analysis [3472/1] additional mathematics · [3472/2] no topics paper 1 paper 2 2006 2007 2008 2009...

59 SOALAN ULANG!JI SPM 2013

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Additional Mathematics Analysis

[3472/1][3472/2]

NO TOPICSPAPER 1 PAPER 2

2006 2007 2008 2009 2010 2011 2012 2006 2007 2008 2009 2010 2011 2012

1 Functions 1,2 1,2,3 1,2,3 1,2,3 1,2,3 1,2,3 1,2,3 2 - - - - - -

2Quadratic

Equations3 4 4 4 5 5 5 - - - 2a,2c - - 2

3Quadratic

Functions4,5 5,6 5,6 5,6 4,6 4,6 4,6 - - 2 2b - - -

4Simultaneous

Equation- - - - - - - 1 1 1 1 1 1 1

5Indices and

Logarithms6,7,8 7,8 7,8 7,8 7,8 7,8 7,8 - - - - - - -

6Coordinate

Geometry12 13,14 13,14 15 13,14 13,14 13,14 9 2 10 9 5 5 10

7 Statistics 24 22 22 24 22 22 22 6 5 5 - 6 6 4

8Circular

Measures16 18 18 12 17 17 17 10 9 9 10 11 11 9

9 Differentiation17,

18,19

19,

20

19,

20

19,

20

20,

21

19,

20,21

19,

20,21-

4a,

4b7a

3a,

7a8

4b,

88

10Solution of

Triangles- - - - - - - 13 15 14 12 13 13 14

11 Index Number - - - - - - - 15 13 13 13 15 15 13

12 Progressions 9,109,

10,11

9,

10,11

9,

10,11

9,

10,11

9,

10,11

9,

10,113 6 3 6 3 3 3

13 Linear Law 11 12 12 12 12 12 12 7 7 8 8 7 7 7

14 Integration 20,21 21 2118,

20,2119 - - 8

4c,

107b,7c 3b,7 4 4a -

15 Vectors 15, - - - - 15,16 15,16 - - - - - 9 5

16Trigonometric

Functions15 17 17 16,17 18 18 18 4 3 4 4 2 2 6

17

Permutations

And

Combinations

22 22 23 22 23 23 23 - - - - - - -

18 Probability 23 24 24 23 24 24 24 - - - - - - -

19Probability

Distributions25 25 25 25 25 25 25 11 11 11 11 10 10 11

20Motion Along

A Straight Line- - - - - - - 12 12 12 15 12 12 12

21Linear

Programming- - - - - - - 14 14 15 14 14 14 15

TOTAL 25 25 25 25 25 25 25 15 15 15 15 15 15 15


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Additional Mathematic Paper 1 [3472/1]

Answer all questions.Jawab semua soalan.

1 !e Cartesian graph above represents a relation of set A = {1, 3, 5, 6} and set B = {1, 2, 3, 4}. State Graf pada rajah di atas menunjukkan hubungan set A = {1, 3, 5, 6} dan set B = {1, 2, 3, 4}. Nyatakan (a) the image of 1, imej bagi 1,

(b) whether the relation is a function or not. samada hubungan ini merupakan fungsi atau tidak. [2 marks]

Answer : (a) ______________________________ (b) ______________________________

2 Given that , "nd .

Diberi fungsi , cari . [2 marks]

Answer : ________________________________

3 Find the value of p if the quadratic equation p(x2 + 1) = 6x has two equal roots.

[2 marks]

Answer : ________________________________

4 Given that , "nd the range of values of x which satisfy the inequality

Diberi , cari julat nilai x yang memuaskan ketaksamaan [2 marks]

Answer : ________________________________

5 Given that , "nd the value of k and of m.

Diberi , cari nilai k dan nilai m. [3 marks]

Answer : (k) _____________________________ (m) _____________________________


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6 Solve the equation .

Selesaikan persamaan . [2 marks]

Answer : ________________________________

7 Given that 4 logx 2 + 2 logx 7 – logx 98 = 3, "nd the value of x.

Diberi 4 logx 2 + 2 logx x [3 marks]

Answer : x = _____________________________

8 Given that 11, p + 1, 19 are three consecutive terms of an arithmetic progression and (p + 1) is the sixth term, "nd the value of

dan (p + 1) ialah sebutan keenam, cari nilai (a) p, (b) the "rst term. sebutan pertama. [3 marks]

Answer : (a) p = __________________________

(b) _____________________________

9 Express the recurring decimal 2·133333……. as a fraction in its simplest form.

Ungkapkan perpuluhan jadi semula 2·133333… dalam bentuk pecahan yang termudah. [3 marks]

Answer : (a) m = __________________________

(b) _____________________________

10 !e common ratio of a geometric progression is and the sum to in"nity is 21. State the "rst three terms of the progression.

Nisbah sepunya suatu janjang geometri ialah dan hasil tambah sehingga ketakterhinggaannya ialah 21. Nyatakan tiga sebutan pertama janjang ini. [3 marks]

Answer : ________________________________

11 Given the straight line 4x + 3y − 6 = 0 is parallel to the straight line y = ax + b which passes through the point (−12, 5), "nd the value of a and of b.

cari nilai a dan nilai b. [3 marks]

Answer : (a) _____________________________

(b) _____________________________


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12 !e diagram above shows the straight line graph obtained by plo$ing against x.

Rajah di atas menunjukkan graf garis lurus yang didapati dengan memplotkan melawan x.

(a) Calculate the value of p. Hitungkan nilai p.

(b) Express y in terms of x. Ungkapkan y dalam sebutan x. [4 marks]

Answer : (a) _____________________________

(b) _____________________________

13 Given that O is an origin and M(3, −4). If MP = 2i − 3j,

Diberi O ialah titik asalan dan M(3, −4). Jika MP = 2i − 3j,

(a) Express OM in terms of i and j. Ungkapkan OM dalam sebutan i dan j.

(b) Find | OP |. Cari | OP |. [3 marks]

Answer : (a) _____________________________

(b) _____________________________

14 Given that a = 2i + (p + 1)j and b = –3i + 6j, "nd the value of p in each of the following cases :

Diberi a = 2i + (p + 1)j dan b i j, cari nilai p bagi setiap kes yang berikut :

(a) a + b is parallel to x-axis. a + b selari dengan paksi-x

(b) a is parallel to b. a selari dengan b. [4 marks]

Answer : (a) p = __________________________

(b) p = __________________________


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15 Find the equation of the curve which has the gradient function x2(x + 3) and passes through the point (2, 14).

Cari persamaan lengkung yang mempunyai fungsi kecerunan x2(x + 3) dan melalui titik (2, 14). [3 marks]

Answer : ________________________________

16 Given , "nd the value

Diberi , cari nilai

(a)

(b) k if

k jika [4 marks]

Answer : (a) ______________________________

(b) k = ___________________________

17 Given , "nd the value of f ″ (1).

Diberi , cari nilai bagi f ″ (1). [3 marks]

Answer : ________________________________

18 !e volume of the liquid in a container, V cm3 is given by V = 2x3 + 4x2 + 5, where x cm is the depth of the liquid in the container. Given that V increases at a rate of 32 cm3s−1, "nd the rate of increase of x when x = 2.

Isipadu cecair dalam sebuah bekas V cm3 diberi oleh V = 2x3 + 4x2

Diberi bahawa V bertambah dengan kadar 32 cm3s−1, cari kadar pertambahan x pada ketika x = 2. [4 marks]

Answer : ________________________________

19 Find the equation of the normal to the curve at the point (2, 8).

Cari persamaan normal kepada lengkung pada titik (2, 8). [4 marks]

Answer : ________________________________

20 !e right diagram shows a semi circle with centre O and diameter POR = 8 cm. Given that the arc length of PQ is 3·87 cm, calculate

Rajah kanan menunjukkan sebuah semi bulatan berpusat O dengan diameter POR = 8 cm. Diberi panjang lengkok PQ ialah 3·87 cm, hitungkan

(a) the value of θ in radian, nilai θ dalam radian, (b) the area of sector OQR, luas sektor OQR. (Use/Gunakan π = 3·142) [4 marks] Answer : (a) _____________________________

(b) _____________________________


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21 Given that cos A = and sin B = where A and B are obtuse angles, "nd the value of

Diberi kos A = dan sin B = dengan sudut A dan B ialah sudut cakah, cari nilai

(a) sin 2A, (b) cos (B − A). [4 marks]

Answer : (a) _____________________________

(b) _____________________________

22 Five le$ers from the word T S U N A M I is to be arranged in a row begins with a vowel. Find the probability that the arrangement begins with the le$er .

Lima daripada huruf-huruf dalam perkataan disusun sebaris bermula dengan suatu vokal. Cari kebarangkalian jika susunan itu bermula dengan huruf . [3 marks]

Answer : ________________________________

23 !e mean of a set of 25 numbers is 24.

(a) If every number in the set is added by 2, determine the new mean of the set. Jika setiap nombor dalam set itu ditambah dengan 2, tentukan min baru bagi set itu.

(b) If two numbers k and k + 2 are taken out from the set, the new mean is 22, "nd the value of k. Jika dua nombor k dan k + 2 dikeluarkan daripada set itu, min baru bagi set itu ialah 22, cari nilai k. [4 marks]

Answer : (a) ______________________________

(b) k = ___________________________

24 In a chess competition, Peter plays "ve games. !e probability that Peter wins one of the games is . Calculate the probability that Peter won

Dalam satu pertandingan catur, Peter bermain lima permainan. Kebarangkalian bahawa Peter memenangi mana-mana satu permainan ialah . Hitungkan kebarangkalian bahawa Peter memenangi

(a) three games a'er playing four games, tiga permainan selepas empat permainan,

(b) the "rst game and the last game. permainan pertama dan permainan terakhir. [4 marks]

Answer : (a) _____________________________

(b) _____________________________

25 X is a continuous random variable with X ~ N(μ, 25). If P(X < 9) = 0·7257, "nd the value of μ.

[4 marks]

Answer : ________________________________


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Additional Mathematic Paper 2 [3472/2]Section A / [40 marks / ]

Answer all questions from this sectionJawab semua soalan dalam bahagian ini.

1 Solve the equation 2p + k = p2 + k2 − 5 = 5 [6 marks]

2 A function is de"ned by , for all values of x except x = h, and m is a constant.

Satu fungsi ditakri%an oleh , bagi semua nilai x kecuali x = h, dan m ialah pemalar.

(a) Determine the value of h. Tentukan nilai h.

(b) Given that 2 maps onto itself under the function g, "nd Diberi nilai 2 dipetakan kepada dirinya sendiri di bawah fungsi g , cari

(i) the value of m, nilai m,

(ii) the value of

nilai bagi

(iii) the value of p if g(2p) = 5 [7 marks]

3 !e straight line x + 4y = p is normal to the curve y = (2x – 1)2 + 1 at the point Q.

2 + 1 pada titik Q.

Find / Cari

(a) the coordinates of Q, [4 marks] koordinat-koordinat Q,

(b) the value of p, [1 mark] nilai p,

(c) the equation of the tangent at the point Q. [2 marks] persamaan tangen pada titik Q.

4 In Diagram 1, the equation of the straight line PQR is 2x – y + 4 = 0. If PQ : QR = 2 : 3, "nd

Dalam Rajah 1, persamaan garis lurus PQR ialah

(a) the coordinates of R, [3 marks] koordinat-koordinat R,

(b) the equation of perpendicular bisector PR, [4 marks] persamaan pembahagi dua sama serenjang PR.

DIAGRAM 1


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5 Solve each of the following equations : Selesaikan setiap persamaan berikut :

(a) [3 marks]

(b) log5 3x – log5 (x – 2) = 1 [3 marks]

6 (a) !e sum of the "rst n terms of an arithmetic progression is given by Sn

Hasil tambah n sebutan pertama suatu janjang aritmetik diberi oleh Sn Find / Cari

(i) the "rst term and common di(erence, sebutan pertama dan beza sepunya,

(ii) the sum of the all the terms from the 5th term to the 10th term. [5 marks]

(b) Given that the 5th and the 8th terms of a geometric progression are 2 and respectively, "nd the common ratio of this progression.

Diberi sebutan kelima dan sebutan kelapan suatu janjang geometri masing-masing ialah 2 dan , carikan nisbah sepunya janjang tersebut.

[2 marks]


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Section B / [40 marks / ]

Answer four questions from this sectionJawab empat soalan dalam bahagian ini.

7 Diagram 2 shows two sectors of a circle, sector OCBA with centre O and sector PCOA with centre P. Δ OAP and Δ OPC are two congruent equilateral triangle with sides 8 cm. Find

Rajah 2 menunjukkan dua sektor bulatan iaitu sektor OCBA yang berpusat di O dan sektor PCOA yang berpusat di P. Δ OAP dan Δ OPC adalah dua segi tiga sama sisi yang kongruen dan mempunyai sisi 8 cm. Cari

(a) the re*ex angle AOC in radian, [1 mark] (b) the perimeter of the shaded region, perimeter kawasan berlorek, [3 marks] (c) the area of the major sector OCBA, luas sektor major OCBA, [2 marks]

(d) the area of the shaded region. luas kawasan berlorek. [4 marks] [Use/ Gunakan π = 3.142]

DIAGRAM 2

8 Table 1 shows the values of two variables, x and y which are related by the equation y = ax (x + b) , where a and b are constants.

Jadual 1 menunjukkan nilai bagi dua pembolehubah x dan y yang dihubungkan oleh persamaan y = ax (x + b) , dengan keadaan a dan b ialah pemalar.

(a) Reduce the equation y = ax (x + b) to the linear form. Tukarkan persamaan y = ax (x + b) kepada persamaan bentuk linear. [1 mark]

(b) Plot the graph of against x . / Lukiskan graf melawan x . [4 marks]

(c) Use the graph from (a) to "nd Gunakan graf anda dari (a) untuk mencari

(i) the value of a and of b nilai a dan nilai b,

(ii) the value of y when x = 6 [5 marks]

9 (a) A set of game score x1 , x2 , x3 , x4 and x5 has the mean 6 and standard deviation 1·2.

Satu set skor bagi suatu permainan x1 , x2 , x3 , x4 dan x

(i) Find the sum of the squares of the score, Σx2 Cari hasil tambah kuasa dua skor itu, Σx2 [2 marks]

(ii) If each score is multiplied by 3 and 2 is added to it, "nd the mean and variance of the new score. Jika setiap skor itu didarabkan dengan 3 dan ditambah dengan 2, cari min dan varians bagi set skor yang baru. [3 marks]

TABLE 1 / JADUAL 1


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(b) Table 2 shows the marks obtained by a group of students in a particular test. Jadual 2 menunjukkan markah yang diperoleh sekumpulan pelajar dalam satu ujian.

(i) Without drawing an ogive, "nd the median marks. [3 marks] Tanpa melukis ogif, cari median bagi markah-markah itu.

(ii) Calculate the mean marks. [2 marks] Hitungkan markah min.

10 Given the quadratic function f(x) = − 4x2 + 8x − 1. Diberi fungsi kuadratik f(x) = − 4x2 + 8x − 1.

(a) Express the quadratic function f(x) in the form p(x + q)2 + r , where p, q and r are constants. Determine whether the function f(x) has a minimum or maximum value and state its value.

Ungkapkan fungsi kuadratik f(x) dalam bentuk p(x + q)2 + r dengan keadaan p, q dan r ialah pemalar. Tentukan sama ada fungsi f(x) mempunyai nilai maksimum atau nilai minimum dan nyatakan nilainya. [3 marks]

(b) Sketch the graph of the function f(x) for the domain − 1 ≤ x ≤ 3. State the corresponding range for this domain. Lakarkan graf fungsi f(x) untuk domain − 1 ≤ x ≤ 3. Nyatakan julat yang sepadan dengan domain ini. [3 marks]

(c) Find the range of values of k such that f(x) = kx has no real roots.

Cari julat nilai k supaya f(x) = kx tidak mempunyai punca nyata. [4 marks]

11 (a) Given that ABCD is a parallelogram where BC = 2i + 3j and CD = − 2i – 2j, "nd

Diberi ABCD ialah sebuah segi empat selari dengan BC = 2i + 3j dan CD = − 2i j, cari

(i) AC,

(ii) the unit vector in the direction of AB. / vektor unit pada arah AB. [4 marks]

TABLE 2 / JADUAL 2

(b) Diagram 3 shows a triangle ABC where CP = λ CB.

Given that AB = a and AC = b . Rajah 3 menunjukkan sebuah segi tiga ABC dengan CP = λ CB

Diberi AB = a dan AC = b .

(i) Show that AP = λ a + (1 − λ )b

Tunjukkan bahawa AP = λ a + (1 − λ )b

(ii) If BP = − 3a + 3b, "nd the value of λ.

Jika BP = − 3a + 3b, cari nilai λ. [6 mark]

DIAGRAM 3


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Section C / Bahagian C[40 marks / 40 markah]

Answer two questions from this sectionJawab dua soalan daripada bahagian ini.

12 A particle moves along a straight line from a "xed point O. Its displacement, S m is given by S = 9t2 – t3 , where t is the time in seconds a'er leaving the point Q.

(Assume that motion to the right is positive)

Suatu zarah bergerak di sepanjang suatu garis lurus bermula dari satu titik tetap O. Sesarannya, S m, diberi oleh S = 9t2 – t3 , dengan keadaan t ialah masa, dalam saat, selepas meninggalkan titik Q.

(Anggapkan gerakan ke arah kanan sebagai positif)

Find / Cari

(a) the distance travelled the 2nd second, [2 marks] jarak yang dilalui dalam saat kedua,

(b) the time at which the velocity of the particle is uniform, masa ketika zarah mencapai halaju seragam, [3 marks]

(c) the value of t when the particle passes the point Q again, nilai t apabila zarah itu melalui titik Q semula, [2 marks]

(d) the range of values of t during which the particle is moving towards the point Q a'er instantaneous rest. julat nilai t apabila zarah menuju ke titik Q selepas rehat seketika. [3 marks]

13 Table 3 shows the price of a set of football jersey which has been worn by a particular team at Lion Championship tournament in the year 2003 and 2004.

(a) Calculate the values of x, y and z. Hitung nilai x, y dan z. [4 marks]

(b) Given that the index number of Adibas jersey is 120 in the year 2003 based on the year 2002, "nd the price in the year 2002. [2 marks]

(c) If the price of Nikas jersey increases at a constant rate every year, determine the price in the year 2005 to the nearest RM.

terdekat. [4 marks]

TABLE 3 / JADUAL 3


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14 Given that x and y are positive integers with the following conditions : Diberi x dan y ialah dua integer positif dengan keadaan berikut :

I : !e value of x is more than the value of y by 5 or more

II : !e minimum value of 2x + 3y is 60

III : !e maximum value of 4x + 3y is 5 times the minimum value of 2x + 3y

(a) Write down three inequalities which satisfy the above conditions. Tulis tiga ketaksamaan bagi setiap keadaan di atas. [3 marks]

(b) Construct and shade the region R that satis"es all of the above conditions. Bina dan lorekkan rantau R yang memuaskan syarat-syarat di atas. [3 marks]

(c) By using your graph from (b), answer the following questions : Dengan menggunakan graf anda dari (b), jawab soalan-soalan berikut :

(i) Find the minimum value of the sum of the integers. Cari nilai minimum bagi hasil tambah integer tersebut. [1 mark]

(ii) Given that x and y represents the number of item M and item N respectively which was sold by the company. Find the total maximum sale obtained if the price of one unit item M and N are RM15 and RM30 respectively.

Diberi bahawa x dan y masing-masing mewakili bilangan barang M dan barang N yang dijual oleh sebuah syarikat.

[3 marks]


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15 (a) Diagram 4 shows a triangle PQR. Calculate QPR. Rajah 4 menunjukkan sebuah segi tiga PQR. Hitungkan QPR. [3 marks]

DIAGRAM 4 / RAJAH 4

DIAGRAM 5 / RAJAH 5

(b) In Diagram 5, ABC and BCD are two triangles which a$ached at BC. Calculate

(i) the length of BC, / panjang BC,

(ii) BCD,

(iii) the area of whole diagram, / luas seluruh rajah. [7 marks]

analysis [3472/1] additional mathematics · [3472/2] no topics paper 1 paper 2 2006 2007 2008 2009...

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