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RI2019 [Turn over

CANDIDATE NAME

CLASS 19

MATHEMATICS 9758/01PAPER 1

3 hoursCandidates answer on the Question PaperAdditional Materials: List of Formulae (MF26)

READ THESE INSTRUCTIONS FIRST

Write your name and class on all the work you hand in.Write in dark blue or black pen.You may use an HB pencil for any diagrams or graphs.Do not use staples, paper clips, glue or correction fluid.

Answer all the questions.Write your answers in the spaces provided in the question 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.The use of an approved graphing calculator is expected, where appropriate. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise.Where unsupported answers from a graphing 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.The total number of marks for this paper is 100.

FOR EXAMINER’S USE Q1 Q2 Q3 Q4 Q5 Q6

Q7 Q8 Q9 Q10 Q11 Total

This document consists of 6 printed pages. RAFFLES INSTITUTIONMathematics Department

RAFFLES INSTITUTION2019 YEAR 6 PRELIMINARY EXAMINATION

www.KiasuExamPaper.com

578

2

H2 MA 9758/2019 RI Year 6 Preliminary Examination Paper 1

1 A curve C has equation

3a

y bx cx

,

where a, b and c are constants. It is given that C has a stationary point 1.2, 6.6 and it also passes through the point 2.1, 4.5 .

(i) Find the values of , and a b c , giving your answers correct to 1 decimal place. [4]

(ii) One asymptote of C is the line with equation x = 0. Write down the equation of the otherasymptote of C. [1]

2 Two variables u and v are connected by the equation 1 1 1

20u v . Given that u and v both

vary with time t, find an equation connectingd d, , and .d du v

u vt t

Given also that u is

decreasing at a rate of 2 units per second, calculate the rate of increase of v when u = 60 units.

r [4]

3

Fig. 1 Fig. 2

Fig. 1 shows a square sheet of metal with side a cm. A square x cm by x cm is cut from each corner. The sides are then bent upwards to form an open box as shown in Fig. 2. Use differentiation to find, in terms of a, the maximum volume of the box, proving that it is a maximum. [6]

a

x


579

3


4 A curve C has parametric equations

2 21 , 2 1 .x t y t

(i) Find the coordinates of the point A where the tangent to C is parallel to the x-axis. [4]

(ii) The line y x d intersects C at the point A and another point B. Find the exactcoordinates of B. [4]

(iii) Find the area of the triangle formed by A, B and the origin. [1]

5

The diagram shows the graph of Folium of Descartes with cartesian equation 3 3 3 ,x y axy

where a is a positive constant. The curve passes through the origin, and has an oblique asymptote with equation y x a .

(i) Given that (0, 0) is a stationary point on the curve, find, in terms of a, the coordinates ofthe other stationary point. [5]

(ii) Sketch the graph of3 3 3 ,x y a x y

including the equations of any asymptotes, coordinates of the stationary points and the point where the graph crosses the x-axis. [3]

(iii)

y x a

O


580

4


6 (a) Given that 2

1

1 1 2 16

n

r

r n n n

, find an expression for

2

11

n

r n

r r

, simplifying

your answer. [3]

(b) (i) Use the method of differences to find 1

+1 ee

n

rr

r r

. [3]

(ii) Hence find 1

2

e eer

n

r

r r

. [2]

7 (a) The complex numbers 3 2i , z, 4 6i are the first three terms in a geometric progression.Without using a calculator, find the two possible values of z. [4]

(b) (i) The complex number w is such that iw a b , where a and b are non-zero real

numbers. The complex conjugate of w is denoted by *w . Given that 2

*w

wis a real

number, find the possible values of w in terms of a only. [4]

(ii) Hence, find the exact possible arguments of w if a is positive. [2]

8 (a) Find the exact value of m such that

13 2

2 2 20 0

1 1 d d9 1

m

x xx m x

. [5]

(b) (i) Use the substitution sin 2u x to show that

1

4 3 3 3 5

0 0

1sin 2 cos 2 d d2

x x x u u u

. [4]

(ii) Hence find the exact value of 4 3 3

0

sin 2 cos 2 dx x x

. [2]


581

5


9 (i) Write down 2 21 dv

a v

, where a is a positive constant. [1]

(ii) In the motion of an object through a certain medium, the medium furnishes a resisting force proportional to the square of the velocity of the moving object. Suppose that a body falls vertically through the medium, the model used to describe the velocity, v ms–1 of the body at time t seconds after release from rest is given by the differential equation

22 2d ,

10 da v

a vt

where a is a positive constant.

(a) Show that

20

20e 1

e 1

t

a

t

a

v a

. [8]

(b) The rate of change of the displacement, x metres, of the body from the point of release is the velocity of the body. Given that 2a , find the value of xwhen t = 1, giving your answer correct to 3 decimal places. [3]

10 The curve 1C has equation 2 2 25, 0x y y .

The curve 2C has equation243

3y x

x

.

(i) Verify that 3, 4 lies on both 1C and 2C . [1]

(ii) Sketch 1C and 2C on the same diagram, stating the coordinates of any stationary points, points of intersection with the axes and the equations of any asymptotes. [4]

The region bounded by 1C and 2C is R.

(iii) Find the exact volume of solid obtained when R is rotated through 2 radians about thex-axis. [6]

The region bounded by 1C , the x-axis and the vertical asymptote of 2C , where 3x , is S.

(iv) Write down the equation of the curve obtained when 1C is translated by 3 units in the negative x-direction.

Hence, or otherwise, find the volume of solid obtained when S is rotated through 2radians about the vertical asymptote of 2C . [4]


582

6


11 Path integration is a predominant mode of navigation strategy used by many animals to return home by the shortest possible route during a food foraging journey. In path integration, animals continuously compute a homebound global vector relative to their starting position by integrating the angles steered and distances travelled during the entire foraging run. Once a food item has been found, the animal commences its homing run by using the homebound global vector, which was acquired during the outbound run.

(a) A Honeybee’s hive is located at the origin O. The Honeybee travels 6 units in thedirection 2 2 i j k before moving 15 units in the direction 3 4i k . The Honeybee isnow at point A.

(i) Show that the homebound global vector AO

is 7 4 16 i j k . Hence find the exact distance the Honeybee is from its hive. [3]

(ii) Explain why path integration may fail. [1]

A row of flowers is planted along the line 3 2, 2

5x

y z

.

(iii) The Honeybee will take the shortest distance from point A to the row of flowers.Find the position vector of the point along the row of flowers which the Honeybeewill fly to. [4]

(b) To further improve their chances of returning home, apart from relying on the pathintegration technique, animals depend on visual landmarks to provide directionalinformation. When an ant is displaced to distant locations where familiar visuallandmarks are absent, its initial path is guided solely by the homebound global vector,h , until it reaches a point D and begins a search for their nest (see diagram). During thesearching process, the distance travelled by the ant is 2.4 times the shortest distance backto the nest.

Let an ant’s nest be located at the origin O. The ant has completed its foraging journey and is at a point with position vector 4 3i j . A boy picks up the ant and displaces it 4 units in the direction i . Given ( 4 3 ji ) as the initial path taken by the ant before it begins a search for its nest, find the value of which gives the minimum total distance travelled by the ant back to the nest. [4][It is not necessary to verify the nature of the minimum point in this part.]

Nest

Path travelled by undisplaced ant

Nest

Path travelled by displaced ant

Ant displaced

Shortest distance to nest (dotted arrow)

Actual path travelled by ant during searching process


583

RI2019 [Turn over

CANDIDATE NAME

CLASS 19

MATHEMATICS 9758/02

PAPER 2 3 hoursCandidates answer on the Question Paper Additional Materials: List of Formulae (MF26)

READ THESE INSTRUCTIONS FIRST

Write your name and class on all the work you hand in.Write in dark blue or black pen.You may use an HB pencil for any diagrams or graphs.Do not use staples, paper clips, glue or correction fluid.

Answer all the questions. Write your answers in the spaces provided in the question 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.The use of an approved graphing calculator is expected, where appropriate. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise.Where unsupported answers from a graphing 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.The total number of marks for this paper is 100.

FOR EXAMINER’S USE SECTION A: PURE MATHEMATICS

Q1 Q2 Q3 Q4 Total

TOTAL SECTION B: PROBABILITY AND STATISTICS

100

Q5 Q6 Q7 Q8 Q9 Q10 Total

This document consists of 7 printed pages.

RAFFLES INSTITUTIONMathematics Department

RAFFLES INSTITUTION2019 YEAR 6 PRELIMINARY EXAMINATION


584

2


Section A: Pure Mathematics [40 Marks]

1 The function f is defined as follows.

21f : ,xx

for , 0x x .

(i) Sketch the graph of f ( )y x . [1]

(ii) If the domain of f is further restricted to x k , state with a reason the least value of kfor which the function f 1 exists. [2]

In the rest of this question, the domain of f is , 0x x , as originally defined.

A function h is said to be an odd function if h( ) h( )x x for all x in the domain of h. The function g is defined as follows.

2g : ,3 1x

x m

for , 0x x .

(iii) Given that g is an odd function, find the value of m.

(iv) Using the value of m found in part (iii), find the range of fg.

2 (a) The curve y = f(x) passes through the point (0, 81) and has gradient given by13d 1 15 .

d 3y

y xx

Find the first three non-zero terms in the Maclaurin series for y. [4]

(b) Let2

24 3g( )

(1 )(1 )x x

xx x

.

(i) Express g( )x in the form 21

1 1 (1 )A B

x x x

, where A and B are constants

to be determined.

The expansion of g( )x , in ascending powers of x, is 2 3

0 1 2 3 ... ... r

rc c x c x c x c x .

(ii) Find the values of 0 1 2, , and c c c and show that 3 3c .

(iii) Express rc in terms of r .


585

3

H2 MA 9758/2019 RI Year 6 Preliminary Examination Paper 2 [Turn over

3 Referred to an origin O, the position vectors of three non-collinear points A, B and C are a,b and c respectively. The coordinates of A, B and C are 2,4,2 , 1,3,1 and 0,1, 2respectively.

(i) Find a b c b . [1]

(ii) Hence

(a) find the exact area of triangle ABC,

(b) show that the cartesian equation of the plane ABC is 3 2 7 16x y z .

(iii) The line 1l has equation 1 11 26 1

r for .

(a) Show that 1l is parallel to the plane ABC but does not lie on the plane ABC.

(b) Find the distance between 1l and plane ABC.

(iv) The line 2l passes through B and is perpendicular to the xy-plane. Find the acute anglebetween 2l and its reflection in the plane ABC, showing your working clearly.

4 (a) The non-zero numbers a, b and c are the first, third and fifth terms of an arithmetic series respectively.

(i) Write down an expression for b in terms of a and c. [1

(ii) Write down an expression for the common difference, d, of this arithmeticseries in terms of a and b. [1

(iii) Hence show that the sum of the first ten terms can be expressed as

5 9 .4

c a [2]

(iv) If a, b and c are also the fourth, third and first terms, respectively of ageometric series, find the common ratio of this series in terms of a and b andhence show that

2 3(2 )b a a b .

(b) The nth term of a geometric series is 122sinn

.

(i) Find all the values of , where 0 2 , such that the series is convergent.

(ii) For the values of found in part (i), find the sum to infinity, simplifying your ans

Section B: Probability and Statistics [60 Marks]


586

4


5 (a) The probability that a hospital patient has a particular disease is p. A test for thedisease has a probability of 0.99 of giving a positive result when the patient has the disease and a probability of 0.95 of giving a negative result when the patient does not have the disease. A patient is given the test.

For a general value of p, the probability that a randomly chosen patient has the disease given that the result of the test is positive is denoted by f ( )p .

Find an expression for f ( )p and show that f is an increasing function for 0 1.p Explain what this statement means in the context of this question. [5

(b) For events A and B, it is given that P( ) 0.7A and P( ) .B k

(i) Given that A and B are independent events, find P( ') A B in terms of k.

(ii) Given instead that A and B are mutually exclusive events, state the range ofvalues of k.

Find P( | ' ) B A in terms of k.

6 Five objects , , ,a b c d and e are to be placed in five containers , , ,A B C D and ,E with one in each container. An object is said to be correctly placed if it is placed in the container of the same letter (e.g. a in A) but incorrectly placed if it is placed in any of the other four containers. Find

(i) the number of ways the objects can be placed in the containers so that a is correctlyplaced and b is incorrectly placed, [2]

(ii) the number of ways the objects can be placed in the containers so that both a and bare incorrectly placed, [3]

(iii) the number of ways the objects can be placed in the containers so that there are atleast 2 correct placings. [3]


587

5


7 Based on past records, at government polyclinics, on average each medical consultation lasts 15 minutes with a standard deviation of 10 minutes.

(i) Give a reason why a normal distribution, with this mean and standard deviation,would not give a good approximation to the distribution of consultation duration. [2]

From recent patient and doctor feedback, a polyclinic administrator claims that the average consultation duration at government polyclinic is taking longer. State suitable null and alternative hypotheses to test this claim.

(ii) Given that the null hypothesis will be rejected if the sample mean from a randomsample of size 30 is at least 18, find the smallest level of significance of the test. Stateclearly any assumptions required to determine this value.

The administrator suspects the average consultation duration at private clinics is actually less than 15 minutes. A survey is carried out by recording the consultation duration, y, in minutes, from 80 patients as they enter and leave a consultation room at a private clinic.The results are summarized by

( 15) 50y , 2( 15) 555y .

(iii) Calculate unbiased estimates of the population mean and variance of the consultationduration at private clinics. Determine whether there is sufficient evidence at the 5%level of significance to support the administrator’s claim.

8 A biased cubical die has its faces marked with the numbers 1, 3, 5, 7, 11 and 13. The random variable X is defined as the score obtained when the die is thrown, with probabilities given by

P( X r ) = kr , 1,3,5,7,11,13,r where k is a constant.

(i) Show that P(X = 3) = 340

. [3]

(ii) Find the exact value of Var(X).

The die is thrown 15 times and the random variable R denotes the number of times that a score less than 10 is observed.

(iii) Find P(R

(iv) Find the probability that the last throw is the 8th time that a score less than 10 isobserved.


588

6


9 Rroar Tyre Company develops Brand R tyres. The working lifespan in kilometres of a Brand R tyre is a random variable with the distribution N(64000, 80002).

(i) Find the probability that a randomly selected Brand R tyre has a working lifespan ofat least 70000 km. [1]

(ii) Rroar Tyre Company wishes to advertise that 98% of their Brand R tyres haveworking lifespans of more than t kilometres. Determine the value of t, correct to thenearest kilometre. [2]

Ssoar Tyre Company, a rival company, develops Brand S tyres. The working lifespan in kilometres of a Brand S tyre is a random variable with the distribution N(68000, 2 ).

(iii) A man selects 50 Brand S tyres at random. Given that = 7500, find the probabilitythat the average of their working lifespans exceeds 70000 km.

(iv) Using = 8000, find the probability that the sum of the working lifespans of 3randomly chosen Brand R tyres is less than 3 times the working lifespan of arandomly chosen Brand S tyre.

(v) Find the range of , correct to the nearest kilometre, if there is a higher percentageof Brand S tyres than Brand R tyres lasting more than 50000 km.

(vi) State clearly an assumption needed for your calculations in parts (iii), (iv) and (v).


589

7


10 (a) With the aid of suitable diagrams, describe the differences between the least square linear regression line of y on x and that of x on y. [2]

(b) The government of the Dragon Island Country is doing a study on its population growthin order to implement suitable policies to support the aging population of the country.The population sizes, y millions in x years after Year 2000, are as follows.

x 9 10 11 12 14 15 16 17 18Population

size, y5.05 5.21 5.32 5.41 5.51 5.56 5.60 5.65 5.67

(i) Draw a scatter diagram of these values, labelling the axes clearly. Use yourdiagram to explain whether the relationship between x and y is likely to be wellmodelled by an equation of the form y ax b , where a and b are constants.

(ii) Find, correct to 6 decimal places, the value of the product moment correlationcoefficient between

(a) ln x and y,

(b) 2x and y.

(iii) Use your answers to part (ii) to explain which of lny a b x or 2y a bx isthe better model.

(iv) It is required to estimate the year in which the population size first exceed 6.5millions. Use the model that you identified in part (iii) to find the equation of asuitable regression line, and use your equation to find the required estimate.

Comment on the reliability of this estimate.

(v) As the population size in Year 2013 is not available, a government statisticianuses the regression line in part (iv) to estimate the population size in 2013. Findthis estimate.

(vi) It was later found that in Year 2013, the population size was in fact 5.31 millions.Comment on this figure with reference to the estimate found in part (v), providinga possible reason in context.


590

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RAFFLES INSTITUTION 2019 YEAR 6 PRELIMINARY EXAMINATION


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Page 15 of 17


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Page 17 of 17

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