TAMPINES JUNIOR COLLEGE JC2 Preliminary Examination 2010

MATHEMATICS Higher 2 9740/02

Wednesday 15 September 2010 3 hoursAdditional materials: Answer paper

Cover PageList of Formulae (MF15)Graph Paper

Name: _______________________

Class: _______________________

At the end of the examination, place the completed cover page on top of your answer scripts and fasten all your work securely together with the string provided.

READ THESE INSTRUCTIONS FIRST

Write your name and class on all the work you hand in, including the Cover Page.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, paper clips, highlighters, glue or correction fluid.Do not write anything on the List of Formulae (MF15).

Answer all the questions. Begin each answer on a fresh page of paper.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 required 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.

For Candidate’s Use

For Examiner’s Use

Question Number

Marks Obtained

123456789

101112

Total

Calculator Model:

This document consists of 7 printed pages including this page.

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Section A: Pure Mathematics [40 marks]

1 Express in partial fractions. [1]

Let

By using the method of differences, find, in terms of N, an expression for , where , simplifying your result. [3]

Deduce the limit of as . [1]

2 The complex number z satisfies the relations and .Illustrate both of these relations on a single Argand diagram. [3]

Hence find(i) the maximum value of , [1](ii) the range of values of arg . [2]

3 The parametric equations of a curve are

, ,

(a) Find and hence determine the exact x-coordinates of the points for which the

tangents to the curve are parallel to the x-axis. [4]

Given that the Cartesian equation of the curve is .

The finite region bounded by the curve, x-axis, y-axis and the line where , , is rotated through 2 radians about the x–axis.

(b) Find the value of k such that the volume of the solid generated is units . [4]

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4 The figure below shows a cuboid positioned on level ground so that it rests on one of its vertices, O. The vectors i and j are on the ground.

Given that , and

(i) Show that the position vector of X is . [1](ii) State the height of X above the ground. Hence find the angle between OX and the

level ground, giving your answer to nearest 0.1o. [3](iii) Find the equation of plane BDX in the form of [3](iv) Find the acute angle between planes BDX and OBDC, giving your answer to

nearest 0.1o [3]

5 Given that , show that . [1]

(i) By repeated differentiation of this result, find the Maclaurin’s expansion for y up to and including the term in . [5]

(ii) By writing down the Maclaurin’s expansion of , show that

[2]

(iii) Use the series in part (ii) for , find the exact value of . [3]

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Section B: Statistics [60 marks]

6 A bag contains eleven small discs, which are identical except that 6 of the discs are blank and five of the discs are numbered, using the numbers 1, 2, 3, 4 and 5. The bag is shaken and four discs are taken one at a time without replacement.

(i) Show that the probability that the disc numbered “3” is not taken is . [1]

(ii) Find the probability that exactly two numbered discs are taken, given that the disc numbered “3” is taken. [3]

7 [In this problem, you may assume that the mean and variance of the score on the uppermost

face for each rolling of a fair die are respectively 3.5 and .]

A fair die is rolled n times, where n > 50.

The probability that the sum of the scores on the uppermost face is more than 365 is at least 0.05. By using a suitable approximation, show that n satisfies the inequality

Hence find the least value of n. [4]

8 12 friends decide to celebrate the completion of their last paper of a major examination with a sumptuous dinner.

(i) Find the number of ways they can form themselves into 3 groups of 4 people each for the purpose of taking taxis to the restaurant. [2]

Among these 12 friends are Alex, Benjamin and Charles.

(ii) Find the number of ways these 12 friends can be seated around a round table if Alex must sit beside Benjamin and Charles must not sit next to Alex. [3]

After the dinner, 4 friends from this group decide to go for a late night movie. The cinema presents them some publicity postcards produced by the movie company. There are 3 identical postcards featuring the lead actress and 2 identical postcards featuring an action group scene.

(iii) Find the number of ways the cards can be distributed if these 4 friends select one postcard each. [2]

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9 An ice-cream shop provides two types of paper cups, regular or large, for its customers. Each customer picks a cup according to his appetite, fills it with ice-cream of flavours of his choice. The mass of each cup together with its ice-cream content is measured at the cashier with a weighing machine, and the customer is charged at a rate of $2 per 100g measured.

Let X and Y be the respective mass, in grammes, of the regular and large cups with their ice-cream content. It is found that both X and Y independently follows a normal distribution, with parameters given in the table below:

Mean Standard DeviationX, the mass of a regular cup with ice-cream content 200 30Y, the mass of a large cup with ice-cream content 350 60

A family of six, consisting of a couple and four boys, enters the ice-cream shop. The couple decides to share ice-cream in a large cup and each of the boys independently takes a regular cup of ice-cream.

(i) Show that the probability that one of the boys pay more than $5 for his regular cup of ice-cream and the other three pay less than $4 each is 0.0239. [2]

(ii) Find the probability that total cost of the children’s regular cups of ice-cream exceeds twice that of the parents’ large cup of ice-cream. [3]

The mass, in grammes, of an empty regular cup is known to follow a normal distribution with mean 30g and standard deviation 5g. The ice-cream content, in grammes, in a regular cup also follows independently a normal distribution. Find, with adequate justification, the variance of the ice-cream content in a regular cup. [2]

10 In the production of plastic sheets, scratches occur at random and independently at an average of scratches per plastic sheet.

(i) Show that, if is sufficiently small, the probability that a plastic sheet has at least 2

scratches is approximately . [2]

500 plastic sheets produced by manufacturer A, for which the value of is 0.12, are randomly selected and inspected for quality control. A plastic sheet will be rejected if it has at least 2 scratches. Let Y denote the number of plastic sheets rejected.

(ii) State the exact distribution of Y. Find, by using a suitable approximation, the probability that at most 6 plastic sheets are rejected. [4]

Another manufacturer B independently produces plastic sheets with an average of 0.5 scratches per plastic sheet. 25 plastic sheets from manufacturer A and 20 plastic sheets from manufacturer B are randomly selected.

(iii) Using a suitable approximation, find the probability that there are a total of at least 11 scratches found in the 45 plastic sheets selected. [4]

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11 The table shows the number of 1st, 2nd, 3rd and 4th year students in a college. The students are distributed into 60 classes of 24 students each.

Year Number of students1st 4082nd 3843rd 3364th 312

Total 1440

At the beginning of the year, the school implemented a new web-based learning portal for all its students.

(a) At the end of the year, a sample of 120 students is to be chosen to take part in a survey that aims to investigate students’ opinions on this new portal.

(i) The school generates a list of the students arranged according to their classes and, within each class, in descending order of the students’ performance in the end-of-year examination. Explain why systematic sampling using this list as the sampling frame would be inappropriate. [2]

(ii) Suggest how a stratified sample can be obtained and state one advantage of adopting this sampling scheme in this context. [3]

(b) Based on records maintained over many years prior to the implementation of this new web-based learning portal, it is known that the mean mark obtained by the students in the end-of year examinations is 63.

A random sample of 12 students who have used the new portal is taken and their marks, x, in the end-of-year examination are summarized as:

, .

(i) Calculate unbiased estimates of the mean and variance of a student’s marks in the end-of-year examination. [3]

(ii) Test, at 5% level of significance, whether the new portal is effective in increasing students’ mean marks in the end-of-year examination. State an assumption made for the test to be valid. [5]

Justify whether your conclusion would be changed if a wrong test-statistic were used in (ii) instead. [1]

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12 [A sheet of graph paper is provided for use in this question.]

Alice obtains cash from an ATM (cash machine). She suspects that the rate at which she spends cash is affected by the amount of cash she withdrew at her previous visit to an ATM.

To investigate this she deliberately varies the amounts she withdraws. She records, for each visit to an ATM, the amount, $x, withdrawn, and the number of hours, y, until her next visit to an ATM. The following is her data for the 2-month period from April to May.

Withdrawal 1 2 3 4 5 6 7 8x 40 100 120 150 30 90 80 130y 56 195 94 270 52 196 214 286

(i) Calculate the equation of the regression line of y on x and sketch this line on the scatter diagram of the data. [4]

(ii) Alice made one withdrawal immediately before going on a weekend visit to Kuala Lumpur, Malaysia. Identify the most likely withdrawal, giving a reason. [2]

By removing the exceptional case in (ii), the product-moment correlation coefficient of the remaining data is computed to be 0.949.

(iii) Write down the equation of the corresponding regression line of y on x. Hence estimate the number of hours, correct to nearest integer, until Alice’s next visit to the ATM if she withdraws $50. Comment on the reliability of the estimate. [4]

(iv) Alice also fits a model of the form to her data. Estimate, using this model, the number of hours, correct to nearest integer, until Alice’s next visit to the ATM if she makes the minimum withdrawal of $20 and explain why the model would probably not be appropriate in this context. [3]

In the months of November and December, Alice collected data for her 8 withdrawals. Alice then combined all her data for the months of April, May, November and December, with the exceptional case in (ii) removed, and computed the resulting product-moment correlation coefficient to be 0.426.

(v) Comment, in context, on the significance of this value. [1]

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