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SERANGOON JUNIOR COLLEGE

2010 JC2 PRELIMINARY EXAMINATION

MATHEMATICS

Higher 2 9740/2

Wednesday 25 August 2010

Additional materials: Writing paper

List of Formulae (MF15)

TIME : 3 hours

READ THESE INSTRUCTIONS FIRSTWrite your name and class on the cover page and on all the work you hand in.Write in dark or black pen on both sides of the paper.You may use a soft pencil for any diagrams or graphs.Do not use staples, paper clips, highlighters, glue or correction fluid.

Answer 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 end of the examination, fasten all your work securely together.

Total marks for this paper is 100 marks.

This question paper consists of 9 printed pages and 1 blank page.

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Section A: Pure Mathematics [40 marks]

1 The computer company call “Orange” manufactures the latest electronic gadget in town called the iBoard with 3 different storage capacities namely 16GB, 32GB and 64GB. The profit earned from each unit sold is as shown in the table below.

Storage Capacity 16GB 32GB 64GBProfit $x $y $z

Within the first week after it was officially launched, the sales from 3 of its outlets for the three different storage capacities is as shown below.

The total profit collected from outlets B and C are $38 750 and $8750 respectively. If the total profit earned due to both the sales of 16GB and 32GB iBoard is equal to 12 times the total profit earned from the sales of the 64GB iBoard, find the value of x, y and z.

[4] Find the total profit collected from outlet A. [1]

2 A curve C has parametric equations

x = a sin2 t, y = a cos t, 0

2t

where a > 0.

(i) Sketch the curve. [2]

(ii) Find the equation of the normal at the point P where t = 3

. [3]

(iii) Using a non-calculator method, determine whether the normal at P will meet C again. [3]

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Storage Capacity 16GB 32GB 64GBOutlet A 75 120 20Outlet B 180 230 70Outlet C 45 50 10

C(3,0)0 2

1

x

A(4, 3)

B(1, – 1) 1)

y

3

3

The diagram above shows the graph of . On separate diagrams, sketch the graphs of

(i) , [3]

(ii) , [2]

(iii) . [3]

showing in each case, the coordinates of the points corresponding to A, B, C and the equations of the asymptotes.

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fy x

f 1 2y x

fy x

2f 1y x

y

x1 2 ln 2

C

R0

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4 (a) (i) Show that [2]

(ii) A curve C is defined by the parametric equations

The region R, which is bounded by the curve C, the line ,

and the x – axis is as shown below.

Find the exact area of R. [4]

(b) The region S is enclosed by the curve D with equation and the line .

Find the volume generated when S is rotated through radian about the x – axis, giving your answer correct to 2 decimal places. [3]

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ln

ln , 0.

x t t

y t t t

1x 2 ln 2x

22 4y x

y x

2

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5 A disease is found to be present in a protected reserve containing 35 orangutans. The rate at which the number of infected orangutans, x, is increasing at any time t is proportional to the product of the number of infected orangutans and the number that have yet to be infected at that instant. Initially there were 5 animals infected.

Form a differential equation that describes this model and show that

[6]

Deduce the total number of infected orangutans after a long period of time and represent the solution to this model on an appropriate graph. [4]

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35

3535

,1

kt

ktAe

xAe

where >0 is to be found. A

R

R’

1

5

Tuesday

R

R’

R1

5

R’

R

R’

R

R’

R

Wednesday

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Section B: Statistics [60 marks]

6 A survey on dining experience was undertaken in a small town with 10 three-star restaurants, 60 two-star restaurants and 30 one-star restaurants.

The 100 restaurants in the small town are numbered from 1 to 100. A sample of 10 restaurants is selected by randomly choosing 10 numbers that are assigned to the restaurants.

(i) Suggest one disadvantage of this sampling method. [1]

(ii) Suggest a better method of sampling and explain briefly how this could be done.

[3]

7 In a certain country, the probability that it rains on a given Tuesday is . For each of the next 2 days, Wednesday and Thursday, the conditional probability that it rains, given that it rained the previous day is and the conditional probability that it rains, given that it

did not rain the previous day is . The situation is illustrated in the uncompleted tree diagram below.

(i) Complete the tree diagram to represent all the possible outcomes up to Thursday.[2]

For and , find(ii) the probability that it rains on a Thursday, [2]

(iii) the probability that it rains on at least two days given that it rains on a Thursday.[3]

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1

5

1

3

2

3

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8 Paul ordered 9 plates of sushi, namely 3 plates of unagi sushi, 2 plates of uni sushi and 4 plates of ebi sushi.

(i) Find the number of ways Paul can eat the 9 plates of sushi. [2]

(ii) Find the number of ways Paul can eat the 9 plates of sushi such that the first plate

and last plate are different types of sushi. [4]

(iii) Find the number of ways of arranging the nine plates of sushi on a round table

such that no two plates of ebi sushi are placed together. [2]

9 Miss Curious wants to determine if there is any correlation between the amount of preparation and the results obtained in a recently concluded exam.

She asked her friends how much time they spent preparing for the exam (x), with their exam scores (y), and recorded her findings in the table below.

(i) Give a sketch of the scatter diagram for the data and find the equation of the least squares regression line of y on x. [2]

(ii) State, with a reason, which of the following would be an appropriate model to represent the above data (where a and b are constants and b > 0).

[2]

(iii) For the appropriate model chosen, find the values of a and b. [1]Explain how this model is a better one than the equation found in part (i). [1]

(iv)Obtain a good estimate of the score of a student who spent 8 hours studying for the exam and comment on the reliability of your answer. [2]

10 You are an Intelligence Quotient (IQ) expert. While reading the newspaper, you become interested in a newspaper advertisement that reads as follows:

[Turn OverIncrease the IQ of your children by 10 points in just 16 weeks!

Subscribe now to Dr. Dune’s Drill (DDD) program and astound your children’s friends, teachers and

grandparents!Assure a university education for your children

(and security for you in your old age).

by a

x A : xy a be B : lny a b x C :

x (hour) 10 15 22 27 38 46 53 64

y (score) 11 40 51 56 61 62 64 66

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As a concerned IQ expert, you would like to investigate the validity of this advertisement. You know that for the general population of children, the mean IQ is 100. Through close contacts in the industry, you confirmed that the scientific study as stated in the advertisement is valid.

(i) Test at 5% significance level, whether the mean IQ points of children who participated in the DDD program has increased. State any assumptions that you have to make in carrying out the test. [6]

(ii) What do you understand by 5% significance level in this context? [1]

Suppose that the population standard deviation is now known and a larger sample size of 50 children is taken.

(iii) Find the range of values of the sample mean in terms of if the conclusion at 5% significance level is now different from that concluded in part (i). [4]

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Increase the IQ of your children by 10 points in just 16 weeks!

Subscribe now to Dr. Dune’s Drill (DDD) program and astound your children’s friends, teachers and

grandparents!Assure a university education for your children

(and security for you in your old age).

9

11 In a certain country, it is found that on average the number of pairs of twins born weekly are 2.

(i) Find the most probable number(s) of pairs of twins that are to be born weekly.[1]

(ii) Find the probability of having at most 7 pairs of twins to be born in a two-week period. [2]

(iii) Assuming there are 26 two-week periods in a year, estimate the probability that there are less than 73 two-week periods with at most 7 pairs of twins born in 3 years. [3]

(iv) Using a suitable approximation, find the least number of consecutive weeks such that the probability of having at most 20 pairs of twins born falls below half. You may assume that the number of weeks is more than 5. [3]

(v) Find the probability that the mean number of pairs of twins born in 50 weeks is less than 1.8. [2]

12 An ornithologist, who studies the behavior of birds, captures one male and one female hornbill from a forest in Osaka, Japan. The masses of hornbills in that forest are assumed to follow normal distributions with male hornbills having mean 3500g and standard deviation 150g while female hornbills having mean 3000g and standard deviation .

(i) It is found from research that 5% of the female hornbills from the forest have

masses exceeding 3.2kg. Show that . [2]

(ii) Find the probability that the difference in mass between two randomly chosen male hornbills is at least 0.1kg. [3]

(iii) Find the probability that the mass of 5 randomly chosen female hornbills exceeds twice the mass of 2 randomly chosen male hornbills. [3]

(iv) Five male hornbills are randomly chosen. Find the probability that the fifth male hornbill is the third hornbill with mass exceeding 3.6kg. [3]

End of Paper

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