SULIT 1 3472/23472/2 SULIT PEPERIKSAAN SELARAS AKHIR TAHUN SEKOLAH-SEKOLAH MENENGAH NEGERI MELAKA Kelolaan PEJABAT PELAJARAN DAERAH JASIN * ALOR GAJAH * MELAKA TENGAH Dengan kerjasama : JABATAN PELAJARAN NEGERI MELAKA TINGKATAN 4 2008 ADDITIONAL MATHEMATICS Paper 2 2 2 1 hours JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU 1. This question paper consists of three sections : Section A, Section B and Section C. 2. Answer all question in Section A , four questions from Section B and two questions from Section C. 3. Give only one answer / solution to each question.. 4. Show your working. It may help you to get marks. 5. The diagram 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 2 to 3. 8. A booklet of four-figure mathematical tables is provided. 9. You may use a non-programmable scientific calculator. Kertas soalan ini mengandungi 11 halaman bercetak 3472/2 Form Four Additional Mathematics Paper 2 2008 2 2 1 hours SULIT 3472/2 [ Lihat sebelah 3472/2 SULIT 2 The following formulae may be helpful in answering the questions. The symbols given are the ones commonly used ALGEBRA 1 x = a ac b b 2 4 2 ÷ ± ÷ 2 a m × a n = a m + n 3 a m ÷ a n = a m - n 4 (a m ) n = a nm 5 log a mn = log a m + log a n 6 log a n m = log a m - log a n 7 log a m n = n log a m 8 log a b = a b c c log log 9 T n = a + (n-1)d 10 S n = ] ) 1 ( 2 [ 2 d n a n ÷ + 11 T n = ar n-1 12 S n = r r a r r a n n ÷ ÷ = ÷ ÷ 1 ) 1 ( 1 ) 1 ( , (r = 1) 13 r a S ÷ = · 1 , r <1 CALCULUS 1 y = uv , dx du v dx dv u dx dy + = 2 v u y = , 2 du dv v u dy dx dx dx v ÷ = , 3 dx du du dy dx dy × = 4 Area under a curve = } b a y dx or = } b a x dy 5 Volume generated = } b a y 2 t dx or = } b a x 2 t dy 5 A point dividing a segment of a line (x , y) = , 2 1 \ | + + n m mx nx | . | + + n m my ny 2 1 6. Area of triangle = ) ( ) ( 2 1 3 1 2 3 1 2 1 3 3 2 2 1 1 y x y x y x y x y x y x + + ÷ + + 1 Distance = 2 2 1 2 2 1 ) ( ) ( y y x x ÷ + ÷ 2 Midpoint (x , y) = \ | + 2 2 1 x x , | . | + 2 2 1 y y 3 2 2 y x r + = 4 2 2 xi yj r x y . + = + GEOM ETRY SULIT 3472/2 [ Lihat sebelah 3472/2 SULIT 3 STATISTICS TRIGONOMETRY 1 Arc length, s = ru 2 Area of sector , A = 2 1 2 r u 3 sin 2 A + cos 2 A = 1 4 sec 2 A = 1 + tan 2 A 5 cosec 2 A = 1 + cot 2 A 6 sin2A = 2 sinAcosA 7 cos 2A = cos 2 A – sin 2 A = 2 cos 2 A-1 = 1- 2 sin 2 A 8 tan2A = A A 2 tan 1 tan 2 ÷ 9 sin (A± B) = sinAcosB ± cosAsinB 10 cos (A± B) = cos AcosB sinAsinB 11 tan (A± B) = B A B A tan tan 1 tan tan ± 12 C c B b A a sin sin sin = = 13 a 2 = b 2 +c 2 - 2bc cosA 14 Area of triangle = C ab sin 2 1 1 x = N x ¿ 2 x = ¿ ¿ f fx 3 o = N x x ¿ ÷ 2 ) ( = 2 _ 2 x N x ÷ ¿ 4 o = ¿ ¿ ÷ f x x f 2 ) ( = 2 2 x f fx ÷ ¿ ¿ 5 M = C f F N L m ( ( ( ( ¸ ( ¸ ÷ + 2 1 6 100 0 1 × = P P I 7 1 1 1 w I w I ¿ ¿ = 8 )! ( ! r n n P r n ÷ = 9 ! )! ( ! r r n n C r n ÷ = 10 P(AB)=P(A)+P(B)-P(A· B) 11 p (X=r) = r n r r n q p C ÷ , p + q = 1 12 Mean , µ = np 13 npq = o 14 z = o µ ÷ x SULIT 3472/2 [ Lihat sebelah 3472/2 SULIT 4 `Section A (40 marks) Answer all questions 1. Solve the simultaneous equations : 3y – 2x = x 2 + 2y – x = 14. (5 marks) 2. A function f(x) = 1 + x k , x ≠ 0, where k is a constant. a) find f -1 (2 marks) b) Calculate the possible values of k such that 12f 2 (-1) + f -1 (2) = 0. (4 marks) 3. Score 1 3 6 x y 14 Number of Students 1 1 2 3 1 1 Table 1 Table 1 show the score distribution of nine students in a top spinning competition held in Malacca recently. The scores are arranged in the ascending order. Given that the mean score is 8 and the third quartile is 11. Find a) the value of x and y (5 marks) b) inter-quartile range of its distribution (2 marks) 4. Given that the quadratic function f(x) = -3x 2 + kx + 1 has a maximum value of 4. a) Without using differentiation method, find the possible values of k. (3 marks) b) By taking the positive value of k, sketch the graph of f(x) = -3x 2 + kx + 1 for 0 ≤ x ≤ 3. (3 marks) SULIT 3472/2 [ Lihat sebelah 3472/2 SULIT 5 5. a) Given y = x(3 – x). Find the value of x which satisfies the equation 12 2 2 + + dx dy x dx y d y = 0. (4 marks) b) The gradient function of curve is 3x + . 2 x The equation of normal to the curve at point (k , 4) is 7y + x – 3 = 0. Find the possible values of k. (4 marks) 6. a) Given log a 3 = x and log a 5 = y. Express log a ( 3 45 a ) in term of x and y. (2 marks) b) Solve 16(8 2x+1 ) = 1. (3 marks) c) Without using a calculator, evaluate log 27 81 + m m log . (3 marks) Section B (40 Marks) Answer four questions from this Section. 7. A point P moves along the arc of a circle with centre A(1, 2). The arc passes through Q(0, -3) and R(6, m). a) Find i. The equation of the locus of point P ii. The possible values of m (6 marks) b) The tangent to the circle at point Q intersects the x-axis at point K. Find the area of OQK (4 marks) SULIT 3472/2 [ Lihat sebelah 3472/2 SULIT 6 8. Diagram 1 shows a triangle PQR with points P(0, 4) and Q(4, -2). Diagram 1 Given that the gradients of PQ, QR and PR are -3m, 4m and 2 1 m respectively. Find a) the value of m (2 marks) b) the equation of PR and QR (4 marks) c) the coordinate of R (2 marks) d) the area of triangle PQR (2 marks) 9. The Diagram 2 shows a sector OAB of a circle centre 0, radius 5 cm. BN is perpendicular to OA. Diagram 2 Given that BN = 3 cm, calculate i. angle of BON in radian (4 marks) ii. the perimeter of the shaded region BNA (3 marks) iii. the area of shaded region BNA (3 marks) y P(0, 4) 0 Q(4, -2) x R SULIT 3472/2 [ Lihat sebelah 3472/2 SULIT 7 10. Diagram 3 A piece of wire of length 240 cm is bent into the shape as shown in the diagram. a) Express y in term of x (2 marks) b) Hence, shows the area, A cm 2 enclosed is given by A= 2880x -540x 2 . (4 marks) c) Find the value of x and y for which A is maximum and state its maximum area. (4 marks) 11. Table 2 shows the harvest done on a farm which cultured a very popular variety of mushroom called ‘shitake’. The data was collected from 100 polybags and their masses in grams were recorded. Mass (gm) No. of polybags 1 – 5 0 6 – 10 12 11 – 15 20 16 – 20 27 21 – 25 16 26 – 30 13 31 – 35 10 36 - 40 2 Table 2 B y cm E 13x 13x A y cm C 24x D SULIT 3472/2 [ Lihat sebelah 3472/2 SULIT 8 a) Calculate its variance. (4 marks) b) Construct its cumulative frequency table. By using a scale of 2 cm to represent 5 grams on x axis and 2 cm to represent 5 polybags on y axis, draw an Ogive to show its mass distribution. From your graph, find the percentage of the polybags which has its mass between 6 to 24 gm. (6 marks) Section C (20 Marks) Answer two questions from this Section. 12. The Table 3 shows the price indices and percentages usage of 4 types of raw materials A, B, C, and D which are the main ingredients in making a certain type of cake. Raw material Price index for year 1998 based on year 1996 Percentage of usage (%) A 125 20 B 105 30 C 120 40 D x 10 Table 3 a) Calculate i) the price of raw material C in the year 1996 if its price in 1998 was RM 5.40 (2 marks) ii) the price index of A in the year 1998 based on 1994 if its price index for the year 1996 based on 1994 is 108. (3 marks) b) If the composite index for the cost of cake production for the year 1998 based on 1996 was 116. Calculate i) the value of x (3 marks) ii) the price of a 1-kg cake in the year 1998 if the corresponding price in the year 1996 was RM 42.50. (2 marks) SULIT 3472/2 [ Lihat sebelah 3472/2 SULIT 9 13. Diagram 4 shows the bar chart for the monthly sales of five essential items sold at a sundry shop. Table 4 shows their prices in the years 2000 and 2006, and the corresponding price indices for the year 2006 taking 2000 as the base year. tems Price in the year 2000 Price in the year 2006 Price Index for the year 2006 based on the year 2000 Cooking Oil x RM2.50 125 Rice RM1.60 RM2.00 125 Salt RM0.40 RM0.55 y Sugar RM0.80 RM1.20 150 Flour RM2.00 z 120 Table 4 a) Find the values of i) x, ii) y, iii) z. (3 marks) b) Find the composite price index for the given essential items in the year 2006 based on the year 2000. (2 marks) Sugar Rice Salt Cooking Oil Flour 10 20 30 40 50 60 70 80 90 100 units Diagram 2 Diagram 4 SULIT 3472/2 [ Lihat sebelah 3472/2 SULIT 10 c) Calculate the corresponding total monthly sales for those essential items for the year 2006, if the total monthly sales in the year 2000 was only RM 150. (2 marks) d) The composite index for the year 2008 based on the year 2000 if the monthly sales of those essential items increased by 20% from the year 2006 to the year 2008. (3 marks) 14. Diagram 5 The diagram 5 shows a quadrilateral ABCD such that DAB Z is acute. Calculate a) DAB Z (3 marks) b) BCD Z (2 marks) c) the area, in 2 cm , of the quadrilateral ABCD (5 marks) SULIT 3472/2 [ Lihat sebelah 3472/2 SULIT 11 15. Diagram 6 The Diagram 6 shows a quadrilateral PQRS. PS is the longest side in PQS A and the area of PQS A is 2 14cm . Calculate a) SPQ Z (4 marks) b) the length of QS (2 marks) c) the length of RS (4 marks) END OF QUESTIONS 3 4 Area under a curve = y dx a b or dy dy du dx du dx = x dy a b 5 Volume generated = dx or y 2 a b a b = x 2 dy GEOM ETRY 1 Distance = 2 Midpoint (x . 2 y . y) = 1 . m n m n 6. r <1 1 r CALCULUS dy dv du u v dx dx dx du dv v u u dy dx dx .log a n n log a mn = n log a m a (r n ) a(1 n ) 1 r .n 9 10 11 12 Tn = a + (n-1)d Sn = 4 (am) n = a nm 5 6 7 loga mn = log am + log a n loga n [2a n ) d ] ( 1 2 T n = ar n-1 Sn = m = log am . (r 1) r 1 1 r a 13 S . y) = 1 ( x1 2 ) 2 y1 y 2 ) 2 x ( 5 A point dividing a segment of a line nx mx my 2 ny 1 2 (x . 2 v dx v 1 y = uv . The symbols given are the ones commonly used ALGEBRA 1 x= b ac b 4 2a 2 8 logab = log c b log c a 2 am an = a m + n 3 am an = a m . Area of triangle = 1 ( x1 y 2 x 2 y3 3 y11 ) x 2 y 1 x 3 y 2 1 y 3 ) x ( x 2 x x 2 .SULIT 2 3472/2 The following formulae may be helpful in answering the questions. y1 y 2 2 2 3 4 r x 2 y 2 r xi yj x2 2 y 3472/2 [ Lihat sebelah SULIT . p + q = 1 Mean . A = 3 sin 2A + cos 2A = 1 4 sec2A = 1 + tan2A 5 cosec A = 1 + cot A 6 sin2A = 2 sinAcosA 7 cos 2A = cos2A – sin2 A = 2 cos2A-1 = 1.2 sin 2A 8 tan2A = 2 tan A 1 tan A 2 9 sin (A = sinAcosB cosAsinB B) 1 2 r 2 10 cos (A = cos AcosB B) sinAsinB 11 tan (A = B) tan A tan B 1 A tan B tan 2 2 12 a b c sin A sin B sin C 13 14 a2 = b2 +c2 .SULIT 3 STATISTICS 3472/2 1 x = x N 7 8 9 2 _2 2 x = fx f ( x x ) N 2 3 = = x N w I I 11 w1 n! n Pr (n )! r n! n Cr (n )!r! r x 10 2 P(A B)=P(A)+P(B)-P(A B) p (X=r) = n Cr p rq n r . s = r 2 Area of sector .2bc cosA Area of triangle = 1 ab sin C 2 3472/2 [ Lihat sebelah SULIT . = np 4 = f ( x x) f 2 = fx f x 2 11 5 M = 1 N F 2 L C fm 12 13 14 6 P I 1 100 P0 npq x z= TRIGONOMETRY 1 Arc length. a) Without using differentiation method. b) By taking the positive value of k. Score Number of Students 1 1 3 1 Table 1 Table 1 show the score distribution of nine students in a top spinning competition held in Malacca recently. (3 marks) 3472/2 [ Lihat sebelah SULIT . Find a) the value of x and y b) inter-quartile range of its distribution Given that the quadratic function f(x) = -3x2 + kx + 1 has a maximum value of 4. The scores are arranged in the ascending order. x (3 marks) (5 marks) (2 marks) 6 2 x 3 y 1 14 1 4. Solve the simultaneous equations : 3y – 2x = x + 2y – x = 14. 1 x (2 marks) b) Calculate the possible values of k such that 12f 2(-1) + f -1(2) = 0. 2 (5 marks) 2. x ≠ where k is a constant.SULIT 4 `Section A (40 marks) Answer all questions 3472/2 1. sketch the graph of f(x) = -3x2 + kx + 1 for 0 ≤ ≤3. (4 marks) 3. A function f(x) = a) find f -1 k 0. Given that the mean score is 8 and the third quartile is 11. . find the possible values of k. a3 (2 marks) (3 marks) ) = 1. The equation of the locus of point P ii. Find the possible values of k. m c) Without using a calculator. (4 marks) 6. a) Find i. (3 marks) Section B (40 Marks) Answer four questions from this Section. The possible values of m b) The tangent to the circle at point Q intersects the x-axis at point K. Find the area of OQK (4 marks) (6 marks) 3472/2 [ Lihat sebelah SULIT . 2). m). A point P moves along the arc of a circle with centre A(1. 7. The equation of normal to the curve at point (k . 12 2 dx dx (4 marks) b) The gradient function of curve is 3x + 2 .SULIT 5. The arc passes through Q(0. a) Given log a 3 = x and log a 5 = y. a) Given y = x(3 – x). evaluate log 27 81 + log m. Express log a ( b) Solve 16(8 2x+1 45 ) in term of x and y. 4) is x 7y + x – 3 = 0. 5 3472/2 Find the value of x which satisfies the equation y d2 y dy x = 0. -3) and R(6. -2) Diagram 1 Given that the gradients of PQ. radius 5 cm. y R P(0. 4m and a) the value of m b) the equation of PR and QR c) the coordinate of R d) the area of triangle PQR x 1 m respectively. 4) and Q(4. angle of BON in radian ii. Diagram 2 Given that BN = 3 cm. -2). the perimeter of the shaded region BNA iii. 4) 0 Q(4.SULIT 8. The Diagram 2 shows a sector OAB of a circle centre 0. calculate i. QR and PR are -3m. BN is perpendicular to OA. the area of shaded region BNA (4 marks) (3 marks) (3 marks) 3472/2 [ Lihat sebelah SULIT . Find 2 (2 marks) (4 marks) (2 marks) (2 marks) 9. 6 3472/2 Diagram 1 shows a triangle PQR with points P(0. 40 No.SULIT 10. The data was collected from 100 polybags and their masses in grams were recorded. Table 2 shows the harvest done on a farm which cultured a very popular variety of mushroom called ‘shitake’. a) Express y in term of x b) Hence. 2 2 (2 marks) (4 marks) (4 marks) 11. of polybags 0 12 20 27 16 13 10 2 Table 2 3472/2 [ Lihat sebelah SULIT . shows the area. Mass (gm) 1–5 6 – 10 11 – 15 16 – 20 21 – 25 26 – 30 31 – 35 36 . A cm enclosed is given by A= 2880x -540x . A 13x B 7 3472/2 13x E y cm y cm C 24x Diagram 3 D A piece of wire of length 240 cm is bent into the shape as shown in the diagram. c) Find the value of x and y for which A is maximum and state its maximum area. SULIT a) Calculate its variance. (3 marks) b) If the composite index for the cost of cake production for the year 1998 based on 1996 was 116. B. draw an Ogive to show its mass distribution. By using a scale of 2 cm to represent 5 grams on x axis and 2 cm to represent 5 polybags on y axis. (6 marks) Section C (20 Marks) Answer two questions from this Section. find the percentage of the polybags which has its mass between 6 to 24 gm. C.50. and D which are the main ingredients in making a certain type of cake. (2 marks) 3472/2 [ Lihat sebelah SULIT . From your graph. 8 3472/2 (4 marks) b) Construct its cumulative frequency table. 12. The Table 3 shows the price indices and percentages usage of 4 types of raw materials A. Raw material A B C D Price index for year 1998 based on year 1996 125 105 120 x Percentage of usage (%) 20 30 40 10 Table 3 a) Calculate i) the price of raw material C in the year 1996 if its price in 1998 was RM 5.40 (2 marks) ii) the price index of A in the year 1998 based on 1994 if its price index for the year 1996 based on 1994 is 108. Calculate i) the value of x (3 marks) ii) the price of a 1-kg cake in the year 1998 if the corresponding price in the year 1996 was RM 42. 9 3472/2 Diagram 4 shows the bar chart for the monthly sales of five essential items sold at a sundry shop. Cooking Oil Rice Salt Sugar Flour 10 20 30 40 50 60 70 80 90 100 Diagram 2 Diagram 4 units tems Price in the year 2000 Price in the year 2006 RM2.50 RM2.55 RM1.00 Table 4 a) Find the values of i) x. (3 marks) b) Find the composite price index for the given essential items in the year 2006 based on the year 2000.80 RM2.60 RM0. iii) z.40 RM0.SULIT 13.20 z Price Index for the year 2006 based on the year 2000 125 125 y 150 120 Cooking Oil Rice Salt Sugar Flour x RM1. and the corresponding price indices for the year 2006 taking 2000 as the base year. Table 4 shows their prices in the years 2000 and 2006. ii) y.00 RM0. (2 marks) [ Lihat sebelah SULIT 3472/2 . of the quadrilateral ABCD 2 (3 marks) (2 marks) (5 marks) 3472/2 [ Lihat sebelah SULIT . Diagram 5 The diagram 5 shows a quadrilateral ABCD such that DAB is acute. (3 marks) 14. if the total monthly sales in the year 2000 was only RM 150.SULIT 10 3472/2 c) Calculate the corresponding total monthly sales for those essential items for the year 2006. Calculate a) DAB b) BCD c) the area. in cm . (2 marks) d) The composite index for the year 2008 based on the year 2000 if the monthly sales of those essential items increased by 20% from the year 2006 to the year 2008. SULIT 15. 11 3472/2 Diagram 6 The Diagram 6 shows a quadrilateral PQRS. Calculate a) SPQ b) the length of QS c) the length of RS (4 marks) (2 marks) (4 marks) END OF QUESTIONS 3472/2 [ Lihat sebelah SULIT . PS is the longest side in PQS and the area of PQS is 14cm 2 .