Name : Form :. 0 SMKA NAIM LILBANAT 0 KOTA BHARU KELANTAN. SEKOLAH BERPRESTASI TINGGI PEPERIKSAAN PERCUBAAN SPM 04 ADDITIONAL MATHEMATICS Kertas ½ Jam 47/ ½ Jam Arahan:. This question paper consists of three sections: Section A, Section B and Section C.. Answer all questions in Section A, an four questions from Section B and an two questions from Section C.. Write our answers on the paper sheets provided. A B C Untuk Kegunaan Pemeriksa Soalan Markah Penuh 7 6 4 7 8 6 7 7 8 9 4 Markah Diperoleh JUMLAH 0 47/ Kertas soalan ini mengandungi 8 halaman bercetak.
Section A [40 marks] (Answer all questions). Solve the following simultaneous equations : Selesaikan persamaan serentak berikut: and [ marks]. (a) Skatch the graph of 4sin for 0 Lakarkan graf bagi 4sin untuk 0 [ marks] Hence, b using the same aes, sketch a suitable straight line to find the number of the solutions of the equation sin for 0. Seterusna, dengan menggunakan paksi ang sama, lakar satu garis lurus ang sesuai untuk mencari bilangan penelesaian bagi persamaan sin untuk 0. [4 marks]. The gradient function of a curve which passes through the point (, ) is k -, where k is a constant. The equation of the tangent to the curve at that point is 0. Fungsi kecerunan bagi satu lengkung ang melalui titik (, ) ialah k -, dengan keadaan k ialah pemalar. Persamaan tangen kepada lengkung pada titik itu ialah - + - 0. (a) Find the value of k. Cari nilai k. Find the equation of the curve. Cari persamaan bagi lengkung itu. [ marks] [ marks] 47/
4. A circle is divided into n sectors such that the angles of the sectors form an arithmetic progression. Sebuah bulatan dibahagikan kepada n sektor dengan keadaan sudut-sudut sektor itu membentuk suatu janjang aritmetik. (a) If the smallest angle and the largest angle are and 7 respectivel, find the value of n. Jika sudut ang terkccil dan sudut ang terbesar masing-masing ialah dan 7. cari nilai n. Using the value of n obtained in 4(a), find Dengan menggunakan nilai n ang diperoleh di 4(a), cari [ marks] (i) the common difference of the progression. beza sepuna janjang itu. (ii) the value of the tenth angle. nilai bagi sudut ang kesepuluh. [4 marks]. Diagram shows a triangle PQR where A is a point on PQ, B is a point on PR and C is a point on AR. Rajah 8 menunjukkan sebuah segi tiga PQR di mana A ialah satu titik pada PQ, B ialah satu titik pada PR dan C ialah satu titik pada AR. R C B Q A Diagram / Rajah P It is given that: Diberi bahawa: (a) QA AP, AC CR Epress in terms of Ungkapkan dalam sebutan (i) (ii) (iii) RA BC CQ a and b a dan b RB a,, PB BR,, PA b [ marks] Hence, show that B, C and Q lie on a straight line. Seterusna, tunjukkan bahawa B, C dan Q berada pada satu garis lurus. [ marks] 47/
6. Table 6 shows the distrubution of the scores of 60 students in a test. Jadual 6 menunjukkan taburan skor bagi 60 orang murid dalam satu ujian. Score / Skor Frequenc/ Kekerapan 0 6 0 40 4-0 Table 6/ Jadual 6 (a) Find the mean of the scores. Cari min bagi skor tersebut. [ marks] Without drawing an ogive, find the interquartile for scores. Tanpa melukis ogif, cari julat antara kuartil bagi skor tersebut. [ marks] Section B [40 marks] (Answer an four questions from this section) 7. Use graph paper to answer this question. Table 7 shows the values of two variables, and, obtained from an eperiment. The variables and are related b the equation p q, where p and q are constants. Jadual 7 menunjukkan nilai-nilai bagi dua pembolehubah dan ang diperolehi daripada satu eksperimen. Pembolehubah dan di hubungkan oleh persamaan p q... 0.0 6.98.96 7.44.0. Table 7/ Jadual 7 (a) (c) Construct a table for the values of log and log Bina sebuah jadual bagi nilai-nilai log dan log [ marks] Plot log against log b using a scale of cm to 0. units on log -ais and cm to 0. unit on the log - ais. Hence, draw the line of best fit. Plot log melawan log, dengan menggunakan skala cm kepada 0. unit pada paksi- log dan cm kepada 0. unit pada paksi-log. Seterusna, lukis garis penuaian terbaik. From our graph in 7, find the value of Dari graf anda di 7, cari nilai bagi, (i) p and q. (ii) when =.7. nilai apabila =.7 [ marks] [ marks] 47/
4 8. Diagram 8 shows a straight line = k touching the maimum point A of the curve Rajah 8 menunjukkan garis lurus = k menentuh titik maksimum A bagi lengkung 6. 6. k P A = k (a) Find the value of k. Cari nilai k. [ marks] Q = 6- O (, 0) Diagram 8/ Rajah 8 Find the area, in unit, of region Q. Cari luas, dalam unit, bagi rantau Q. [ marks] (c) Calculate the volume generated, in terms of π, when the region P is revolved 60 about the -ais. Hitung isi padu janaan, dalam sebutan π, apabila rantau P dikisarkan melalui 60 pada paksi-. [4 marks] 9. Solution b scale drawing is not accepted. Penelesaian secara lukisan berskala tidak diterima. Diagram 9 shows a triangle ABC. Rajah 9 menunjukkan segi tiga ABC. C (4, 8) O A(, ) B(, ) Diagram 9/ Rajah 9 (a) Calculate the area, in unit, of the triangle ABC. Hitung luas, dalam unit, segi tiga ABC. Find the equation of the perpendicular bisector of BC. Can persamaan pembahagi dua serenjang BC. [ marks] [4 marks] (c) The perpendicular bisector of BC intersects line AB at a point D. Find the coordinates of point D. Pembahagi dua sama serenjang BC bersilang dengan garis AB di titik D. Cari koardinati bagi titik D. [ marks] (d) A point P(, ) moves such that PC = units. Find the equation of the locus of point P. Satu titik P(, ) bergerak dengan keadaan PC = unit. Cari persamaan lokus bagi titik P. [ marks] 47/
. Diagram shows a sector AOB of a circle with centre O and a radius of 4 cm. Point C lies on OB such that AC = OC and OC = cm. Rajah menunjukkan sektor AOB bagi satu bulatan berpusat O dan berjejari 4 cm. Titik C terletak pada OB dengan keadaan AC = OC dan OC = cm. A Calculate/ Hitung, (a) the value of θ, in radian. nilai θ, dalam radian. [ marks] O C Diagram / Rajah B the perimeter, in cm, of the shaded region. perimeter, dalam cm, kawasan berlorek. [ marks] (c) the area, in cm, of the shaded region. luas, dalam cm, kawasan berlorek. [4 marks]. (a) In a surve carried out in a particular district, it is found that three out of five families have less than four children. If 9 families are chosen at random from that district, calculate the probabilit that at least seven families have less than four children. Dalam suatu tinjauan di sebuah daerah tertentu, didapati tiga daripada lima keluarga mempunai kurang daripada empat orang anak. Jika 9 keluarga dipilih secara rawak dari daerah itu, hitung kebarangkalian bahawa sekurang-kurangna tujuh keluarga mempunai kurang daripada empat orang anak. [4 marks] 00 students from a school sat for a test. The marks obtained are normall distributed with a mean of and a standard deviation of. Seramai 00 orang murid dari sebuah sekolah menduduki satu ujian. Markah ang diperoleh bertaburan secara normal dengan min dan sisihan piawai. (i) Find the number of students who pass the test if the passing mark is 4. Cari bilangan murid ang lulus ujian itu jika markah lulus ialah 4. (ii) If.87% of the students obtained distinction, find the minimum mark needed to obtain a distinction. Jika.87% daripada murid-murid itu mendapat kepujian, cari markah minimum ang diperlukan untuk mendapat kepujian. [6 marks] 47/
6 SECTION C [0 marks] (Answer an two questions from this section). Diagram shows a quadrilateral ABCD. Given the area of triangle ABD is 9.7 cm. Rajah menunjukkan sisi empat ABCD. Diberi luas segi tiga ABD ialah 9.7 cm. A 80 o 8 cm Calculate/ Hitung, (a) the length of AB, in cm. panjang AB, dalam cm [ marks]. B D the length of BD, in cm panjang BD, dalam cm. (c) BCD, [ marks] [ marks].6 cm 7 cm (d) CBD [ marks] C Diagram / Rajah. Table shows the prices, the price indices and the weightage of four materials, A, B, C and D, used in the production of a model car. Jadual menunjukkan harga, indeks harga dan peratusan bagi empat bahan A, B, C dan D, ang digunakan dalam menghasilkan model sejenis kereta. Price (RM) for the ear Harga (RM) bagi tahun Price inde for the ear 006 based 004 Indeks harga bagi tahun 006 004 006 berasaskan tahun 004 A 0 60 Material Bahan Weightage Peratusan (%) B 40 00 0 C 800 0 0 D 60 40 Table / Jadual (a) Calculate the values of and. Hitung nilai dan nilai. [ marks] 47/
7 Calculate the composite inde for the ear 006 based on the ear 004 Hitungkan indeks gubahan bagi tahun 006 berasaskan tahun 004. [ marks] (c) The price of material C is epected to decrease b 0% in the ear 008 compared to the ear 006. Find the price inde of C for the ear 008 based on the ear 004. Harga bahan C dijangka akan menurun sebanak 0% pada tahun 008 berbanding dengan tahun 006. Cari indeks harga C bagi tahun 008 berasaskan tahun 004. [4 marks] 4. Use graph paper to answer this question. Guna kertas graf untuk menjawab soalan ini A compan wishes to purchase two tpes of computers, A and B, for its staff. It plans to bus computers of tpe A which cost RM 000 each and computers of tpe B which cost RM 000 each. The purchase of the computers is subject to the following constraints: Sebuah sarikat ingin membeli dua jenis komputer, A dan B, untuk pekerjana. Sarikat itu bercadang membeli unit komputer A ang setiapna berharga RM000 dan unit komputer B ang setiapna berharga R M 000. Pembelian komputer itu adalah berdasarkan kekangan ang berikut: I : The compan allocated RM00 000 for the purchase of the computers. Sarikat itu memperuntukkan RM00 000 untuk membeli komputer itu. II : The total number of computers purchased cannot eceed 600. Jumlah bilangan komputer ang dibeli tidak boleh melebihi 600 unit. Ill: The ratio of the number of computers of tpe B purchased to the number of computers of tpe A purchased must be at least :. Nisbah bilangan komputer B ang dibeli kepada bilangan komputer A ang dibeli mestilah sekurangkurangna :. (a) Write three inequalities, other than 0 and 0, which satisf all the above constraints. Tulis tiga ketaksamaan, selain 0 dan 0, ang memenuhi semua kekangan di atas. [ marks] Using a scale of cm to 0 computers on both aes, construct and shade the region R which satisfies all the above constraints. Dengan menggunakan skala cm kepada 0 unit komputer pada kedua-dua paksi, bina dan lorek rantau R ang memenuhi semua kekangan di atas. [ marks] 47/ (c) (i) (ii) If 0 computers of tpe A are purchased, find the range of the number of computers of tpe B purchased. Jika 00 unit komputer A telah dibeli, cari julat bagi bilangan komputer B ang dibeli. If the cost of maintenance is RM00 for a computer of tpe A and RM00 for a computer of tpe B, find the maimum total cost of maintenance that has to be borne b the compan. Jika kos penelenggaraan ialah RM00 untuk seuint komputer A dan RM00 untuk seunit komputer B, cari jumlah kos penelenggaraan maksimum ang akan ditanggung oleh sarikat itu. [4 marks]
8. A particle moves along a straight line and passes through a fied point O at a velocit 6 ms -. Its acceleration, a m s -, t s after passing through O is given b a = 6(t - 4). Satu zarah bergerak di sepanjang suatu garis lurus dan melalui titik tetap O dengan halaju 6 m s -. Pecutanna, a m s -, selepas melalui O diberi oleh a = 6(t - 4). [Assume motion to the right is positive.) [Anggapkan gerakan ke arah kanan sebagai positif.] (a) Find its velocit, v m s -, in term of t. Cari halajuna, v m s -, dalam sebutan t Find the values of t when the particle reverses its direction of motion. Cari nilai-nilai t ketika zarah itu bertukar arah gerakanna. (c) Find the minimum velocit, in m s -, of the particle. Cari halaju minimum, dalam m s - zarah itu. (d) Calculate the distance, in m, travelled during the third second. Hitung jarak, dalam m, vang dilalui dalam saat ang ketiga. [ marks] [ marks] [ marks] [ marks] END OF QUESTIONS PAPER. KERTAS SOALAN TAMA 47/
SEK. MEN. KEB. AGAMA NAIM LILBANAT PEPERIKSAAN PERCUBAAN SPM 04 SKEMA PERMARKAHAN MATEMATIK TAMBAHAN KERTAS NO SOLUTIONS MARKS TOTAL. = - or -( - ) (- ) -( -)= - or 7 0 - + = 0 9 ( )( ) 0 or, = 7, = (a) 4 0 - shape of sin curve ½ ccle for 0 7 maimum = and minimum = - straight line 4 No. of solutions = 47/
47/ (a) k d d k : ), ( k d ) ( c : ) (, gantian 6 4(a) (i) (ii) 7 l a or 4 n 60 7 n S n 4 d 7 d T 4.9 or 9 ) 4 9( T 7 (a) (i) PA RP RA b a 6 (ii) RC BR BC b) (6a 4 a b a (iii) AQ CA CQ b b) (6a 4 b 9 a 9
. (i) a b BC 9 9 CQ a b a b 9[ a b] 9 CQ BC 8 6. (a) f N 60.7 Q (60) 4. 6.6 Q (60) 7 4 0. 7.67 Julat antara quartile = 7.67.6 =.4 7 47/
7 (a) log 0.8 0.0 0.40 0.48 0.4 0.70 log.48..04 0.86 0.70 0.40 Both aes correct (at least plotting point) Plotting all 6 points correct line of the best fit - correct. (c) (i) log (q )log log p log p Y int ercept.86 p. 86 7.44 (ii) q gradient 0.40..07 0.70 0.0 q.07 When =.7, log = 0.4 log = 0.96 = 9. 8 (a) d d 6 0 Ganti (, k) : k = 6() - = 9 Area of Q 6 0 d ( ) (0) 0 or 0 47/
47/ (c) Volume of P.4 48.6 4 4 4 8 ) (6 8d 0 4 0 0 0 9 (a) (c) (d) Area of ABC 4 6) 8 (0 8) 80 (4 8 4 Mid point BC 7, 8, 4 Gradient BC 4 8 m 7) ( ) (, D ) (4, D 4 0 6 8 8) ( 4) ( PC
4 (a) 4 cos (4)() 0.864 0.97.4 0.7 80 rad Perimter 9 4(0.7) 6.78 cm (c) o Area = (4) (0.7) ()(4) sin 0.97 ).646 7.604 4.0 cm (a) p or q or n 9 P(X 7) 9C 7 (0.6) 0.8 7 (0.4) 9C 8 (0.6) 8 (0.4) 9C 9 (0.6) 9 (0.4) 0 (i) n 00,, P(X 4) 4 P Z P(Z ) 0.84 Bilangan Murid = 0.84 X 00 = 40 / 4 (ii) P z.87 Pz 0.87.0 6 47/
(a) (AB)(8)sin 80 o 9.7 9.7 AB o 8 sin 80 BD 8 ()(6) cos 80 o 7.8 8.667 cm (c).6 cos BCD 0.0669 7 8.667 (.6)(7) BCD 86.6 o or o ' 86 (d) sin CBD sin 86.6 7 8.667 sin CBD 0.808 o ' CBD.69 o or o ' 4. (a) Q Use I 0 Q 480, o 0 I06/ 04 (60) 00(0) 0() 0() 60 0 4 (c) I08/ 06 70 I I 08/ 04 0 I Use 08/ 06 06/ 04 I08/ 04 70 0 0 47/
6 4 (a) 00 ` 600 Draw correctl all three straight line which involves and. Region shaded correctl (c) (i) 0, 0 (ii) Use 00 00 for point in the shaded region (400, 00) : 00(400) 00(00) RM40,000. (a) v 6t 4 dt t c 6, 4t c v t 4t 6 V= 0 t 4t 6 0 t (t )(t 6) 0 t 8t 0, t 6 (c) V min, a = 0, t = 4 V (4 ) 4(4) 6 min ms (d) S S t 4t 6dt t t 6t 7 (9) 6() (8 (4) 6() 47/
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