TEM4SEK JUNCR COLLEGE, SINGAPORE

fUl I—I Prelimirtary Examination 2011 - - Higher 2

MATHEMATICS 9740/01 Paper 1

15 September 2011

Additional Materials: Answer paper 3 hours List of Formulae (MF15)

READ THESE INSTRUCTIONS FIRST

Write your Civics group and name on all the work that 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 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. At the end of the examination, fasten all your work securely together.

This document consists of 4 printed pages.

TEMASEX JlMOFtOOtlESE. SNGAKME

l&21&ul£ [Turn over

1 It is given that u\ 2 and ur*\ u, = 2(r + 1) where r e Z*

By using the method of difference, find un in terms of n. [5]

«>0

2 Given that a is a positive constant, solve the inequa!;ty \x-2a\<2x + a

Hence solve the inequality \x + 4| < 2 - 2x.

[4]

[2]

Functions f and g are defined by

tixt+ljl-x* , xeR,-l<x<l,

g:*»-* lnx , « R * .

(i) Show that composite function gf exists and find the range of gf in exact form. [4]

(ii) Describe the transformations such that the graph of y = g(x) is mapped onto the graph ofy = ln (2x -1 ) .

It is given that S. » -—-. t j r ( r - l )

(I) Write down the values of S 2 . and S*.

(ii) Make a conjecture for S„ in terms of n, for n e Z", n £ 2.

(iii) Prove your conjecture by the method of induction.

(iv) Determine if the series J\—T-—r converges. t$r{r-\)

A graphic calculator is not to be used in answering this question.

Two complex numbers z and w are such that z • — 3̂ + i and

(i) the fourth roots of z,

(ii) the modulus and argument of w.

I = 2i. Find

[2]

[1]

[1]

[4]

[1]

[4]

[4]

A curve has parametric equations given by •

x = 2 0 - s i n 2 0 and y = 2-cos20.

(i) Show that ^ = cot0. [31 dx

(ii) Find the equation of the normal to the curve at the point where 8 - —. [3] 4

(iii) For the part of the curve where 0 < 6 < 2K , find the coordinates of the point where the tangent is parallel to the y-axis. [3]

TJC/MA 9740/Prellmlnarv Exam 2011

3

7 Given that y - In (cosx), show that — f « -1 - — . [2] Ax* \dxj

Determine the first two non-zero terms in the Maclaurin series for y. [4]

Use this series to find an approximation for In 2, in terms of n. [3]

8 The vector equations of lines /i and h are given by

r - 71 + 3j+ 7k + M.I + k), and

r - 3j+ 3k + p(qi + 2k), respectively, where A, fl and q e R .

It is given that l2 passes through the points A and B with position vectors 3j+ 3k and gl+3j+5k respectively.

(I) If the acute angle between l\d / 2 is 60° , find the exact values of q. [3]

(Ii) If <7 = 1, find the position vector of the point C on /| such that A is the foot of perpendicular from C to / j . [4]

(ill) Let 7* be any point or (|, If the area of triangle ABT is constant for any X, find the value of q. [2]

(a) Find the sum of all the integers between 1102 and 2011 (inclusive) which are not divisible by 3. [5]

(b) A geometric progression G has first term a and common ratio r. A sequence H is formed by squaring the terms of G. Prove that H is also a geometric progression. The sum to infinity of G is S. Find the value of r such that the sum to infinity of His2S2. [4]

10 (a) On a diagram, shade the region bounded by the curve y • — and the lines y • 2x x

and y • 3x for x 2 0 . Label all the intersection points clearly. [2]

Find the exact area of the shaded region. [3]

(b) Find the volume of the solid formed by rotating the circle ;cJ + y J - 4 y + 3 = 0 about the x-axis. Give your answer correct to one decimal place. [4]

[Turn over

TJC/MA 97«/Prellmlnarv Exam 2011

11 (a) The graph of y - ft*) is shown below. The curve cuts the y-axis at ̂ 0»~̂ j 811(1

the ;t-axis at (1, 0) and (7, 0). Sketch the graph of y «= — ~ . showing clearly the I (X)

main relevant features of the curve. . [3]

(b) The curve C has equation y • X + P - , wherep is a non-zero constant.

*(* + 3)

(i) State the equations of the asymptotes. [2]

(ii) Show that if C has 2 stationary points, then p < 0 or p > 3. [4] (iii) Given p = 4, sketch the curve C, showing clearly the equations of the

asymptotes and the coordinates of the axial intercepts and stationary points. [2]

12 The line / passes through the points (2, - 1 , -1) and (6, 1, 3). The equation of the plane pt is given by r • (i + 2j + 2k) - 6.

(i) Write down a vector equation of / .

(ii) Show that the position vector of the point of intersection of / and pi is 41 + k. [3]

A plane p2 contains / and meets the plane p\n a line m. Given that the line m is perpendicular to / , show that m is parallel to 2i + 2j -3k. [2] Hence write down a vector equation of pi. [1 ]

A plane p 3 has equation x - 7 y - 4 z + it(;c+2y + 2 z - 6 ) = 0. Explain why m lies on pi for any constant k. Hence or otherwise, find a cartesian equation of the plane, in which both m and the point (0, 0, 2) lie. [4]

so so so End of Paper ca eg cs

TJC/MA 9740/Prellmlnary Exam 2011

TEMASEK JLNCR COLLEGE, SINGAPORE

I—I Preliminary Examination 2011 . £ J = _ Higher 2

MATHEMATICS 9740/02 Paper 2

19 September 2011

Additional Materials: Answer paper 3 hours Graph paper List of Formulae (MF15)

READ THESE INSTRUCTIONS FIRST

Write your Civics group and name on all the work that 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 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. At the end of the examination, fasten all your work securely together.

This document consists of 6 printed pages.

CTJC2011 [Turn over

2

Section A: Pure Mathematics [40 marks] ry.

1 The nth term of a sequence is denoted by a , . The sequence is defined by

a .»i B Pa. +<7 for all n £ 1, where p and q are constants.

The first three terms of the sequence are given by a, = 0, a 2 • 2 and ai = 7 .

(i) Find the values of p and q.

(ii) Express a, in terms of n, for n > 3.

[2]

[4]

2 (a) Given f ( * ) • •

that of y « f(x). justifying your answer clearly.

, identify which of the following graphs, A, B and C is

[2]

c i

c

I I

(b) A point P(x, y) moves on the curve y = tan x in such a way that the x-coordinate

of P increases at a constant rate of 4 units per second. Find the rate of change

with respect to time of the gradient of the curve when x = —. 4

[4]

3 (a) Find J i * fr,

(b) Using integration by parts, find the exact value of J o xjlx + l dx .

[4]

[4]

4 Given that a = 1 + i , and the complex numbers z and w are such that

| z - a | « | a | and a r g ^ ^ j - ^ j i - a r g a .

Sketch on a single Argand diagram the loci of points representing z and w. [5]

Hence find

(1) the least value of | w + 31 , [2]

(ii) the greatest value of arg(z + 3). [3]

; TJC/MA 9740/Preliminary Exam 2011

3

At time I = 0, there are 8000 fish in a lake. At time t days the birth-rate of fish per day is

equal to one-fortieth ( ^ ) o f the number N of fish present. Fish are taken from the lake

at the rate of 150 per day. dN

Modelling W as a continuous variable, show that 4 0 — • N -6000. [2] di

Solve the differential equation to find N in terms of t. Find the time taken for the population of fish in the lake to increase to 12000. [5]

When the population of fish has reached 12000, the number of fish taken from the lake is increased from 150 per day to r per day.

Write down, in terms of r, the new differential equation satisfied by yV. [ 1 ]

Find the maximum value of r *o that the population of fish in the lake will not decrease.

[2]

Section B^Statistics [60 marks]

(a) How many different nine-digit numbers can be formed from five "0"s and four "1 "s such that no three "0"s are together? [4]

(b) Six people sit at a round table with eight identical chairs. Find the number of seating arrangements if

(I) there are no restrictions, [2]

(II) the seats are numbered. [1]

The heights of a randomly chosen boy and a randomly chosen girl in a junior college are denoted by B and G respectively. The random variables B and G are independent normal variables with means and standard deviations as shown in the table.

Mean (cm) Standard deviation (cm)

Boys(B) 170 4

Girls (G) 155 5

(i) Explain briefly why the assumption of a normal distribution with the above mean and standard deviation for the heights of the boys may be reasonable even though such a height cannot possibly be negative. [1]

(ii) Find the probability that 3 times the height of a randomly chosen girl will exceed the total height of 2 randomly chosen boys by more than 120 cm. [3]

(iii) Find the value of k such that P(|G -155| < k) = 0.85 [3] (iv) Three boys are chosen at random. Find the probability that the first chosen boy is

taller than 165 cm and the other two boys are shorter than 165 cm. [2] [Turn over

TJC/MA 9740/Prellminary Exam 2011

i

4

Nine balls numbered from 1 to 9 are placed into a bag. The odd-numbered balls are red in colour and the even-numbered balls are blue in colour.

Three balls are drawn from the bag without replacement. For each ball drawn, the number on the ball represents the score. For any blue ball drawn, the score on the next ball drawn will be doubled. [For example, if balls numbered 3, 4, 5 are drawn in this order, the total score is3 + 4 + 5 x 2 » 1 7 ]

Find the probability that

(i) all balls are red, [2]

(ii) there is at least one ball of each colour, [2]

(iii) the total score is more than 18, given that all balls are red, [3]

(iv) the total score is even. [3]

9 (a) A junior college intends to make changes to its school uniform and decides to obtain the views from students on the design of a new uniform. Mr Tong obtains a sample by selecting 50 students randomly from the school register.

(i) Name the sampling method used. [1]

(11) Comment on the appropriateness of the use of this method for this situation.

m (iii) Describe briefly another sampling method which can be used by the school

and state one advantage of the method used in this context. [3]

(b) A machine is set to fill .jars with 270 g of kaya. Over a period of time it has been established that the standard deviation of the mass of kaya in a jar is 0.8 g.

A setting on the filling machine is altered and the operator suspects that the mean mass of kaya filled by the machine has decreased. A random sample of 10 jars of kaya were measured and the masses in grams were found to be

269, 271, 270," 268, 269. 270. 268, 270. 271, 269.

(1) Assuming that the standard deviation may have been altered, perform an appropriate test at the 10% level of significance, on whether the operator's suspicion about the machine is justifiable. State any assumptions make in carrying out the test. [5]

(ii) Assume that the standard deviation of the mass of kaya in a jar is unaltered. State what test you would use in this case, giving any assumptions that you need to make. [You are not required to carry out this test]. [1]

TJC/MA 9740/Prelimlnary Exam 2011

5

10 (a) The scatter diagram shows a sample of size 6 of bivariate data, together with the regression line of yonx.

State and giving a reason, whether, for the data shown, the regression line of y on x is the same as the regression line of x on y. [1]

Given that y = -8x+86 is the regression line of y on x for the 6 pairs of

observations with V x = 60.

(i) Find the value of the mean of y. [2]

(ii) State, with a reason, the regression line of y on x after a new pair of observation (10,6) is added. [2]

(b) A car is travelling along a stretch of road with speed v (km/h) when the brakes are applied. The car comes to rest after'travelling a further distance of d m. The values of d for 10 different values of v are given in the table, correct to 2 decimal places.

V 35 40 45 50 55 60 65 70 75 80

d 4.90 5.30 5.93 6.45 7.00 8.50 12.00 17.00 22.10 25.90

It is given that the value of the linear product moment correlation coefficient for this data is 0.919, correct to 3 decimal places. The scatter diagram for the data is shown below.

(1) Calculate the product moment correlation coefficient between v and 4d .

What does this indicate about the scatter diagram of the points (v, -Jd •)? [2]

(ii) Which regression line is more appropriate: 4d on v or d on v? State the

reason and find the equation of the appropriate regression line. [2]

(iii) Estimate the distance a car travels after its brake is applied when the car's speed is lOOkm/h. Comment on the reliability of your answer. [2]

[Turn over TJC/MA 9740/Preliminary Exam 2011

11 (a) X is a discrete random variable with mean 20.5 and variance 3.2. A random sample of 60 observations of X is taken, find th- probability that the sample mean lies between 19 and 20. [3]

(b) Analysis of the scores in football matches in a local league suggests that the number of goals scored in a randomly chosen match may be modelled by the Poisson distribution with mean 2.7. State, in the context of the question, one assumption for the Poisson distribution to be a valid model. [1]

(I) Find the probability that a randomly chosen match will end with at least two goals scored. [1]

(ii) On one afternoon, 12 matches are played in the league. Find the probability that there are fewer than 8 matches in which there are at least 2 goals scored. [2]

T is the total number of goals scored in n matches. State the distribution of T and its parameters). [1]

Using a suitable approximation, find the least value of n such that the probability of T being greater than 100 is more than 0.9. [4]

00

BO so BO End of Paper w w w

TJC/MA 9740/Preliminary Exam 2011

x u u u

2011 Preliminary Exam H2 Maths 9740 Paper 1 Solutions

1 Method of Difference

£(*•-*)- *Rr+n

=> u . -u , = ( « - l X " + 2) M. n -2 + 2 = n(n + l)

Thus u.*n(n + l) [Alternative Methyl

Summing up to n terms, we obtain umti — M, «/»(n + 3) = n J +3/i

«„, - n 1 * n + 2 « ( n + l)(n + 2) Thus M„=«(n + 1)

Inequalities

To find intersection point, 2x + a = -(x - 2a)

x • — 3

From the graph, for |x - 2a| < 2x + a, a

x>-3

Replace x by -x and let a - 2 in the above inequality,

|(-x) - 2(2)| < 2(-x) + 2 becomes |x + 4| < 2 - 2x ^ . 2 2 Thus -x>-=>x<--

TJC/MA8740/P1/Preliminary Exam 2011/SohJtlons

3 Functions & Transformations (D

>--f(x)

Since R f - (0 , 2] c R*-D, , gf exists.

Method 1 (mapping method)

(-1, 1) (0,2] - L . . ( -« , In 2]-Range of gf

Method 2 (graphical method)

gfl&r)- l n ( 2 v ' l - x J ) - b 2 + ^ l n ( l - x , ) , - l < x < J

y - gfl»

Range of gf-(-oo, In 2]

(ii) A translation of 1 unit along the positive x-axis followed by a scaling of

factor ^ along the x-axis.

OR

A scaling of factor along the x-axis followed by a translation of — unit

along the positive x-axis,

TJC/MAB740/P1/Prellmlnary Exam 2011/SohJtani

Mathematical Induction (0 s I m l

1 " 2(1) * 2

5, • — +• » — ' 2 3(2) 3

4 - 3 + 4{3)"4

( , , ) t . £ = i . . » 3

( H , ) Let / \e the statement 5. » — where n I Z * , « * 2. n

When « - 2, LHS - 5, = - from (i)

— ¥4 .-. /»i if true Assume Pt be true for some k e Z*, A: £ 2, i.e. 5, • .

y . — - — + tir(r-l) ^ r ( r - l ) (* + !)*

k-l 1

( * - l X * + Q - l (* + !)*

(* + i)* * + i * + i Thus Pt*\s true.

Since Pi is true, and Pt.\s true if Pt is true, by the method of mathematical induction, PH is true for all n e Z', n 2 2.

As n -» oo, 5 ' • • I » 1 . Thus > — converges to 1.

TJC/MA9740/P1/Preliminary Exam 2011/Solutlont

Complex Numbers (0 |* |«>/377«2

argz * *-tan"

r ' - 2 * | e ^ ~ ' | , i t - - 2 , - 1 . 0 , 1

i i . ,2 , i , 1 . i ,!!. . 2 « e » , 24e » , 2 ' e " , 2*t»

bi . j = > - i a r g ( w ) - 1 0 a r g ( 2 ) » - |

_ . . , . . 47* * ,*, Principal arg(w) - — — loir m-—

- 2 i

2e

= 2e"t? 2e

w « 2 n e ^ ~ - 2 u e " ^

••• H " 2 * ™d a r g ( w ) » - y

TJDMA97«0/Pt/Pre<nnlnary Exam 2011/SoftiDon i 4

6 Tangents & Normals (Parametric Equations) (i) — = 2-2cos26> , i ^ - 2 s i n 2 0

60 d0 dy 2sin2fl dx 2-2cos20

_ 4sinflcosfl 4sin2f?

C O t f ?

^ At the point where 0= —, x = — - 1 , y = 2, — = cot—= 1 4 2 dx 4

Equation of the normal: y-2 = -(jr——+1)

y = l + --x

^ When the tangent is parallel to they-axis, — is undefined dx

For C 0 S ^ to be undefined, s i n # » 0 sine?

=> 0 = rt for 0 < 0 < 2/r When 0 = n, x = 2/r, y = 1 .'. the tangent is parallel to the y-axis at the point (2rt, 1).

7 Maclaurin's Series y - In (cosx) dy -sinx — = -tanx dx cosx ^ = - s e c 2 x = - ( l + t a n 2 x ) - - 1 -B9

2

dx3 d x [ d x 2 y

d 4 y dx4

= -2 fd 2 y ' d V ^ l ,dx 2 1 + d x

Whenx = 0 . y « 0 , ^ - 0 , ^ - - 1 , ^ - 0 , ^ - 2 . dx dx2 dx3 dx4

y " 2! " 4! + 12

Taking x = —, 4

In 4'

1 . _ it rt — In2«

2

=> in 2 « — + 32 3072 16 1536

TJC/MA9740/P1 /Preliminary E x a m 2011/Solutlon«

(I) Vectors Given acute angle between l\d is 60*,

(1)

cos 60° =

Apply modulus to ensure an acute angle is obtained regardless of value of q.

1^2) = - ^ # 7 4

( f l + 2 ) 2 = i ( ^ + 4 )

q2 + Sq + 4 - 0

Marker's comments: The modulus on the scalar product was missing even when the angle was given to be acute in the question. Among answers which applied the modulus, the modulus "disappeared" right after the scalar product had been evaluated to be q + 2. Many students made careless mistakes in expanding 2(q2 + 4) and (q + 2)2.

(ii) '1 m Since C lies on /,, &C = 3 0 for some X Obtain OC in terms of one

unknown X only.

JC=UC-Ua = o

4+AJ

Since AC 1 /„ AC • d 2 =0 , ,4(0,3,3) [1*1

0 • 0

-5

.-. o~c-

(hi) Since the area of ABT is constant for all k, l\d /; are parallel. V

c = * o

X A

k • H 9 = 2

r

TJCVMAJV40/P1/Preliminary E x a m 2011/Sowtoni 6

[Alternative solution] T

Since flies on l\, UT = 3 0

J)

for some X

Area of MBT =-\

f 7 + y l] 0 X 0

A

l4-4? + ( 2 - * M 2

. | 1 4 - 4 o + ( 2 - a M |

Since the area is a constant independent of X, QHSflB .-. 9 - 2

Marker's comments: Quite a number of students left the part unanswered. Some used the wrong formula eg

•̂1 AB || AT\, J AB*AT |. For students who attempted to use the cross-product formula,

many did not bow to proceed after obtaining the area. Many students made mistakes in 0

1 evaluating — 14-4<7 + (2-<7)A 2

; some wrong answers were — (14 - 4o + (2 - q)X) and

TJC/MA9740/P1/Prellmlnary Exam 2011/Solutlons

9 AP&GP (a) Sum of all integers between 1102 and 2011 inclusive

= 2 0 1 1 - 1 1 0 2 , l ( l l 0 2 ^ 2 0 | l ) , l 4 1 6 4 1 s

1104,1107,2010 is an AP with first term 1104 and common difference 3 2010-1104 + (n - l )3 => n«303

Sum of integers between 1102 and 2011 that are divisible by 3

= ^ ( 1 1 0 4 + 2010)

= 471771

.. required sum- 1416415-471771 -944644

(b) Let (J* and H„ be the nth terra of G and H respectively.

- 2 = ^ = r * > a constant which is independent of n

Therefore, H is a geometric progression.

l - r J [,1-r,

( . - r ) ' = 2 ( W )

3 r J - 2 r - l = 0

r•—— or r- 1 (rejected since |rj< 1)

i

TJC/MA974WP ^Preliminary E x a m 201 PMUt fOM 3

10 Applications of Integration (Area & Volume) (a)

- 2 x , A » - I ( V J ) ( 3 V 5 ) + ^ f d , - I ( V S ) ( 2 V 5 )

- 3 ( l n 3 - l n 2 )

(b)

0

x ' + y - 4 y + 3 « 0 = > x a + ( y - 2 ) 1 - I

Volume- 2^( / ry , 2 - / ry 2 J )dx where y, =2 + V l - x J

and y 2 = 2 - V l - * 2 .

39.5 using GC

dx

TJC/MA97407P1/Prelimlnary E x a m 2011/Solutlont 9

11 Graphing Techniques & Graph o f f '(x)

fO

(a)

: l i

- > v °

- 2 N

! l 3 5 7

\X- 1 x - 7

(b)(1) Horizontal asymptote: y - 0 Vertical asymptotes: x - 0, x - -3

(U) d , ( , ' + 3 x ) - ( x + p)(2x + 3) dx" x*(x + 3)*

_ (x J +3x)-(2x J +3x + 2/7x + 3p) = x*(x + 3) J

_-x 2 -2/?x-3/> x J(x + 3)J

dx ^ ( x + 3)* =»x l + 2/« + 3/>»0

Since C has two stationary points, discriminant > 0 =*4p , -12p>0 p ( p - 3 ) > 0 p < 0 or p > 3 (shown)

(iii)

( - i \ - l )

x - - 3 JT-0

TJCAIA9740/P1/Pfel iminafy Exam 20J ITSolutnna 10

t

12 Vectors (0

A directional vector of / is 6̂ (2 *j

' 2 1 1 - -1 »2 1

.3; r l A r2*> '2>

• Equation of / is r = -1 + a 1 . a c t

- 1 . ,2, (ID (2\

-1 +a 1 I • 2 -

-2 + 8a = 6 . - .a-I

Thus the position vector of point of intersection • ( 2> '2 4*1

-1 + 1 • • 0

A I ' . (ill)

A vector // to m

Alternative method

[21 [1*1 r-2] r 2*1 1 X 2 • -2 2 :. m //

.2, 2̂ . 3 .

{ 2 1 [1*1 [2*1 '1 2 • 2 = 0 and 2 • 1 -3; ,2, -3 .2,

- 0

(Iv) [2*1 ^ 1 is parallel to vectors l and 2

,2,

Hence, equation of the plane 11, is r ' f 2 ] [2*1 r 2 \

-1 l + t* 2 v2, -̂3

'4> [2*1 Equation of line m is r « 0 + 0 2

l-3>

Substitutes-4 + 2/?,>-= 2p and z - 1-3/9 into the equation of n ,

LHS- 4 + 2 /9-14 /9-4 + 12/9 + *(4 + 2/5 + 4/? + 2-6 / 5-6) • 0 for any P and k

Thus every point on m is on the plane FI, for all k. Hence m lies in n , for all k.

Since (0, 0, 2) lies on n , , substitutex - 0, y - 0 , r - 2 into the equation of n , -8 + *(4-6) = 0 =>/c--4 .. equation of plane is x - 7y - 4r - 4(x + 2y + Iz - 6) = 0

i.e. x + 5y + 4r - 8 TJC/MA9740/P1/Preliminary Exam 2011/SoluIlon» TJC/MAS740/P1'Pre fcrtr.ary Exam 2011/ScJullona

1

J ~ L

TEMASEK JUNIOR C O L L E G E . SINGAPORE

Preliminary Examination 2011 Higher 2

MATHEMATICS Paper 2

Addit ional Mater ials: A n s w e r paper G r a p h paper List of F o r m u l a e ( M F 1 5 )

9740/02

19 S e p t e m b e r 2011

3 h o u r s

READ T H E S E INSTRUCTIONS FIRST

Write your C i v i c s group a n d n a m e on all the work that you h a n d In.

Write in dark blue or black p e n on both s i d e s of the paper .

Y o u m a y u s e a soft penci l for a n y d i a g r a m s or g r a p h s .

D o not u s e s t a p l e s , paper c l ips , highlighters, g lue or correct ion fluid.

A n s w e r al l the q u e s t i o n s .

G i v e n o n - e x a c t numer ica l a n s w e r s correct to 3 signif icant f igures, or 1 dec ima l p l a c e In the c a s e of a n g l e s in d e g r e e s , u n l e s s a different level of a c c u r a c y Is speci f ied In the ques t ion .

Y o u a r e e x p e c t e d to u s e a graph ic calculator .

Unsuppor ted a n s w e r s from a graph ic ca lculator a r e a l lowed u n l e s s a quest ion speci f ical ly s ta tes o therwise .

W h e r e unsuppor ted a n s w e r s from a graphic calculator are not a l lowed in a quest ion , you a r e required to present the mathemat ica l s t e p s us ing mathemat ica l notations a n d not calculator c o m m a n d s

Y o u a r e reminded of the n e e d for c lear presentat ion In your a n s w e r s .

T h e n u m b e r of m a r k s Is given in b r a c k e t s [ ) at the e n d of e a c h quest ion or part quest ion .

At the e n d of the examinat ion , fas ten all your work s e c u r e l y together.

T h i s document c o n s i s t s of 6 printed p a g e s .

m i JUMK» course, SKGATOW

warn M * G M CMM Vti&tiNk [ T u r n o v e r

[ S o l u t i o n ]

S e c t i o n A : P u r e M a t h e m a t i c s [ 40 m a r k s |

The nth term of a sequence is denoted by a,. The sequence is defined by

a„, » pa. + q for all n 2i 1, where p and q are constants.

The first three terms of the sequence are given by a, - 0, a2 - 2 and ay - 7.

(I) Find the values of p and q.

(II) Express a„ in terms of n, for n it 3.

I

(0 at-pa}+q

« - 2 •7 -2p + 2

5

[2]

[4]

(li) Given a.., = - a „ + 2, nil.

Oj « 2 5

a , - - a , + 2

-f(2) + 2

* , - 0 2 ) + 2) + 2

"(fT(2)*f(2)+2

- 2 1 +

-iter-1 , for n 21 '>

3

2 (a) Given f \x) = ——. identify which of the following graphs. A, B and C is

that ofy • f(x), justifying your answer clearly. [2]

(b) A point P{x, y) moves on the curve y • tan x in such a way that the x-coordinate of P increases at a constant rate of 4 units per second. Find the rate of change

with respect to time of the gradient of the curve when x - —. [4]

(Solution]

(«) Given f ( x ) - 1

( l + * 2 ) Since f (0) • I but the gradient of the graph C at x • 0 is undefined, graph C cannot be the graph ofy - fi».

Since f (x) * 0 for all real x, y - f(x) has no turning points. Therefore graph A cannot be the graph ofy - f(x).

(Or) When x -* oo, f '(*) 0. Graph A cannot be the graph ofy - tTx).

Hence, Graph B is the graph ofy - f(x).

(b) y » t a n x =» — = sec 2x. dx

Given that — • 4 . Let G - — d/ dx

dG dC dx dr dx d/

- ( 2 s e c J x t . n x ) x ^

W h e n x - ^ , ^ . 2 ( 7 2 ) 2 ( . ) ( 4 ) - 1 6

correct — , apply chain rule. d |

4

3 (a) Find f , * dx. 1 x -2x + 3

(b) Using integration by parts, find the exact value of J xV2x +1 dx

(Solution]

J x J - 2 x + 3 1 x 3 - 2 x + 3 1 r 2x-2 r 1 . = - —: dx + —: dx 2 J x"-2x + 3 J x*-2x + 3 1

= - l n | x " - 2 x + 3 | + f 1 rdx

2 J ( - . ^ ( ^ 1. . j . . . 1 . i f x - 0

= - l n x -2x + 3 + -wtan — 7 - - +c 2 ' 72 I V 2 J

[4]

[4]

(Ml]

(b) | W 2 x + l dx (2x + l)»

= 1(373-0) - 1 (2x + l ) J

2>/3 1

5

4 Given that a = I + i , and the complex numbers z and >v are such that

| r - o | = | a | and arg^-^-^-j = arga.

Sketch on a single Argand diagram the loci of points representing z and M .

Hence find

(I) the least value of | w + 31,

(II) the greatest value of arg(z + 3). [Solution]

a - l + i , l a l - V l + l "72 and arga = — 4

org w-2\>r

2i J 4

arg(w-2) = - + arg(2i) 4

. it It lit =>arg (w-2 ) - - + - - T

[5]

[2]

[3]

(1) In MBF, \AF] - 5, ZBFA = n/4.

n| • j ] " ^ ^

(11) Greatest value of arg(z + 3) - ZDAE =ZDAC+ZCAE

.06+14.04-34.1 + u n UJ + u n UJ

At time / - 0, there are 8000 fish in a lake. At time / days the birth-rate offish per day is

equal to one-fortieth ( ^ ) o f the number /V of fish present. Fish are taken from the lake

at the rate of 150 per day.

Modelling N as a continuous variable, show that 40— = N - 6000. [2] dt

Solve the differential equation to find /V in terms of /. Find the time taken for the population of fish in the lake to increase to 12000. [5] When the population of fish has reached 12000, the number of fish taken from the lake is increased from 150 per day to r per day.

Write down, in terms of r, the new differential equatipn satisfied by N. [1]

Find the maximum value of r so that the population of fish in the lake will not decrease. [2]

[Solution)

Birth-rate of fish per day - — N ^ 7 40

Rate of fish removed from the lake per day - 150

• — = —/V-150 => 40—-/V-6000 d/ 40 dt

40 f 1 o W - f l d f J A'-6000 J

4 0 l n | t f - 6 0 0 0 | - / + c

| iV-6000 | - e*>

N- 6000 + Ae« , where A-te"

When / - 0, N - 8000, A - 2000 i

.-. N -6000 + 2000 e3*

When N - 12000, t - 40 ln3 - 43.9 days

When N - 12000, number of fish taken from the lake - r per day. New di fferential equation is — • — /V - r

dt 40 d/V

For population of fish in the lake not to be decrease, — 2: 0. di

When N - 12000, — - —(12000) - r - 300 -/• 2:0 dt 40 =»rS300

The maximum value of r is 300. |

7

Section B: Statistics |60 marks]

(a) How many different nine-digit numbers can be formed from five "0"s and four " 1 "s such that no three "0"s are together? [4]

(b) Six people sit at a round table with eight identical chairs. Find the number of seating arrangements if

(I) there arc no restrictions,

(II) the seats are numbered. [Solution|

(a) Case 1: All "0"s separated.

Case 2: One "00" , three "0"s separated.

Case 3: Two "00"s, one "0",

Therefore total number of ways = 51.

1 way.

3 ! — •20 ways 3! 7

2! 30 ways.

[2]

[1]

Alternative method: (0000. 0)

; $ j - S - A ' "126-5-20-20-30-51

(00000)] ^(000.00) ]^(0O0.0 .0) ]

( 8 - 0 ! (b) (I) Number of ways = L ~ ^ L - 2520.

(II) Number of ways - (8)-20160.

8

The heights of a randomly chosen boy and a randomly chosen girl in a junior college are denoted by B and G respectively. The random variables B and G are independent normal variables with means and standard deviations as shown in the table.

Mean (cm) Standard deviation (cm) Boys (B) 170 4 Girls (G) 155 5

(I) Explain briefly why the assumption of a normal distribution with the above mean and standard deviation for the heights of the boys may be reasonable even though such a height cannot possibly be negative. [ 1 ]

(II) Find the probability that 3 times the height of a randomly chosen girl will exceed the total height of 2 randomly chosen boys by more than 120 cm. [3]

(Ui) Find the value of* such that P(|G-155|<*)«0.85 [3]

(lv) Three boys are chosen at random. Find the probability that the first chosen boy is taller than 165 cm and the other two boys are shorter than 165 cm. [-2]

[Solution)

(I) It is reasonable because P(Height of a boy < 0) * 0 which is negligible.

(11) G-N(155,5 J), B~N(I70,4'). E (3G- f l , - £ , )»3x155-2x170-125 , Var (3G-5 , -B , ) -3 1 x5 J +2x4 J -257 3G - 5, - fl2 ~ N(l 25,257) P(3G-B,-5,>120)-0.622

(ill) 0-/^(155,25) P(|G-155|<*) = 0.85

=> P(-*<G-155<*) = 0.85

=> P ^ - j < Z < j j - 0 . 8 5

=> P ^ Z < - j j = 0.075

. . . . i = _|.4395

=• * - 7.20

(OR) P(|G-155|<*) = 0.85 =>P(-*<G-155<*) = 0.85

=>P(G<!55-*)-0.075

=5155-*-147.802

=> * = 7.20

(lv) Required probability -P (B > 165)x[p(fl < 165)]1

= 0.00998

9

Nine balls numbered from 1 to 9 are placed into a bag. The odd-numbered balls are red in colour and the even-numbered balls are blue in colour.

Three balls are drawn from the bag without replacement. For each ball drawn, the number on the ball represents the score. For any blue ball drawn, the score on the next ball drawn will be doubled. [For example, i f balls numbered 3, 4, 5 are drawn in this order, the total score is 3 + 4 + 5 x 2 - 1 7 ]

Find the probability that

(I) all balls are red, [2]

(II) there is at least one ball of each colour, [2]

(Ui) the total score is more than 18, given mat all balls are red, [3]

(lv) the total score is even. [3] [ S o l u t i o n !

o r - x i x - = —-0.119 9 8 7 42

(II) P (at least one ball of each colour) - 1 - P( no R) - P( no B)

I I 9 ' .3 U .

Alternative method:

P (at least one ball of each colour) - P( {2R1B} or {1R2B})

. 2 A l ) + U A 2 J 5 0.833

C) 6

(111) P (Total score > 18 | all balls are red) - P ({5,7,9), {3,7,9} | all balls are red) 4 , - 0 . 2 (OR) • ( " i " H ! - w ' . ! . , 3

ZS\3 5

Alternative method: Ix3!+lx3!

_ P( Total score>18 and all balls are red ) m P( {5,7,9}, {3,7,9} ) g a 0 2

P(all balls are red) " P(aU balls are red) " _5_ 42

(lv) P (Total score is even) = P((E,E,E).(E,E.O).(E,0, E) , (0 ,O.E) ) _ (4x3x2) + (4x3x5) + (4x5x3) + (5x4x4)

9x8x7 2 8 4

~63~9

Alternative method: P (Total score is even) -1 - P( ( 0 ,0 ,0 ) , (O, E, E), (O, E, O), (E, O, O))

{ J '(5x4x3)+(5x4x3) + (5x4x4) + (4x5x4) > | 4 9x8x7 1*9

11

9 (a) A junior college intends to make changes to its school uniform and decides to obtain the views from students on the design of a new uniform. Mr Tong obtains a sample by selecting 50 students randomly from the school register.

(i) Name the sampling method used. [ l ]

(II) Comment on the appropriateness of the use of this method for this situation. [ I ]

(III) Describe briefly another sampling method which can be used by the school and state one advantage of the method used in this context. [3]

(b) A machine is set to fill jars with 270 g of kaya. Over a period of time it has been established that the standard deviation of the mass of kaya in ajar is 0.8 g.

A setting on the filling machine is altered and the operator suspects that the mean mass of kaya filled by the machine has decreased. A random sample of 10 jars of kaya were measured and the masses in grams were found to be

269, 271, 270, 268, 269, 270, 268, 270, 271, 269.

(I) Assuming that the standard deviation may have been altered, perform an appropriate test at the 10% level of significance, on whether the operator's suspicion about the machine is justifiable. State any assumptions make in carrying out the test. [5]

(II) Assume that the standard deviation of the mass of kaya in a jar is unaltered. State what test you would use in this case, giving any assumptions that you need to make. [You are not required to carry out this test]. [1]

[Solution]

(a) (I) Simple Random Sampling method.

(II) Simple Random Sampling is not appropriate as there is a possibility that the sample is not a good representation of the student population. For example, the sample could have an unevenly large number of one gender and less of the other.

(ill) Stratified sampling could be used where the sample is obtained by dividing the student population according to gender. The students are selected randomly from each gender and the number of students selected from each gender should be proportional to the sizes of each gender in the school. The advantage of this method ensures a good representative sample of the student population as each gender is proportionally represented.

( Note: Quota Sampling is not acceptable as the sampling frame is available. I

12

(b) (0 From GC, x -269 .5 . J-1.08012

270

/ / , : / / < 270

Assumption: The mass of kaya in a jar follows a normal distribution since a1 is unknown and n =10 is small.

Test Statistic: X~{L-i.

Level of significance: 10% Assuming Ho is true, using GC, />-value • 0.0886

Since p-value - 0.0886 < 0.10, we reject //0and conclude that there is sufficient evidence at 10% level of significance that the mean mass of kaya in a jar has decreased.

(II) If standard deviation is unaltered, a Z-test is used and we assume that the mass of kaya in a jar to be normally distributed since sample size It— 10 is small.

13

10 (a) The scatter diagram shows a sample of size 6 of bivariate data, together with the regression line of y on x.

State and giving a reason, whether, for the data shown, the regression line of y on x is the same as the regression line of x ony. [1]

Given that y - -8x + 86 is the regression line of y on x for the 6 pairs of

observations with £ x » 60.

(I) Find the value of the mean ofy. [2]

(II) State, with a reason, the regression line of y on x after a new pair of observation (10, 6) is added. [2]

(b) A car is travelling along a stretch of road with speed v (km/h) when the brakes are applied. The car comes to rest after travelling a further distance of dm. The values of d for 10 different values of v are given in the table, correct to 2 decimal places.

V 35 40 45 50 55 60 65 70 75 80 d 4.90 5.30 5.93 6.45 7.00 8.50 12.00 17.00 22.10 25.90

It is given that the value of the linear product moment correlation coefficient for this data is 0.919, correct to 3 decimal places. The scatter diagram for the data is shown below.

0)

(ID

(ill)

Calculate the product moment correlation coefficient between v and What does this indicate about the scatter diagram of the points (v, -Jd)? [2]

Which regression line is more appropriate: -Jd on v or d on v? State the reason and find the equation of the appropriate regression line. [2]

Estimate the distance a car travels after its brake is applied when the car's speed is 1 OOkm/h. Comment on the reliability of your answer. (2]

14

(Solution)

(a) No, the regression line of y on x is not the same as the regression line of x ony since the points are not collinear.

(I) Regression line of y on x: y = -8x + 86,

r-9-io o

=> y--8(10)+ 86 - 6

(II) Since (10, 6) is (x , y ) , the regression line ofy on x remains the same.

I (b) (I) r j . - 0.947.

This indicates that the points (v, -Jd ) are lie close to a straight line with positive gradient

(II) Since i j - - 0.947 is closer to 1 than 0.919, the regression line of Jd on v is more appropriate than the regression line of a1 on v.

Equation of the regression line of -Jd on v : Jdm- 0.53154 + 0.06564v. i.e. yfd - - 0.532 + 0.0656v.

(Hi) Whenv-100, To 7--0.53154+ 0.06564(100) = 7.09554 => d~ 50.35 m

The estimate is not reliable as it is an extrapolation, v - 100 lies outside the data range.

15

11 (a) A" is a discrete random variable with mean 20.5 and variance 3.2. A random sample of 60 observations of X is taken, find the probability that the sample mean lies between 19 and 20. [3]

(b) Analysis of the scores in football matches in a local league suggests that the number of goals scored in a randomly chosen match may be modelled by the Poisson distribution with mean 2.7. State, in the context of the question, one assumption for the Poisson distribution to be a valid model. [1]

(I) Find the probability that a randomly chosen match will end with at least two goals scored. [1]

(II) On one afternoon, 12 matches are played in the league. Find the probability that there are fewer than 8 matches in which there are at least 2 goals scored. [2]

T is the total number of goals scored in n matches. State the distribution of T and its parameters). [1]

Using a suitable approximation, find the least value of n such that the probability of T being greater than 100 is more than 0.9. [4]

[Solution] (a) Given E(A) - 20.5 and \u{X) - 3.2

Since n - 60 is large, by Central Limit Theorem, - m Xi+Xl •....» X„ m N ( 2 Q ^ 3^2} t p p r o x i m a t e , v

60 60 P(19<7< 20)-0.0152

i

(b) Assumption: Each goal is scored independently of one another. OR The mean number of goals scored is proportional to the duration of the

match. OR The goals are scored at a constant rate.

(I) Let AT be the number of goals scored in a match X P(XZ2)= 1 -P(XZ 1)-0.751

(II) Let Y be the number of matches in which there are at least 2 goals scored r~B(12, 0.75133960) P ( y < 8 ) - P ( r s 7 ) - 0.155

T-Xi+Xi + Xi+.-.+X, r~P0(2.7n) Assuming that 2.7n > 10, T~ N(2.7n, 2.7») approximately

P(7"> 100)*P(r> 100.5) > 0.9

Using GC, when n - 42, ?(T > 100.5) - 0.88713 when n- 43, P(f> 100.5) - 0.92616

Thus the least value of n is 43

16

Alternative solution: Since n is large, by Central Limit Theorem T - X, + X2 + Xj +....+ X. - N(2.7n. 2.7n) approximately For P(7"> 100) > 0.9

Using GC, when n -42 , P(7/> 100) - 0.89587 when n-43, P(7/> 100) - 0.93244

Thus the least value of n is 43

w w w End of Paper eg c* cu