ANGLO-CHINESE JUNIOR COLLEGE

MATHEMATICS DEPARTMENT

MATHEMATICS

Higher 2

Paper 2 26 August 2015

JC 2 PRELIMINARY EXAMINATION

Time allowed: 3 hours

Additional Materials: List of Formulae (MF15)

READ THESE INSTRUCTIONS FIRST

Write your Index number, Form Class, graphic and/or scientific calculator model/s on the cover page.

Write your Index number and full name on all the work you hand in.

Write in dark blue or black pen on your answer scripts.

You may use a soft pencil for any diagrams or graphs.

Do not use 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 the 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.

This document consists of 6 printed pages.

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9740 / 02

Anglo-Chinese Junior College

H2 Mathematics 9740: 2015 JC 2 Preliminary Examination Paper 2

Page 2 of 6

ANGLO-CHINESE JUNIOR COLLEGE

MATHEMATICS DEPARTMENT

JC2 Preliminary Examination 2015

MATHEMATICS 9740

Higher 2

Paper 2

Calculator model: _____________________

Arrange your answers in the same numerical order.

Place this cover sheet on top of them and tie them together with the string provided.

Question No. Marks

1 /9

2 /9

3 /11

4 /11

5 /3

6 /5

7 /7

8 /6

9 /9

10 /10

11 /10

12 /10

Summary of Areas for Improvement

Knowledge (K) Careless

Mistakes (C)

Read/Interpret

Qn wrongly (R) Formula (F) Presentation (P)

/ 100

Index No: Form Class: ___________

Name: _________________________

Anglo-Chinese Junior College

H2 Mathematics 9740: 2015 JC 2 Preliminary Examination Paper 2

Page 3 of 6

Section A: Pure Mathematics [40 marks]

1 (i) Differentiate 2

1

2 1x with respect to x. [1]

(ii) Hence find

2

22

d2 1

xx

x . [4]

The curve C has equation 22 1

xy

x

, where 0x .

(iii) The region bounded by C and the line 2y x is rotated 2 radians about the x-axis.

Find the exact volume of the solid obtained. [4]

2 The complex number z satisfies the equation i 3

12 i

z

z

.

(i) Using an algebraic method, find the purely imaginary number that satisfies the given

relation. [2]

(ii) Sketch the locus of the points representing z, labelling the coordinates of the y-intercept.

[2]

(iii) Describe the locus of the points representing w such that 4iw a , where a is a non-

zero constant. Hence find the exact value of a such that there is exactly one value of z

that satisfies i 3

12 i

z

z

and 4iz a . [3]

For this value of a, find the exact value of z that satisfies the above conditions, giving

your answer in the form ix y where , x y . [2]

3 (i) Sketch the curve with equation 1

, 0y x xx

, stating the equation(s) of any

asymptotes and the coordinates of any turning points and any points of intersection with

the axes. [3]

The functions f and g are defined as follows

1f : , , ,x x a x b x a

x a

1g : , 0

lnx x

x , 1x

where a and b are constants and 0a .

(ii) Using part (i) or otherwise, state the smallest value of b, in terms of a, such that f is

one-one. [2]

(iii) Explain why the composite function gf exists and find the range of gf. [3]

(iv) Given that a = 2, define f-1

in a similar form. [3]

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Anglo-Chinese Junior College

H2 Mathematics 9740: 2015 JC 2 Preliminary Examination Paper 2

Page 4 of 6

4 Planes 1p and 2p have the following equations:

1p : 3 2 6 2x y z

2p : 1 2 2 2 3 2s t t s t r i j k , ,s t

(i) Explain why 1p and 2p are parallel and distinct planes. Hence find the shortest distance

between these two planes. [5]

(ii) The line l has equation (5 ) ( 5 3 ) ( ) r i j k , where and ,

are real constants.

(a) Find conditions on and such that l only intersects one plane but not the

other. [3]

(b) The angle between l and 1p is 22°. Find the two possible values of . [3]

Section B: Statistics [60 marks]

5 At a railway station, serious delays occur at random such that the expected number of serious

delays per week is 4.3 and the variance is 2.56. State, with a reason, whether the number of

serious delays may be modelled by a Poisson distribution.

Find the probability that, in a period of 60 weeks, not more than 4 serious delays occur on

average per week. [3]

6 In a junior college there are 750 Year 2 students in the science faculty. These students are

grouped according to 4 major subject combinations, namely SA, SB, SC and SD. During a

science talk, the students are seated according to combination in four different venues. The

number of students in each combination and their venues are given in the table below.

Combination SA SB SC SD

Number of Year 2 students 195 180 225 150

Venue LT1 LT2 LT3 LT4

The teacher-in-charge of this talk intends to obtain a sample of 100 students for a survey. She

selects 25 students from the last occupied row in each venue for the survey.

(i) Name the sampling method described and state a reason, in the context of the question,

why this sampling method is not desirable. [2]

(ii) Suggest a method of obtaining a more representative sample and describe how it may be

carried out. [3]

7 (a) A teacher wants to set a class test for her class of 25 students. She plans to give each

student the same questions, but have each student's questions appear in a different order.

Find the least number of questions she must set. [2]

(b) Sally has 12 bars of chocolates in four different flavours as summarised in the table

below.

flavour white milk dark hazelnut

number 4 3 4 1

Sally intends to give one bar of chocolate each to three of her best friends in school.

(i) Find the number of ways she can do it. [3]

(ii) Sally remembers that one particular friend amongst the three has a nut allergy, and

should not be given the hazelnut chocolate bar. Find the number of ways she can

now give her friends the chocolates. [2]

Anglo-Chinese Junior College

H2 Mathematics 9740: 2015 JC 2 Preliminary Examination Paper 2

Page 5 of 6

8 A particular brand of paper cups is found to have capacity that is normally distributed with

mean 500 ml and standard deviation 45 ml.

(i) Andy and twenty of his friends are each given one paper cup to fill a 10-litre tank with

water. Each person completely fills his paper cup with water once, and then empties the

water into the tank. Find the probability that the tank is completely filled after the last

person empties his cup. [2]

(ii) Andy decides to attempt to fill the tank with water using his one cup. He does so by

making 21 trips from the tap to the tank, each time filling his cup completely at the tap

before pouring the contents into the tank. Show that the probability that he manages to

fill the tank is 0.70163, correct to 5 significant figures. [1]

(iii) In a game, each of 60 people is given one paper cup and one 10-litre tank to fill with

water. If every person makes 21 trips from the tap to his tank, using a suitable

approximation, find the probability that at most 45 of the tanks will be filled. [3]

9 John and Jane are working on a research project about the heights of 17-year-old boys from

Taz, a minority ethnic community. They come across an internet website claiming that the

average height of 17-year-old Taz boys is 170 cm.

(a) John believes that the average height quoted on the website is too low. He assumed that

the heights of 17-year-old Taz boys follow a normal distribution and measured the

heights of eight randomly chosen boys. The data collected, measured in cm, are as

follows:

171.8 167.4 174.5 169.4 171 175.5 170.4 173.5

Carry out a test of John’s belief at the 5% significance level, defining any symbols you

used in the hypotheses. [4]

(b) Jane finds out that the standard deviation of the heights of 17-year-old Taz boys is

4.2 cm. She decides to carry out a one-tail test to determine whether the claim on the

website is valid by measuring the heights of a random sample of n 17-year-old Taz

boys. Their mean height is found to be 168.6 cm.

(i) State appropriate hypotheses for the test. [1]

(ii) Given that the null hypothesis is not rejected at 5% level of significance, find the

range of values of n. State, giving a reason, whether it is necessary to assume that

the heights of Taz 17-year old boys are normally distributed. [4]

10 A confectionary bakes chocolate chip cookies in batches. To each large batch of cookie

dough, 600 chocolate chips are added and thoroughly mixed into the dough to make 150

cookies.

(i) State, in this context, two conditions for the number of chocolate chips in one cookie to

be well modelled by a Poisson distribution. [2]

For the remainder of this question, assume that the conditions in (i) are met.

(ii) Find the probability that, out of four randomly chosen cookies from the confectionary,

exactly two have five chocolate chips each and the other two have more than five

chocolate chips each. [2]

(iii) The confectionary sells the cookies in boxes of 30 each. Using a suitable approximation,

find the probability that there are at least two cookies in a box with at most one

chocolate chip. [3]

(iv) The baker needs to add n more chocolate chips per batch of cookie dough in order to be

at least 95% certain that a randomly chosen cookie has at least two chocolate chips.

Express this information as an inequality in n, and hence find the smallest possible

integer value of n. [3]

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Anglo-Chinese Junior College

H2 Mathematics 9740: 2015 JC 2 Preliminary Examination Paper 2

Page 6 of 6

11 (a) In MCJC, 75% of the students are right-handed. Given that a student is left-handed, the

probability that he takes mathematics as a subject is 4

7. Find the probability that a

student is left-handed and does not take mathematics. [2]

If half of the right-handed students take mathematics, determine if being right-handed

and taking mathematics are independent events. [2]

(b) In a game, a player chooses a card from a complete deck of poker cards. If the card

chosen is a picture card, the player wins the game. If the card chosen is any number card

other than the Ace, the player loses the game. If an Ace card is chosen, the player puts

the card aside and takes a second card. He wins if the second card is a picture card, loses

if it is a number card, and picks a third card if it is an Ace. He continues to do this until

he either gets a picture card, in which case he wins, or a number card and he loses.

Find the probability that

(i) a player wins the game on picking the second card, [2]

(ii) a player wins the game, [2]

(iii) only one card is chosen, given that the player wins the game. [2]

[A deck of poker cards is made up of 4 Aces, 12 picture cards (Jacks, Queens and Kings)

and 36 number cards.]

12 The sales manager of a retailer of KSI sunblock is monitoring the effects of its television

advertising campaign. Over the last 7 weeks, different durations of television air time, x

minutes, were used and the corresponding number of sales, y, in hundreds of bottles of KSI

sunblock, were recorded as follows:

x 8 11 15 18 20 22 25

y 11.1 15.2 19.9 22.8 24.1 25.0 26.1

(i) Draw a scatter diagram for the data, labelling the axes clearly. [1]

(ii) Using the scatter diagram, explain if each of the following can model the given data set.

(A): 2 y a bx , where a and b are positive constants

(B): ln y a b x , where a is negative constant and b is positive constant

(C): b

y ax

, where a and b are positive constants. [3]

Calculate the least squares estimates of a and b, and the product moment correlation

coefficient for the model that best fits the data set. [2]

(iii) Give an interpretation of b in the context of this question. [1]

(iv) The advertising costs $2000/min and each bottle of KSI sunblock yields a profit of $20.

Estimate the weekly profit when 10 mins of TV time is purchased. Comment on the

reliability of the estimate. [3]

- End of Paper -

2015 ACJC H2 Mathematics Prelim Paper 2 solutions

1(i)

22 2

d 1 4.

d 2 1 2 1

x

x x x

1(ii)

2

2 22 2

22

22

2

1

2

1 4 d d

42 1 2 1

1 11 d

4 2 14 2 1

1 1 d

84 2 1 1

2

2tan 2 .

84 2 1

x xx x x

x x

xx

xx

xx

xx

xx c

x

1(iii) 2

2

2 , , 02 1

22 , , 0

8 1

1 10, 0 or , .

2 2 2

xx y y x

x

yx y y x

y

y x y x

The line intersects C at 0,0 and 1 1

,2 2 2

.

2 21

220

1

231

2

0

3

d2 1 2

2tan 2

8 124 2 1

1 2 1

328 2 24 2

2 2

32 12

2 1unit .

4 8 3

x xx

x

x xx

x

2(i) i 31

2 i

z

z

Let iz k where k

i i 3 2 i( i)k k

2 23 ( 1)i 2k k

2 2 2( 3) ( 1) (2 )k k

2 29 2 1 4 4k k k k

6 6k

1k

2(ii) 3 i i 2 iz z

(3 i) 2iz z

2(iii) The locus of points representing w such that 4iw a is a circle with centre (0, 4)

and radius a units.

To have exactly one value of z satisfying the 2 conditions, the perpendicular bisector

should be tangent to the circle (see sketch in 2(ii))

Gradient of bisector 1 2

13 0

1

⇒ bisector makes an angle of 4 rad. with the horizontal axis

Consider right-angled triangle ADE:

sin4 4 ( 1)

a

5 5 2 or

22a

z 5 5

cos i 4 sin4 42 2

5 3

i2 2

5 3 ,

2 2x y

Alternative method for finding a:

Re(z)

Im(z)

O

(3, −1)

C(0, 2)

B(1.5, 0.5)

E(0, 4)

A(0, −1)

D

a

Triangles ABC and ADE are similar right-angled triangles with common angle BAC

sinBC DE

BACAC AE

2 2(0 1.5) (2 0.5)

3 5

a

5 9

3 2

5 5 2 or

22

a

3(i) 1y x

x

Asymptotes: y x , x = 0

3(ii) f( )y x is a translation of

1y x

x a units in the positive x-direction. Therefore the

turning points are ( 1, 2)a and ( 1, 2)a . Hence 1b a .

3(iii) f gR 2, 0,1 1, D

Hence gf exists.

1gf ln2

R 0,

3(iv) 1f( ) 2 , 3

2x x x

x

2

12 , 3

2

( 2) 1

2

y x xx

x

x

2

2

2 2

2

( 2) 4 5

(4 ) 5 2 0

4 (4 ) 4(1)(5 2 ) 4 4

2 2

4 43,

2

x y x x

x y x y

y y y y yx

y yx x

21 4 4

f : , 22

x xx x

4(a) Proving parallel & distinct planes

Normal vectors of 1p and 2p are respectively

1

32

6

n 2

2 20 31 2

n ×

3

26

Since 1 2 n n , the normal vectors are parallel and hence the planes are parallel as

well.

Furthermore, 3(1) 2( 2) 6(0) 7 2 , so the point (1, −2, 0) is on 2p but not on 1p .

Hence the two planes are distinct.

Finding shortest distance

Method 1:

Since 32

(0) ( 1) 3(0) 1 , the point (0, 1,0)A is on 1p .

The point (1, 2,0)B is on 2p .

1 0 12 1 1

0 0 0AB

Shortest distance between planes = |AB⃗⃗⃗⃗ ⃗ . n̂1|

1 311 . 2

9 4 36 0 6

3 2 0

7

5

7

Method 2:

Express equations of both planes in scalar product form dr.n where n is a unit

normal vector and d is the shortest distance between the origin and the plane.

1p : 32 2

6

r. ⇒ 31 22

9 4 36 9 4 366

r.

⇒ 2

7r.n where

312

7 6

n

2p : 3 1 3

2 2 2 76 0 6

r. .

⇒ 31 ( 1)( 7)

29 4 36 9 4 366

r.

⇒ 1r.n where 312

7 6

n

Since both planes are on the same side of the origin,

shortest distance between both planes 2

17

5

7

4(b)(i)

l: 55 3

1r

,

Since the planes are parallel, for l to intersect one but not the other, we need l to be

contained in exactly one plane at one time.

1p : 3 0 32 1 2 2

6 0 6

r. .

2p : 3 1 32 2 2 7

6 0 6

r. .

l parallel to 1p and 2p ⇒ 3

3 2 01 6

.

⇒ 0

l contained in 1p ⇒ 5 35 2 2

6

.

⇒ 2 15 10

6

⇒ 23

6

l contained in 2p ⇒ 5 35 2 7

6

.

⇒ 7 15 10

6

⇒ 3 236

0, and or 3

4(b)(ii)

2

33 21 6

sin 229 1 9 4 36

.

2

30.37461

7 10

Using GC, 5.69 or 5.69 (3 s.f.)

5 Number of serious delays is not modelled by a Poisson distribution since the mean is

not equal to the variance.

Let X be the random variable denoting the number of serious delays per week.

E(X) = 4.3, Var(X) = 2.56

Since 60 is large, n by Central Limit Theorem,

2.56

N 4.3, 60

X approximately

4 0.0732 (3 s.f.) P X

Alternative Method:

Let 1 2 60 W X X X .

Since 60 is large, n by Central Limit Theorem,

N 258,153.6 W

240 0.0732 3 s.f. P W

6(i) Quota sampling.

Disadvantage:

The 20 boys seated on the last row of LT may belong to the same class OR

the 20 boys seated on the last row of LT may be latecomers so they may have missed

part of the talk OR

Since the sampling frame is available, it is more appropriate to use simple random

sampling or systematic sampling.

6(ii) Stratified Sampling

SA

(LT1)

SB

(LT2)

SC

(LT3)

SD

(LT4)

Number of Year 2

students

selected in each LT

26 24 30 20

Select the number of students randomly from each strata (i.e. subject combination)

according to the table shown above.

7(a) ! 25

5

n

n

7(b)(i) Method 1

Case 1 (all distinct): 4

3 24P

Case 2 (all the same): 3

1 3C

Case 3 (AAB in any order): 3 3

1 2

3!27

2!C C

Total = 24 + 3 + 27 = 54 ways

Method 2

Case 1 (nobody gets hazelnut): 3 3 3 27

Case 2 (someone gets hazelnut): 3

1 3 3 27C

Total = 27 + 27 = 54 ways

7(b)(ii) Method 1

No. of ways the particular friend gets the hazelnut bar

= 3 3

2 12! 9C C

(the other two friends get two distinct bars or two of the same bars)

Hence no. of ways that friend DOES NOT get the hazelnut bar

= total no. of ways without restriction – 9

= 54 – 9

= 45.

Method 2

Case 1 (nobody gets hazelnut): 3 3 3 27

Case 2 (someone gets hazelnut): 2

1 3 3 18C

Total = 27 + 18 = 45 ways

8(i) Let X be the capacity of a randomly chosen paper cup.

Then 2N 500,45X .

2

1 2 21 N 21 500,21 45 N 10500,42525X X X .

1 2 21P 10000 0.992X X X (3 s.f.).

8(ii) Method 1:

2 221 N 10500, 21 45 N 10500,945X

P 21 10000 0.70163X (shown)

Method 2:

10000

P 21 10000 P 0.7016321

X X

(shown)

8(iii) Let Y be the number of tanks, out of 60, that will be filled.

Then B 60,0.70163Y .

For approximation, check that

60 0.70163 42.0978 5

60 1 0.70163 17.9022 5

np

nq

Therefore,

N 42.0978,12.5607Y approximately.

P 45 P 45.5

0.831 (3 s.f.)

Y Y

9(a) From GC, Unbiased estimate of population mean, 171.6875x

Unbiased estimate of population variance, 2 22.7121s

Let X be the random variable for the height of a randomly chosen Taz 17-year old boy.

Let be the population mean height of Taz 17-year old boys.

To test : 170 oH

Against 1 : 170 H at 5% level of significance

c.c.

Under oH , 170

(7)

8

XT t

s

Value of test statistic : 171.6875 170

1.762.7121

8

t

p-value = 0.0609 > 0.05

Do not reject oH .

There is insufficient evidence at 5% level of significance that the mean height of Taz 17-

year old boys is more than 170cm i.e.

There is insufficient evidence that John’s belief is justified at 5% level of significance.

9(b)(i) To test : 170 oH

Against 1 : 170 H at 5% level of significance

9(b)(ii) Under oH , X ~

4.2N 170,

25

Test statistic, 170

(0,1)4.2

XZ N

n

Value of test statistic : 168.6 170

4.2

z

n

For oH not to be rejected at 5% level of significance

168.6 170

1.644854.2

n

24.4n

Thus range of values of n is {1 24, } n n

The assumption that the heights are normally distributed is necessary in order that X

will be normally distributed.

This is because X will not be approximately normal by Central Limit Theorem since n

is small.

10(i) The mean number of chocolate chips in one cookie is a constant. AND

Either: The chocolate chips are randomly distributed such that the occurance of

chocolate chips in a cookie is independent of each other, OR

The chocolate chips are randomly distributed such that the number of chocolate chips

in one cookie is independent of the number in another cookie.

10(ii) Let C be the number of chocolate chips in one cookie.

Then oP 4C .

Required probability = 2 2 4!

P 5 P 52!2!

C C

= 0.00677 (3 s.f.)

10(iii) P 1 0.091578C

Let X be the number of cookies in one box with at most one chocolate chip.

Then B 30,0.091578X .

For approximation, check that

30 0.091578 2.74734 5np .

Hence oP 2.74734X approximately.

P 2 1 P 1

0.75980

0.760 (3 s.f.)

X X

10(iv) Now

o

600P

150

nC

.

Given: P 2 0.95C

600

150

1 P 1 0.95

P 1 0.05

600 1 0.05

150

n

C

C

ne

From G.C., 111.58n

Hence least integer n is 112.

11(a) Method 1

Given: 4

P7

M L .

P 4

P 7

M L

L

4

P P7

M L L

4 1 1

P7 4 7

M L .

Hence 1 1 3

P P P .4 7 28

M L L M L

Method 2 (Using tree diagram drawn from next part of information)

1 3 3

P P x P ' | .4 7 28

M L L M L

To determine whether being right-handed and taking mathematics are independent

events

3 1 1 4 29

P4 2 4 7 56

M , 1

P2

M R

Since P PM R M , being right-handed and taking mathematics are NOT

independent events.

ALTERNATIVELY,

R

L

M

M’

M

M’

Since it is given that 4 29

P P7 56

M L M , being left-handed and taking

mathematics are NOT independent events.

Thus, being right-handed and taking mathematics are NOT independent events.

ALTERNATIVELY,

3 1 1 4 29P( )

4 2 4 7 56 M

3 1 3P( )

4 2 8 M R ,

29 3 87P( ) P( )

56 4 224 M R

Since P( ) P( ) P( ) M R M R , being right-handed and taking mathematics are NOT

independent events.

11(b)(i) Required probability = P (ace, picture)

= 4 12

52 51

= 4

221 or 0.0181 (3 s.f.)

11(b)(ii) Required probability

= P (picture) + P (ace, picture) + P (ace, ace, picture) +

P (ace, ace, ace, picture) + P (ace, ace, ace, ace, picture)

= 12 4 12 4 3 12 4 3 2 12 4 3 2 1 12

52 52 51 52 51 50 52 51 50 49 52 51 50 49 48

=1

4.

11(b)(iii) Required probability =

P (wins on first card picked)

P (player wins the game)

= 12 1

52 4

= 12

13 or 0.923 (3 s.f.)

12(i)

12(ii)

x

y

Model A

y

Model B

x

x (mins)

y (sales)

11.1

8 25

26.1

The scatter diagram in (i) shows that as x increases, y increases by decreasing amounts

(or at a decreasing rate). This is consistent with model B. Furthermore, 0.997r for

Model B, which suggests a strong positive linear relationship between ln x and y.

Model A is not appropriate as it suggests that as x increases, y increases by increasing

amounts (or at an increasing rate).

Model C is not appropriate since it suggests that as x increases, y decreases, which is

not consistent with the scatter diagram in (i).

Equation of least square regression line of y on ln x is

y = –17.326 + 13.698 ln x

a = –17.3 (3 s.f.) , b = 13.7 (3 s.f.)

0.997r (3 s.f.)

12(iii) For every unit increase in ln x, the corresponding increase in sales of bottles of

sunblock is 1369.

12(iv) When x = 10, y = –17.326 + 13.698 ln 10

= 14.214

Number of bottles sold = 1421

Thus weekly profit = (1421)(20) –10 (2000)

= $8420

Estimate is reliable since 0.997 1 r and x = 10 is within the range of sample data,

8 25 x .

y

Model C

x