Hak Cipta MPSM Pulau Pinang 2012 SULIT
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1 1449/2 NAMA : KELAS : MAJLIS PENGETUA SEKOLAH MALAYSIA CAWANGAN PULAU PINANG MODUL PENILAIAN SPM /2 MATHEMATICS Kertas 2 September 1 2 jam Dua jam tiga puluh minit 2 JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU 1. Tulis nombor kad pengenalan dan angka giliran anda pada petak yang disediakan. 2. Kertas soalan ini adalah dalam dwibahasa. 3. Soalan dalam bahasa Inggeris mendahului soalan yang sepadan dalam bahasa Melayu. 4. Calon dibenarkan menjawab keseluruhan atau sebahagian soalan sama ada dalam bahasa Inggeris atau bahasa Melayu. 5. Calon dikehendaki membaca maklumat di halaman belakang kertas soalan ini. Untuk Kegunaan Pemeriksa Kod Pemeriksa : Bahagian A B Soalan Markah Penuh Jumlah Markah Diperoleh Kertas soalan ini mengandungi 33 halaman bercetak. Hak Cipta MPSM Pulau Pinang 2012
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5 5 1449/2 On the graph in the answer space, shade the region which satisfies the three inequalities. y + x 2, y x - 2 and y < 2. Pada graf ruang jawapan, lorekkan rantau yang memuaskan ketiga-tiga ketaksamaan y + x 2, y x - 2 dan y < 2. [3 marks] [3 markah] Answer / Jawapan:
6 /2 2 Diagram 2 shows a right prism. The base is a horizontal rectangle. The right-angled triangle is the uniform cross-section of the prism. is the midpoint of. Rajah 2 menunjukkan sebuah prisma tegak. Tapak ialah segiempat tepat yang mengufuk. Segitiga bersudut tegak ialah keratan rentas seragam prisma itu. ialah titik tengah Diagram 2 Rajah 2 (a) Name the angle between the line and the base. Namakan sudut antara garis dengan tapak. (b) Calculate the angle between the line and the base. Hitungkan sudut antara garis dengan tapak. Answer / Jawapan (a) [3 marks] [3 markah] (b)
7 7 1449/2 3 Solve the following quadratic equation. Selesaikan persamaan kuadratik berikut. Answer / Jawapan : [ 4 marks ] [ 4 markah]
8 8 1449/2 4 Calculate the value of x and of y that satisfy the following simultaneous linear equations : Hitungkan nilai x dan nilai y yang memuaskan persamaan linear serentak berikut : [ 4 marks ] [ 4 markah ] Answer / Jawapan :
9 9 1449/2 5 Diagram 5 shows a right prism with rectangular base on a horizontal plane. Rightangle triangle is the uniform cross-section of the prism. A solid right cylinder with diameter 3 cm and length 21 cm is taken out from the right prism. Rajah 5 menunjukkan sebuah prisma tegak dengan tapak yang berbentuk segi empat tepat mengufuk. Segitiga bersudut tegak ialah keratan rentas seragam prisma itu. Sebuah pepejal berbentuk silinder tegak dengan diameter 3 cm dan panjang 21 cm dikeluarkan dari prisma tersebut. Using Diagram 5 Rajah 5 22 π, calculate the volume, in cm 3, of the remaining solid. 7 Menggunakan 22 π, hitungkan isipadu, dalam cm 3, pepejal yang tinggal. 7 [4 marks] [4 markah] Answer / Jawapan:
10 /2 6 (a) State whether the following statement is true or false. Nyatakan sama ada pernyataan berikut adalah benar atau palsu. 3 and 4 are factors of 8 3 dan 4 adalah faktor bagi 8 (b) Write down two implications based on the following compound statement: Tuliskan dua implikasi berdasarkan pernyataan majmuk berikut : is an even number if and only if x can be divided by 2 x ialah nombor genap jika dan hanya jika x boleh dibahagi tepat dengan 2 (c) Make a general conclusion by induction for the sequence of number 2, 3, 6, 11, which follows the following pattern. Buat satu kesimpulan umum secara aruhan bagi urutan nombor 2, 3, 6, 11, yang mengikut pola berikut. Answer /Jawapan:.. =... [5marks] [5 markah] (a)... (b) Implication 1/ Implikasi 1 :.... Implication 2 / Implikasi 2 :... (c)
11 /2 7 In the diagram 7 below, O is the origin. Points P and Q lie on the y-axis and x-axis respectively. QT is parallel to y-axis and PQ is parallel to RS. Dalam Rajah 7 di bawah,o ialah titik asalan. Titik P dan titik Q terletak pada paksi-y dan paksi-x masing-masing. TQ adalah selari dengan paksi-y dan PQ adalah selari dengan RS. Diagram 7 Rajah 7 The equation of the straight line PQ is 2y + 3x = 7. Persamaan bagi garis lurus PQ ialah 2y + 3x = 7. ( a ) State the equation of the straight line QT. Nyatakan persamaan bagi garis lurus QT. ( b ) Find the equation of the straight line RS. Cari persamaan bagi garis lurus RS dan. Answer / Jawapan [5 marks] [5 markah] (a) (b)
12 /2 8 A box contains one red marble, one green marble, one yellow marble, and one white marble. Two marbles are draw from the box, one after the other, without replacing the first one drawn. Sebuah kotak mengandungi satu biji guli merah, satu biji guli hijau, satu biji guli kuning dan satu biji guli putih. dua biji guli dipilih satu demi satu secara rawak dari kotak itu dan guli yang dipilih tidak dimasukkan semula ke dalam kotak tersebut. (a) (b) List all the possible outcomes of the event in this sample space. Senaraikan semua kesudahan peristiwa yang mungkin dalam ruang sampel ini. List all the outcomes of the events and find the probability that, Senaraikan semua kesudahan peristiwa dan cari kebarangkalian bahawa, (i) (ii) one of the marbles chosen is green. satu daripada guli yang dipilih adalah berwarna hijau. the first marble is white or the second marble is yellow are chosen, guli pertama berwarna putih atau guli kedua berwarna kuning dipilih, [5 marks] [5 markah] Answer / Jawapan (a) (b) (i) (ii)
13 /2 9 Diagram 9 shows quadrant OAB, quadrant OEC and sector OCD are with common centre O. Given that OC = OE = 7 cm and OA = 12 cm. Rajah 9 menunjukkan sukuan bulatan OAB, sukuan bulatan OEC dan sektor OCD dengan titik pusat sepunya di O. Diberi OC = OE = 7 cm dan OA = 12 cm. Diagram 9 Rajah 9 Using = 7 22, calculate Dengan menggunakan = 22, hitung 7 (i) (ii) the perimeter, in cm, of the whole diagram, perimeter, dalam cm, seluruh rajah, the area, in cm 2, of the shaded region luas, dalam cm 2 kawasan berlorek,. [6 marks] [6 markah]
14 /2 Answer / Jawapan : (a) (b)
15 /2 10 Diagram 10 shows the speed-time graph of a lorry and a car for a period of 60 seconds. Rajah 10 menunjukkan graf laju-masa sebuah lori dan sebuah kereta dalam tempoh 60 saat. Diagram 10 Rajah 10 ABCD represents the movement of the lorry and PQ represents the movement of the car. ABCD mewakili pergerakan lori dan PQ mewakili pergerakan kereta. (a) (b) State the maximum speed of the lorry. Nyatakan laju maksimum lori itu. Calculate the value of, if the rate of change of of speed of lorry in first second is Hitungkan nilai, jika kadar perubahan laju dalam tempoh saat itu ialah (c) The difference in distance travelled by the lorry and a car during 60 seconds is 78m. Find the uniform speed of a car. Beza diantara jarak yang dilalui oleh lori dan kereta tersebut dalam tempoh 60 saat ialah 78m. Cari laju seragam kereta itu. [6 marks] [6 markah]
16 /2 Answer / Jawapan : (a) (b) (c)
17 /2 11 Given that the inverse matrix of is 6 q 5 3 Diberi bahawa matriks songsang bagi ialah 6 q 5 3 (a) Find the value of and of. Carikan nilai dan nilai. (b) Write the following simultaneous linear equations as matrix equation. Tuliskan persamaan linear serentak berikut dalam bentuk persamaan matriks. Hence, using matrix method, calculate the value of and of. Seterusnya, dengan menggunakan kaedah matriks, hitung nilai dan nilai. Answer / Jawapan : (a) [ 6 marks ] [ 6 markah ] (b)
18 /2 Section B Bahagian B [48 marks] [48 markah] Answer any four questions from this section Jawab mana-mana empat soalan dalam bahagian ini 12 (a) Complete the table 12 in the answer space on page 19, for the equation by writing down the values of when and.. [ 2 marks ] Lengkapkan Jadual 12 di ruang jawapan pada halaman 19, bagi persamaan dengan menulis nilai-nilai apabila dan [2 markah] (b) this part of the question, use the graph paper provided on page 20. You may use a flexible curve rule. By using a scale of 2 cm to 1 unit on the x axis and 2 cm to 5 units on the y axis, draw the graph of for and [4 marks] Untuk ceraian soalan ini, gunakan kertas graf yang disediakan pada halaman 20. Anda boleh menggunakan pembaris fleksibel. Dengan menggunakan skala 2 cm kepada 1 unit pada paksi dan 2 cm kepada 5 unit pada paksi, lukiskan graf bagi untuk dan [4 markah] (c) From the graph in 12(b), find Dari graf di 12(b), cari (i) the value of when nilai apabila (d) (ii) the value of when nilai apabila Draw a suitable straight line on the graph in 12(b) to find the values of equation for and State these values of. [2 marks] [ 2 markah ] which satisfy the [4 marks] Lukis satu garis lurus yang sesuai pada graf di 12(b) untuk mencari nilai-nilai memuaskan persamaan untuk dan Nyatakan nilai-nilai ini. yang [4 markah]
19 1449/2 19 Answer / Jawapan : (a) Table 12 Jadual 12 (b) Refer graph on page 20 Rujuk graf pada halaman 20 (c) (i) y =. (ii) x =. (d) The equation of the straight line : Persamaan garis lurus : x =,...
20 /2 Graph for Question 12 Graf untuk Soalan 12
21 /2 13 Diagram 13.1 shows points K, H and straight line y = x drawn on a Cartesian plane. Rajah 13.1 menunjukkan titik K, H dan garis lurus y = x dilukis pada suatu satah Cartesan. y 7 6 H 5 y = x 4 3 K x 1 2 Diagram 13.1 Rajah 13.1 (a) Transformation R is a rotation of 90 anticlockwise about the centre K. Transformation P is a reflection in the straight line y = x. State the coordinates of the image of point H under each of the following transformations: Penjelmaan R ialah putaran 90 arah lawan jam pada pusat K. Penjelmaan P ialah pantulan pada garis lurus y = x Nyatakan koordinat imej bagi titik H di bawah setiap penjelmaan berikut: (i), (ii). [4 marks] [4 markah]
22 /2 (b) Diagram 13.2 shows two heptagon, ABCDEFG and PQRSTUV drawn on square grids. Rajah 13,2 menunjukkan dua heptagon, ABCDEFG dan PQRSTUV dilukis pada grid segi empat sama. y U G F V S A B C R D E P T Q x Diagram 13.2 Rajah 13.2 (i) PQRSTUV is the image of ABCDEFG under combined transformation NM. Describe in full, the transformations: PQRSTUV ialah imej bagi ABCDEFG di bawah gabungan penjelmaan NM. Huraikan selengkapnya penjelmaan: (a) M, (b) N. (ii) Is given that ABCDEFG represents a region of area 25.5 m 2. Calculate the area, in m 2, of the region represented by the shaded region. Diberi bahawa ABCDEFG mewakili suatu kawasan yang mempunyai luas 25.5 m 2. Hitung luas, dalam m 2, kawasan yang diwakili oleh kawasan berlorek. [8 marks] [8 markah]
23 /2 Answer / Jawapan : (a) (i) (ii) (b) (i) (a) (b) (ii)
24 /2 14 Diagram 14 shows the score marks obtained by 45 students in a Sejarah Test. Rajah 14 menunjukkan markah yang diperolehi oleh sekumpulan 45 orang murid dalam satu ujian Sejarah Diagram 14 Rajah 14 Based on the data in Diagram 14, complete Table 14 in the answer space on page 25. [4 marks] Berdasarkan data di Rajah 14, lengkapkan Jadual 14 di ruang jawapan pada halaman 25. [4 markah] (a) Calculate the estimated mean mark of the group. Hitungkan min anggaran markah bagi kumpulan tersebut. [3 marks] [3 markah] (b) this part of the question, use the graph paper provided on page 26 By using a scale of 2 cm to 5 marks on the horizontal axis and 2 cm to 1 student on the vertical axis, draw a frequency polygon for the data. [4 marks] Untuk ceraian soalan ini, gunakan kertas graf yang disediakan di halaman 26 Dengan menggunakan skala 2 cm kepada 5 markah pada paksi mengufuk dan 2 cm kepada 1 murid pada paksi mencancang, lukis satu poligon kekerapan bagi data tersebut. [4 markah] (c) Based on the frequency polygon in 14(b), calculate the number of students who obtained more than 25 marks. [1 mark] Berdasarkan poligon kekerapan di 14(b), hitung bilangan murid yang mendapat lebih daripada 25 markah. [1 markah]
25 /2 Answer / Jawapan: (a) Marks Markah Midpoint Titik Tengah Frequency Kekerapan 7 (b) (c) Refer graph on page 25. Rujuk graf pada halaman 25. (d)
26 /2 Graph for Question 14 Graf untuk Soalan 14
27 /2 15 (a) The Diagram 15.1 shows a solid right prism with rectangular base on a horizontal plane. The surface is the uniform cross-section of the prism. Rectangle ia a horizontal plane and rectangle is an inclined plane. and are the vertical edges. Rajah 15.1 menunjukkan sebuah pepejal berbentuk prisma tegak dengan tapak segi empat tepat terletak di atas tapak mengufuk. Permukaan ialah keratan rentasseragam prisma itu. Segi empat tepat ialah satah mengufuk dan segi empat tepat ialah satah condong. Tepi dan adalah sisi tegak. Diagram 15.1 Rajah 15.1 Draw to full scale, the plan of the solid. Lukis dengan skala penuh, pelan pepejal itu. [3 marks] [3 markah]
28 /2 Answer / Jawapan : (a)
29 /2 (b) A solid right prism with uniform cross-sectional area is removed from the solid in Diagram The remaining solid is as shown in the Diagram Rectangle is a horizontal plane. and are vertical edges. Given that and. Draw to full scale, Sebuah prisma tegak dengan keratan rentas seragam dikeluarkan daripada pepejal dalam Rajah Pepejal yang tinggal adalah seperti dalam Rajah Segi empat adalah satah mengufuk. dan adalah satah condong. Diberi bahawa dan. Lukiskan dengan skala penuh. Diagram 15.2 Rajah 15.2 (i) the elevation of the solid on a vertical plane parallel to as viewed from. [4 marks] dongakan pepejal itu pada satah mencancang yang selari dengan sebagaimana dilihat dari. (ii) the elevation on a vertical plane parallel to as viewed from dongakan pepejal itu pada satah mencancang yang selari dengan sebagaimana dilihat dari [4 markah] [5 marks] [5 markah]
30 1449/2 30 Answer / Jawapan : (b) (i) (ii)
31 /2 16 J(63 N, 82 W), K(63 N, 24 W), L and M are four points on the surface of the earth. JL is the diameter common parallel of latitude of 63 N. J(63 U, 84 B), K(63 U, 24 B), L dan M adalah empat titik di atas permukaan bumi. JL ialah diameter selarian latitud sepunya 63 U. (a) State the location of L. [3 marks] Nyatakan kedudukan bagi L. [3 markah] (b) Calculate the shortest distance, in nautical mile, from point J to point L measured along the surface of the earth. [2 marks] Hitung jarak terpendek, dalam batu nautika, dari titik J ke titik L diukur sepanjang permukaan bumi. [2 markah] (c) Calculate the distance, in nautical mile, from J due to east K measured along the common parallel of latitude. [3 marks] Hitung jarak, dalam batu nautika, dari J arah timur ke K diukur sepanjang selarian latitud sepunya. [3 markah] (d) An aeroplane took off from K and flew due south to M. The average speed of the flight was 520 knots and the time taken was 6 hours. Sebuah kapal terbang berlepas dari K dan terbang arah ke selatan ke M. Purata laju penerbangan itu ialah 520 Knot dan masa yang diambil untuk penerbangan itu ialah 6 jam. Calculate Hitung (i) the distance, in nautical mile, from K to M measured along the meridian, jarak, dalam batu nautika, dari K ke M diukur sepanjang meridian, (ii) the latitude of M. latitud bagi M. [4 marks] [4markah]
32 /2 Answer / Jawapan : (a) (b) (c) (d) END OF ASSESSMEN MODULE MODUL PENILAIAN TAMAT
33 /2 INFORMATION FOR CANDIDATES MAKLUMAT UNTUK CALON 1. This question paper consists of two sections: Section A and Section B. Kertas soalan ini mengandungi dua bahagian: Bahagian A dan Bahagian B. 2. Answer all questions in Section A and any four questions from Section B. Jawab semua soalan dalam Bahagian A dan mana-mana empat soalan daripada Bahagian B. 3. Write your answers in the spaces provided in the question paper. Tulis jawapan anda pada ruang yang disediakan dalam kertas soalan ini. 4. Show your working. It may help you to get marks. Tunjukkan kerja mengira anda. Ini boleh membantu anda untuk mendapatkan markah. 5. If you wish to change your answer, cross out the answer that you have done. Then write down the new answer. Jika anda hendak menukar jawapan, batalkan jawapan yang telah dibuat. Kemudian tulis jawapan yang baru. 6. The diagrams in the questions provided are not drawn to scale unless stated. Rajah yang mengiringi soalan tidak dilukis mengikut skala kecuali dinyatakan. 7. The marks allocated for each question and sub-part of a question are shown in brackets. Markah yang diperuntukkan bagi setiap soalan dan ceraian soalan ditunjukkan dalam kurungan. 8. A list of formulae is provided on pages 2 to 4. Satu senarai rumus disediakan di halaman 2 hingga A booklet of four-figure mathematical tables is provided. Sebuah buku sifir matematik empat angka disediakan. 10. You may use a scientific calculator. Anda dibenarkan menggunakan kalkulator saintifik. 11. Hand in this question paper to the invigilator at the end of the examination. Serahkan kertas soalan ini kepada pengawas peperiksaan pada akhir peperiksaan.
34 1449/2 Skema Matematik Kertas 2 Question Marks Scheme Sub mark 1 Total mark Line y = 2 correctly drawn Region correctly shaded N2 3 2(a) Note : 1. Shade any region which satisfy any two inequalities award N1 2. Deduct 1 mark if y = 2 is drawn as a solid line from N2 P (b) or N
35 1449/2 Question Marks Scheme Sub mark 3 or equivalent Total mark or equivalent N1N1 4 4 or equivalent 4 5 N1 N N1 4 Note 1. Accept for K mark 4 6 (a) False//Palsu P1 1 6 (b) If x is an even number then x can be divided by 2 // Jika x ialah nombor genap maka x boleh dibahagi tepat dengan 2. P1 If x can be divided by 2 then x is an even number // Jika x boleh dibahagi tepat dengan 2 maka x ialah nombor genap. P1 2 6 (c) K
36 1449/2 Question Marks Scheme Sub mark 7 (a) Total mark N1 2 7 (b) P1 or N (a) { RG, RY, RW, GR, GY, GW, YR, YG, YW, WR, WG, WY} P1 1 8 (b)(i) { RG, GR, GY, GW, YG, WG} N1 2 8 (b)(ii) { RY, GY, WR, WG, WY} N1 2 Note : Accept correct answer without working N1 5 9 (a) N1 3 3
37 1449/2 Question Marks Scheme Sub mark Total mark 9 (b) N (a) 36 P (b) N (c) K2 award N (a) P1 P1 2 4
38 1449/2 Question Marks Scheme Sub mark 11 (b) P1 Total mark N1 N1 Note: 4 1. award Do not accept solution not using matrix method (a) (b) 5
39 1449/2 Question Marks Scheme Sub mark Graph Axes drawn in correct directions with uniform scales for and. All 7 points and *2 points correctly plotted or curve passes through all the points for and P1 K2 Total mark Note: 1. 7 or 8 points correctly plotted, award 2. Ignore curve out of range Smooth and continuous curve without any straight line passing through all 9 correct points using the given scales 4 x 4. and 16 y 16. N (c) (i) P1 12(c) (ii) P (d) K2 N1 N (a)(i) P2 Note : 13(a)(ii) P2 Note : 4 6
40 1449/2 Question Marks Scheme Sub mark Total mark 13(b)(i) M = Reflection in the line Note: Reflection only award P1 P2 N= Enlargement, scale factor 3 at centre U P3 Note: 1. Enlargement, scale factor 3 or enlargement at centre U award P2 2. Enlargement only award P1 13(b)(ii) K2 or (K2) 204 N1 Note : Marks Markah Midpoint Titik Tengah Frequency Kekerapan I 7 4 II 12 7 III 17 8 IV 22 9 V 27 5 VI 32 4 VII 37 5 VIII
41 1449/2 Question Marks Scheme Sub mark Class Interval : ( I until VIII ) Midpoint : ( I until VIII ) Frequency : ( I until VIII ) 14 (b) K2 P1 P1 P2 4 Total mark N (c) Graph Axes drawn in the correct directions with uniform scales for and P1 8 points correctly plotted K2 Note : 6 or 7 points correctly plotted, award Continuous straight line and passes all 8 correct points. N1 4 8
42 1449/2 Question Marks Scheme Sub mark Total mark 14 (d) * (a) 12 Correct shape with rectangles All solid lines Measurements correct to vertices of rectangles and cm (one way) and all angles at the N1 3 9
43 1449/2 Question Marks Scheme Sub mark 15 (b)(i) Total mark Correct shape with heksagon All solid lines and trapezium Measurements correct to cm (one way) and all angles at the vertices of rectangles N (b)(ii) Correct shape with heksagon All solid lines Note : Ignore straight line and rectangle and joined with dashed line Measurements correct to vertices of rectangles cm (one way) and all angles at the N
44 1449/2 Question Marks Scheme Sub mark Total mark 16 (a) ( 63 N, 98 E) P2 2 Note : 63 N or 98 E or 63 or 98 correct award P1 16(b) 3240 N1 2 16(c) K Note : 3 N1 16 (d)(i) 3120 N (d)(ii) Latitude of M = N
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