/ August 0 MAJLIS PENGETUA SEKOLAH MALAYSIA NEGERI KEDAH DARUL AMAN MODUL PENINGKATAN PRESTASI TINGKATAN LIMA 0 MATEMATIK TAMBAHAN KERTAS MODUL jam 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. Kertas soalan ini mengandungi halaman bercetak dan halaman kosong. / Additional Mathematics Paper [Lihat halaman sebelah
/ August 0 The following formulae may be helpful in answering the questions. The symbols given are the ones commonly used. ALGEBRA b b ac log c b. x. log a b a log c a m n m n. a a a 9. T n a ( n ) d m n m n. a a a. ( a ) m n mn a. T n n 0. S n [ a ( n ) d ] n 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. y = uv, dy dx dv u dx du v dx CALCULUS Area under a curve = b a = b a y dx or x dy du dv v u u dy. y =, dx dx v dx v. Volume of revolution = b y dx or a = b x dy a. dy dx dy du du dx GEOMETRY. Distance = ( x x ) ( y y ). Mid point x ( x, y ) = x y, y. Area of triangle = ( ) ( ) x y x y x y x y x y x y. r x y. Division of line segment by a point nx ( x, y ) = mx ny my, m n m n. rˆ xi yj x y / Additional Mathematics Paper
/ August 0. STATISTICS x x I N W i W I i i. fx x f n P r n! ( n r )!. ( x x ) = N x 9 N x n C r n! ( n r )! r!. f ( x x ) = f fx f x 0 P(AB) = P(A) + P(B) P(AB) n P ( X = r ) = C r p r q n r, p + q = N F Mean, = np. m = L + C fm npq Q. I 00 Z = Q 0 X TRIGONOMETRY. Arc length, s = r. sin ( A B ) = sin A cos B cos A sin B. Area of sector, A = 9. 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 sin A b sin B c 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
/ August 0 Section A Bahagian A [ 0 marks ] [ 0 markah ] Answer all questions. Jawab semua soalan. Solve the simultaneous equations xy and x (y) 0. [ marks] Selesaikan persamaan serentak xy dan x (y) 0. [ markah] cm cm cm Diagram / Rajah Diagram shows part of a structure made up of rectangular blocks. The first column has one block. For each of the other columns, the number of blocks is doubled the previous column. (a) Find the number of blocks in the th columns, (b) Calculate (i) the total volume of the blocks if there are 0 columns of blocks. [ marks] (ii) the total cost of the 0 columns of blocks if each block cost RM 0 0. [ marks] Rajah menunjukkan susunan suatu struktur yang terdiri daripada blok yang berbentuk segi empat tepat. Lajur pertama mempunyai satu blok. Bagi setiap lajur berikutnya, bilangan blok adalah dua kali ganda daripada lajur sebelumnya. (a) Carikan bilangan blok bagi lajur ke. (b) Hitungkan (i) jumlah isipadu blok jika terdapat 0 lajur bagi struktur itu. [ markah] (ii) jumlah kos bagi 0 lajur blok jika setiap blok RM 0 0. [ markah] / Additional Mathematics Paper
/ August 0 (a) Sketch the graph of y cosx for 0 x. [ marks] (b) Hence, using the same axes, sketch a suitable straight line to find the number of solutions for the equation x cosx0 for 0 x. State the number of solutions. [ marks] (a) Lakar graf bagi y cosx untuk 0 x. [ markah] (b) Seterusnya, dengan menggunakan paksi yang sama, lakar satu garis lurus yang sesuai untuk mencari bilangan penyelesaian bagi persamaan x cosx 0 untuk 0 x. Nyatakan bilangan penyelesaian itu. [ markah] (a) Given that a set X has score x, x, x... x 0. The mean and standard deviation of set X are 0 and respectively. Find x and x for set X. [ marks] x 0 (b) Another set Y has score x,, x x.... Find the mean and variance for set Y. [ marks] (a) Diberi bahawa set X mempunyai skor x, x, x... x 0. Min dan sisihan piawai ialah 0 dan masing-masing. Carikan x and x bagi set X. [ markah] x 0 (b) Satu lagi set Y mempunyai skor x,, x x.... Carikan min dan varians bagi set Y. [ markah] / Additional Mathematics Paper [Lihat halaman sebelah
/ August 0 (a) Given that log Klog V, express V in terms of K. [ marks] (b) Given that the function f : x k mx, and (i) the value of k and of m, f : x x, calculate (ii) the value of p if f ( p). [ marks] (a) Diberi bahawa log Klog V, ungkapkan V dalam sebutan K. [ markah] (b) Diberi bahawa fungsi f : x k mx, dan (i) nilai-nilai bagi k dan m, f : x x, hitungkan (ii) nilai bagi p jika f ( p). [ markah] (a) Given that (b) 9x f( x) x A curve has a gradient function of, find f '( x ). [ marks] (,) is parallel to the straight line yx 9, find (i) the value of k, (ii) the equation of the normal to the curve at point (,). kx x, the tangent to the curve at the point [ marks] (a) Diberi bahawa 9x f( x) x (b) Fungsi kecerunan suatu lengkung ialah adalah selari kepada garis lurus yx 9, cari, cari f '( x ). [ markah] kx x, tangen pada lengkung di titik (,) (i) nilai bagi k, (ii) persamaan normal pada lengkung di titik (,). [ markah] / Additional Mathematics Paper
/ August 0 Use graph paper to answer this question. Section B Bahagian B [ 0 marks ] [ 0 markah ] Answer four questions from this section. Jawab empat soalan daripada bahagian ini. Gunakan kertas graf untuk menjawab soalan ini. x y 0 0 0 Table / Jadual Table shows the values of two variables, x and y, obtained from an experiment. Variables x p and y are related by the equation y kx x, where k and p are constants. k y (a) Plot against x, using a scale of cm to unit on the x axis and cm to unit x y on the - axis. Hence, draw the line of best fit. [ marks] x (b) Use the graph in (a) to find the value of (i) k (ii) p (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 dengan keadaan k dan p ialah pemalar. y p k kx x, y (a) Plot melawan x, dengan menggunakan skala cm kepada unit pada paksi x dan x y cm kepada unit pada paksi-. Seterusnya, lukis garis lurus penyuaian terbaik. x [ markah] (b) Gunakan graf di (a) untuk mencari nilai (i) k, (ii) p, (iii) y apabila x =. [ markah] / Additional Mathematics Paper [Lihat halaman sebelah
/ August 0 P T Q R M S Diagram / Rajah Diagram shows a quadrilateral PQRT and a triangle RST. M is a midpoint of TR. PQ =9x, PT =y, PQ =TS and PT = QR. (a) Express the following vectors in terms of x and y. (i) TR (ii) PS (iii) MS Hence, Show that P, M and S are collinear. [ marks] (b) It is given that x and y, find PS. [ marks] Rajah menunjukkan sebuah sisiempat PQRT dan segitiga RST. M ialah titik tengah TR. PQ = 9x, PT = y, PQ =TS dan PT = QR. (a) Ungkapkan vektor yang berikut dalam sebutan x dan y (i) TR (ii) SR (iii) MS Seterusnya, tunjukkan bahawa P, M dan S adalah segaris. [ markah] (b) Diberi bahawa x dan y, cari PS. [ markah] / Additional Mathematics Paper
9 / August 0 9 y y = x + K A y = x B O x Diagram 9/Rajah 9 Diagram 9 shows the straight line y = x + intersecting the curve y = x at the points K. Find (a) the coordinates of K, [ marks] (b) the area of the shaded region B, [ marks] (c) the volume generated, in terms of π, when the shaded region A is revolved through 0 o about the y-axis. [ marks] Rajah 9 menunjukkan garis lurus y = x + yang menyilang lengkung y = x pada titik K. Cari (a) koordinat K, [ markah] (b) luas rantau berlorek B, [ markah] (c) isipadu janaan, dalam sebutan π, apabila rantau berlorek A dikisarkan melalui 0 o pada paksi-y. [ markah] / Additional Mathematics Paper [Lihat halaman sebelah
0 / August 0 0 B A O C Diagram 0 / Rajah 0 Diagram 0 shows a semicircle ABC with centre O and a sector ABO with centre A. The radius of semicircle ABC and sector ABO is cm. [ Use π = ] Calculate (a) the value of, in radian, (b) the perimeter, in cm, of shaded region, (c) the area, in cm, of the shaded region. [ marks] [ marks] [ marks] Rajah 0 menunjukkan sebuah semi bulatan ABC dengan pusat O dan sector ABO dengan pusat A. Jejari bagi semi bulatan ABC dan sektor bulatan ABO ialah 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
/ August 0 (a) In an examination, % of the candidates passed Mathematics. If a sample of candidates is chosen at random, find the probability that (i) (ii) all the candidates passed Mathematics, at least candidates failed Mathematics. [ marks] (b) The body mass of 00 students in a school follows a normal distribution with a mean of kg and a standard deviation of 0 kg. (i) If a student is chosen at random, find the probability that his body mass is between 0 kg and 0 kg. (ii) Calculate the number of students whose body mass are between 0 kg and 0 kg. [ marks] (a) Dalam suatu peperiksaan, % calon lulus Matematik. Jika calon dipilih secara rawak, cari kebarangkalian bahawa (i) (ii) kesemua calon itu lulus Matematik, sekurang-kurangnya calon gagal Matematik. [ markah] (b) Jisim badan 00 pelajar sebuah sekolah adalah mengikut taburan normal dengan min kg dan sisihan piawai 0 kg. (i) Jika seorang pelajar dipilih secara rawak, carikan kebarangkalian bahawa jisim badannya berada di antara 0 kg dan 0 kg. (ii) Hitung bilangan pelajar yang mempunyai jisim badan di antara 0 kg dan 0 kg. [ markah] / Additional Mathematics Paper [Lihat halaman sebelah
/ August 0 Section C Bahagian C [ 0 marks ] [ 0 markah ] Answer any two questions from this section. Jawab mana-mana dua soalan daripada bahagian ini. A particle moves along a straight line from a fixed point O. Its velocity, v ms -, is given by v = kt t, where k is a constant and t is the time, in seconds, after leaving the point O. The velocity of the particle is maximum when t = s. [Assume motion to the right is positive.] Find (a) the value of k, [ marks] (b) the value of t when the particle passes O again, [ marks] (c) the time, in seconds, when the particle stops instantaneously, [ marks] (d) the distance travelled, in m, by the particle in the first seconds. [ marks] Suatu zarah bergerak di sepanjang suatu garis lurus dari satu titik tetap O. Halajunya, v ms -, diberi oleh v = kt t, dengan keadaan k ialah pemalar dan t ialah masa, dalam saat, selepas meninggalkan titik O. Halaju zarah itu adalah maksimum pada t = s [Anggapkan gerakan ke arah kanan sebagai positif.] Cari (a) nilai k, [ markah] (b) nilai bagi t apabila zarah itu melalui titik O semula, [ markah] (c) masa, dalam saat, apabila zarah berhenti seketika, [ markah] (d) jarak yang dilalui, dalam m, oleh zarah itu dalam tujuh saat pertama. [ markah] / Additional Mathematics Paper
/ August 0 Diagram shows a quadrilateral KLMN. N cm cm K M 0 cm L Diagram /Rajah Calculate (a) the length, in cm, of KM,. [ marks] (b) (c) KMN, LKM, [ marks] [ marks] (d) the area, in cm, of quadrilateral KLMN. [ marks] Rajah menunjukkan sisiempat KLMN. Hitungkan (a) panjang, dalam cm, KM, [ markah] (b) KMN, [ markah] (c) LKM, [ markah] (d) luas, dalam cm, bagi sisiempat KLMN. [ markah] / Additional Mathematics Paper [Lihat halaman sebelah
/ August 0 Table shows the prices and the price indices of five components P, Q, R, S and T needed to produce a certain type of digital camera. Pie chart shows the relative quantity of components needed in producing the camera. Component Komponen Price (RM) per unit Harga (RM) per unit 00 0 Price index for the year 0 based on the year 00 Indeks harga pada tahun 0 berasaskan tahun 00 P 0 0 Q 0 x 0 R 0 00 0 S 0.0 T 00 0 y Table / Jadual T 0º P 0º º Q R 0º S Pie chart / Carta pai / Additional Mathematics Paper
/ August 0 (a) Find the value of x and y. [ marks] (b) Calculate the composite index for the production cost of the camera in the year 0 based on the year 00. [ marks] (c) The price of each component increases by 0% from the year 0 to the year 0. Given that the production cost of the camera in year 00 is RM00, calculate the corresponding cost in year 0. [ marks] Jadual menunjukkan harga dan indeks harga bagi lima komponen P, Q, R, S dan T yang diperlukan untuk menghasilkan sejenis kamera digital. Carta pai menunjukkan kuantiti relatif bagi komponen yang diperlukan dalam penghasilan kamera digital itu. (a) Cari nilai x dan y. [ markah] (b) Hitungkan indeks gubahan bagi kos penghasilan kamera digital itu pada tahun 0 berasaskan tahun 00. [ markah] (c) Harga setiap komponen meningkat 0% dari tahun 0 ke tahun 0. Diberi kos penghasilan kamera digital itu dalam tahun 00 ialah RM00, hitungkan kosnya yang sepadan pada tahun 0. [ markah] / Additional Mathematics Paper [Lihat halaman sebelah
/ August 0 Use a graph paper to answer this question. A factory produces two types of pillow, type A and type B. In a day, it can produce x pillows of type A and y pillows of type B. The time taken to produce a pillow of type A is 0 minutes and a pillow of type B is 0 minutes. The production of the pillow per day is based on the following constraints. I. : The time taken to make pillows of type A is not more than the time taken to make pillows of type B. II. : The total number of pillows produced is not more than 00. III. : The number of pillows of type B must exceed the number of pillows of type A by at most 00. (a) Write three inequalities, other than which satisfy all the above constraints. [ marks] (b) (c) By using the scale of cm to 00 pillows on both axes, construct and shade the region R which satisfies all the above constraints. [ marks] Use graph from (b), to find (i) the maximum number of pillows of type A if 0 of pillows of type B produced. (ii) maximum profit that can be obtained, if the profit of selling pillow A is RM0 and pillow B is RM 00. [ marks] / Additional Mathematics Paper
/ August 0 Gunakan kertas graf untuk menjawab soalan ini. Sebuah kilang menghasilkan jenis bantal, jenis A dan jenis B. Dalam satu hari, kilang itu boleh menghasilkan x biji bantal jenis A dan y biji bantal jenis B. Masa yang diambil untuk menghasilkan sebiji bantal jenis A ialah 0 minit dan sebiji bantal jenis B ialah 0 minit. Pengeluaran bantal dalam satu hari adalah berdasarkan kekangan yang berikut: I: Masa yang diambil untuk membuat bantal A tidak melebihi masa yang diambil untuk membuat bantal jenis B. II: III: Jumlah bantal yang dihasilkan tidak melebihi 00 bji. Bilangan bantal jenis B mesti melebihi bilangan bantal jenis A selebih-lebihnya 00 biji. (a) Tulis tiga ketaksamaan, selain, yang memenuhi semua kekangan di atas. [ markah] (b) (c) Menggunakan skala cm kepada 00 biji bantal pada kedua-dua paksi, bina dan lorek rantau R yang memenuhi semua kekangan di atas. [ markah] Gunakan graf anda daripada (b) untuk mencari (i) bilangan maksimum bantal jenis A jika 0 biji bantal jenis B dihasilkan. (ii) keuntungan maksimum yang boleh diperolehi, jika keuntungan jualan bagi bantal A ialah RM0 dan bantal B ialah RM 00. [markah] END OF QUESTION PAPER KERTAS SOALAN TAMAT / Additional Mathematics Paper [Lihat halaman sebelah
/ August 0 THE UPPER TAIL PROBABILITY Q(z) FOR THE NORMAL DISTRIBUTION N(0,) KEBARANGKALIAN HUJUNG ATAS Q(z) BAGI TABURAN NORMAL N(0, ) z 0 9 9 Minus / Tolak 0.0 0. 0. 0. 0. 0.000 0.0 0.0 0. 0. 0.90 0. 0. 0. 0.09 0.90 0. 0.9 0. 0. 0.0 0. 0.090 0.0 0. 0.0 0. 0.0 0.9 0.00 0.0 0.0 0.0 0. 0. 0. 0. 0.9 0.9 0. 0. 0. 0.9 0. 0.9 0. 0. 0.9 0.0 0. 0. 0. 0.9 0. 0. 0 0 9 9 0 9 0. 0. 0. 0. 0.9 0.0 0. 0.0 0.9 0. 0.00 0.09 0.9 0.090 0. 0.0 0. 0. 0.0 0. 0.9 0. 0. 0.0 0. 0.9 0. 0.9 0.00 0. 0.9 0. 0. 0.9 0. 0. 0. 0. 0.99 0. 0. 0. 0.0 0.9 0.0 0.0 0. 0. 0.9 0. 0. 0. 0. 0. 0. 0 0 9 0 0 9 9 0 9.0.... 0. 0. 0. 0.09 0.00 0. 0. 0. 0.09 0.09 0.9 0. 0. 0.09 0.0 0. 0.9 0.09 0.09 0.0 0.9 0. 0.0 0.090 0.09 0.9 0. 0.0 0.0 0.0 0. 0.0 0.0 0.09 0.0 0. 0.0 0.00 0.0 0.00 0.0 0.90 0.00 0.0 0.09 0.9 0.0 0.09 0.0 0.0 9 0 9 0 0 9.....9 0.0 0.0 0.0 0.09 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.00 0.0 0.0 0.0 0.0 0.0 0.00 0.009 0.09 0.0 0.00 0.09 0.00 0.0 0.0 0.09 0.0 0.09 0.0 0.00 0.0 0..0 0.0 0.00 0.0 0.0 0.0 0.0 0.00 0.09 0.09 0.0 0.0 0.09 0.0 0 9.0... 0.0 0.09 0.09 0.00 0.0 0.0 0.0 0.00 0.0 0.00 0.0 0.00 0.0 0.0 0.09 0.00990 0.00 0.0 0.0 0.009 0.00 0.0 0.0 0.0099 0.09 0.0 0.09 0.009 0.09 0.00 0.0 0.009 0.0 0.0 0.0 0.00 0.0 0.0 0.00 0.00 0 0 0 0 0 9 0. 0.000 0.009 0.00 0.00 0.00 9 0.00 0.009 0.00 0.00 0.009 9.....9 0.00 0.00 0.00 0.00 0.00 0.000 0.00 0.00 0.00 0.00 0.00 0.000 0.00 0.000 0.00 0.000 0.00 0.00 0.00 0.009 0.00 0.00 0.000 0.00 0.00 0.009 0.000 0.009 0.009 0.009 0.00 0.009 0.009 0.00 0.00 0.000 0.009 0.000 0.000 0.009 0.009 0.00 0.00 0.0099 0.00 0.000 0.00 0.00 0.009 0.009 0 9 9 9 0 9.0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000 0.000 0.0000 0 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
9 / August 0 HALAMAN KOSONG / Additional Mathematics Paper [Lihat halaman sebelah
0 / August 0 HALAMAN KOSONG / Additional Mathematics Paper
/ Additional Mathematics Paper Ogos, 0 MODUL PENINGKATAN PRESTASI TINGKATAN TAHUN 0 ANJURAN MAJLIS PENGETUA SEKOLAH MALAYSIA (KEDAH) ADDITIONAL MATHEMATICS MARKING SCHEME Paper MODUL. / Additional Mathematics Paper
/ PROGRAM PENINGKATAN PRESTASI AKADEMIK SPM 0 Marking Scheme Module Additional Mathematics Paper Question Solution/ Marking Scheme Answer Marks (a) B : m = or n = (a) m = AND n = (b) many to one or = = p p B : f ( ) B : f = p p = (a) B : y = (a) ( ) g = (b) B : ( k ) = (b) k = p p = 0 or B : ( )( ) p = or p = B : ( p) ( )( p ) p = AND p = (a) B : p = or q = (a) p = AND q = (b) x = / Additional Mathematics Paper
/ Question Solution/ Marking Scheme Answer Marks B : or x =, x = x B: (x )( x+ ) B : B : log + log h + log log log or log h or log k or log k + a + b B: + x = ( x ) B : ( ) () or 9 B : + ( ) B : ( ) ( ) + d d + = or d = B : ( ) d or ( ) d + + / Additional Mathematics Paper
/ Question Solution/ Marking Scheme Answer Marks 0 (a) B: m (a) m = AND m = = m 9 (b) B: S 9 = 9 (b) 9. B: r = (a) B: xy = ax + b (a) (b) B : ( ) = * + b or ( ) = * + b (b) b = B : x + y = x + y B : ( ) ( ) ( ) ( x ) + ( y ) or ( x 0) + ( y ) x y x y 0 + + = B: ( ) + ( ) + ( k ) ( ) + ( k ) + ( ) = 0 OR = k k = / Additional Mathematics Paper
/ Question Solution/ Marking Scheme Answer Marks (a) i + j (b) B : or + (b) i + j B : + m = 0 OR + m = 0 B : m 0 + + = 0 m = B : OB = B : x = or x = 9 B : sin x sinx + = 0 or sinx = or sinx = x = AND x = 9 B : ( x) sin = sinx (a).0rad (b) B: ( ) *(.0) ( )( 0.9) B: ( ) *(.0) or ( )( 0.9) (b). / Additional Mathematics Paper
/ Question Solution/ Marking Scheme Answer Marks 9 (b) B: (a) 0 0 + k (b) k = = 0 B : h () = B : h ( x) = x x + B : k = or x = B : x B : B : ( ) ( x + ) or 9 ( ) x h x dx = 0 ( x + ) x k = 0 (a) (b) B: + or (b) (a) 9 (b) B : C C C (b) 0 B : p( z>k ) = 0.00 B : P(z > 0. ) = 0. k =. END OF MARKING SCHEME / Additional Mathematics Paper