PEPERIKSAAN PERCUBAAN SPM TINGKATAN LIMA TAHUN Kertas 2. Dua jam tiga puluh minit JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU

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1 47/ Matematik Tambahan Kertas September 0 ½ jam SMK KAMPUNG GELAM Melaka PEPERIKSAAN PERCUBAAN SPM TINGKATAN LIMA TAHUN 0 MATEMATIK TAMBAHAN Kertas Dua jam tiga puluh minit JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU. This question paper consists of three sections : Section A, Section B and Section C Kertas soalan ini mengandungi tiga bahagian : Bahagian A, Bahagian B dan Bahagian C.. Answer all questions in Section A, four questions from Section B and two questions from Section C. Jawab semua soalan dalam Bahagian A, empat soalan daripada Bahagian B, dan dua soalan daripada Bahagian C.. Give only one answer/solution to each question. Bagi setiap soalan, berikan satu jawapan / penyelesaian sahaja. 4. Show your working. It may help you to get marks. Tunjukkan langkah-langkah penting dalam kerja mengira anda. Ini boleh membantu anda untuk mendapatkan markah. 5. The diagrams in the questions provided are not drawn to scale unless stated. Rajah yang mengiringi soalan tidak dilukiskan mengikut skala kecuali dinyatakan, 6. 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. 7. A list of formulae is provided on pages and. Satu senarai rumus disediakan di halaman dan. 8. A booklet of four-figure mathematical tables is provided. Buku sifir matematik empat angka boleh digunakan. 9. You may use a non-programmable scientific calculator. Anda dibenarkan menggunakan kalkulator saintifik yang tidak boleh diprogram. Kertas soalan ini mengandungi 7 halaman bercetak 47/ [ Lihat sebelah

2 The following formulae may be helpful in answering the questions. The symbols given are the ones commonly used. Rumus-rumus berikut boleh digunakan untuk membantu anda menjawab soalan.. Simbol-simbol yang diberi adalah yang biasa digunakan. ALGEBRA b b 4ac log c b 8 log a b = a log c a a m a n = a m + n a m a n = a m - n 4 (a m ) n = a nm 5 log a mn = log a m + log a n 6 m log a n = log a m - log a n 7 log a m n = n log a m CALCULUS ( Kalkulus) 9 T n = a + (n-)d 0 n S n = [a ( n ) d] T n = ar n- n n a( r ) a( r ) S n = r r a S r, r <, (r ) y = uv, dy d dv u d du v d du dv v u u d y, d d, v dy v dy d dy du Distance (Jarak ) = du d ( y ) ( y ) Midpoint (Titik tengah ) (, y) = y, y 4 r r y i yj y GEOM ETRY 4 Area under a curve ( Luas dibawah lengkung ) b = y d or (atau ) a b = dy a 5 Volume generated ( Isipadu janaan ) b = y d or a b = a dy 5. A point dividing segment of a line (Titik yang membahagi suatu tembereng garis) n m (,y) =, m n 6. Area of triangle (Luas segitiga ) = ny my m n ( y y y ) ( y y y ) 47/ [ Lihat sebelah

3 STATISTICS ( STATISTIK ) = = = N f f ( ) N = N _ wi I w n n! P r ( n r)! n n! C r ( n r)! r! 0 P(A B)=P(A)+P(B)-P(A B) 4 = f ( ) f = f f P(X=r) = Mean, µ = np r C p q n r n r, p + q = N F 5 m = L C fm npq 4 z = Q I 00 Q 6 0 TRIGONOMETRY Arc length, s = r (Panjang lengkok, s = j ) Area of sector, A = r ( Luas sektor, L = j ) sin A + cos A = 4 sek A = + tan A 5 cosec A = + cot A 6 sina = sinacosa 7 cos A = cos A sin A = cos A- = - sin A tan A 8 tana = tan A 9 sin (A B) = sinacosb cosasinb (sin (A B) = sinakosb kosasinb) 0 cos (A B) = cos AcosB sinasinb (kos (A B) = kos AkosB sinasinb ) tan A tan B tan (A B) = tan Atan B a sin A b sin B c sin C a = b +c - bc cosa ( a = b +c - bckosa ) 4 Area of triangle (Luas segitiga) = absin C 47/ Lihat sebelah

4 4 Section A Bahagian A [40 marks] [40 markah] Answer all questions in this section Jawab semua soalan dalam bahagian ini. Solve the simultaneous equation Selesaikan persamaan serentak ( y y) y. [6 marks] ( y y) y. [6 markah] Company Supper made a profit of RM in the year 008. The company's profit grew by 0.5% from year to year. Syarikat Supper mendapat keuntungan RM pada tahun 008. Keuntungan syarikat itu meningkat sebanyak 0.5% setiap tahun. Calculate Hitung (a) the profit, to the nearest RM, for the year 0. keuntungan, kepada RM terdekat, untuk tahun 0. [ marks] [ markah] (b) the minimum value of n such that the profit in the n th year will eceed RM 00,000. nilai minimum n dengan keadaan keuntungan tahunannya pada tahun ke-n akan melebihi RM 00,000. [ marks] [ markah] (c) the total profit, to the nearest RM, made by the company from year 008 to 0. jumlah keuntungan, kepada RM terdekat, yang diperoleh oleh syarikat itu dari tahun 008 hingga 0. [ marks] [ markah] 47/ Lihat sebelah

5 5 (a) Sketch the graph of y tan for 0. [4 marks] Lakarkan graf bagi y tan untuk 0. [4 markah] (b) Hence, using the same aes, sketch a suitable graph to find the number of Solutions to the equation tan for 0. State the number of solutions. [ marks] Seterusnya, dengan menggunakan paksi yang sama, lakarkan satu graf yang sesuai untuk mencari bilangan penyelesaian bagi persamaan tan untuk 0. Nyatakan bilangan penyelesaian itu. [ markah] 4 The point R lies on the curve y ( 4 5). It is given that the gradient of the normal at R is. 8 Titik R terletak pada lengkung y ( 4 5). Diberi bahawa kecerunan normal pada R ialah. 8 Find Cari (a) the coordinates of R. koordinat R. (b) the equation of tangent at the point R. persamaan tangen pada titik R. [4 marks] [4 markah] [ marks] [ markah] 47/ Lihat sebelah

6 6 5. Table 5 (i) shows the cumulative frequency distribution of the marks of 40 students. Jadual 5 (i) menunjukkan taburan kekerapan longgokan markah bagi 40 orang murid. Score Skor Number of students Bilangan murid < 0 < 0 < 40 < 50 < Table 5 (i) Jadual 5 (i) (a) Based on table 5(i), copy and complete the Table 5 (ii) Berdasarkan Jadual 5(i), salin dan lengkapkan Jadual 5 (ii). Score Skor Number of students Bilangan murid Table 5 (ii) Jadual 5 (ii) [ mark] [ markah] (b) (i) State the modal class. Nyatakan kelas mod. (ii) Using a scale of cm to 0 marks on the horizontal ais and cm to students on the vertical ais, draw a histogram to represent the frequency distribution of the marks in table 5 (ii). Hence, find the mode mark. Dengan menggunakan skala cm kepada 0 markah pada paksi mengufuk dan cm kepada orang murid pada paksi mencancang, lukis sebuah histogram bagi mewakili taburan kekerapan markah dalam jadual 5 (ii). Seterusnya, cari markah mod. [5 marks] [5 markah] 6. In Diagram 6, OAB is a triangle. P lies on OA such that OP : PA = : and Q lies 47/ Lihat sebelah

7 7 on OB such that OQ : QB = :. R is the point of intersection of the line AQ and BP. Dalam Rajah 6, OAB ialah sebuah segi tiga. P terletak pada OA dengan keadaan OP : PA = : dan Q terletak pada OB dengan keadaan OQ : QB = :. R ialah titik persilangan bagi garis AQ dan BP. Q B R It is given = and = OA a OB b Diberi = dan = (a) Find the following vectors. Cari vector yang berikut. (i) BP O OA a OB b Diagram 6 Rajah 6 P A (ii) AQ (b) Using BR = hbp and AR = kaq terms of, where h and k are constants, epress BR in [ marks] [markah] Dengan menggunakan BR = hbp dan AR = kaq dengan keadaan h dan k ialah pemalar, ungkapkan BR dalam sebutan (i) h, a and b h, a dan b (ii) k, a and k, a dan b b Hence, find the values of h and k, Seterusnya, cari nilai h dan nilai k [5 marks] [5 markah] 47/ Lihat sebelah

8 8 Section B Bahagian B [40 marks] [40 markah] Answer four questions from this section. Jawab empat soalan daripada bahagian ini. 7. Use graph paper to answer this question. Gunakan kertas graf yang disediakan untuk menjawab soalan ini. Table 7 shows the values of two variables, and y, obtained from an eperiment. Variables and y are related by the equation k y h, where h and k are constants. Jadual 7 menunjukkan nilai-nilai bagi dua pembolehubah, dan y, yang diperoleh daripada satu eksperimen. Pembolehubah dan y dihubungkan oleh persamaan k y h dengan keadaan h dan k adalah pemalar y Table 7 Jadual 7 (a) Based on table 7, construct a table for the values of log y and (+). [ marks] 0 Berdasarkan Jadual 7, bina satu jadual bagi nilai nilai log y dan (+). [ markah] (b) Plot log y against ( + ), using a scale of cm to 0.5 unit on the (+)-ais 0 and cm to 0. on the log y -ais. 0 Hence, draw the line of best fit. [ marks] Plotkan log y melawan ( + ), dengan menggunakan skala cm kepada unit pada paksi- ( + ) dan cm kepada 0. unit pada paksi- log y. 0 Seterusnya, lukiskan garis lurus penyuaian terbaik. [ markah] (c) Use your graph in 7 (b) to find the value of Gunakan graf anda dari 7(b) untuk mencari nilai 0 (i) k (ii) h [5 marks] [5markah] 47/ Lihat sebelah

9 8. Solution by scale drawing is not accepted. Penyelesaian secara lukisan berskala tidak diterima. 9 In Diagram 8, the straight line BC has equation y = 0. BC intersects the y ais at point S and O is the origin. Dalam Rajah 8, garis lurus BC mempunyai persamaan y = 0. BC memotong paksi-y pada titik S dan O ialah titik asalan y A(-, h) B O S y++6=0 5 C (9, ) Diagram 8 Rajah 8 (a) It is given that S lies on the straight line BC such that BS : BC = : 4. Find the coordinates of B, Diberi bahawa S terletak pada garis lurus BC dengan keadaan BS : BC = : 4. Cari koordinat B., (b) Given the area of triangle OAB is 5 unit, find the value of h. Diberi luas segitiga OAB adalah 5 unit, cari nilai h, [ marks] [markah] [ marks] [ markah] (c) A point P moves such that its distance from point S is half its distance from point A. Find the equation of the locus of P. [4 marks] Satu titik P bergerak supaya jaraknya dari titik S ialah separuh jaraknya dari titik A. Cari persamaan lokus P. [4 markah] 47/ Lihat sebelah

10 9. Diagram 9 shows a sector OPQ, centre O and a radius of cm. SPT is another sector of a circle with centre S and a radius of 5 cm, which is inscribed in the sector OPQ such that ST is perpendicular to OQ. Rajah 9 menunjukkan sebuah sektor OPQ, berpusat O dan berjejari cm. SPT adalah sebuah sektor lain berpusat S dan berjejari 5 cm terterap dalam sektor OPQ dengan keadaan ST adalah berserenjang dengan OQ. 0 [Use/Guna π =.4] Diagram 9 Rajah 9 Calculate Hitung (a) the value of θ, in radians, nilai θ, dalam radian, (b) the perimeter, in cm, of the shaded region, perimeter, dalam cm, bagi kawasan berlorek, (c) the area, in cm, of the shaded region. luas, dalam cm, bagi kawasan berlorek. [ marks] [ markah] [4 marks] [4 markah] [4 marks] [4 markah] 0. Diagram 0 shows part of the curve y = f() which has gradient function 4 ( ) Rajah 0 menunjukkan sebahagian daripada lengkung y = f(,) yang mempunyai fungsi 47/ Lihat sebelah.

11 kecerunan 4. ( ) Diagram 0 Rajah 0 The curve intersects the straight line y = at point P(, ). Lengkung itu bersilang dengan garis lurus y = pada titik P(, ). Find Cari (a) the equation of the curve, persamaan lengkung tersebut, (b) the area of the shaded region, luas rantau berlorek, [ marks] [ markah] [4 marks] [4 markah] (c) the volume generated, in terms of π, when the region which is bounded by the curve, the -ais and the straight lines = and =, is revolved through 60 about the -ais. [ marks] isipadu yang dijanakan, dalam sebutan π, apabila rantau yang dibatasi oleh lengkung itu, paksi- dan garis lurus = dan =, dikisarkan melalui 60 pada paksi-. [ markah]. (a) In a certain year, 80% of the students in a university graduated with honours. If 0 of the students are selected at random, find the probability that 47/ Lihat sebelah

12 Pada suatu tahun, 80% daripada pelajar di sebuah universiti lulus dengan kepujian. Jika 0 orang daripada pelajar itu dipilih secara rawak, cari kebarangkalian bahawa (i) only one of the students did not graduate with honour. hanya seorang pelajar tidak lulus dengan kepujian. (ii) not more than two students did not graduate with honour. tidak lebih daripada dua orang pelajar tidak lulus dengan kepujian. [4 marks] [4 markah] (b) The masses of packets of coffee powder produced by a factory have a normal distribution with a mean of 00 g and a variance of 400 g. Jisim serbuk kopi dalam peket yang dihasilkan oleh sebuah kilang adalah bertaburan secara normal dengan min 00 g dan varians 400 g. (i) Find the probability that a packet of coffee selected randomly is more than 5 g. Cari kebarangkalian bahawa sepeket serbuk kopi yang dipilih secara rawak adalah melebihi 5 g. (ii) A random of 500 packets of coffee powder is chosen. Given that from the sample 4 packets of coffee powder from the sample have a mass of more than m g. Find the value of m. Satu sampel rawak 500 peket serbuk kopi dipilih. Diberi 4 peket serbuk kopi daripada sampel itu mempunyai jisim melebihi m g. Cari nilai m. [6 marks] [6 markah] Section C Bahagian C 47/ Lihat sebelah

13 [0 marks] [0 markah] Answer two questions from this section. Jawab dua soalan daripada bahagian ini.. Table shows the prices in the year 009 and 00 of four types of local food. Jadual menunjukkan harga untuk tahun 009 dan 00 bagi empat jenis makanan tempatan. Food Makanan Price per kilogram (RM) Harga per kilogram(rm) Price inde for the year 00 based on the year 009 Indeks harga pada tahun 00 berdasarkan tahun 009 Weightage Pemberat P.80.0 a 4 Q R.00 b 0 S c m (a) Find the value of Cari nilai (i) a, (ii) b, (iii) c. Table Jadual [ marks] [ markah] (b) The composite inde for the price of the food in the year 00 based on the year 009 is Calculate the value of m. [ marks] Indeks gubahan bagi harga makanan tersebut pada tahun 00 berdasarkan tahun 009 ialah Hitung nilai m. [ markah] (c) The total cost for the food in the year 009 is RM55. Calculate the total cost in the year 00. [ marks] Jumlah kos makanan tersebut pada tahun 009 ialah RM55. Hitung jumlah kos bagi tahun 00. [ markah] (d) The price inde for Q in the year 0 based on the year 009 is. Calculate the price inde for Q in the year 0 based on the year 00. [ marks] Indeks harga bagi Q pada tahun 0 berdasarkan tahun 009 ialah. Hitung indeks harga bagi Q pada tahun 0 berasaskan tahun 00. [markah]. A particle moves along a straight line and passes through a fied point O. Its velocity, v ms, is given by v = t 0t + 4, where t is the time, in seconds, after passing 47/ Lihat sebelah

14 4 through O. Suatu zarah bergerak di sepanjang suatu garis lurus melalui satu titik tetap O. Halaju zarah itu, v ms, diberi oleh v = t 0t + 4, dengan keadaan t ialah masa dalam saat selepas melalui O. [Assume motion to the right is positive]. [Anggapkan gerakan ke arah kanan sebagai positif] Find Cari (a) the initial velocity, in ms. halaju awal, dalam ms. (b) the minimum velocity, in ms. halaju minimum, dalam ms (c) the range of t during which the particle moves to the left. julat nilai t ketika zarah bergerak ke arah kiri. (d) The total distance, in m, travelled by the particle in the first 5 seconds. Jumlah jarak, dalam m, yang dilalui oleh zarah dalam 5 saat pertama. [ mark] [markah] [ marks] [markah] [ marks] [ markah] [4 marks] [4 markah] 4. Use the graph paper provided to answer this question. Gunakan kertas graf yang disediakan untuk menjawab soalan ini. Machine A(minutes) Machine B (minutes) Model Mesin A (minit) Mesin B (minit) 47/ P 8 5 Lihat sebelah Q 8 8

15 5 Table 4 Jadual 4 Table 4 shows time taken by machine A and machine B to produce two types of toys of model P and model Q. The factory production is based on the following constraints Jadual 4 menunjukkan masa yang digunakan oleh mesin A dan mesin B untuk menghasilkan dua jenis alat permainan iaitu model P dan model Q. Kilang telah menetapkan syaratsyarat seperti berikut I II III Machine A operates at least 70 minutes per day. Mesin A beroperasi sekurang-kurangnya 70 minit sehari. Machine B operates at most 800 minutes per day. Mesin B beroperasi tidak melebihi 800 minit sehari The ratio of the number of model Q toy to the number of model P toy is not more than 5 : 8. Nisbah bilangan alat permainan model Q terhadap bilangan alat permainan model P tidak melebihi 5 : 8. The factory produced model P toy and y model Q toy. Kilang itu mengeluarkan buah alat permainan model P dan y buah alat permainan model Q. (a) Write three inequalities, other than 0 and y 0, which satisfy all the above constraints. Tuliskan tiga ketaksamaan selain daripada 0 dan y 0 yang memuaskan kesemua syarat di atas. [ marks] [ markah] (b) Using the scale of cm to 0 toys on the -ais and cm to 0 toys on y-ais, construct and shade the region R which satisfies all of the above inequalities. Dengan menggunakan skala cm kepada 0 alat permainan pada paksi- dan cm kepada 0 buah alat permainan pada paksi-y, bina dan lorekkan rantau R yang memuaskan kesemua ketaksamaan itu [ marks ] [ markah] (c) Using your graph from (b), find Dengan menggunakan graf anda dari (b), cari (i) the maimum and minimum number of model P toys produced if 5 model Q toys was produced. bilangan maksimum dan minimum alat permainan model P yang dihasilkan jika 5 buah alat permainan model Q dihasilkan. (ii) the maimum total amount gained by the factory if one model P and model Q toy is sold at the price of RM0 and RM0 respectively. jumlah jualan maksimum yang diperolehi oleh kilang itu jika sebuah alat permainan model P dan model Q masing-masing dijual dengan harga RM0 dan RM0. [4 marks] [4markah] 47/ Lihat sebelah

16 5. Diagram 5 shows triangles LRM and LNM. Rajah 5 menunjukkan segitiga LRM dan LNM. 6 Diagram 5 Rajah 5 It is given that RL = cm, LN = 6 cm and MR = 8 cm. Diberi bahawa RL = cm, LN = 6 cm dan MR = 8 cm. Calculate Hitung NLM 0 NLM 0 o o dan and LMN 85 LMN 85 o o (a) the length, in cm, of LM, panjang, dalam cm, bagi LM, (b) RLN, (c) the area, in cm, of quadrilateral RLNM. luas, dalam cm, bagi sisiempat RLNM. [ marks] [ markah] [ marks] [ markah] [4 marks] [4 markah] END OF QUESTION PAPER KERTAS SOALAN TAMAT 47/ Lihat sebelah

17 7 MARKING SCHEME FOR ADDITIONAL MATHEMATICS TRIAL SPM 0 PAPER NO. Solution Mark Scheme ( y) y y y ( y) y either two correct y or y Eliminate or y ( y ) y (y ) y Solve the quadratic equation Using factoristion y( 7 y 8) 0 or equivalent 8 y,0 7 7, 7 [6] (a) a = RM60 000, r =.05 (b) (c) T = n n > 7. The minimum value of n is (.05 ) [7] 47/ Lihat sebelah

18 NO. 8 Solution Mark Scheme (a) (b) Aes and Shape of tan 0 cycles for 0 Modulus y are shown 0 y or equivalent Straight line drawn correctly Number of solutions = 8 [7] 4(a) dy d 8(4 5) (b) m of tangent = 8 8(4 5) = 8 =, y = R= (, ) y = 8( ) y = [6] 47/ Lihat sebelah

19 9 NO. 5(a) Score Skor Number of students Bilangan murid Solution Mark Scheme (b) (i) Modal class = (ii) Refer to graph Uniform scale and correct aes Histogrm correctly (all the bar correctly) Find the mode from the histrogram Mode = 4 ( 0..5) [6] 6. (a) (i) BP BO OP or AQ AO OQ BP a b (ii) AQ b a 4 (b) (i) h a b (ii) BR BA AR BR BO OA k AQ ( k) a k b 4 k h or k h 4 h, k (both) [8] 47/ Lihat sebelah

20 0 NO. Solution Mark Scheme 7(a) Log y (b) Uniform scale + one of the points plotted correctly 6 Points plotted correctly Line of best fit (c) (i) log0 y ( )log0 k log0 h log 0 k 0.05 k =.604 (ii) log 0 h 0.9 h =.549 [0] 8(a) (b) S(0, ) 5 b a 9, 4 4 B (, ) 0, (c) h 5 h = 9 PS = PA ( y ) ( ) ( y 9) 4[( + (y + ) = ( + ) + (y 9) + y 4 + 4y 49 = 0 [0] 47/ Lihat sebelah

21 47/ Lihat sebelah NO. Solution Mark Scheme 9(a) (b) (c) sin rad ) ( (0.675) 5(.46) (0.675) 5(.46).46 Perimeter ors S rad PST )(0.675) ( or ) )(.46 (5 (5)(6.45) )(0.675) ( ) )(.46 (5 ) (5)( [0] 0 (a) (b) (c) c y )() ( ) 4( ) ( () y c ) )( ( ) ( A ) (() ) ()() (4)() or d A or 4 unit 5 8 A A units ) (() 6 ) (() )() ( ) ( [0]

22 47/ Lihat sebelah NO. Solution Mark Scheme (a) (b) (i) (0.)(0.8) 0 ) ( 9 C P (ii) (0.8) (0.) (0.8) (0.) (0.8) C C C (i) or z P (ii) 79.8 m ) 0 00 ( ) ( m m P m z P [0]

23 NO. (a) (i) a =.. 00 = 75 Solution Mark Scheme (ii). b = 00 = 0. =.60 (iii). 00 = 6 c =. = 5.00 (b) 7.6m = 45.6 m= 6 0() (4) + 0() + 9 = 08.4, 00 = (c) = = 0 (d) [0] 47/ Lihat sebelah

24 4 NO. (a) 4 Solution Mark Scheme (b) t 0 = 0 t = 5 V min = (c) (t 4)(t 6) < 0 4 < t < 6 (d) 4 0 ( t 0t 4) dt or ( t 0t 4) dt ( t 0t 4) dt + ( t 0t 4) dt 5 4 t 5t 4t 4 0 t + 5t 4t [0 47/ Lihat sebelah

25 NO. 5 Solution Mark Scheme 4 (a) 8 + 8y 70 atau setara 5 + 8y 800 8y 5 atau setara atau setara (b) - Lukis dengan betul sekurang-kurangnya satu garis lurus daripada ketaksamaan yang melibatkan dan y Lukis dengan betul ketiga-tiga garis lurus rantau berlorek (c) (i) Bil. maksimum = 6 Bil. minimum = 40 (ii) Gunakan 0 + 0y untuk titik dalam rantau 0(76) + 0(47) RM 70 [0] 47/ Lihat sebelah

26 NO. 6 Solution Mark Scheme 5(a) LNM 75 LM 6 sin 75 sin85 LM 5.88 (b) ()(5.88) cos RLN RLM 5.8 RLN 5.8 (c) A = Area RLM = ()(5.88)sin 5.8 A = Area LMN = (6)(5.88)sin 0 Total area = A A = 4.4 cm [0] 47/ Lihat sebelah

27 7 Number of Students score/skor 47/ Lihat sebelah

28 8 Log y / Lihat sebelah

29 9 y / Lihat sebelah

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