∫a 2 b−a where yr = f ( a + rh) and h = n b

f ( x) dx ≈

Regression line of y on x : y=a+bx where b =

∑ (x − x )( y − y ) ∑ (x − x ) i

i

2

i

a = y − bx

Trigonometry

sin( A ± B) = sin A cos A ± cos A sin B cos( A ± B) = cos A cos B ∓ sin A sin B

tan( A ± B ) =

tan A ± tan B 1 ∓ tan A tan B

cos 2 A = cos 2 A − sin 2 A = 2 cos 2 A − 1 = 1 − 2 sin 2 A sin 3 A = 3 sin A − 4 sin 3 A cos 3 A = 4 cos 3 A − 3 cos A

 A+ B  A− B sin A + sin B = 2 sin  cos   2   2   A+ B  A− B sin A − sin B = 2 cos  sin   2   2   A+ B  A− B cos A + cos B = 2 cos  cos   2   2   A+ B  A− B cos A − cos B = −2 sin  sin   2   2 

3

)

papercollection

4

CONFIDENTIAL*

1. An auditor of a credit card company knows that, on average, the monthly balance of any given customer is µ 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 RM a and RM156.00 is 0.6826. Find the sample mean and a. [5] 2. A six-sided die is biased such that there is an equal chance of scoring each of the numbers from 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. Predecessor(s) Duration Activity Description (days) A House selection and training I – 19 B 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.

Week \ Day 1st

Monday 9.10

Tuesday 8.40

2nd 3rd 4th

8.60 8.30 8.20

7.70 7.30 7.50

Price / RM per kg Wednesday Thursday 6.90 7.10 7.60 7.40 7.30

(a) Plot the above data as a time series. (b) Find the 5–point moving averages for the data given (c) Compute the seasonal index using the multiplicative model. (d) Deseasonalise the data.

4

6.90 7.10 6.90

Friday 7.60 6.80 8.10 7.60 [2] [3] [3] [3]

papercollection

5

CONFIDENTIAL*

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 Score / y

80

130

190

60

180

150

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 variables x and y, the initial tableau was: P 1 0 0 0

x -3 1 1 3

y -5 1 2 2

s 0 1 0 0

t 0 0 1 0

u 0 0 0 1

v 0 10 14 18

a) State the three non-trivial inequalities in x and y and state the objective function. b) Apply one iteration of the simplex method by increasing y.

[4] [2]

What point does the new tableau represent? Explain how you know that the optimum point has not been reached.

[1] [1]

5

papercollection

CONFIDENTIAL*

6

9. The following stemplot shows the masses, in gm of mangoes harvested in a particular Stem Leaf 3 2 3 4 5 7 4 0 8 8 9 5 2 2 2 4 7 9 day. 6 1 1 2 4 7 7 9 8 0 6 9 9 (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. (b) Without calculation, will the confidence interval be bigger or smaller if the confidence level is increased to 98%? Explain your reason. (c) What is the minimum sample size to ensure that the estimated proportion is within 0.1 of the actual proportion with probability 0.90?

[4] [2] [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. monthly turnover, T Town Population,P(0 (RM’ 000) 000’s) Arau 11.1 3.2 Bember 7.6 12.4 Camelon 5.2 13.3 Dinding 9.0 19.3 Ehsan 17.9 8.1 Fama 11.8 4.8 (a) Represent these data on a scatter diagram with T on the vertical axis. [3] (b) Which town’s shop might appear to be underachieving given the populations of the towns? [1] You may assume that ΣP = 37.9, ΣT = 85.8, ΣP 2 = 264.69, ΣT 2 = 1286, ΣPT = 574.25. (c) Find the equation of the regression line of T on P. [6] (d) Estimate the monthly turnover that might be expected if a shop were opened in Gersang, a town with a population of 68 000. [2]

6

papercollection

7

CONFIDENTIAL*

(e) Why might the management of Pencilshop 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 Nasi lemak Fried mee Fried rice Egg sandwich

Price 2.00 1.20 1.30 1.60

2003 Value 780 696 598 928

Price 2.50 1.50 1.80 2.10

Value 1200 1110 936 840

(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]

7

papercollection

CONFIDENTIAL*

8

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 , µ ml and standard deviation 4 ml. Find the range of values of mean , µ , 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]

8

papercollection

9

CONFIDENTIAL*

MARKING SCHEME PEPERIKSAAN PERCUBAAN STPM SABS Mathematics S Paper 2 ( 950 / 2 ) Question 1

Scheme Standard error of the sample mean,

s =

σ n

=

56 49

= 8

B1

P( µ − s < X < µ + s) = 0.6826 µ + 8 =156 µ = 156 – 8 ∴ µ = 148 a = µ − s = 148 – 8 ∴ a = 140

M1 A1

A1 M2 A2

2. (a)

2. (b) 3. (a)

Marks B1

(b) Σ xP(x) = 1(1 + 2 + 3 + 4 + 5 + 18)/8 = 33/8

M2 A1 D1(sequence and flow, relax on the arrows) D1(allow one mistake, and with arrows, and 3 dummies) D1 (all correct )

3. (b)

85, A-C-H-J-L

3. (c)

The total float for the tennis game is 26 – 13 – 9 = 4 days . No delay. The total float for the squash game is 26 – 13 – 10 = 3 days. Project will be delayed by 4 – 3 = 1 day.

9

M1(attempt) A1 allow one mistake, A1 all EST,EFT,LS T,LFTcorrec t A1critical path M1 A1 A1

papercollection

CONFIDENTIAL*

Question 4. (a)

10

Scheme

4. (b)

10

Marks

papercollection

11

CONFIDENTIAL*

Question 4 (c)

Scheme

Marks

(d)

5

Mean =

∑x

M

+ ∑ xF

, Mean =

(80 × 52.9 ) + (100 × 61.4 )

n M + nF 80 + 100 4232 + 6140 10372 Mean = , Mean = = 57.622 180 180 2 Male : ∑ ( x M ) = n × σ 2 + µ 2

[

]

∑ ( x ) = 80 × [5.3 + 52.9 ] = 226 120 Female : ∑ ( x ) = 100 × [4.1 + 61.4 ]= 378 677 2

2

2

M

2

2

F

11

2

M1

A1

B1 B1

papercollection

12

CONFIDENTIAL*

Question 5

Scheme Standard deviation =

=

 ∑ xM 2 + ∑ xF 2   180 

 604797   10372   −   180   180 

Marks   10372  2 −    180  

M1

2

= 6.2978 = 6.298 or 6.30

A1

6(a)

B1

B1

6(a) B1

M1

A1

(b)

The correlation coefficient is rather small. So there is no linear relationship between for the score obtained and the time spent in practising the piano.

12

A1

papercollection

CONFIDENTIAL*

Question 7

8(a)

13

Scheme X represents the number of stamps with one flower X ∼ B ( 100 , 0.1 ) Normal Approximation : mean=10 , variance = 9 15.5 − 10   4.5 − 10 ≤Z≤ P (5 ≤ X ≤ 15 ) = P   3 3   = P (− 1.833 ≤ Z ≤ 1.833 ) = 1 − 2 × (0.0334 ) = 0. 9332 x + y ≤ 10 x + 2y ≤ 14 3x + 2y ≤ 18 The objective function: P = 3x + 5y

(b)

Marks B1 B1 M1 Standardiz e M1 Continuity correction A1 B1 B1 B1 B1 M1

A1

9. (a)

The tableau represents the point (0, 7). The optimum point has not been reached because there is a negative entry in the row that corresponds to the objective function. 1351 Mean = = 56.29 24 2

83379  1351  Standard deviation = −  =17.475 24  24  9.(b)

52 + 54 = 53 2 x + x7 40 + 48 Q1= 6 ⇒ Q1 = = 44 2 2 x + x 19 62 + 64 Q 3 = 18 ⇒ Q3 = = 63 2 2 1 Semi-interquartile Range = × [63 − 44 ] = 9.5 2

Median =

13

A1 A1 B1 B1 for 83379 M1 A1 B1

M1 A1

papercollection

CONFIDENTIAL*

Question 9.(c)

14

Scheme Lower boundary = 44 − 1.5 × ( 63 − 44 ) =15. 5 Upper boundary = 63 + 1.5 × ( 63 − 44 ) =91. 5 ∴Outlier is 99.

Marks M1 l.b.&u.b. A1 outlier D1 Box D1 Whiskers

9(c)

, Q 2 − Q1 = 9 Q 3 − Q 2 = 10 Since , Q 3 − Q 2 > Q 2 − Q1 the distribution is skewed to the right.

B1

10 (a) B1 ( 0.92 )

B1 (1.645)

M1

A1 (b)

M1 A1

14

papercollection

15

CONFIDENTIAL*

Question 10 (c)

11(a)

Scheme

Marks

T ( Monthly turnover, RM’000 ) D1 (scale & label) D1(plot allow one mistake) D1(all correct)

P ( Population 0’000 )

11(b) 11(c)

Bember SPT = 574.25 – (37.9x85.8)/6 = 32.28 SPP = 264.69 – (37.9)2 / 6= 25.288 b = 32.28/ 25.288 = 1.2765 a = 85.8/6- 1.2765(37.9/6) = 6.2369 T = 6.24 + 1.28P

A1 M1 M1 M1 A1 M1A1 A1

15

papercollection

CONFIDENTIAL* (d) (e) 12 (a) Question 12

P = 6.8 giving T = 14.917

16 hence £14 900

P = 17.2 which lies outside the set of values used to obtain the equation

M1 A1 B1

value = (price)x(quantity), ∴ quantity =value/ price

Scheme

Marks

B1 (4086 or 3164)

16

papercollection

CONFIDENTIAL*

17 B1(3164, or 3002)

M1 A1 A1

17