PROGRAM PENINGKATAN AKADEMIK. Aktiviti-aktiviti yang dijalankan sepanjang tahun: 1. Kem Kecemerlangan Akademik Diamond In The Rough(15-17-oktober-2015). B) Meneruskan pelaksanaan kursus peningkatan prestasi matematik peringkatan SRP dan SPM yang telah dimulakan pada tahun 1990. C)Meneruskan penerbit buku kerja matematik sekolah rendah (Tahun 1-6)yang telah diterbitkan mulai tahun 1990.
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1 September, 0 / Additional Mathematics Kertas September 0 jam 0 minit PROGRAM PENINGKATAN PRESTASI AKADEMIK SPM 0 MATA PELAJARAN ADDITIONAL MATHEMATICS 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.. Answer all questions in Section A, four questions from Section B and two questions from Section C.. Give only one answer/solution to each question.. Show your working. It may help you to get your marks.. The diagrams provided are not drawn according to scale unless stated.. The marks allocated for each question and sub - part of a question are shown in brackets.. You may use a non-programmable scientific calculator.. A list of formulae is provided in page and. This question paper consists of printed pages. / Additional Mathematics Paper [Lihat halaman sebelah
2 September, 0 The following formulae may be helpful in answering the questions. The symbols given are the ones commonly used. ALGEBRA b b ac log. x. c b log a b a log c a m n m n. a a a. T n a ( n ) d m n m n. a a a n 0. S n [ a ( n ) d ] m n mn. ( a ) a n. T ar. log a mn log a m log a n n n a ( r ) a ( r ). S n, r r r m. log a log a m log a n n. a S, r < r n. log m nlog m a a n CALCULUS. y = uv, dy dx dv u dx du v dx Area under a curve = b a y dx or du dv v u u dy. y =, dx dx v dx v. dy dx dy du du dx = b x dy a. Volume of revolution = b y dx or a = b x dy a GEOMETRY. Distance = ( x x ) ( y y ). Mid point x ( x, y ) = x y, y. Division of line segment by a point nx ( x, y ) = mx ny my, m n m n. Area of triangle = ( ) ( ) x y x y x y x y x y x y. r. rˆ x y xi yj x y / Additional Mathematics Paper [Lihat halaman sebelah
3 September, 0 STATISTICS.... x x N fx x f ( x x ) = N f ( x x ) = f x N fx f x x I n P r n C r W i I i Wi n! ( n r )! n! ( n r )! r! 0 P(AB) = P(A) + P(B) P(AB) P ( X = r ) = n C r p r q n r, p + q = N F Mean, = np. m = L + C fm npq Q. X I 00 Z = Q 0 TRIGONOMETRY. Arc length, s = r. sin ( A B ) = sin A cos B cos A sin B. Area of sector, A =. cos ( A B ) = cos A cos B sin A sin B r. sin ² A + cos² A = tan A tan B 0 tan ( A B ) = tan A tan B. sec ² A = + tan ² A tan A tan A = tan A. cosec ² A = + cot ² A a b c sin A sin B sin C. sin A = sin A cos A a² = b² + c² bc cos A. cos A = cos ² A sin ² A = cos ² A = sin ² A Area of triangle = sin ab C / Additional Mathematics Paper [Lihat halaman sebelah
4 September, 0 Section A Bahagian A [ 0 marks ] [ 0 markah ] Answer all questions. Jawab semua soalan.. Solve the simultaneous equations xy and x y. Selesaikan persamaan serentak xy dan x y. [ marks] [ markah]. The quadratic equation Find (a) the value of h and of k, (b) the range of x if Persamaan kuadratik > k. Cari (a) nilai h dan nilai k, (b) julat nilai x jika x x 0 has roots h and k, where h > k. x x always positive. x x 0 [ marks] mempunyai punca-punca h dan k, dengan keadaan h x x sentiasa positif. [ markah]. (a) Sketch the graph of y sin x for 0 x. [ marks] (b) Hence, using the same axes, sketch a suitable straight line to find the number of solutions to x the equation sin x 0 for 0 x. State the number of solutions. [ marks] (a) Lakar graf bagi y sin x untuk 0 x. [ markah] (b) Seterusnya, dengan menggunakan paksi yang sama, lakar satu garis lurus yang sesuai untuk x mencari bilangan penyelesaian bagi persamaan sin x 0 untuk 0 x. Nyatakan bilangan penyelesaian itu. [ markah] / Additional Mathematics Paper [Lihat halaman sebelah
5 September, 0. B Diagram / Rajah Diagram shows a triangle ABC. Point P lies on the straight line AB such that AP : PB = :. Point R lies on the straight line BC such that BR BC that PC PQ. It is given that AB x and AC y. (a) Express in terms of x and y (i) BC, (ii) PC, A P Q R C. Point Q lies on the straight line PC such (iii) AQ. [ marks] (b) Hence, shows that the points A, Q and R are collinear. [ marks] Rajah menunjukkan suatu segi tiga ABC. Titik P terletak pada garis AB dengan keadaan : PB = :. Titik R terletak pada garis BC dengan keadaan BR BC PC dengan keadaan PC PQ. Diberi bahawa AB x dan AC y (a) Ungkapan dalam sebutan x dan y, (i) BC, (ii) PC, AP. Titik Q terletak pada garis (iii) AQ. [ markah] (b) Seterusnya, tunjukkan titik A, Q dan C adalah segaris. [ markah] / Additional Mathematics Paper [Lihat halaman sebelah
6 September, 0. (a) Table shows the marks obtained by a group of students in a test. Marks Markah Number of students Bilangan murid Table / Rajah Without drawing an ogive, calculate the median mark. [ marks] (b). A set of marks x, x, x, x, x, x has a mean of and standard deviation of. Find the new mean and new standard deviation if each mark is multiplied by and then is added to it. [ marks] (a) Jadual menunjukkan markah yang diperoleh sekumpulan murid dalam satu ujian. Tanpa melukis ogif, hitungkan markah median. [ markah] (b) Min dan sisihan piawai bagi satu senarai markah x, x, x, x, x, x ialah dan masingmasing. Cari nilai min dan sisihan piawai baru jika setiap markah tersebut didarab dengan dan ditambah kepadanya. [ markah] / Additional Mathematics Paper [Lihat halaman sebelah
7 September, 0. B E G I A D F H O Diagram / Rajah Diagran shows part of the arrangement of an infinite series of quadrants. The radius of the quadrant AOB is p cm. D is the midpoint of AO, F is the midpoint of DO, H is the midpoint of FO and so on. (a) Show that the areas of the quadrant AOB, DOE, FOG,... form a geometric progression and hence, state the common ratio of the progression. [ marks] (b) Given AO = 0 cm, (i) determine which quadrant has an area of cm, (ii) find the sum to infinity of the areas, in terms of cm, of the quadrants. [ marks] Rajah menunjukkan sebahagian daripada susunan tak terhingga bagi siri sukuan. Jejari bagi sukuan AOB ialah p cm. D ialah titik tengah bagi AO, F ialah titik tengah bagi DO, H ialah titik tengah bagi FO dan seterusnya. (a) Tunjukkan luas bagi sukuan AOB, DOE, FOG,...bentuk satu janjang geometri dan seterusnya, nyatakan nisbah sepunya bagi janjang ini. [ markah] (b) Diberi AO = 0 cm, (i) tentukan sukuan yang keberapa mempunyai luas cm, (ii) cari hasil tambah hingga tak terhinggaan, dalam sebutan cm, bagi sukuan-sukuan. [ markah] / Additional Mathematics Paper [Lihat halaman sebelah
8 September, 0 Section B Bahagian B [ 0 marks ] [ 0 markah ] Answer four questions from this section. Jawab empat soalan daripada bahagian ini.. Use graph paper to answer this question. Gunakan kertas graf untuk menjawab soalan ini. x y 0 00 Table / Rajah Table shows the values of two variables, x and y, obtained from an experiment. Variables x and y are related by the equation y 0 kx, where h and k are constants. h (a) Plot log0 y against x, using a scale of cm to unit on the x -axis and cm to 0. unit on the log0 y -axis. Hence, draw the line of best fit. [ marks] (b) Use your graph in (a) to find the value of (i) h, (ii) k, (iii) y when x. [ marks] Jadual menunjukkan nilai-nilai bagi dua pembolehubah, x dan y, yang diperoleh daripada satu eksperimen. Pembolehubah x dan y dihubungkan oleh persamaan keadaan h dan k ialah pemalar. (a) Plot log0 y 0 kx, dengan y melawan x, dengan menggunakan skala cm kepada unit pada paksi- x dan cm kepada 0. unit pada paksi - log0 y. Seterusnya, lukis garis lurus penyuaian h terbaik. (b) Gunakan graf di (a) untuk mencari nilai [ markah] (i) h, (ii) k, (iii) y apabila x. [ markah] / Additional Mathematics Paper [Lihat halaman sebelah
9 September, 0. B A A A O A 0 o D C Diagram / Rajah Diagram shows a sector OBC with centre O and a sector ADC with centre D. AD is perpendicular to OB and the length of OB is cm. It is given DC = cm. [ Use π =. ] Calculate (a) the value of, in radians, [ marks] (b) the perimeter, in cm, of the shaded region, [ marks] (c) the area, in cm of shaded region. [ marks] Rajah menunjukkan sebuah sektor OBC dengan pusat O dan sektor ADC dengan pusat D. AD berserenjang dengan OB dan panjang OB ialah cm. Diberi DC = cm. [ Guna π =. ] Hitung (a) nilai, dalam radian, (b) perimeter, dalam cm, kawasan berlorek, (c) luas, dalam cm, kawasan berlorek. [ markah] [ markah] [ markah] / Additional Mathematics Paper [Lihat halaman sebelah
10 0 September, 0. Solution by scale drawing is not accepted. Penyelesaian secara lukisan berskala tidak diterima. y R(,) Q 0 x P(-,-) Diagram / Rajah Diagram above shows a straight line PR. Point Q divides the straight line PR internally in the ratio PQ : QR = : (a) Find the coordinates of Q, [ marks] (b) Calculate the area of triangle POR, [ marks] (c) Find the equation of the straight line that passes through Q and perpendicular to PR, [ marks] (d) Point T moves such that its distance from P is always twice its distance from R. Find the equation of the locus of T. [ marks] Rajah menunjukkan suatu garis lurus PR. Titik Q membahagi dalam garis lurus PR dengan nisbah PQ : QR = : (a) Cari koordinat titik Q, (b) Hitung luas segi tiga POR, (c) Cari persamaan garis lurus yang melalui Q dan berserenjang dengan PR, [ markah] [ markah] [ markah] (d) Titik T bergerak dengan keadaan jaraknya dari P adalah sentiasa dua kali dari R. Cari persamaan lokus bagi T. [ markah] / Additional Mathematics Paper [Lihat halaman sebelah
11 September, 0 0. y y x P y = x + O x Diagram 0 / Rajah 0 Diagram 0 above shows a shaded region bounded by the curve y x and the straight line y = x +. Find (a) the coordinates of P, (b) the area of the shaded region, [ marks] [ marks] (c) the volume of revolution, in terms of π, when the region bounded by the curve, and the straight line is rotated through 0 o about the y-axis. [ marks] Rajah 0 menunjukkan kawasan berlorek yang dibatasi oleh lengkung y x dan garis lurus y = x +. Cari (a) koordinat P, [ markah] (b) luas rantau yang berlorek, [ markah] (c) isipadu kisaran, dalam sebutan π, apabila rantau yang dibatasi oleh lengkung dan garis lurus, dikisarkan melalui 0 o pada paksi-y. [ markah] / Additional Mathematics Paper [Lihat halaman sebelah
12 September, 0 (a) In a survey carried out in a school, it is found that 0% of the students have their breakfast in the school canteen. If 0 students from the school are selected at random, find the probability that (i) exactly three students have their breakfast in the school canteen, (ii) at least two students have their breakfast in the school canteen. [ marks] (b) The masses of a group of boys have a normal distribution with a mean of kg and a standard deviation of kg. If a boy is selected randomly from this group, find (i) the probability that his mass less than 0 kg. (ii) the value o f m if 0 % of the boys have mass more than m kg. [ marks] (a) Dalam satu kajian yang dijalankan di sebuah sekolah,didapati 0% daripada murid-murid mengambil sarapan pagi di kantin sekolah. Jika 0 murid daripada sekolah itu dipilih secara rawak, hitung kebarangkalian bahawa (i) tepat tiga orang murid mengambil sarapan pagi di kantin sekolah (ii) sekurang-kurangnya dua orang murid mengambil sarapan pagi di kantin sekolah. [ markah] (b) Jisim sekumpulan budak lelaki adalah mengikut taburan normal dengan min kg dan sisihan piawai kg. Jika seorang budak lelaki dipilih secara rawak daripada kumpulan ini, cari (i) kebarangkalian jisimnya kurang daripada 0 kg. (ii) nilai m jika 0% dari budak lelaki itu mempunyai jisim melebihi m kg. [ markah] / Additional Mathematics Paper [Lihat halaman sebelah
13 September, 0 Section C Bahagian C [ 0 marks ] [ 0 markah ] Answer two questions from this section. Jawab dua soalan daripada bahagian ini.. A particle moves in a straight line and passes through a fixed point O with a velocity of ms -. Its acceleration, a m s, t s after passing through O is given by a = t. The particle stops when t = p s. [Assume motion to the right is positive] Find (a) the initial acceleration, in ms -, of the particle, (b) the maximum velocity of the particle, (c) the value of p, (d) the total distance travelled by the particle in the first seconds. [ marks] [ marks] [ marks] [ marks] Satu zarah bergerak di sepanjang suatu garis lurus dan melalui satu titik tetap O dengan halaju m s. Pecutannya a m s, t s selepas melalui O diberi oleh a = t. Zarah itu berhenti pada masa t = p s. [Anggapkan gerakan ke arah kanan sebagai positif] Cari (a) pecutan awal, dalam ms -, zarah itu, (b) halaju maksimum bagi zarah itu, (c) nilai untuk p, (d) jumlah jarak yang dilalui oleh zarah dalam saat pertama. [ markah] [ markah] [ markah] [ markah] / Additional Mathematics Paper [Lihat halaman sebelah
14 September, 0. Diagram shows a quadrilateral PQRS. The area of triangle QRS is cm and QRS is an acute angle. Q cm R cm cm P 0 S Calculate Diagram Rajah (a) QRS, (b) the length, in cm, of QS, (c) PQS. (d) the area, in cm, of quadrilateral PQRS. [ marks] [ marks] [ marks] [ marks] Rajah menunjukkan sebuah segiempat PQRS. Luas segitiga QRS ialah cm dan QRS ialah sudut tirus. Hitungkan (a) QRS, [ markah] (b) panjang, dalam cm, QS, (c) PQS. (d) luas, dalam cm, bagi sisi empat PQRS. [ markah] [ markah] [ markah] / Additional Mathematics Paper [Lihat halaman sebelah
15 September, 0. Use graph paper to answer this question. A travel company offers a package of special missions to Kuala Lumpur. The delegation consist of x children and y adults. The management has set the following constraints for the delegation. I) A delegation is limited to 0 members only. II) The price for an adult is RM0 and RM0 for a child under the age of years. The collection payment of all travelers at least RM 00. III) The ratio of the number of children to the number of adults is :. (a) Write three inequalities, other than x 0 and y 0 which satisfies all the above constrains. [ marks] (b) Using a scale of cm to 0 people on both axes construct and shade the region R that satisfies all the above constrains. [ marks] (c) Using the graph constructed in (b), find (i) the minimum number of child who can follow the trip. (ii) the minimum profit obtained if the profit for one adult and one child is RM 0 and RM 0 respectively. [ marks] / Additional Mathematics Paper [Lihat halaman sebelah
16 September, 0 Sebuah syarikat pelancongan menawarkan satu pakej rombongan khas ke Kuala Lumpur. Rombongan ini terdiri daripada x orang kanak-kanak dan y orang dewasa. Pihak pengurusan telah menetapkan kekangan seperti berikut bagi rombongan itu. I: Rombongan itu dihadkan kepada 0 ahli sahaja. II: Harga bagi seorang dewasa ialah RM0 dan RM0 untuk seorang kanak- kanak berumur tahun ke bawah. Kutipan bayaran kesemua pelancong sekurang kurangnya RM 00. III: Nisbah bilangan kanak-kanak kepada bilangan orang dewasa ialah : (a) Tulis tiga ketaksamaan, selain x 0 dan y 0 yang memenuhi semua kekangan di atas. [ markah] (b) Dengan menggunakan skala cm kepada 0 orang pada kedua-dua paksi, bina dan lorek rantau R yang memenuhi semua kekangan di atas. [ markah] (c) Berdasarkan graf anda, cari (i) bilangan minimum kanak-kanak yang boleh mengikuti rombongan itu. (ii) keuntungan maksimum yang diterima jika keuntungan bagi seorang dewasa ialah RM 0 dan seorang kanak-kanak ialah RM 0 [ markah] / Additional Mathematics Paper [Lihat halaman sebelah
17 September, 0. Table shows the prices and the price indices of four ingredients T, U, V, and W, used to make bread. Diagram shows the relative quantity of the ingredients used. Ingredient Bahan Price (RM) Harga (RM) 00 0 Price index in the year 0 based on year 00 Indeks harga pada tahun 0 berasaskan tahun 00 T 0 0 U p V q 0 0 W 0 Table / Jadual T W o 0 o V o U Diagram / Rajah (a) Find the values of p and q. [ marks] (b) Calculate the composite index of the cost of making bread in the year 0 based on year 00. [ marks] (c) The cost of making bread increased by % from year 0 to year 0. Calculate (i) the composite index in the year 0 using the year 00 as the base year. (ii) the cost of making the bread in the year 0 if the cost in year 00 is RM0. [ marks] / Additional Mathematics Paper [Lihat halaman sebelah
18 September, 0 Jadual menunjukkan harga dan indeks harga bagi empat bahan T, U, V, dan W yang digunakan untuk membuat roti. Rajah menunjukkan kuantiti relatif bagi penggunaan bahanbahan itu. (a) Cari nilai bagi p dan q. [ markah] (b) Hitung nombor indeks gubahan bagi kos pembuatan roti itu pada tahun 0 berasaskan tahun 00. [ markah] (c) Harga untuk membuat roti telah meningkat sebanyak % dari tahun 0 hingga 0. Hitungkan (i) nombor indeks gubahan pada tahun 0 dengan menggunakan tahun 00 sebagai tahun asas. (ii) kos untuk membuat roti pada tahun 0 jika kosnya pada tahun 00 ialah RM0. [ markah] END OF QUESTION PAPER KERTAS SOALAN TAMAT THE UPPER TAIL PROBABILITY Q(z) FOR THE NORMAL DISTRIBUTION N(0,) / Additional Mathematics Paper [Lihat halaman sebelah
19 September, 0 KEBARANGKALIAN HUJUNG ATAS Q(z) BAGI TABURAN NORMAL N(0, ) z 0 Minus / Tolak f ( z) exp z Q ( z) k f ( z) dz f (z) O Q(z) Example / Contoh: If X ~ N(0, ), then P(X > k) = Q(k) Jika X ~ N(0, ), maka P(X > k) = Q(k) z / Additional Mathematics Paper [Lihat halaman sebelah
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