Yishun Junior College 2010 Preliminary Exam H2 Maths 9740 Paper 2

Section A: Pure Mathematics [40 marks]

1 A curve has parametric equations , for .

(i) Find in terms of t and deduce that the curve is an increasing function. [2]

(ii) Find the equation of L1, the tangent to the curve at the point .

Hence, find the coordinates of point P on the curve at which L1 passes through the origin O. [4]

(iii) The line L2 is another tangent to the curve which is parallel to L1. Find the equation of L2. [3]

(iv) The line L2 cuts the y-axis at Q. Find the exact area of triangle OPQ. [2]

2 (i) The equation 2x2 – x – ln(x + 1) = 0 has 2 real roots and , where <.Find the values of and , giving any non-exact answers correct to 3 decimal places. [1]

(ii) A sequence of positive real numbers satisfies the recurrence

relation xn+1 = for .

Prove algebraically that, if the sequence converges, then it converges to . [3]

(iii) If x1 = 2, write down the values of x2 and x3 .

Sketch the graphs of y = and y = x on the same axes.

Illustrate on your diagram how the sequence will converge to starting with x1 = 2. [3]

3 (a) Solve the equation , expressing each of the roots in the form , where a and b are real. [4]

Hence write down the solutions of the equation . [2]

(b) On a single, clearly labelled diagram, sketch the loci of w and z defined by and . [2]

(i) State the minimum value of . [1](ii) Find the exact values of the modulus and argument of w for which

.

Hence, express w in the form , where x and y are exact real values.[3]

4 The plane p1 has equation x + y – 2 z = 4.

(i) A plane p2 with equation 2x + ay + bz = 4 (where a and b are constants) is

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Yishun Junior College 2010 Preliminary Exam H2 Maths 9740 Paper 2

parallel to p1, find the values of a and b and also the exact distance between p1

and p2 . [3]

(ii) Another plane p3 contains a point with coordinates (2, 4, 1) and a line with

equation r = , where s . Show that the equation of p3 can be

expressed as 3x – y + z = 3.

Find also a vector equation of the line l where p3 meets p1. [4]

(iii) The plane p4 has equation 5x + y + z = where and are constants. (a) If all three planes p1 , p3 and p4 intersect in the line l , find the values of

and . [2]

(b) Deduce the geometrical relationship of the three planes p1, p3 and p4

if = 3 and ≠ 11. [1]

Section B: Statistics [60 marks]

5 A Residents’ Committee wants to propose improvements to the recreational facilities in Punggol. In order to find out the needs of the adults and children of both genders, a survey is to be carried out on a sample of 120 residents who are at least 5 years old. Given that of the 6525 male residents who are at least 5 years old, there are 1450 children. On the other hand, there are 7975 females who are at least 5 years old, out of which 2175 are children. Describe how the sample can be obtained using quota sampling. [1]

Name and describe another method of sampling in which each group is represented proportionately. [3]

State an advantage of quota sampling over the above method. [1]

6 A recent research study indicates that 16% of the total population are aged 60 years or more and that 18% of the total population have a measurable hearing defect. Furthermore, 65% of those aged 60 years or more have a measurable hearing defect.

Find the probability that a randomly chosen person from the population

(i) has a measurable hearing defect, given that he is less than 60 years old, [3]

(ii) is either aged 60 years or more, or has a measurable hearing defect, or both. [2]

State, with a reason, whether or not the events ‘a person is 60 years old or more’ and ‘a person has a measurable hearing defect’ are independent. [1]

Find, correct to 3 decimal places, the probability that, if two persons are chosen at random from the population, at least one of them will be aged 60 years or more and at least one of them will have a measurable hearing defect. [4]

7 A manufacturer claims that the breaking strength of a climbing rope is normally distributed with mean 1702 N and standard deviation 105 N. A random sample of 10 climbing ropes is tested and the mean breaking strength of the sample is N. A test is then carried out at the 5% significance level to determine whether the manufacturer’s

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Yishun Junior College 2010 Preliminary Exam H2 Maths 9740 Paper 2

claim is valid. Given that the null hypothesis is rejected in favour of the alternative hypothesis, find the range of possible values of . [4]

The manufacturer believes that adding a special chemical to the ropes will increase the mean breaking strength and change the standard deviation. A random sample of 10 such ropes is found to have a mean breaking strength of 1724 N and a standard deviation of 35 N. Carry out a significance test at the 5% level to decide whether this result provides sufficient evidence to confirm the manufacturer’s belief that the mean breaking strength has increased. [4]

8 An experiment carried out to see how the intensity of radiation from a particular radioactive source, I, varies with time, t. In the following table, the values of t may be considered to be exact, while the values of I are subject to experimental errors.

I 22.5 25.0 28.0 30.5 38.0 40.5 42.5 48.0 54.5 55.0 70.0t 44.0 42.0 33.5 28.0 18.0 13.6 15.0 10.3 9.0 6.3 4.0

(i) Sketch the scatter diagram for the given data. [2]

(ii) State, with a reason, which of the following models is more appropriate to fit the data points:

A: I = a +bt2, where a > 0 and b < 0;B: I = atb, where a > 0 and b < 0. [2]

(iii) For the appropriate model, find the product-moment correlation coefficient for the transformed data. Find an estimate for a, correct to 3 decimal places. [2]

(iv) Hence, estimate the time when the value of intensity of radiation is 35.0 and comment on the reliability of the estimate found. [2]

9 After production, blocks of butter are wrapped by a machine. Each block is supposed to weigh 220 g but the blocks produced have masses which are normally distributed. A worker then packs every six dozen randomly chosen blocks of butter in a box of mass 175 g. It is known that the total mass of each box of butter follows a normal distribution with mean 16.375 kg and standard deviation 0.149 kg.

(i) Three such boxes of butter are chosen at random. Find the probability that each box of butter weighs more than its mean mass. [1]

(ii) Find the percentage of blocks of butter which weigh at least 220 g. [3]

(iii) To increase the consumers’ confidence, the company wishes to adjust the mean mass of a block of butter such that more than 75 % of the blocks will have a mass of not less than 220 g while the standard deviation remains unchanged.

Find the least value of the mean mass after the adjustment. [3]

10 (a) Find the number of 8-letter code-words that can be formed using the lettersA, B, C, D, E if

(i) there are no restrictions, [1]

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Yishun Junior College 2010 Preliminary Exam H2 Maths 9740 Paper 2

(ii) each vowel (A, E) appears once and each consonant (B, C, D) appears twice, [1]

(iii) each letter occurs at least once and the letters appear in alphabetical order. [3]

(b) At a particular reception, 9 guests are to stand at 3 identical round cocktail tables. How many ways can this be done if there must be at least two people at each table? [3]

11 A multiple choice test consists of 20 questions, each with four possible answers, of which only one is correct. If a student randomly chooses the answer to each question, find the probability of getting at least four but less than nine correct answers. [2]

Suppose that each correct answer is awarded five marks and each incorrect answer carries a penalty of one mark, what is the expected score obtained by a student? [2]

50 students took the test. Using a suitable approximation, find the probability that the mean score is more than 12 marks. [3]

12 At a post office, the number of customers purchasing postal products in a half-hour period during peak hours follows a Poisson distribution with mean 5. During the off- peak period, the number of customers purchasing postal products in a two-hour period is an independent Poisson distribution with mean 10. The peak period for the post office is from 12 pm to 1.30 pm and the post office is open from 8 am to 5 pm.

(i) Find the probability that there are more than 15 customers from 12.30 pm to 1.30 pm. [2]

(ii) Using suitable approximations, find the probability that the total number of customers during off-peak hours is less than thrice the total number of customers during peak hours on a particular day. [5]

~ End of Paper ~

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