SULIT /2 3472/2 Matematik Tambahan Kertas 2 2 ½ jam 2009 SEKOLAH-SEKOLAH MENENGAH ZON A KUCHING

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1 SULIT 1 347/ 347/ Mtemtik Tmbhn Kerts ½ jm 009 SEKOLAH-SEKOLAH MENENGAH ZON A KUCHING PEPERIKSAAN PERCUBAAN SIJIL PELAJARAN MALAYSIA 009 MATEMATIK TAMBAHAN Kerts Du jm tig puluh minit JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU 1. This question pper consists of three sections : Section A, Section B nd Section C.. Answer ll question in Section A, four questions from Section B nd two questions from Section C. 3. Give only one nswer / solution to ech question.. 4. Show your working. It my help you to get mrks. 5. The digrm in the questions provided re not drwn to scle unless stted. 6. The mrks llocted for ech question nd sub-prt of question re shown in brckets.. 7. A list of formule is provided on pges to A booklet of four-figure mthemticl tbles is provided. 9. You my use non-progrmmble scientific clcultor. Kerts soln ini mengndungi 11 hlmn bercetk 347/ ZON A KUCHING 009 SULIT

2 SULIT 347/ The following formule my be helpful in nswering the questions. The symbols given re the ones commonly used. 1 x b ± b 4c ALGEBRA 8 log b log log c c b m n m + n 3 m n m n 4 ( m ) n mn 5 log mn log m + log n 6 m log n log m log n 7 log m n n log m 9 T n + (n 1)d 10 n S n [ + ( n 1) d] 11 T n r n 1 1 n n ( r 1) (1 r ) S n r 1 1 r 13 S 1 r, r <1, (r 1) 1 y uv, 3 dy dx dv u dx du v + du v dx dv u u dy y, dx dx, v dx v dy dx dy du du dx CALCULUS 4 Are under curve b y dx or b x dy 5 Volume generted b π y dx or b π x dy GEOM ETRY 1 Distnce Midpoint x + (x, y) 1 x 3 4 r x + y xi + yj r x + y ( x y 1 x ) + ( y1 ) y, 1 + y 5 A point dividing segment of line nx1 + mx ny1 + my (x, y), m + n m + n 6. Are of tringle 1 ( ) ( ) x y + x y + x y x y + x y + x y 347/ ZON A KUCHING 009 SULIT

3 SULIT 3 347/ STATISTICS 1 x N x 3 σ x fx f ( x x ) N x N x w1 I1 I w1 n n! P r ( n r)! n n! C r ( n r)! r! 10 P(A B) P(A) + P(B) P(A B) 4 σ f ( x x) f fx f x 11 P(X r) 1 Men µ np r C p q n r n r, p + q 1 1 N F 5 m L + C fm Q I 100 Q σ npq 14 z x µ σ TRIGONOMETRY 1 Arc length, s rθ Are of sector, A 3 sin A + cos A 1 4 sec A 1 + tn A 5 cosec A 1 + cot A 6 sin A sina cosa r θ 7 cos A cos A sin A cos A 1 1 sin A tn A 8 tn A 1 tn A 1 9 sin (A ± B) sina cosb ± cosa sinb 10 cos (A ± B) cosa cosb m sina sinb tn A ± tn B 11 tn (A ± B) 1m tn Atn B 1 sin A b sin B c sin C 13 b + c bc cos A 1 14 Are of tringle bsin C 347/ ZON A KUCHING 009 SULIT

4 SULIT / SECTION A [40 mrks] Answer ll questions in this section. 1 Solve the simultneous equtions x y nd x + xy 0 0. Give your nswer correct to 3 deciml plces. [5 mrks] Digrm 1 shows circle with centre O. 300 O S R DIAGRAM 1 P T Q PTQ is tngent to the circle t T nd PQ OQ 0 cm. Clculte () the length of the rc STR, [4 mrks] the re of the shded region. [4 mrks] 3 Tble 1 shows the frequency distribution of scores of group of plyers in gme. Score Number of plyers w 6 TABLE 1 It is given tht the medin of the distribution is 17. () Clculte the vlue of w. Hence, clculte the vrince of the distribution. [4 mrks] 347/ ZON A KUCHING 009 SULIT

5 SULIT 5 347/ 4 () 3 If the volume of cube decreses from 15 cm to 14.4 cm 3, find the smll chnge in the sides of the cube. 3x + 4 Given tht f ( x), find the vlue of f (). 3 x 5 () Prove tht sin x sin x cot x. [ mrks] Sketch the grph of y sin x for 0 x π. Hence, using the sme xes, sketch suitble stright line to find the number of solutions of the eqution x π for 0 x π. Stte the number of solutions. sin x [6 mrks] 6 () cm 5 cm 8 cm... DIAGRAM A piece of wire is cut into 15 prts which re bent to form circles s shown in Digrm. The rdius of ech circle increses by 3 cm consecutively. Clculte (i) the rdius of the lst circle, [ mrks] the re of the lst circle. [1 mrk] Digrm 3 shows rectngulr geometric pttern. A M B P N D Digrm The first rectngle is ABCD nd followed by MBNP nd so on. The length nd width of the next rectngle is hlf of the length nd width of the previous rectngle. Given tht AB 30 cm nd BC 0 cm. Find the perimeter of the seventh rectngle. 347/ ZON A KUCHING 009 SULIT C DIAGRAM 3

6 SULIT 6 347/ SECTION B [40 mrks] Answer four questions from this section. 7 Use grph pper to nswer this question. Tble shows the vlues of two vribles x nd y which re relted by nd q re constnts. y pq x+, where p () x y TABLE Convert x+ y pq to liner form of Y mx + c. [ mrks] Plot log 10 y ginst ( x + ) by using scle of cm to 1 unit on the Y-xis nd cm to 1 unit on the X-xis. Hence, drw the line of best fit. [4 mrks] (c) From the grph in, find the vlue of p nd of q. [4 mrks] 8 Digrm 4 shows tringle OPQ. The point R lies on OP nd the point S lies on PQ. The stright line QR intersects the stright line OS t point T. Q T S O R P DIAGRAM 4 Given OP : OR 4 : 3, PQ : PS : 1, OP 1x % nd OQ 9y. % () Express, in terms of x nd / or y, (i) QR, OS. If OT hos nd QT k QR, where h nd k re constnts, find the vlues of h nd k. [5 mrks] (c) Given tht x 3 % units, y 5 % units nd POQ 90 o, find PQ. [ mrks] 347/ ZON A KUCHING 009 SULIT

7 SULIT 7 347/ 9 () In certin re, 30% of the trees re rubber trees. (i) If 8 trees in the re re chosen t rndom, find the probbility tht t lest two of the trees re rubber trees. If the vrince of the rubber trees is 315, find the number of rubber trees in the re. [5 mrks] The msses of the children in the Primry One in the school hve norml distribution with men 33.5 kg nd vrince 5 kg. 150 of the children hve msses between 30 kg nd 36.5 kg. Clculte the totl number of children in Primry One in tht school. [5 mrks] 10 Solution by scle drwing is not ccepted. In Digrm 5, point T lies on the perpendiculr bisector of AB. y D A (1, 4) T B(3, ) C O DIAGRAM 5 x () Find the eqution of stright line AB. [ mrks] A point P moves such tht PA AB. Find the eqution of locus of P. (c) Locus of P intersects the x-xis t points Y nd Z. Stte the coordintes of Y nd Z. (d) Find the x-intercept of CD. [ mrks] 347/ ZON A KUCHING 009 SULIT

8 SULIT 8 347/ 11 () Given tht curve hs grdient function px + x such tht p is constnt. y 6 x is the eqution of tngent to the curve t the point (, q). Find the vlue of p nd of q. Digrm 6 shows the curve y (x 3) nd the stright line y x + intersect t point (1, 4). y y x + Clculte O (1, 4) y (x 3) DIAGRAM 6 x (i) the re of the shded region, [4 mrks] the volume of revolution, in terms of π, when the region bounded by the curve, the x-xis, the y-xis nd the stright line x is revolved through 360 bout the x-xis. 347/ ZON A KUCHING 009 SULIT

9 SULIT 9 347/ SECTION C [0 mrks] Answer two questions from this section. 1 A prticle moves long stright line nd psses through fixed point O. Its velocity, v ms 1, is given by v t 6t + 5, where t is the time, in seconds, fter pssing through O. (Assume motion to the right is positive). Find () the initil velocity, in ms 1, the minimum velocity, in ms 1, [1 mrk] (c) the rnge of vlues of t t which the prticles moves to the left, [ mrks] (d) the totl distnce, in m, trvelled by the prticle in the first 5 seconds. [4 mrks] 13 In Digrm 7, ABC is tringle. BMC nd AM re stright lines. A B M DIAGRAM 7 C () Clculte (i) AMB, the re, in cm, of tringle ABC. [7 mrks] A new tringle A B M is formed with A B AB, B M BM nd B A M BAM, find the length of A M. 347/ ZON A KUCHING 009 SULIT

10 SULIT / 14 Use the grph pper provided to nswer this question. A fctory produces two types of school bgs M nd N using two types of mchines A nd B. Given tht mchine A requires 0 minutes to produce bg M nd 30 minutes to produce bg N while mchine B requires 5 minutes to produce bg M nd 40 minutes to produce bg N. The mchines produce x units of M nd y units of N in prticulr dy ccording to the following conditions. I : Mchine A is operted for not more thn 8 hours. II : Mchine B is operted for t lest 4 hours. III : The number of units of bg M produced is not more thn twice the number of units of bg N. () Write the three inequlities for the bove conditions. (c) Using scle of cm to units for both xes, construct nd shde the region R which stisfies ll the bove conditions. Use the grph constructed in 14, to find (i) the mximum number of units of bg M tht cn be produced if the fctory produces 1 units of bg N. the mximum profit obtined if the profit from one unit of bg M nd bg N re RM 18 nd RM 0 respectively. [4 mrks] 347/ ZON A KUCHING 009 SULIT

11 SULIT / 15 Tble 3 shows the price indices nd percentge of usge of four components, P, Q, R nd S, which re the number of prts in the mking of n electronic device. Item Price index for the yer 000 bsed on the yer 1997 Percentge of usge (%) P 15 0 Q R x 30 S () Clculte TABLE 3 (i) the price of Q in the yer 1997 if its price in the yer 000 is RM 50.40, the price index of P in the yer 000 bsed on the yer 1994 if its price index in the yer 1997 bsed on the yer 1994 is 10. [5 mrks] The composite index number for the cost of production in the yer 000 bsed on the yer 1997 is 1. Clculte (i) the vlue of x, the price of n electronic device in the yer 1997 if the corresponding price in the yer 000 ws RM 88. [5 mrks] END OF QUESTION PAPER 347/ ZON A KUCHING 009 SULIT

SULIT 47/ The following formulae may be helpful in answering the questions. The symbols given are the ones commonly used. x b ± b 4ac a ALGEBRA 8 log

SULIT 47/ The following formulae may be helpful in answering the questions. The symbols given are the ones commonly used. x b ± b 4ac a ALGEBRA 8 log SULIT 47/ 47/ Matematik Tambahan Kertas ½ jam 009 SEKOLAH-SEKOLAH MENENGAH ZON A KUCHING PEPERIKSAAN PERCUBAAN SIJIL PELAJARAN MALAYSIA 009 MATEMATIK TAMBAHAN Kertas Dua jam tiga puluh minit JANGAN BUKA