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1 / Additional Mathematics Kertas September 00 jam 0 minit PERSIDANGAN KEBANGSAAN PENGETUA-PENGETUA SEKOLAH MENENGAH NEGERI KEDAH DARUL AMAN PEPERIKSAAN PERCUBAAN SPM 00 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 9 printed pages. / [Lihat halaman sebelah

2 The following formulae may be helpful in answering the questions. The symbols given are the ones commonly used. ALGEBRA. x = b b ac a. log log a b log. m a n m n a a 9. T n a ( n ) d. m n m n n a a a 0. S n [ a ( n ) d ] n. m mn n ( a ) a. T n ar c c b a. log a mn log a m log a n m. log log m log n a a a n. log a m n log a m. y = uv,. y =. dy dx dy dx dv u dx n du v dx du dv v u u dy, dx dx v dx v dy du du dx.. S n n a ( r ) r a S, r < r CALCULUS Area under a curve = b a = b a n a ( r ), r r y dx or x dy. Volume of revolution = b y dx or a = b a x dy 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 x y xi yj. rˆ x y / [Lihat halaman sebelah

3 . x x N STATISTICS I W i W I i i. fx x f n P r n! ( n r )!.. ( x x ) = N f ( x x ) = f N F. m = L + C fm Q. I 00 Q 0 x 9 N fx f x x n C r n! ( n r )! r! 0 P(AB) = P(A) + P(B) P(AB) P ( X = r ) = Mean, = np npq Z = X n C r p r q n r, p + q = 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 Area of triangle = sin = cos ² A ab C = sin ² A / [Lihat halaman sebelah

4 Section A Bahagian A [ 0 marks ] [ 0 markah ] Answer all questions. Jawab semua soalan.. Solve the following simultaneous equations : Selesaikan persamaan serentak berikut : y x y 0 0 xy 0. [ marks] [ markah]. The table below shows the ages of a group of members in a club. Jadual di bawah menunjukkan umur bagi sekumpulan ahli dalam suatu kelab. Age ( year ) Umur (Tahun) Number of members Bilangan ahli k Find the maximum value of k if the modal age is. Given k =, calculate the value of (i) the mean, (ii) the variance, (iii) the median. [ mark] [ marks] Cari nilai maksimum bagi k jika mod umur ialah. Diberi k =, hitungkan nilai bagi (i) min, (ii) varians, (iii) median. [ markah] [ markah] / [Lihat halaman sebelah

5 . Given that the equation of a curve is dy y x x and dx at point P and point Q. Find the coordinates of P and of Q. the equation of the normal to the curve at these points. [ marks] Diberi bahawa persamaan suatu lengkung ialah dy dan pada titik P dan titik Q. dx Cari y x x koordinat titik P dan titik Q, persamaan normal kepada lengkung pada titik-titik itu. [ markah]. Sketch the graph of y sin x for 0 x. [ marks] Hence, using the same axes, sketch a suitable straight line to find the number of solutions for the equation x sin x for 0 x. State the number of solutions. [ marks] Lakar graf bagi y sin x untuk 0 x. [ markah] Seterusnya, dengan menggunakan paksi yang sama, lakar satu garis lurus yang sesuai untuk mencari bilangan penyelesaian x bagi persamaan sin x untuk 0 x. Nyatakan bilangan penyelesaian itu. [ markah] / [Lihat halaman sebelah

6 . A A A cm cm cm The diagram above shows the first three of an infinite series of quadrants. The radius of the first quadrant is cm and its area is represented by A. The radius of the second quadrant is twice the radius of the first quadrant and its area is represented by A. The radius of the third quadrant is twice the radius of the second quadrant and its area is represented by A. The radius of each subsequent quadrant is double the measurements of its previous one. State the common ratio. Find the area of the sixth quadrant. (c) Calculate the sum of the area from the third quadrant to the sixth quadrant, in terms of. [ marks] Rajah di atas menunjukkan tiga suku bulatan pertama untuk suatu siri tak terhingga bagi suku bulatan. Panjang jejari suku bulatan pertama ialah cm dan luasnya diwakili oleh A. Panjang jejari suku bulatan kedua ialah dua kali panjang jejari suku bulatan pertama dan luasnya diwakili oleh A. Panjang jejari suku bulatan ketiga ialah dua kali panjang jejari suku bulatan kedua dan luasnya diwakili oleh A. Jejari suku bulatan yang berikutnya adalah dua kali ganda ukuran jejari suku bulatan sebelumnya. Nyatakan nisbah sepunya. Cari luas suku bulatan keenam. (c) Hitung hasil tambah luas dari suku bulatan ketiga hingga suku bulatan keenam dalam sebutan. [ markah] / [Lihat halaman sebelah

7 . D B E O A C The diagram above shows triangle OCD. It is given that OA a, CD b, AB is parallel to CD and OA OC. Express in terms of a and / or b : (i) OD (ii) AB Given AE a kb and BE hbd, where h and k are constants. Find the value of h and of k. [ marks] [ marks] Rajah di atas menunjukkan segitiga OCD. Diberi bahawa OA a, CD b, AB adalah selari dengan CD dan OA OC. Ungkapkan dalam sebutan a dan / atau b : (i) OD (ii) AB Diberi AE a kb dan BE hbd, dengan keadaan h dan k ialah pemalar. Cari nilai h dan nilai k. [ markah] [ markah] / [Lihat halaman sebelah

8 Section B Bahagian B [ 0 marks ] [ 0 markah ]. Answer four questions from this section. Jawab empat soalan daripada bahagian ini. x y The table shows the values of two variables, x and y, obtained from an experiment. Variables x and y are related by the equation y = A kx +, where A and k are constants. Based on the table above, construct a table for the values of log 0 y. [ mark] (c) Plot log 0 y against x, using a scale of cm to unit on the x-axis and cm to 0. unit on the log 0 y-axis. Hence draw the line of best fit. Use your graph in to find the value of (i) x when y = 0, (ii) A, (iii) k. [ marks] [ marks] Jadual menunjukkan nilai-nilai bagi dua pembolehubah, x dan y, yang diperoleh daripada satu eksperimen. Pembolehubah x dan y dihubungkan oleh persamaan y = A kx +, dengan keadaan A dan k adalah pemalar Berdasarkan jadual di atas, bina satu jadual bagi nilai-nalai log 0 y. [ markah] Plot log 0 y melawan x, dengan menggunakan skala cm kepada unit pada paksi-x dan cm kepada 0. unit pada paksi- log 0 y. Seterusnya, lukis garis lurus penyuaian terbaik. (c) Gunakan graf di untuk mencari nilai (i) x apabila y = 0, (ii) A, (iii) k. / [Lihat halaman sebelah [ markah] [ markah]

9 9. y P y² = y = x A B O Q R x The diagram shows part of a curve y x which intersects the straight line y = at point P. Point Q and point R are on the x-axis such that PR is parallel to the y-axis. Find the coordinates of P and of Q. Calculate the area of the shaded region A. (c) The shaded region B is revolved through 0 o about the x-axis. Find the volume of revolution, in terms of π. [ marks] [ marks] [ marks] Rajah menunjukkan lengkung y x yang menyilang garis lurus y = pada titik P. Titik Q dan titik R terletak pada paksi-x dengan keadaan PR adalah selari dengan paksi-y. Carikan koordinat titik P dan titik Q. Hitungkan luas rantau berlorek A. (c) Rantau berlorek B dikisarkan melalui 0 o pada paksi-x. Cari isipadu kisaran, dalam sebutan π. [ markah] [ markah] [ markah] / [Lihat halaman sebelah

10 0 9. O T cm S P R Q 0 cm The diagram shows a circle with centre O and radius 0 cm touching a rectangle PRST at P. SRQ is a straight line and TS = cm. Use π =. and give the answers correct to two decimal places. Calculate POQ, in radian, the length, in cm, of the major arc PQ, (c) the area, in cm, of the shaded region. [ marks] [ marks] [ marks] Rajah menunjukkan sebuah bulatan berpusat O dengan jejari 0 cm menyentuh segiempat tepat PRST di P. SRQ ialah garis lurus dan TS = cm. Guna π =. dan beri jawapan betul kepada dua tempat perpuluhan. Hitung POQ, dalam radian, panjang, dalam cm, lengkok major PQ, (c) luas, dalam cm, kawasan berlorek. [ markah] [ markah] [ markah] / [Lihat halaman sebelah

11 0. y y = x + P Q(0, ) x O R Solution by scale drawing is not accepted. The diagram shows two perpendicular lines y = x + and PQR intersecting each other at point P. Find the equation of the line PQR Find the coordinates of P. (c) It is given that PR = PQ, find the coordinates of R. (d) A point S( x, y ) moves such that its distance from point R is always half its distance from point P. (i) Find the equation of the locus of S. (ii) Hence, determine whether this locus intercepts the y-axis or not. [ marks] [ marks] [ marks] [ marks] Penyelesaian secara lukisan berskala tidak diterima. Rajah menunjukkan dua garis lurus serenjang y = x + dan PQR yang bersilang pada titik P. Cari persamaan garis PQR. Cari koordinat P. (c) Diberi bahawa PR = PQ, cari koordinat R. (d) Suatu titik S( x, y ) bergerak dengan keadaan jaraknya dari titik R adalah sentiasa setengah daripada jaraknya dari titik P. (i) Cari persamaan lokus S. (ii) Seterusnya, tentukan sama ada lokus ini memintas paksi-y atau tidak. [ markah] [ markah] [ markah] [ markah] / [Lihat halaman sebelah

12 . The masses of duck eggs from a farm have a normal distribution with a mean of 0 g and a standard deviation of g. Find the probability that a duck egg chosen randomly from this farm has a mass of more than g. [ marks] The farm produces 000 eggs daily and the eggs are graded as follow: Grade A B C Mass, x ( g ) x > 9 x 9 x < (i) Calculate the number of eggs that belong to grade C. (ii) Given that 00 of the eggs produced daily have a mass of less than m g. Find the value of m. [ marks] Jisim telur itik dari sebuah ladang adalah mengikut satu taburan normal dengan min 0 g dan sisihan piawai g. Cari kebarangkalian bahawa sebiji telur itik yang dipilih secara rawak dari ladang ini berjisim melebihi g. [ markah] Ladang ini menghasilkan 000 biji telur setiap hari dan telur itu diberi gred seperti berikut : Gred A B C Jisim, x ( g ) x > 9 x 9 x < (i) Hitung bilangan telur gred C. (ii) Diberi bahawa 00 daripada telur yang dihasilkan setiap hari mempunyai jisim kurang daripada m g. Cari nilai m. [ markah] / [Lihat halaman sebelah

13 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. The velocity of the particle, v ms -, is given by v = t t, where t is the time, in s, after leaving O. [Assume motion to the right is positive.] Find (c) (d) the initial velocity, in ms -, of the particle, the range of t during which the particle moves to the left, sketch the velocity-time graph of the motion of the particle for 0 t, the total distance, in m, travelled by the particle in the first seconds. [ mark] [ marks] [ marks] [ marks] Suatu zarah bergerak di sepanjang suatu garis lurus melalui satu titik tetap O. Halaju zarah itu, v ms -, diberi oleh v = t t, dengan keadaan t ialah masa, dalam s, selepas melalui O. [Anggapkan gerakan ke arah kanan sebagai positif] Cari halaju awal, dalam ms -, bagi zarah itu, [ markah] julat bagi t semasa zarah itu bergerak ke arah kiri, [ markah] (c) lakarkan graf halaju-masa gerakan zarah itu untuk 0 t, (d) jumlah jarak yang dilalui, dalam m, oleh zarah itu dalam saat yang pertama. [ markah] [ markah] / [Lihat halaman sebelah

14 . The table shows the price indices and respective weightages, in the year 009 based on the year 00, of four materials, P, Q, R and S in the production of a type of moisturizing cream. Material Bahan Price index in the year 009 based on year 00 Indeks harga pada tahun 009 berasaskan tahun 00 Weightage Pemberat P Q 0 m R 0 S 0 m + If the price of material P in the year 00 was RM0.00, calculate its price in the year 009. Given that the composite index for the production cost of the moisturizing cream in the year 009 based on the year 00 is 0. Find (i) the value of m. (ii) the price of the moisturizing cream in the year 00 if its price in the year 009 is RM0.00. (c) Given that the price of material Q is estimated to increase by % from the year 009 to the year 00, while the others remain unchanged. Calculate the composite index of the moisturizing cream in the year 00 based on the year 00. [ marks] [ marks] [ marks] [ marks] / [Lihat halaman sebelah

15 Jadual di atas menunjukkan indeks harga dan pemberat masing-masing bagi tahun 009 berasaskan tahun 00 bagi empat bahan P, Q, R dan S dalam penghasilan suatu jenis krim lembapan. Jika harga bagi bahan P pada tahun 00 ialah RM0.00, hitungkan harga bagi bahan tersebut pada tahun 009. Diberi bahawa indeks gubahan bagi kos penghasilan krim lembapan itu pada tahun 009 berasaskan tahun 00 ialah 0. Cari (i) nilai m, (ii) harga bagi krim lembapan pada tahun 00 jika harganya pada tahun 009 ialah RM0.00. [ markah] [ markah] [ markah] (c) Diberi bahawa harga bagi bahan Q dijangka akan naik sebanyak % dari tahun 009 ke tahun 00, sementara lain-lain bahan harganya kekal. Hitung indeks gubahan bagi krim lembapan itu pada tahun 00 berasaskan tahun 00. [ markah] / [Lihat halaman sebelah

16 . Use graph paper to answer this question. The manager of a warehouse intends to order some chairs. The warehouse needs x units of office chairs and y units of dining chairs. The purchase of these chairs is based on the following constraints: I : The number of dining chairs is at least 00 units. II : The total numbers of chairs is not more than 00 units. III (c) : An office chair takes up four units of storage space while a dining chair occupies one unit of storage space. The maximum storage space available is 00 units. Write three inequalities, other than x 0 and y 0, which satisfy all the above constraints. Using a scale of cm to 00 units on both axes, construct and shade the region R which satisfies all of the above constraints. Use your graph in to find (i) the maximum number of dining chairs that could be ordered, if the manager plans to order only 0 units of office chairs, [ marks] [ marks] (ii) the maximum total profit if the profit from an office chair is RM0.00 and from a dining chair is RM.00. [ marks] / [Lihat halaman sebelah

17 Gunakan kertas graf untuk menjawab soalan ini. Seorang pengurus gudang ingin membeli dua jenis kerusi, iaitu kerusi pejabat dan kerusi dewan makan. Gudang tersebut memerlukan x unit kerusi pejabat dan y unit kerusi dewan makan. Pembelian kerusi-kerusi tersebut adalah berdasarkan kekangan berikut: I : Bilangan kerusi dewan makan sekurang-kurangnya 00 unit. II : Jumlah kerusi tidak melebihi 00 unit. III : Satu kerusi pejabat memerlukan empat unit ruang menyimpan dan satu kerusi dewan makan memerlukan satu unit ruang menyimpan. Ruang menyimpan maksimum yang boleh dibekal ialah 00 unit. Tuliskan tiga ketaksamaan, selain x 0 dan y 0, yang memenuhi semua kekangan di atas. Menggunakan skala cm kepada 00 unit pada kedua-dua paksi, bina dan lorek rantau R yang memenuhi semua kekangan di atas. [ markah] [ markah] (c) Gunakan graf anda di untuk mencari (i) (ii) bilangan maksimum kerusi dewan makan yang boleh dibeli jika pengurus tersebut bercadang untuk membeli hanya 0 unit kerusi pejabat, keuntungan maksimum keseluruhannya jika keuntungan yang diperoleh dari satu unit kerusi pejabat ialah RM0.00 dan dari satu unit kerusi dewan makan ialah RM.00. [ markah] / [Lihat halaman sebelah

18 . The diagram shows a parallelogram PQRS. T is a point on RS such that RT = 9 cm. Given that the area of triangle SQR is. cm. Find the length, in cm, of TQ, QTR, (c) the length, in cm, of ST, (d) the area, in cm, of quadrilateral PQTS. [ marks] [ marks] [ marks] [ marks] S T 9 cm R cm P Q Rajah di atas menunjukkan segiempat selari PQRS. T ialah satu titik pada RS dengan keadaan RT = 9 cm. Diberi bahawa luas segitiga SQR ialah. cm. Cari panjang, dalam cm, TQ, QTR, (c) panjang, dalam cm, ST, (d) luas, dalam cm, sisiempat PQTS. [ markah] [ markah] [ markah] [ markah] END OF QUESTION PAPER KERTAS SOALAN TAMAT / [Lihat halaman sebelah

19 THE UPPER TAIL PROBABILITY Q(z) FOR THE NORMAL DISTRIBUTION N(0,) 9 KEBARANGKALIAN HUJUNG ATAS Q(z) BAGI TABURAN NORMAL N(0, ) z Minus / Tolak f ( z) exp z Q ( z) k f ( z) dz f (z) z / O [Lihat halaman sebelah Q(z) Example / Contoh: If X ~ N(0, ), then P(X > k) = Q(k) Jika X ~ N(0, ), maka P(X > k) = Q(k)

20 MARKING SCHEME ADDITIONAL MATHEMATICS PAPER SPM TRIAL EXAMINATION 00 N0. SOLUTION MARKS x 0 y P y 0 y y K Eliminate x y 0y 0 y y 0 y or y x or x K Solve quadratic equation k = P Mid point,,,, P (i) Mean fx f. 0 K Use formula and calculate (ii) Varian fx f x K Use formula and calculate (iii) Median, m N F (0) L C 0. fm.0 K Use formula and calculate

21 N0. SOLUTION MARKS y x x dy x x dx x x 0 x x 0 x, x y x y K Equate and solve quadratic equation, and, Equation of normals : mnormal y x y x or equivalent y x y x or equivalent K Use m normal to form equations y y sin x P Modulus sine shape correct. P Amplitude = [ Maximum = and Minimum = -] O - x P Two full cycle in 0 x - x y P Shift down the graph

22 N0. SOLUTION MARKS x sin x or x y For equation Draw the straight line x y K Sketch the straight line Number of solutions =. Common ratio, r = A OR T ar K (c) S S.. 0 K Use S or S K Use S - S

23 N0. SOLUTION MARKS K for using vector (i) triangle for a(i) or OD OC CD a(ii) a b (ii) AB OB OA OD OA a b a b OR AB CD b [ K ] AE AB BE b h OD b ha b a kb ha h b K for using vector triangle and BE K h h k h K for equating coefficients correctly

24 N0. SOLUTION MARKS x log 0 y correct values of log y log 0 y K Plot log 0 y vs x Correct axes & uniform scale points plotted correctly 0. Line of best-fit 0 x (c) (i) log 0 y = ( k log 0 A ) x + log 0 A x =. P (ii) y-intercept = log 0 y K A =.9 gradient = k log 0 A K gradient k = * log A 0 = 0. 0

25 N0. SOLUTION MARKS P(, ) P Q(, 0 ) P A = 0 ( y ) dy K use x dy = = y y 0 OR equivalent K correct limit K integrate correctly (c) V = x dx K integrate y dx = x x K correct limit K integrate correctly = 0

26 N0. SOLUTION MARKS 9 cos POQ 0 POQ =. rad. K Use ratio of trigonometry or equivalent (c) ( π. ) rad PQ = 0 ( π. ) =. cm 0 = 9. cm P K Use s P r Area of trapezium POQR = ( 0) 9. =. cm * K Area of sector POQ = (0) (.) = cm K Use formula A r Area of shaded region =. =. cm K 0

27 N0. SOLUTION MARKS 0. Equation of str. line PQR : m = K y = x + x + = P(, ) x + K solving simultaneous equation (c) ( x) ( ) 0 ( y) () or K Use the ratio rule R(, ) (d) (i) ) ( x ) ( y ) x y ( ( ) K Use distance formula [ x x + + y + y + ] = x + x + + y y + x + y x + y + = 0 (ii) Substitute x = 0, y + y + = 0 b ac = () ()() = < 0 No real root for y, The locus does not intercept the y-axis. K Substitute x = 0 and use b ac to make a conclusion if b ac = - 9

28 0 N0. SOLUTION MARKS 0, 0 P ( X ) P ( Z ) = P ( Z.) K Use Z = X = 0.0 = 0.9 K Use Q(Z) = or K (c) 00 = Q( Z ) = 0.0 Z =. P K Find value of Z m 0. m = 0. g K K m Use Use negative value 0 0

29 N0. SOLUTION MARKS - ms - v < 0 t - t - < 0 (t ) (t +) < 0 K K 0 < t < (c) v P (for shape ) t P min(,-9), (,) &(0,-) must be seen - (d) Total dis tan ce vdt 0 vdt t t t t t t 0 K for and 0 K (for Integration; either one) ( ) ( ) 0 ( ) ( ) ( ) 0 = 0 ( ) K (for use and summation) m

30 0 N0. SOLUTION MARKS P K P 09 RM (i) 0 ( ) ( 0m) ( 0 ) 0m 0 m K 0 0m 0 0m K (use formula) m (ii) P 0 00 RM 0 0 K RM (c) 0 + (0 0.) = K I 0 / 0 ( ) ( ) ( 0 ) ( 0 ) K

31 0 N0. SOLUTION MARKS y 00 x + y 00 x + y 00 y x + y = (00,00) R 00 y = x + y = x At least one straight line is drawn correctly from inequalities involving x and y. All the three straight lines are drawn correctly K (c) (i) 0 Region is correctly shaded (ii) Maximum point (00, 00) Maximum profit = 0(00) + (00) K = RM 00

32 0 N0. SOLUTION MARKS TQ = 9 + (9)()cos o K TQ =. cm sinqtr sin. 0 K QTR = o (c). = ( RS)( )sin o K RS = ST = 9 (or ST + 9 in formula of area) K (d) = cm 0 Area QTR (9)() sin =. cm K Area of quadrilateral PQTS = (.). K =. cm 0 END OF MARKING SCHEME