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INNOVA JUNIOR COLLEGE JC 2 PRELIMINARY EXAMINATION 1 in preparation for General Certificate of Education Advanced Level Higher 2

CANDIDATE NAME Civics Group

INDEX NUMBER

Mathematics

9740/02

Paper 2 01 July 2011 Additional materials:

Answer Paper Graph paper List of Formulae (MF15) ( MF15)

3 hours

READ THESE T HESE INSTRUCTIONS FIRST Do not open this booklet until you are told to do so. Write your name, class and index number on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, staples, paper paper clips , highlight ers, glue or co rrection f luid. Answer Answer all the all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. You are expected to use a graphic calculator. Unsupported answers from a graphic calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphic calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. The number of marks is given in brackets [ ] at the end of each question or part question. At the the end end of the examinatio examination, n, fasten fasten all your your work work secur securely ely together. together.

This document consists of 7 printed pages and 1 blank page.

Innova Junior College

[Turn over

2 Section A: Pure Mathematics [40 marks]

1

2

The curve with equation y = x ( x − a ) , where a > 0 is transformed by a translation of a units in the positive x-direction, followed by a stretch with scale factor

parallel to the y-

axis, where b > 1 , followed by a reflection about the x-axis. (i)

Find the equation of the new curve in the form y = f ( x ) . Sketch the new curve, indicating the coordinates of the points of intersection with the axes.

(ii)

[5]

The volume of the solid of revolution formed when the region bounded by the x2

axis and the curve y = x ( x − a ) is rotated completely about the x-axis is denoted by S . The volume of the solid of revolution formed when the region bounded by the x-axis and the curve y = f ( x ) is rotated completely about the x-axis is denoted by T . Express T in terms of S .

2

[2]

(

)

Find the roots of the equation z 3 + 4 − 4 3 i = 0 , expressing each root in the form r eiθ , where r > 0 and − π < θ ≤ π .

[4]

Two of the roots are denoted by z1 and z2 such that 0 < arg( z1 ) < arg( z2 ) . Show the points P, Q, and R representing z1 , z2 and z1 + z2 respectively on an Argand diagram

3

with origin O. Hence find the exact value of arg( z1 + z2 ) .

[3]

Describe briefly the geometrical relationship between O, P, R, and Q.

[1]

(i)

(

Given that 1 + y 2

d y

) d x = x + 2 y and y = 1 when x = 0 , find Maclaurin’s series

for y up to and including the term in x3 .

(ii)

Denote the answer to part (i) by f ( x ) and let g ( x ) = 1 + coefficient of x 2 in the series expansion of g( x) .

IJC/2011/JC2

[6]

9740/02/July 2011

4 3

3 f ( x ) . Find the x 2 [3]

3 4

(a)

Show that the substitution z = 3 x − y d y d x

2

= −2 ( 3 x − y ) − 2 to d z d x

Hence d y d x

reduces the differential equation

find

the

2

= 5 + 2 z .

particular

[2]

solution

2

= −2 ( 3 x − y ) − 2 given that y = −

5 2

of

the

differential

when x = 0 .

[5]

Sketch the graph of the particular solution for −0.7 ≤ x ≤ 0.2 . (b)

equation

[2]

The population of a community is known to increase at a rate proportional to the size of the population at time t years. When t 0, the population is A. If the =

population doubles in 6 years, find the number of years it takes for the population to be tripled, giving your answer correct to 1 decimal place.

[7]

Section B: Statistics [60 marks]

5

Olympus Junior College has a total student population of 500 with the following breakdown according to the students’ nationality.

Nationality

Singapore

Malaysia

Thailand

Vietnam

No of Students

350

105

10

35

The principal wishes to select a sample of 40 students to find out their views on the level of cohesiveness between the various nationalities in the college. (i)

He intends to choose a sample by selecting 10 students from each country. Give a reason why the sample formed is not a random sample. [1]

(ii)

Describe how stratified sampling can be used by the principal to obtain a random sample of 40 students. [2]

IJC/2011/JC2

9740/02/July 2011

[Turn over

4

6

A random sample of 6 students is taken from those sitting examinations in Mathematics and Social Studies, and their marks, x and y, each out of 100, are given in the table.

Mathematics mark ( x)

52

75

39

90

65

70

Social Studies mark ( y)

45

70

50

72

55

60

(i) (ii) (iii)

Draw the scatter diagram for these values, labelling the axes clearly.

Find the value of the product moment correlation coefficient between x and y. [1] Find the equation of the regression line of (a)

7

[2]

y on x,

(b) x on y.

[2]

An office telephone switchboard handles both internal and external calls which occur at random times. On average, there are 3 internal calls arriving in any ten-minute period and 5 external calls arriving in any thirty-minute period. Internal and external calls occur independently. (i)

Find the probability that exactly one internal call and at least two external calls arrive during a twenty-minute period.

(ii)

Using a suitable approximation, find the probability that there are at least 30 calls arriving during a one-hour period.

IJC/2011/JC2

[2]

9740/02/July 2011

[4]

5

8

Find the number of three-letter code-words that can be formed from the word

(a)

CARTOON .

[3]

A group of eleven people consisting of four married couples and three children

(b)

visits a mini-theatre at Science Centre to watch a show. The layout of the seats in the mini-theatre is shown below.

Screen 1st row 2nd row 3rd row

Find the number of possible seating arrangements if

9

(i)

all the eleven people do not mind where they sit,

[1]

(ii)

the three children sit in the 2nd row, and each man sits next to his wife. [2]

At a chess competition, a set of 3 games are played between two players John and Mary. Each game is won by either John or Mary. Past records have shown that: •

The probability of John winning the first game is 0.65.

•

For each game after the first, the probability of John winning a game is 0.86 if he has won the preceding game.

•

For each game after the first, the probability of Mary winning a game is 0.72 if she has won the preceding game.

Find the probability that (i) (ii) (iii)

IJC/2011/JC2

Mary wins the second game,

[2]

John wins the third game,

[3]

John wins the first game, given that he loses the third game.

[3]

9740/02/July 2011

[Turn over

6 10

A budget airline charges for breakfast on its early morning flights. On average 8% of its passengers on its early morning flights order breakfast. Using suitable approximations, find (i)

the probability that, out of 35 randomly selected passengers on one of its early morning flights, more than 5 order breakfast,

(ii)

[3]

the probability that, out of 130 randomly selected passengers on one of its early morning flights, more than 10 but at most 40 order breakfast.

[4]

The plane has a capacity of 150 passengers and the early morning flights are always fully booked. Each passenger orders at most one breakfast set. If the airline wishes to be at least 98% certain that the plane will have sufficient breakfast sets for all passengers who order them, find the minimum number of breakfast sets that should be carried on each flight.

11

[2]

The head circumference of 3-year-old boys is known to be normally distributed with mean 49.7 cm. A nutritionist claims that boys who have been fed on a special organic diet will have a larger mean head circumference than other boys. A random sample of twelve 3-year-old boys who have been fed on this organic diet is selected, and the head circumference, x cm, of each boy is measured. It is found that

∑ ( x − 50 ) = 5.4 , (i)

∑ ( x − 50 )

2

= 30.58 .

Find unbiased estimates of the population mean and variance.

[3]

(ii)

Carry out a test at the 10% significance level to examine the nutritionist’s claim. [4]

(iii)

Find the least significance level at which this sample would indicate that the nutritionist’s claim is true. [1]

(iv)

Explain, in the context of the question, the meaning of ‘at the 10% significance level’. [1]

(v)

IJC/2011/JC2

Explain why the Central Limit Theorem does not apply in this context.

9740/02/July 2011

[1]

7

12

The masses, in grams, of potatoes and tomatoes sold in a market have independent normal distributions with means and standard deviations as shown in the following table.

(i)

Mean

Standard deviation

Potatoes

275

180

Tomatoes

87

20.5

The random variable X denotes the mean mass of n randomly chosen potatoes.

(

)

Find the value of n such that P X > 280 = 0.4617 .

[3]

(ii)

Find the probability that sum of the mass of 2 randomly chosen potatoes exceeds five times the mass of a randomly chosen tomato by at least 100 grams. [3]

(iii)

Potatoes cost $1.50 per kilogram and tomatoes cost $2 per kilogram. Find the probability that the total cost of 10 potatoes and 20 tomatoes is less than $8. [5]

(iv)

A random sample of 12 tomatoes is taken. Find the probability that between 4 and 8 inclusive have a mass of less than 80 grams each. [2]

IJC/2011/JC2

9740/02/July 2011

[Turn over

8

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IJC/2011/JC2

9740/02/July 2011

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INNOVA JUNIOR COLLEGE JC 2 PRELIMINARY EXAMINATION 1 in preparation for General Certificate of Education Advanced Level Higher 2

CANDIDATE NAME Civics Group

INDEX NUMBER

Mathematics

9740/02

Paper 2 01 July 2011 Additional materials:

Answer Paper Graph paper List of Formulae (MF15) ( MF15)

3 hours

READ THESE T HESE INSTRUCTIONS FIRST Do not open this booklet until you are told to do so. Write your name, class and index number on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, staples, paper paper clips , highlight ers, glue or co rrection f luid. Answer Answer all the all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. You are expected to use a graphic calculator. Unsupported answers from a graphic calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphic calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. The number of marks is given in brackets [ ] at the end of each question or part question. At the the end end of the examinatio examination, n, fasten fasten all your your work work secur securely ely together. together.

This document consists of 7 printed pages and 1 blank page.

Innova Junior College

[Turn over

2 Section A: Pure Mathematics [40 marks]

1

2

The curve with equation y = x ( x − a ) , where a > 0 is transformed by a translation of a units in the positive x-direction, followed by a stretch with scale factor

parallel to the y-

axis, where b > 1 , followed by a reflection about the x-axis. (i)

Find the equation of the new curve in the form y = f ( x ) . Sketch the new curve, indicating the coordinates of the points of intersection with the axes.

(ii)

[5]

The volume of the solid of revolution formed when the region bounded by the x2

axis and the curve y = x ( x − a ) is rotated completely about the x-axis is denoted by S . The volume of the solid of revolution formed when the region bounded by the x-axis and the curve y = f ( x ) is rotated completely about the x-axis is denoted by T . Express T in terms of S .

2

[2]

(

)

Find the roots of the equation z 3 + 4 − 4 3 i = 0 , expressing each root in the form r eiθ , where r > 0 and − π < θ ≤ π .

[4]

Two of the roots are denoted by z1 and z2 such that 0 < arg( z1 ) < arg( z2 ) . Show the points P, Q, and R representing z1 , z2 and z1 + z2 respectively on an Argand diagram

3

with origin O. Hence find the exact value of arg( z1 + z2 ) .

[3]

Describe briefly the geometrical relationship between O, P, R, and Q.

[1]

(i)

(

Given that 1 + y 2

d y

) d x = x + 2 y and y = 1 when x = 0 , find Maclaurin’s series

for y up to and including the term in x3 .

(ii)

Denote the answer to part (i) by f ( x ) and let g ( x ) = 1 + coefficient of x 2 in the series expansion of g( x) .

IJC/2011/JC2

[6]

9740/02/July 2011

4 3

3 f ( x ) . Find the x 2 [3]

3 4

(a)

Show that the substitution z = 3 x − y d y d x

2

= −2 ( 3 x − y ) − 2 to d z d x

Hence d y d x

reduces the differential equation

find

the

2

= 5 + 2 z .

particular

[2]

solution

2

= −2 ( 3 x − y ) − 2 given that y = −

5 2

of

the

differential

when x = 0 .

[5]

Sketch the graph of the particular solution for −0.7 ≤ x ≤ 0.2 . (b)

equation

[2]

The population of a community is known to increase at a rate proportional to the size of the population at time t years. When t 0, the population is A. If the =

population doubles in 6 years, find the number of years it takes for the population to be tripled, giving your answer correct to 1 decimal place.

[7]

Section B: Statistics [60 marks]

5

Olympus Junior College has a total student population of 500 with the following breakdown according to the students’ nationality.

Nationality

Singapore

Malaysia

Thailand

Vietnam

No of Students

350

105

10

35

The principal wishes to select a sample of 40 students to find out their views on the level of cohesiveness between the various nationalities in the college. (i)

He intends to choose a sample by selecting 10 students from each country. Give a reason why the sample formed is not a random sample. [1]

(ii)

Describe how stratified sampling can be used by the principal to obtain a random sample of 40 students. [2]

IJC/2011/JC2

9740/02/July 2011

[Turn over

4

6

A random sample of 6 students is taken from those sitting examinations in Mathematics and Social Studies, and their marks, x and y, each out of 100, are given in the table.

Mathematics mark ( x)

52

75

39

90

65

70

Social Studies mark ( y)

45

70

50

72

55

60

(i) (ii) (iii)

Draw the scatter diagram for these values, labelling the axes clearly.

Find the value of the product moment correlation coefficient between x and y. [1] Find the equation of the regression line of (a)

7

[2]

y on x,

(b) x on y.

[2]

An office telephone switchboard handles both internal and external calls which occur at random times. On average, there are 3 internal calls arriving in any ten-minute period and 5 external calls arriving in any thirty-minute period. Internal and external calls occur independently. (i)

Find the probability that exactly one internal call and at least two external calls arrive during a twenty-minute period.

(ii)

Using a suitable approximation, find the probability that there are at least 30 calls arriving during a one-hour period.

IJC/2011/JC2

[2]

9740/02/July 2011

[4]

5

8

Find the number of three-letter code-words that can be formed from the word

(a)

CARTOON .

[3]

A group of eleven people consisting of four married couples and three children

(b)

visits a mini-theatre at Science Centre to watch a show. The layout of the seats in the mini-theatre is shown below.

Screen 1st row 2nd row 3rd row

Find the number of possible seating arrangements if

9

(i)

all the eleven people do not mind where they sit,

[1]

(ii)

the three children sit in the 2nd row, and each man sits next to his wife. [2]

At a chess competition, a set of 3 games are played between two players John and Mary. Each game is won by either John or Mary. Past records have shown that: •

The probability of John winning the first game is 0.65.

•

For each game after the first, the probability of John winning a game is 0.86 if he has won the preceding game.

•

For each game after the first, the probability of Mary winning a game is 0.72 if she has won the preceding game.

Find the probability that (i) (ii) (iii)

IJC/2011/JC2

Mary wins the second game,

[2]

John wins the third game,

[3]

John wins the first game, given that he loses the third game.

[3]

9740/02/July 2011

[Turn over

6 10

A budget airline charges for breakfast on its early morning flights. On average 8% of its passengers on its early morning flights order breakfast. Using suitable approximations, find (i)

the probability that, out of 35 randomly selected passengers on one of its early morning flights, more than 5 order breakfast,

(ii)

[3]

the probability that, out of 130 randomly selected passengers on one of its early morning flights, more than 10 but at most 40 order breakfast.

[4]

The plane has a capacity of 150 passengers and the early morning flights are always fully booked. Each passenger orders at most one breakfast set. If the airline wishes to be at least 98% certain that the plane will have sufficient breakfast sets for all passengers who order them, find the minimum number of breakfast sets that should be carried on each flight.

11

[2]

The head circumference of 3-year-old boys is known to be normally distributed with mean 49.7 cm. A nutritionist claims that boys who have been fed on a special organic diet will have a larger mean head circumference than other boys. A random sample of twelve 3-year-old boys who have been fed on this organic diet is selected, and the head circumference, x cm, of each boy is measured. It is found that

∑ ( x − 50 ) = 5.4 , (i)

∑ ( x − 50 )

2

= 30.58 .

Find unbiased estimates of the population mean and variance.

[3]

(ii)

Carry out a test at the 10% significance level to examine the nutritionist’s claim. [4]

(iii)

Find the least significance level at which this sample would indicate that the nutritionist’s claim is true. [1]

(iv)

Explain, in the context of the question, the meaning of ‘at the 10% significance level’. [1]

(v)

IJC/2011/JC2

Explain why the Central Limit Theorem does not apply in this context.

9740/02/July 2011

[1]

7

12

The masses, in grams, of potatoes and tomatoes sold in a market have independent normal distributions with means and standard deviations as shown in the following table.

(i)

Mean

Standard deviation

Potatoes

275

180

Tomatoes

87

20.5

The random variable X denotes the mean mass of n randomly chosen potatoes.

(

)

Find the value of n such that P X > 280 = 0.4617 .

[3]

(ii)

Find the probability that sum of the mass of 2 randomly chosen potatoes exceeds five times the mass of a randomly chosen tomato by at least 100 grams. [3]

(iii)

Potatoes cost $1.50 per kilogram and tomatoes cost $2 per kilogram. Find the probability that the total cost of 10 potatoes and 20 tomatoes is less than $8. [5]

(iv)

A random sample of 12 tomatoes is taken. Find the probability that between 4 and 8 inclusive have a mass of less than 80 grams each. [2]

IJC/2011/JC2

9740/02/July 2011

[Turn over

8

BLANK PAGE

IJC/2011/JC2

9740/02/July 2011

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