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VICTORIA JUNIOR COLLEGEPreliminary Examination

Higher 2

MATHEMATICS 9740/ 02 Paper 2

September 2010

3 hoursAdditional materials: Answer paper

Graph paperList of Formulae (MF15)

READ THESE INSTRUCTIONS FIRST

Write your name and CT group 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, 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, 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.

At the end of the examination, fasten all your work securely together.The number of marks is given in brackets [ ] at the end of each question or part question.

This document consists of 6 printed pages

© VJC 2010 VICTORIA JUNIOR COLLEGE [Turn over

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Section A: Pure Mathematics [40 marks]

1 Sketch, on an Argand diagram, the locus representing the complex number z for which

[2]

(i) Given that a is the least possible value of find a. [2]

(ii) The complex number p is such that

State the exact value of . [1]

(iii) Deduce the greatest value of , giving your answer correct to 2 decimal

places. [2]

2 (i) A curve C has equation

The diagram shows the curve C1 with equation and the curve C2 with

equation . The line is an asymptote to C1 and C1 passes through

the points ( 1, 0) and (3, 0).The lines and are asymptotes to C2. C2 passes through the

points ( 3, 0), (3, 0) and . The minimum points on C2 have coordinates

and .

Given that for and the point is the only other

turning point on C, sketch C, indicating clearly the intercepts, the equations of the asymptotes and the coordinates of the turning points. [4]

(ii) Find the x-coordinates of the stationary points on the curve . [3]

O

(–1, –1)

y

x –3 3

y = x – 4y = – x – 4

(1, –1)

O

x = –3

–1 3 x

y

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3 The plane and the line have equations and respectively,

where is a real parameter and a is a constant.

(a) It is given that . Find(i) the acute angle between and the plane [2](ii) the exact perpendicular distance of the point (1, 3, 2) from . [3]

(b) It is given that Find the acute angle between and [3]

4 (i) Obtain the expansion of up to and including the term in . [2]

(ii) Given that x is small such that powers of x above could be ignored, use your answer in part (i) to show that where b is a constant to be found.

[3](iii) State the equation of the tangent to the curve at the origin. On a single

diagram, sketch this tangent and the graph of . You should make clear the relationship between the two graphs for points near the

origin. [3]

5 The functions f, g and h are defined by

(i) Sketch the graph of y = f(x). [2](ii) Define in a similar form. [3]

(iii) Use a non-calculator method to solve [2](iv) State, giving a reason, whether fg exists. [2](v) Find [1]

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[Turn overSection B: Probability and Statistics [60 marks]

6 A circular disc is divided into n equal sectors which are labeled 1, 2, 3, … n –1 and a “skunk”. A game is played by spinning a pointer pivoted at the centre of the disc at most three times. If the result is a “skunk”, the game ends and the player loses all his previous winnings. If the result from a spin is k where k = 1, 2, 3, … or n – 1, then the player wins $k. He continues spinning and adding to his winnings until a maximum of three spins or a “skunk” is spun.

Show that the probability of winning $3 is . [1]

Find, in terms of n, the probability that the player(i) wins nothing when the game ends, [2](ii) wins nothing when the game ends, given that he wins $3 in his first spin. [2]

[There is no need for you to simplify your answer in both cases.]

7 In a statistical investigation, a researcher wants to find out how a person’s maximum walking speed varies with age. He selects a random sample of 12 males of certain ages and measures their individual maximum walking speeds. Their ages, t years and maximum walking speeds, w ms–1 are shown in the table below.

t 20 25 30 35 40 45 50 55 60 65 70 75

w 2.59 2.55 2.85 2.62 2.48 2.43 2.32 2.27 2.34 2.28 2.19 2.10

(i) Draw a scatter diagram for the data. [1](ii) State, giving a reason, whether a regression line of w on t or t on w could be used

to estimate the age of a male who has a maximum walking speed of 2.65 ms –1. (There is no need to do any calculations.) [1]

The researcher decides to study the maximum walking speed of males between the age of 30 and 55 inclusive. It is given that the correlation coefficient for the six data points is .

(iii) State, giving a reason, whether the regression line stated in (ii) is suitable for this study. [1]

(iv) For this study, the variables y is defined by . For the variables y and w,

(a) calculate the product moment correlation coefficient and comment on its value, [2]

(b) calculate the equation of the appropriate regression line, [1](c) determine the best estimate that you can of the maximum walking speed

when the age is 43. [1]

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8 The distribution of the masses of Perayaan balls has a mean of 420 g and a standard deviation of 6 g. Find the probability that 50 randomly selected Perayaan balls have a combined mass which is more than 20.9 kg. [2]State, with a reason, whether it is necessary to assume that the masses of Perayaan balls follow a normal distribution. [1]

The masses of Jubalani balls follow a normal distribution with a mean of 440 g and a standard deviation of 1 g. Find the probability that the combined mass of 50 randomly selected Perayaan balls differs from 50 times the mass of one Jubalani ball by less than 900 g. [3]State the assumption that you have made in arriving at your answer. [1]

9 The number of arrivals per minute at a fast food drive-through outlet has a Poisson distribution with mean . On a weekend evening, . Find the probability that in a 10-minute interval, there will be at most 10 arrivals given that there are more than 5 arrivals. [3]

In view of space constraints, the management wants to control the number of arrivals during the peak period which spans over a 30-minute interval. By using a normal distribution to approximate the Poisson distribution, find, to 4 decimal places, the largest value of such that the probability of having more than 30 arrivals during the peak period is less than 0.05. [5]

10 Annabel has 7 tiles each lettered A, N, N, A, B, E, L respectively. A code-word is formed when some tiles are picked and arranged to form a “word”.

(a) Find the number of different ways in which a 4-letter code-word can be formed (i) if the first letter is N and last letter is E, [3](ii) if there are no restrictions. [5]

(b) Annabel picks up 4 tiles and arranges them in a random order. Find the probability that the tiles spell ANNA. [2]

11 The weekly earnings, in dollars, at two casinos are modeled by independent normal distributions with means and standard deviations as shown in the table.

Mean Earnings Standard Deviationcasino 1 600 000 50 000casino 2 700 000 75 000

(i) Find the probability that in 2 randomly chosen weeks, the total earnings at casino 2 exceed 1 500 000 dollars. [2]

(ii) Find the probability that in a 12-week period, the weekly earnings at casino 1 exceeds $650 000 in at least 3 weeks. [3]

(iii) The government imposes a weekly tax on the earnings at casino 1 and 2 at a rate of 7% and 10% respectively. Find the probability that the tax exceeds $99 000 in

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any given week. Hence, by using a suitable approximation, find the probability that in a year consisting of 52 weeks, the weekly tax received by the government from the two casinos exceeds $99 000 in at least 45 weeks. [6]

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12 (a) Club Gunpla emailed all of its 1600 members to find out which of the four Celestial Being mechs was the most popular. The members were asked to select their favourite mech. 226 members responded and it was concluded that Gundam Exia is the most popular of the four mechs among the club members.

Explain if the sampling method used is a random one. [2]

(b) A large department store wants to find out how much its customers spend on Gundam model kits. From a random selection of 100 transactions, the results are summarized by

, ,

where $x is the amount spent on Gundam model kits in a single transaction.

The distributor claimed that the mean amount a customer spent on Gundam model kits is $40.Test whether the distributor has overstated his claim at the 5% significance level.

[6]

State, giving a reason, whether any assumption is needed for the test to be valid.[1]

(c) In testing the mean breaking strain of a type of fishing line, a researcher measured the breaking strain of 80 fishing lines. He carried out a t-test at the 5% significance level and, based on the sample results, he concluded that the population mean breaking strain is significantly different from

If the researcher had carried out a z-test instead, determine which of the following 2 statements is correct, giving clear reasons to support your claim.

(I) The researcher would have concluded that there is significant evidence at the 5% significance level that the population mean breaking strain is different from

(II) It is not possible for the researcher to conclude, by using only the information given, whether there is significant evidence at the 5% significance level that the population mean breaking strain is different from

. [3]