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1 SULIT 47/ 47/ Matematik Tambahan Kertas ½ jam Ogos 008 SEKTOR SEKOLAH BERASRAMA PENUH BAHAGIAN PENGURUSAN SEKOLAH BERASRAMA PENUH / KLUSTER KEMENTERIAN PELAJARAN MALAYSIA PEPERIKSAAN PERCUBAAN SIJIL PELAJARAN MALAYSIA 008 MATEMATIK TAMBAHAN 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 question in Section A, four questions from Section B and two questions from Section C.. 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 to. 8. A booklet of four-figure mathematical tables is provided. 9. You may use a non-programmable scientific calculator. Kertas soalan ini mengandungi halaman bercetak 47/ SULIT

2 SULIT 47/ The following formulae may be helpful in answering the questions. The symbols given are the ones commonly used. b ± = b 4ac a ALGEBRA 8 log ab = log log c c b a a m a n = a m + n a m a n = a m - n 4 (a m ) n = a nm 5 log a mn = log am + log a n m 6 log a = log am - log a n n 7 log a m n = n log a m 9 T n = a + (n-)d 0 n S n = [a + ( n ) d] T n = ar n- n n a( r ) a( r ) S n = = r r a S = r, r <, (r ) y = uv, u y =, v dy d = dy du dy dv du = u + v d d d du dv v u dy d d, = d v du d CALCULUS 4 Area under a curve b = y d or a = b dy a 5 Volume generated = b π y d or a GEOM ETRY b = π dy a Distance = Midpoint + (, y) = r = + y i + r = 4 yj + y ( y ) + ( y ) y + y, 5 A point dividing a segment of a line n + m ny + my (,y) =, m + n m + n 6. Area of triangle = ( y + y + y ) ( y + y + y ) 47/ SULIT

3 SULIT 47/ STATISTICS = = σ = = 4 σ = = 5 M = P(AB)=P(A)+P(B)-P(AB) p (X=r) =, p + q = Mean, = np 4 z = TRIGONOMETRY Arc length, s = r Area of sector, A = sin A + cos A = 4 sec A = + tan A 5 cosec A = + cot A 6 sina = sinacosa 7 cos A = cos A sin A = cos A- = - sin A 8 tana = 9 sin (AB) = sinacosb cosasinb 0 cos (AB) = cos AcosB sinasinb tan (AB) = a = b +c - bc cosa 4 Area of triangle = [ Lihat sebelah 47/ SULIT

4 4 SECTION A [40 marks] Answer all questions in this section.. Solve the simultaneous equations h + k = 5 and k h = 7. Give your answers correct to three decimal places. [5 marks]. Solution to this question by scale drawing will not be accepted. Diagram shows a straight line PRQ y S P R 0 Q Diagram The points P and Q intersects the y ais and the -ais. R is the midpoint of line PQ. The equation of line PR is + y = 0 (a) Find (i) the coordinates of R. [ marks] (b) (ii) the area of triangle OPS if SR was etended to the origin and PS is parallel to the -ais. [ marks] A point X moves such that its distance from P is always of its distance from Q [ marks]

5 SULIT 5 47/ Diagram shows a histogram representing the distribution of the heights of 0 students in a class. Number of students Calculate (a) the value of f Height (cm) Diagram [ marks] (b) the value of f [ marks] (c) the standard deviation of the number of patients visiting the clinic. [ marks] 4. Diagram shows the particles A and B are projected simultaneously towards each other from the opposite end of a straight tube, 9 m long. A Diagram Particle A travels 47 cm in the st nd rd second, 45 cm in second, 4 cm in the second, etc. Particle B travels 5 cm in the st nd second, 4 cm in second, cm in rd the second, etc. Find how long it takes for both particles to meet. [6 marks] B [ Lihat sebelah 47/ SULIT

6 5. (a) Find the equation of the normal to the curve y at the point (, -). (b) [ marks] A cylindrical tank with a circular base of radius 0.5 m is filled with h m of turpentine. If the turpentine is evaporating at a uniform rate of 0.00 m s -, find the rate of change in the level of turpentine. Leave your answer in terms of. marks] [ 6. (a) Prove the identity cos 4 θ - sin 4 θ = cos θ. [ marks] (b) By sketching the graph of y = sin 4 θ - cos 4 θ and a suitable straight line on the same ais for 0 θ π, state the number of solutions for the equation sin 4 θ - cos 4 θ =. [5 marks] 47/ SULIT

7 SULIT 7 47/ SECTION B [40 marks] Answer four questions from this section. 7. Use the graph paper provided to answer this question. Table shows the values of two variables, and y, obtained from an eperiment. Variables and y are related by an equation y pk, where p and k are constants y Table (a) Plot log 0 y against ( ), by using the scale of cm to unit on the -ais and cm to 0. unit on the y-ais. Hence, draw the line of best fit. [5 marks] (b) Use the graph in 7 (a) to find the value of (i) p, (ii) k [5 marks] [ Lihat sebelah 47/ SULIT

8 8. In the Diagram 4, OP = 8 p, OQ = 0 q and PS = 4 q. Q 0 T S 4 O 8 P Diagram 4 (a) Epress each of the following vectors in terms of p and/or q. (i) OS (ii) QP [4 marks] (b) Given that OT = aos and QT = bop, epress OT in terms of (i) a, p and q (ii) b, p and q [ marks] (c) Hence, find the values of a and b. [ marks] 47/ SULIT

9 SULIT 9 47/ 9. Diagram 5 shows sector AOB and sector OED with centre O and E respectively. OCE is a right angle triangle. (Use =.4). A C B O D Diagram 5 θ E Given that AOB = 50 0, OA = 0 cm, OE = 8 cm and OB : BC = :. Calculate (a) θ in radian, [ marks] (b) perimeter of the shaded region in cm, [4 marks] (c) area of the shaded region in cm. [4 marks] [ Lihat sebelah 47/ SULIT

10 0. (a) Given y = (t )(t + ) and = t +, find dy d in terms of. [ marks] (b) Diagram 6 shows the curve y = ( )( 5) intersects with a straight line y p at the point (, 8). y Find, (, 8) y = + p Diagram 6 (i) the value of p, (ii) the area of the shaded region. [5 marks] O (iii) the volume generated when the region y bounded = ( - )( by - 5) the straight line =, y = + p, the and y-aes is revolved 60 o on the -ais. [ marks]. (a) In one housing area, 0% of the residents are senior citizens. i) If a sample of seven persons is chosen at random, calculate the probability that at least two of them are senior citizens. [ marks] ii) If the variance of the senior citizens is 8, find the number of residents in the housing area. [ marks] b) In a field study, it is found that the mass of a student is normally distributed with a mean of 50 kg and standard deviation 5 kg. i) If a student is selected randomly, calculate the probability that his mass is less than 4 kg. [ marks] ii) Given that % of the students have a mass greater than m kg, find the value of m. 47/ SULIT

11 SULIT 47/ SECTION C [ marks] [0 marks] Answer two questions from this section.. A particle moves along a straight line from a fied point O. Its velocity, V ms -, is given by V = 5t t, where t is the time, in seconds, after leaving the point O. (Assume motion to the right is positive) Find a) the maimum velocity of the particle, [ marks] b) the distance travelled during the fourth second, [ marks] c) the value of t when the particle passes the point O again, [ marks] d) the time between leaving O and when the particle reverses its direction of motion. [ marks] Food Price Inde, I Weightage, w Fish 0 Chicken m Rice 0 5 Meat 05 n Prawn 5 Table Table shows the price indices and weightage of 5 types of food consumed in the year 007 using 006 as the base year. The composite inde of these 5 items in the year 007 using 006 as the base year is 7 and w =. a) Calculate the values of m and n. [4 marks] b) Find the price of a kilogram of rice in the year 007 if its price in the year 006 is RM.50. [ marks] [ Lihat sebelah 47/ SULIT

12 c) Given that the projected rate of change in the prices of all the foods from 007 to 008 is the same as that from 006 to 007. Find (i) (ii) the composite inde number of these foods in the year 008, using the year 006 as the base year. the amount to be paid for these foods in the year 008 if the amount paid for these items in 006 was RM650. [4 marks] 4. An institution offers two types of Mathematics courses, Calculus and Statistic. The number of students taking Calculus course is and the number of students taking Statistic course is y. The number of students taking the Mathematics courses is based on the following constraints: I : II : The ratio of the number of students taking Calculus course to the number of students taking Statistic course is not more than 80 : 0. The total number of students taking Mathematics courses is less than or equal to 80. III : The number of students taking Statistic course is at least 0 IV: The number of students taking Calculus course is more than 0. (a) (b) (c) Write four inequalities, other than 0 and y 0,which satisfy all the above constraints. [4 marks] Using a scale of cm to 0 students on both aes, construct and shade the region R which satisfies all of the above constraints. [ marks] By using your graph from (b),find (i) (ii) the range of the number of students for Calculus course if the number of students for Statistics course is 0. the maimum eamination fee that can be collected if the eamination fees for Calculus and Statistics courses are RM 00 and RM 400 respectively. [ marks] 47/ SULIT

13 SULIT 47/ 5. A E 5 cm D 6 cm B 7 o C Diagram 7 Diagram 7 shows the triangle ABC, where D is a midpoint of the line AC and ABC is an obtuse angle. Triangle CDE is an isosceles triangle such that CD = DE. Given that the length of AC = 0 cm, EC = 6 cm, AB = 5 cm and ACB 7 o. a) Calculate the ABC [ marks] b) Find the area of triangle ABC [ marks] c) If the line CB is etended to point F, find the length of the shortest distance from point A to line CF. [ marks] d) Calculate the CDE [ marks] END OF QUESTION PAPER [ Lihat sebelah 47/ SULIT

14 47/ Matematik Tambahan Kertas ½ jam Ogos 008 SEKOLAH BERASRAMA PENUH BAHAGIAN PENGURUSAN SEKOLAH BERASRAMA PENUH / KLUSTER KEMENTERIAN PELAJARAN MALAYSIA PEPERIKSAAN PERCUBAAN SIJIL PELAJARAN MALAYSIA 008 MATEMATIK TAMBAHAN Kertas Dua jam tiga puluh minit MARKING SCHEME Skema Pemarkahan ini mengandungi halaman bercetak

15 QUESTION NO. SOLUTION MARKS h + k = (i) k - h = (ii) From (i) h = 5 - k (iii) k - (5 - k) = 7 k + 6k - = 0 Eliminate h or k Use k = -b b - 4ac a ()(-) () =.568, Solve the quadratic equation by using the quadratic formula or completing square h = -0.6,.6 5 a i) P( 0, 6 ), Q (4, 0 ) R = (, ) Find coord. P,Q and use midpoint formula ii) OPS = or 4 6 = b) PX = XQ P = ( 0) + ( y 6) = ( 4) + ( y 0) = y 8 48y P P P P

16 a) See 4, 48, 5, 58, 6 and 68 P 4(4) + 5(48) + 8(5) + 9(58) +(6) + (68) 465 b) 4(4 ) + 5(48 ) + 8(5 ) + 9(58 ) + (6 ) + (68 ) 795 c) σ = f f Find the value of min Use s.d formula 8 4 a 47, d = or a = 5, = A = A B d B P n [ (47) + ( n )( ) ] n or [ (5) + ( n )( ) ] n [ (47) + ( n )( ) ] n + [ (5) + ( n )( ) ] = 900 n 47n = 0 Factorize/solve Accept listing method n = a) Find dy d correctly and subtitude = Use mm = y =

17 4 b) V 0.5 h dv 0.5 dh dv dv dh dt dh dt 0.00 (0.5 ) dh dt 50 dh dt 6 6 a) 4 4 cos sin (cos sin )(cos sin ) (cos sin )() cos b) y P P Graph of cos Min/ma Graph of = cos - & Draw a straight line of y P period for 0 Number of solutions = 8 P P

18 5 7 a) log 0 y = log 0 p + ( ) log 0 k P log y b) Refer to the graph Accept listing method Plot all the points correctly Draw the best line fit y-intercept = log 0 p t = 4 gradient = log 0 k p = 5.6 k =

19 Answer for question 7 6 Log 0 y (a) - 456log y A cce

20 7 8 a i) ii) b i) ii) uuur uuur uuur OS OP OS = 8p+4q % % uuur uuur uuur QP QO OP = 8p-0q % % uuur OT a(8 p 4 q) 8ap 4aq % % uuur uuur uuur uuur QT OT OQ bop uuur OT 8bp 0q P 4 c) 8bp 8ap % % b = a 4 aq = 0q % % 4 a = a =, b Comparison Solve Both correct 0 9 a) 5 tan o =.08 rad b) 50 S AB = 0 ( π ) = P 4 Perimeter of OAB = or EC = = 7 Perimeter of shaded region = = 4.76

21 c) Area OAB Area OAB or Area OED 4 Area OED Area area shaded region

22 9 0 a) dy d = t or dt dt dy d = dy dt dt d = t = = = or y b i) ii) p = 6 P Area of shaded region (6 8)() ( 8 5) d Area of triangle Integration L + L 5 iii) Volume ( 6) d 0 (4 4 6) d Use 0 Integrate 0

23 0 a i) ii) b i) ii) 0 a) a = 5 6t 5 6t = 0, t = 5 6 s 75 V ma = 8 ms - // 8.75 // 4 4 a = 0 Substitute t into v b) c) d) S = 5 t t Distance travelled = S 4 S 5 t t = 56 - = 40 S 4 -S 5 m = 0, S=0 t = 5 s 5 t t = 0 Time between = 5 s Accept listing method V=0 0

24 a) n = P 0()+m()+0(5)+05()+5() 7 m = 08 b) Q07 00 = 0 Q Q 07 = 00 Q = c i) I I I = X I I I = = 6.89 ii) Q08 X 00 = Q 08 = Refer to the graph

25 Answer for question 4 (a) I. y II. III. IV (b) Refer to the graph, graph correct graphs correct 90 Correct area (c) ma point ( 50,0 ) 80 i) ii) k = y 70 Ma fees = 00() + 400(59) = RM 7, (,59)

26 5 a) 5 0 sin 7 sin ABC sin ABC ' ABC 65. or 65 4 ABC ' 4.77 or 4 46 b) BAC P Area of triangle ABC (0)(5)sin * c) sin d) CDE (5)(5) cos ' cos CDE 7.74 or

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