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1 SULIT NAMA : KELAS : JABATAN PELAJARAN NEGERI SABAH SIJIL PELAJARAN MALAYSIA 00 7/ EXCEL ADDITIONAL MATHEMATICS PAPER OGOS 00 Jam Dua jam JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU. Tuliskan angka giliran dan nombor kad pengenalan anda pada ruang yang disediakan.. Calon dikehendaki membaca arahan di halaman. Question Full Marks Marks Obtained Total 80 This paper consists of 8 printed pages. [Lihat sebelah SULIT

2 INFORMATION FOR CANDIDATES. This question paper consists of 5 questions.. Answer all questions.. Give only one answer for each question.. Write your answers clearly in the space provided in the question paper. 5. Show your working. It may help you to get marks. 6. If you wish to change your answer, cross out the work that you have done. Then write down the new answer. 7. The diagrams in the questions provided are not drawn to scale unless stated. 8. The marks allocated for each question are shown in brackets. 9. A list of formulae is provided on pages to A booklet of four-figure mathematical tables is provided.. You may use a non-programmable scientific calculator.. This question paper must be handed in at the end of the examination.

3 SULIT 7/ The following formulae may be helpful in answering the questions. The symbols given are the ones commonly used. ALGEBRA. x b b ac a 8. log a logc b b log a c. m n m n a a a 9. T a ( n ) d n. m n m n a a a. ( a ) a m n mn 5. log mn log m log n a a a m 6. log log m log n a a a n 7. log m nlog m a a n n 0. Sn [ a ( n ) d].. Tn S n ar n n n a( r ) a( r ), r r r a. S, r r CALCULUS dy dv du. y uv, u v dx dx dx. Area under a curve b = y dx or a du dv v u u dy. y, dx dx v dx v = b a x dy. dy dy du dx du dx 5. Volume generated = b y dx or a = b x dy a

4 STATISTICS.... x x N x fx f ( x x) x x N N f ( x x) fx x f f I n n P r C r WI i W i i n! n r! n! n r!! r 0. P A B P A PB P A B N F 5. m L c fm Q 6. I 00 Q o. n r n P X r C p q r, p q. Mean, μ = np. npq. Z X r GEOMETRY. Distance. Area of triangle = = x x y y. Midpoint x x y y xy,,. A point dividing a segment of a line nx mx ny my xy,, m n m n ( ) ( ) x y x y x y x y x y x y r x y rˆ xi yj x y

5 SULIT 5 7/ TRIGONOMETRY. Arc length, s r. Area of sector,.. 5. A sin Acos A sec A tan A cosec A cot A 6. sin A sin Acos A 7. cos cos sin r A A A cos A sin A 8. sin( A B) sin Acos B cos Asin B 9. cos( AB) cos Acos B sin Asin B 0. tan A tan B tan ( AB) tan Atan B tan A. tan A tan A.. a b c sin A sin B sinc a b c bc cos A. Area of triangle sin ab C

6 For Examiner s Use Answer all questions. Jawab semua soalan. The relation between P,6,5,7 and Q,,5, 7 pairs of,, 6,, 6,, 5,, 5,5. Hubungan di antara P,6,5,7 dan Q,,5, 7 bertertib,, 6,, 6,, 5,, 5,5. is defined by the ordered ditakrif oleh pasangan State Nyatakan (a) the object of 5, objek bagi 5, (b) the range of the relation. julat bagi hubungan tersebut. [ marks] [ markah] Answer / Jawapan : (a)... (b). Given that f : x x and gf : x x, find the function g. [ marks] Diberi bahawa f : x x dan gf : x x, carikan fungsi g. [ markah] Answer / Jawapan :...

7 SULIT 7 7/. Given that f : x 6x 5 and g : x x, find the function fg. For Examiner s Use Diberi bahawa f : x 6x 5 dan g : x x, cari fungsi fg. [ marks] [ markah] Answer / Jawapan :... Given one of the roots of the quadratic equation x the other root. Diberi salah satu daripada punca persamaan kuadratik x kuasa dua punca satu lagi. px 7 0 is the square of px 7 0 ialah (a) Find the value of p. Cari nilai p. (b) State the roots of the quadratic equation. Nyatakan punca persamaan kuadratik itu. [ marks] [ markah] Answer / Jawapan : (a). (b).....

8 For Examiner s Use 5 Find the range of values of x for which 5xx. [ marks] Cari julat nilai x bagi 5xx. [ markah] 5 Answer / Jawapan : Solve the equation Selesaikan persamaan x x. [ marks] x x. [ markah] 6 Answer / Jawapan :... 7 Solve the equation log ( x 6) logx. [ marks] Selesaikan persamaanlog ( x 6) logx [ markah] 7 Answer / Jawapan :...

9 SULIT 9 7/ 8 Diagram 8 shows the graph of quadratic function f ( x) a( x ) k, where a and k are constants. The graph has a minimum point (, 8). For Examiner s Use Rajah 8 menunjukkan graf fungsi kuadratik f ( x) a( x ) k, dengan keadaan a dan k adalah pemalar. Graf itu mempunyai titik minimum (, 8). f (x) O x (, 8) State Nyatakan (a) the value of k, nilai bagi k, Diagram 8 Rajah 8 (b) the value of a, nilai bagi a, (c) the equation of axis of symmetry. persamaan bagi paksi simetri. [ marks] [ markah] Answer / Jawapan : (a) k =... (b) a =... 8 (c)

10 For Examiner s Use 9 Find the number of terms in the arithmetic progression, 8,, 8,, 57. Cari bilangan sebutan dalam janjang aritmetik, 8,, 8,, 57. [ marks] [ markah] 9 Answer / Jawapan :... 0 The first three terms of a geometric progression are h +, h, h. Tiga sebutan pertama suatu janjang geometri ialah h +, h, h. Find Cari (a) the value of h, nilai h, (b) the common ratio of the progression. nisbah sepunya janjang itu. [ marks] [ markah] 0 Answer / Jawapan : (a) h =. (b)

11 SULIT 7/ In a geometric progression, the first term is and the sum of the first two terms is 7. Find the sum to infinity of the progression. [ marks] Dalam suatu janjang geometri, sebutan pertama ialah dan hasil tambah dua sebutan pertama ialah 7. Cari hasil tambah hingga sebutan ketakterhinggaan bagi janjang itu. [ markah] For Examiner s Use Answer / Jawapan :. A straight line graph is obtained by plotting log 0 y against log 0 x, as shown in Diagram. Given that the equation of graph is log 0 y = log 0 x +. Graf garis lurus diperoleh dengan memplotkan log 0 y melawan log 0 x, seperti yang ditunjukkan di Rajah. Diberi bahawa persamaan graf itu ialah log 0 y = log 0 x +. log 0 y J(, k) K( h, ) O Diagram Rajah log 0 x Find the value of h and of k. Cari nilai h dan nilai k. [ marks] [ markah] Answer / Jawapan : h =.... k =...

12 For Examiner s Use The vertices of a triangle are A(,7), B( h,) and C(0, ). Given that triangle ABC is right-angled at B, calculate the possible values of h. Bucu-bucu sebuah segitiga ialah A(,7), B( h,) and C(0, ). Diberi bahawa segitiga ABC bersudut tepat pada B, hitungkan nilai-nilai yang mungkin untuk h. [ marks] [ markah] Answer / Jawapan :.. Diagram shows two vectors, OP and QO. Rajah menunjukkan dua vektor, OP dan QO. Q(6, 8) y P(, ) O Diagram Rajah x Express PQ in the form x y Ungkapkan PQ dalam bentuk. [ marks] x y. [ markah] Answer / Jawapan :.....

13 SULIT 7/ 5 Given that AB = m and AC = 6, find the value of m and of n if BC n 5. Diberi AB m = dan AC = 6, cari nilai m dan nilai n jika BC n 5. [ marks] [ markah] For Examiner s Use Answer / Jawapan : m=.. n =. 5 6 Diagram 6 shows a triangle ABC. Rajah 6 menunjukkan sebuah segitiga ABC B 5b 8a C The point D lies on AC such that AD: DC = :. Titik D terletak pada AC dengan keadaan AD: DC = :. Express in terms of a and b Ungkapkan dalam sebutan a dan b D Diagram 6 Rajah 6 A (a) AC, (b) BD. [ marks] [ markah] Answer / Jawapan : (a) (b).

14 For Examiner s Use 7 Solve the equation cos x 0 sin x + 5 = 0 for 0 o x 60 o. Selesaikan persamaan kos x 0 sin x + 5 = 0 untuk 0 o x 60 o. [ marks] [ markah] 7 Answer / Jawapan :... 8 It is given that y = Diberi bahawa y = 5 dy u, where u = 6x +. Find dx in terms of x. 5 dy u, dengan keadaan u = 6x +. Cari dx dalam sebutan x. [ marks] [ markah] 8 Answer / Jawapan :.. For Examiner s Use

15 SULIT 5 7/ 9 B 6 cm 0 O Diagram 9 Rajah 9 A Diagram 9 shows the sector OAB with centre O. The reflex angle of AOB is 0 o and the length of the arc AB is 6 cm. Using =., find Rajah 9 menunjukkan sektor OAB dengan pusat O. Sudut reflex AOB ialah 0 o dan panjang lengkok AB ialah 6 cm. Dengan menggunakan =., cari (a) the value of, in radians, nilai bagi, dalam radian, (b) the area, in cm, of sector OAB. luas, dalam cm, bagi sektor OAB. [ marks] [ markah] Answer / Jawapan : (a) =... 9 (b).....

16 For Examiner s Use 0 The volume of water, V m, in a tank is given by V = h ( + h), where h is the height of the water, in m, in the tank. Water leaked from the tank at the rate of m s. Find the rate of change of the height of the water in m s, at the instant when its height is m. [ marks] Isipadu air, V m, dalam sebuah tangki diberi oleh V = h ( + h), dengan keadaan h ialah tinggi air, dalam m, dalam bekas itu. Air bocor dari tangki itu dengan kadar m s. Cari kadar perubahan tinggi air dalam m s, pada ketika tingginya ialah m. [ markah] 0 Answer / Jawapan :.. Given that g ( x) dx, find 0 Diberi bahawa g ( x) dx, cari 0 (a) g ( x) dx, (b) [ x g( x)] dx. 0 0 [ marks] [ markah] Answer / Jawapan : (a)... (b)...

17 SULIT 7 7/ (a) How many five-digit odd numbers can be formed from the digits 5, 6, 7, 8, 9 if no repetition is allowed? Berapakah bilangan nombor ganjil lima digit yang boleh dibentuk daripada digit 5, 6, 7, 8, 9 tanpa ulangan? For Examiner s Use (b) A karate team of 5 members is chosen from girls and 6 boys. Calculate the number of different ways the team can be formed if there is no restriction. Satu pasukan karate yang terdiri daripada 5 orang ahli dipilih daripada orang perempuan dan 6 orang lelaki. Hitungkan bilangan cara yang berlainan pasukan itu boleh dibentuk jika tiada syarat dikenakan. [ marks] [ markah] Answer / Jawapan : (a).... (b)..... A marble is drawn at random from a bag containing white marbles, red marbles and 5 blue marbles. Sebiji guli dikeluarkan secara rawak dari sebuah beg yang mengandungi biji guli putih, biji guli merah dan 5 biji guli biru. Find the probability of Cari kebarangkalian untuk (a) getting a red or blue marble, mendapat sebiji guli merah atau biru, (b) not getting a red marble. tidak mendapat guli merah. [ marks] [ markah] Answer / Jawapan : (a).... (b)

18 For Examiner s Use A set of data consists of five numbers. The sum of the numbers is 0 and the sum of the squares of the numbers is 5. Satu set data mengandungi lima nombor. Hasil tambah bagi nombor-nombor itu ialah 0 dan hasil tambah bagi kuasa dua nombor-nombor itu ialah 5. (a) Find the mean for the five numbers. Cari min bagi lima nombor itu. (b) When a number p is added to this set, the mean is unchanged. Hence, find the variance. Apabila satu nombor p ditambah ke set nombor ini, minnya tidak berubah. Seterusnya, cari varians. [ marks] [ markah] Answer / Jawapan : (a).. (b)......

19 SULIT 9 7/ 5 For Examiner s Use Diagram 5 shows a standard normal distribution graph. Rajah 5 menunjukkan satu graf taburan normal piawai. f(z) 0 k z Diagram 5 Rajah 5 Given P(0 z k ) is 0.5. Diberi P(0 z k ) ialah 0.5. (a) Find P(z k). Cari P(z k). (b) X is a continuous random variable which is normally distributed with a mean of 8 and a variance of. X ialah pembolehubah rawak selanjar bertaburan secara normal dengan min 8 dan varians. Find the value of X when the z-score is k. Cari nilai X apabila skor-z ialah k [ marks] [ markah] 5 Answer / Jawapan : (a).. (b).. END OF QUESTION PAPER KERTAS SOALAN TAMAT

20 No. Suggested solution and mark scheme Sub Marks Total Marks. (a) 5. (b),,5 g( x) x B: g y ( ) ( y) or g(x ) = (x ) + or f x x.. fg B: B: (a) B: x x fg 6 5 x g B: or (b), 9 5. x or x B: x or x B: (x)( x) 0 or ( x)( x) 0 6 x = x B: equivalent. x x x ( ) 7 or (9) or or B: x x = or equivalent. OR using logarithms method: x = (accept.9999) B: 0.00x + (x )(0.77) =.079 or.55x =.50 B: log 0 x + log 0 x = log 0 (accept any base)

21 7. x B: x 6 or x + 6 = x or equivalent B: x log x 6 x 8. (a) k = 8 (b) a = (c) x = 9. 6 or 6 terms or n = 6 B: 8 + (n )(5) = 57 or by listing method: 8,, 8,,, 7,, 7,, 7,, 7,, 7, 5, (a) h = 6 B : h h h h or h = (h )(h + ). (b) r = S =6 B: S = B: r =. h =.,., k = 7(both) B: h = or k = 7 B: = h + or k = () + B: Y = X + B: h h + = 0 B: ( ) h h0 B: PQ = PO + OQ or m = 9, n = (both) B: m + () = 6 or + n = 5 B: BC BA AC or 6 m or equivalent 5 n 6. (a) 8a5b

22 (b) 8 0 a b B: BD 8 a (8a 5 b) or BD 5b ( 8a5 b) B: AD (8 5 ) a b or BD 8 a AC OR ( 8 a 5 b ) CD or 5 BD b CA 7..8 o, 8.9 o or 9, 8 B : sin x =, sin x = ( both) B : ( sin x )(sin x + ) = 0 B : ( sin x ) 0sin x + 5 = (6x +) dy 5 6 dx B: u dy 5 du dx B: u du or 6 9. (a).69 (b) 7.9 B: 6 0. dh dt. (6).69 or B: 8 dh dt dv 6 (.69).69 B: 6 () () dh = 8 dv ( h) h ( h) dh B: (a) (b) 6 9 or r =.6 or dv 6 h 9 h dh B : B :. (a) 7 x 0 xdx B : x! (b) 5 B : 0 C g( x) dx

23 (a) 5 B : (b) 0 7 B : - 0 or (a) 6 (b) 7 B : (a) (b) 6. 6 B : x 8.5 B : k =.5

24 SULIT hours NAMA : KELAS : JABATAN PELAJARAN NEGERI SABAH SIJIL PELAJARAN MALAYSIA 00 7/ EXCEL ADDITIONAL MATHEMATICS Paper SEPTEMBER 00 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 for each question.. Show your working. It may help you to get marks. 5. The diagrams in the questions provided are not drawn to scale unless stated. 6. The marks allocated for each question and sub-part of a question are shown in brackets. 7. A list of formulae is provided on pages to. 8. A booklet of four-figure mathematical tables is provided. 9. You may use a non-programmable scientific calculator. This paper consists of 6 printed pages.

25 SULIT The following formulae may be helpful in answering the questions. The symbols given are the ones commonly used. ALGEBRA. b b ac x a. m n m n a a a. m n m n a a a. ( a m ) a mn 5. log mn log m log n a a a m 6. log log m log n n 7. log m nlog m a a a a a n 8. log a logc b b log a 9. T a ( n ) d n 0. n Sn [ a ( n ) d]. n ar Tn c. n n a( r ) a( r ) Sn, r r r. a S, r r dy dv du. y uv, u v dx dx dx du dv v u u dy. y, dx dx v dx v. dy dy du dx du dx CALCULUS. Area under a curve b = y dx or = a b a x dy 5. Volume generated = = b y dx or a b x dy a.... x x N fx x f ( x x) x x N N f ( x x) fx x f f STATISTICS I n P r n C WI i W i i n! n r! n! r n r!! r 0. P A B P A PB P A B. n r n P X r C p q r, p q r

26 SULIT 5. N F m L c fm 6. Q I 00 Q o. Mean, μ = np. npq. x Z GEOMETRY. Distance. Area of triangle = = x x y y ( ) ( ) x y x y x y x y x y x y. Midpoint x x y y 5. r x y xy,, xi yj 6. rˆ. A point dividing a segment of a line x y nx mx ny my xy,, m n m n TRIGONOMETRY. Arc length, s r. Area of sector, A. sin Acos A. sec A tan A 5. cosec A cot A 6. sin A sin Acos A 7. cos A cos A sin A cos A sin r A 8. sin( A B) sin Acos B cos Asin B 9. cos( AB) cos Acos B sin Asin B 0. tan A tan B tan ( AB) tan Atan B. tan A tan A tan A. a b c sin A sin B sinc. a b c bc cos A. Area of triangle sin ab C

27 SULIT 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 + =0 x + y + xy 0 = 0 [5 marks] [5 markah] Given that f(x) = 5x + and g(x) =. Find Diberi bahawa f(x) = 5x + dan g(x) =. Cari (a) f - (x) [ mark] f (x) [ markah] (b) gf - (x) [marks] gf (x) [markah] (c) h(x) such that hf(x) = + 0x [ marks] h(x) supaya h f(x) = + 0x [ markah] (a) Sketch the graph of y = cos x + for 0x. [ marks] Lakar graf y = cos x + untuk 0x. [ markah] (b) Hence using the same axes, sketch a suitable straight line to find the number of solutions for the equation cos x x 0 for 0 x. State the number of solutions. [ marks] Seterusnya, dengan menggunakan paksi yang sama, lakar satu garis lurus yang sesuai untuk mencari bilangan penyelesaian bagi persamaan cos x x 0 untuk 0x. Nyatakan bilangan penyelesaian itu. [ markah]

28 SULIT 5. w cm Diagram Rajah l cm Diagram shows the arrangement of the first four of an infinite series of similar rectangles. The length and width of the first rectangle is l cm and w cm respectively. The measurements of the length and width of each subsequent rectangle are half that of the previous one. Rajah menunjukkan empat buah segi empat tepat yang pertama bagi suatu siri infinit segi empat tepat yang serupa. Panjang dan lebar segi empat tepat yang pertama ialah l cm dan w cm masing-masing. Panjang dan lebar untuk segi empat tepat yang seterusnya adalah separuh ukuran segi empat tepat sebelumnya. (a) (b) Show that the areas of the rectangle form a geometric progression and state the common ratio. [ marks] Tunjukkan bahawa luas- luas segi empat tepat itu membentuk suatu janjang geometri. Nyatakan nisbah sepunyanya. [ markah] Given that l = 5 cm and w = 8 cm, Diberi bahawa l = 5 cm dan w = 8 cm, (i) determine the number of rectangles with area greater than cm, [ marks] tentukan bilangan segi empat tepat yang luasnya lebih daripada cm, [ markah] (ii) find the sum to infinity of the areas, in cm of the rectangles. [ marks] cari hasil tambah hingga infiniti, luas segi empat tepat itu dalam cm. [ markah] 5. Number of pupils Bilangan Murid Scores Markah Diagram Rajah Diagram shows a frequency polygon which represents the distribution of the scores obtained by 80 pupils in a test. Rajah menunjukkan suatu poligon kekerapan yang mewakili taburan markah yang di perolehi oleh 80 orang pelajar dalam satu ujian.

29 SULIT 6 (a) Complete the frequency distribution table below. [ mark] Lengkapkan jadual taburan kekerapan di bawah. [ markah] Score Markah Cumulative Frequency Kekerapan Longgokan (b) Calculate the median score [ marks] Hitung nilai markah median [ markah] (c) Calculate the variance of the distribution. [ marks] Hitung varians bagi taburan tersebut [ markah] 6. Diagram shows triangles OAB and OAC. The straight lines OB and AC intersect at point K such that AK : AC = :. Given that OA = a and OC = hb where h is a constant. Rajah menunjukkan segitiga-segitiga OAB dan OAC. Garis lurus OB dan AC bertemu di titik K di mana AK : AC = :. Diberi bahawa ialah pemalar. OA = a and OC = hb, dengan keadaan h C B ` Find Cari hb O K a A Diagram Rajah (a) AK in terms of h, a and b AK dalam sebutan h, a and b (b) OK in terms of h, a and b OK dalam sebutan h, a and b Hence, if KB = 0a + 5b, find the value of h. Oleh itu, jika KB = 0a + 5b, cari nilai h. [ marks] [ markah] [ marks] [ markah]

30 SULIT 7 7. Given that Diberi 0 0 Section B Bahagian B [0 marks] [0 markah] Answer only four questions from this section Jawab mana-mana empat soalan daripada bahagian ini f ( x) dx = 7, where f(x) is a linear function. f ( x) dx = 7, yang mana f(x) ialah suatu fungsi linear. (a) Find the value of k when Cari nilai k apabila y 0 0 f x k dx =. [marks] f x k dx =. [markah] y = f(x). P(,9) y = (x- 5) Diagram Rajah O x Diagram shows the graph of the straight line y = f(x) which intersects the curve y = ( x 5 ) at point P(, 9). Rajah menunjukkan graf garis lurus y = f(x) yang bersilang dengan lengkung y = ( x- 5 ) pada titik P(, 9). (b) Find the area bounded by the straight line y = f(x), the curve y = ( x 5 ), the x- axis and the y-axis. [marks] Cari luas yang dibatasi oleh garis lurus y = f(x), lengkung y = ( x 5 ), paksi-x dan paksi-y. [ markah] (c) Calculate the volume generated in terms of, when the area bounded by the line OP, the curve y = ( x 5 ) and the x-axis is revolved through 60 about the x-axis. [ marks] Hitung isipadu yang dijanakan dalam sebutan, apabila rantau yang dibatasi oleh garis lurus OP, lengkung y = ( x 5 ) dan paksi-x dikisarkan melalui 60 pada paksi-x. [ markah]

31 SULIT 8 8. Use graph paper to answer this question. Gunakan kertas graf untuk menjawab soalan ini. x y Table 5 Jadual 5 Table 5 shows the values of the variables x and y obtained from an experiment. Variables x and y are related by the equation y = pq x, where p and q are constants. Jadual 5 menunjukkan nilai-nilai pembolehubah x dan y, yang didapati daripada suatu ujikaji. Pembolehubah x dan y dikaitkan oleh persamaan y = pq x, dengan keadaan p dan q adalah pemalar. (a) Using a scale of cm to unit on the ( x - )-axis and cm to 0. unit on the log y- axis, plot log y against ( x ). Hence, draw the line of best fit. [5 marks] Dengan menggunakan skala cm kepada unit pada paksi-( x ) dan cm kepada 0. unit pada paksi- log y, plotkan log y melawan ( x ). Seterusnya, lukis satu garis lurus penyuaian terbaik. [5 markah] (b) From your graph in 8(a), find the value of Daripada graf anda di 8(a), cari nilai (i) p and of q p dan nilai q (ii) x when y = 5.0. [5 marks] x apabila y = 5.0 [5 markah]

32 SULIT 9 9. Solution by scale drawing is not accepted. Penyelesaian secara lukisan berskala tidak diterima. Diagram 6 shows a rectangle PQRS. Rajah 6 menunjukkan sebuah segi empat tepat PQRS. y R S(-, 5) T Q(6,) P(0,) Diagram 6 Rajah 6 O x Given that the equation of the straight line PR is y x. Point T lies on the straight line PR such that PT : TR = : Diberi persamaan garis lurus PR ialah y x. Titik T terletak pada garis lurus PR dengan keadaan PT : TR = : (a) Find Cari, (i) the equation of the straight line SR [ marks ] persamaan garis lurus SR [ markah ] (ii) the coordinates of T [ marks] koordinat titik T, [ markah ] (iii) the area of triangle PST luas segitiga PST [ marks] [ markah] (b) A point M moves such that its distance from point S is always 5 units. Find the equation of the locus of M. [ marks] Suatu titik M bergerak dengan dengan keadaan jaraknya dari titik S ialah sentiasa 6 unit. Cari persamaan lokus titik M. [ markah ]

33 SULIT 0 0. D A rad N M C B Diagram 7 Rajah 7 Diagram 7 shows two circles with centres at A and B which intersect each other at points C and D. AC and AD are tangents to the circle with centre B. Given that AB is 6 cm and CAD = π radian. Rajah 7 menunjukkan dua buah bulatan yang berpusat A dan B dan bersilang satu sama lain di C dan D. AC dan AD adalah garis tangen kepada bulatan berpusat B. Diberi AB ialah 6 cm dan CAD ialah π radian. [ Use =.. Give your answers correct to three decimal places ] [ Guna =.. Beri jawapan anda betul kepada tiga tempat perpuluhan] Calculate Hitung (a) the length of AC, in cm [ marks ] panjang AC, dalam cm [ markah ] (b) length of arc DNC, in cm [ marks ] panjang lengkok DNC, dalam cm [ markah ] (c) the area of sector ADMC, in terms of [ marks ] luas sektor ADMC, dalam sebutan [ markah ] (d) the area of the shaded region, in terms of [ marks ] luas kawasan berlorek, dalam sebutan [ markah ]

34 SULIT (a) Given that 65% of the candidates who sat for a particular examination passed. If 9 candidates are chosen at random from the candidates who sat for the examination, find the probability that Diberi bahawa, 65% daripada calon yang menduduki suatu peperiksaan tertentu telah lulus. Jika 9 calon dipilih secara rawak daripada calon yang menduduki peperiksaan itu, cari kebarangkalian bahawa (i) exactly 6 candidates passed, tepat 6 orang calon lulus, (ii) at least candidates failed. sekurang-kurangnya orang calon gagal. [5 marks] [5 markah] (b) The time taken to produce a souvenir are normally distributed with a mean of 0 minutes and a standard deviation of 5 minutes. Find the probability of producing a souvenir Masa yang diambil untuk menghasilkan satu cenderamata adalah mengikut taburan normal dengan min 0 minit dan sisihan piawai 5 minit. Cari kebarangkalian menghasilkan satu cenderamata dalam tempoh (i) in not more than 5 minutes, tidak melebihi 5 minit, (ii) between 5 and 5 minutes. antara 5 dan 5 minit. [5 marks] [5 markah]

35 SULIT Section C Bahagian C [0 marks] [0 markah] Answer any two questions from this section. Jawab mana-mana dua soalan daripada bahagian ini.. Table shows the prices and the prices indices of five commodities A, B, C, D and E. Jadual menunjukkan harga dan indeks harga bagi lima komoditi A, B, C, D dan E. Commodity Komoditi Price (RM) for year Harga (RM) bagi tahun Price index for year 00 based on year 005 Indeks harga pada tahun 00 berasaskan tahun 005 A B y C x D E.00 z 95 Table 8 Jadual 8 Diagram 9 shows a bar chart indicating the monthly expenditure (in hundreds RM) of a family for the commodities above for the year 005. Rajah 9 ialah carta bar yang menunjukkan perbelanjaan bulanan (dalam ratus RM) sebuah keluarga bagi komoditi-komoditi di atas dalam tahun 005. Monthly Expenditure Perbelanjaan Bulanan (RM 00) 0 8 Diagram 9 Rajah 9 A B C D E Commodity Komoditi

36 SULIT (a) Find the values of x, y and z. [ marks] Cari nilai x, y dan z. [ markah] (b) Calculate the composite index for the expenditure of the commodities in the year 00 based on the year 005. [ marks] Hitung indeks gubahan bagi perbelanjaan komoditi-komoditi tersebut pada tahun 00 berasaskan tahun 005. [ markah] (c) Calculate the total monthly expenditure for the commodities in the year 00. [ marks] Hitung jumlah perbelanjaan bagi komoditi-komoditi tersebut pada tahun 00. [ markah] (d) If the indices of the commodities is expected to increase by 0% from the year 00 to the year 05, calculate the composite index for the year 05 based on the year 005. [ marks] Jika indeks komoditi-komoditi tersebut dijangka meningkat 0% dari tahun 00 ke tahun 05, hitung indek gubahan tahun 05 berasaskan tahun 005. [ markah] Diagram 0 shows a quadrilateral PQRS. Rajah 0 menunjukkan sebuah sisi empat PQRS. S 6.7 cm 0 R P cm Q Diagram 0 Rajah 0 (a) Calculate the length, in cm, of Hitung panjang, dalam cm, bagi (i) PR, (ii) PS. [5 marks] [5 markah] (b) Point P lies on PR such that PS P S. Calculate Titik P terletak pada PR dengan keadaan PS P S. Hitung (i) RPS, (ii) the area, in cm, of PP S. luas, dalam cm, bagi PP S. [5 marks] [5 markah]

37 SULIT Use graph paper to answer this question. Gunakan kertas graf untuk menjawab soalan ini. A furniture workshop has workers, A and B, doing the job of assembling and then painting desks. The time taken by these workers is as tabulated in Table. Suatu kilang perabot menggaji orang pekerja, A dan B, untuk memasang dan mengecat meja. Masa yang diambil oleh pekerja adalah dijadualkan di Jadual. Worker Pekerja Time taken (minutes) Masa diambil (minit) Assemble Paint Memasang Mengecat A B 0 0 Table Jadual In a week, worker A can complete x desks, while worker B complete y desks. These two workers work under the following constraints: Dalam suatu minggu, pekerja A dapat menyiapkan x buah meja, manakala pekerja B menyiapkan y buah meja. Kedua-dua orang pekerja itu bekerja berdasarkan kekangan berikut: I II The total time taken by the two workers does not exceed 0 hours. Jumlah masa yang diambil oleh dua orang pekerja itu tidak melebihi 0 jam. The minimum total time by the two workers for assembling the desks is hours. Jumlah masa minimum yang diambil oleh dua pekerja untuk memasang meja ialah jam. III The number of desks completed by A must not exceed by 0 that completed by B. Bilangan buah meja yang disiapkan oleh A tidak melebihi 0 buah bilangan yang disiapkan oleh B. (a) Write three inequalities, other than x 0 and y 0, which satisfy all the above constraints. [ marks] Tuliskan tiga ketaksamaan, selain x 0 dan y 0, yang memenuhi semua kekangan di atas. [ markah] (b) Using a scale of cm to 5 desks on the both axes, construct and shade the region R which satisfies all the above constraints. [ marks] Dengan menggunakan skala cm kepada 5 meja pada kedua-dua paksi, bina dan lorekkan rantau R yang memenuhi semua kekangan di atas. [ markah] (c) Use your graph from (b) to answer the following questions:. Guna graf anda dari (b) untuk menjawab soalan berikut: (i) Find the range of the number of desks which completed by worker B if worker A completed 6 desks for that particular week. Cari julat bagi bilangan meja yang disiapkan oleh pekerja B jika pekerja A menyiapkan 6 buah meja pada minggu yang tertentu.

38 SULIT 5 (ii) The profit for each desk sold is RM. Assuming that all the desks made by A and B for that particular week are sold, find the maximum total profit netted. Keuntungan bagi sebuah meja ialah RM. Anggapan bahawa semua meja yang disiapkan oleh A dan B bagi hari yang tertentu telah dijual, cari jumlah keuntungan maximum. [ marks] [ markah] 5. A particle moves along a straight line and passes through a fixed point O. Its velocity, v ms, is given by v = 6 + t t, where t is the time, in seconds, after passing through O. The particle stops instantaneously at a point R. [Assume motion to the right is positive] Suatu zarah bergerak di sepanjang suatu garis lurus dan melalui satu titik tetap O. Halajunya v ms diberi oleh v = 6 + t t di mana t ialah masa dalam saat. selepas melalui O. Zarah itu berhenti seketika di suatu titik R. ( Anggapkan gerakan ke arah kanan sebagai positif) Find Cari (a) the initial velocity [ mark] halaju awalnya [ markah] (b) the acceleration, in ms, of the particle at R, [ mark] pecutan dalam ms - bagi zarah itu di R [ markah] (c) the maximum velocity, in ms, of the particle, [ marks] halaju maksimum, dalam ms - bagi zarah itu [ markah] (d) the total distance, in m, travelled by the particle in the first seconds, after passing through O. [ marks] jumlah jarak dalam m, yang dilalui oleh zarah itu dalam saat pertama selepas melalui O. [ markah] END OF QUESTION PAPER KERTAS SOALAN TAMAT

39 SULIT 6 NO. KAD PENGENALAN ANGKA GILIRAN Arahan Kepada Calon Tulis nombor kad pengenalan dan angka giliran anda pada ruang yang disediakan. Tandakan ( ) untuk soalan yang dijawab. Ceraikan helaian ini dan ikat sebagai muka hadapan bersama-sama dengan buku jawapan. Kod Pemeriksa Bahagian Soalan Soalan Dijawab Markah Penuh Markah Diperoleh (Untuk Kegunaan Pemeriksa) A B C Jumlah

40 SULIT 7 ADDITIONAL MATHS - PAPER y y = x or x = Substitute x or y into x + y + xy 0 = 0 P a b c ( 5) ( 5) (7)( ) x (7) (7)( ) or y (7) x =.08 and N y=.5 and -.9 N x 5 P x ( ) x 9 0 N y 0( ) 5 (5x+) - y - h(x) = x - N 5 6 a b a 0 - Cosine graph shape Amplitude = cycles Translated unit and passing through (0,),(,) P P P P x y N Draw the line x y L No of solutions = N lw, l w, l w,... lw, lw, lw,... r = 6 P A GP with r = N

41 SULIT 8 b(i) b(ii) a = 5(8) or equivalent n 5(8) No. of rectangles = 6 N 58 S 6007 N P 5 5 5a 5b P L = 9.5 or F = 5 or f m = 0 P ( ) 5 0 N Mean =.688 P 7 5c N 6a AC = -a + hb h (-a + hb) or -a + b N 6 6b a + (-a + hb) a + h b 0a + 5b = m(a + h b ) where m is a constant N 7 7 m = 5 N h = N 7 kx 7a 0 7b k = -9 ( x 5) 0 5 N P f ( x) dx ( x 5) dx 0 ( ) 7 + ( ) 6 N 7c 9 (9 ) or x) dx ( P 0 0

42 SULIT 9 7 7c 5 9 ( x) dx ( x 5) dx or equivalent x ( x 5) N 8 8a (refer a diagram ) 8b(i) 8b(ii) log y P Using the correct, uniform scale and axes P All points plotted correctly P Line of best fit P log y = ( x )log q + log p P Use log q = * m or log p = * c q = 0.5 N p =.65 N Use log y = 0.70 to find x- from the graph x =.0 N 0 9 9a(i) 9a(ii) 9a(iii) 9b y 5 = (x +!) or 5 = (-) + c y = x + 6 or equivalent N 5 9 (, ) P R(5,7) P 5 0 7, 0 6, N or equivalent 0 N ( x ) ( y 5) = 5 x + y + x 0y + = 0 N cos or equivalent 0a or N BC = P 0b 6.8 N c.8 N 0d

43 SULIT N 9 6 C 6 (0.65) (0.5) a(i) 0.76 N C (0.65) (0.5) or C 0 (0.65) (0.5) P a(ii) N 5 0 P(Z ) b(i) N P( < Z < ) b(ii) N y = 6.67 P a x =. P z =.80 P b N c N.80 d N PQR = 5 P a(i) 0 8sin5 0 sin N a(ii) PS = (6.7)(7.6)cos N sin 0 b(i) RPS = N PSP = P b(ii) (8.775) sin N

44 SULIT 5 0 G ry = 6 G: (0.00,.00) fx = -.x+ a 0 5 R hx = - 8 x+7 gx = x x + 0y 0 or equivalent N 7x + 8y > 56 or equivalent N x y 0 or equivalent N Draw correctly at least straight line from *the inaqualities which involves x and y b Draw all correctly all three *straight lines N Shade the correct region R N c(i) y 7 N Maximum point (0, ) c(ii) (0) + () or (0 + ) or () RM58 N 5a 6 cms - N Accelerataion = 6t 5b (t - )(t + ) = 0-9 N 5c t = 6 or equivalent N t s = 6t t 5d () () 6() () or 6() () or N 0 0