SULIT /2 3472/2 Matematik Tambahan Kertas 2 2 ½ jam 2010 SEKOLAH-SEKOLAH MENENGAH ZON A KUCHING

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1 SULIT / / Mtemtik Tmhn Kerts ½ jm 00 SEKOLAH-SEKOLAH MENENGAH ZON A KUCHING PEPERIKSAAN PERCUBAAN SIJIL PELAJARAN MALAYSIA 00 MATEMATIK TAMBAHAN Kerts Du jm tig puluh minit JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU. 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.. Give only one nswer / solution to ech question... Show your working. It my help you to get mrks.. The digrm in the questions provided re not drwn to scle unless stted.. The mrks llocted for ech question nd su-prt of question re shown in rckets... A list of formule is provided on pges to.. A ooklet of four-figure mthemticl tles is provided.. You my use non-progrmmle scientific clcultor. Kerts soln ini mengndungi hlmn ercetk / ZON A KUCHING 00 SULIT

2 SULIT / The following formule my e helpful in nswering the questions. The symols given re the ones commonly used. x ± c ALGEBRA log log log c c m n m + n m n m n ( m ) n mn log mn log m + log n m log n log m log n log m n n log m T n + (n )d 0 n S n [ + ( n ) d] T n r n n n ( r ) ( r ) S n r r S r, 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 dx dy du du dx CALCULUS Are under curve y dx or x dy Volume generted π y dx or π x dy GEOM ETRY Distnce Midpoint x + (x, y) x r x + y xi + yj r x + y ( x y x ) + ( y ) y, + y A point dividing segment of line nx + mx ny + my (x, y), m + n m + n. Are of tringle ( ) ( ) x y + x y + x y x y + x y + x y / ZON A KUCHING 00 SULIT

3 SULIT / STATISTICS x N x σ x fx f ( x x ) N x N x w I I w n n! P r ( n r)! n n! C r ( n r)! r! 0 P(A B) P(A) + P(B) P(A B) σ f ( x x) f fx f x P(X r) Men µ np r C p q n r n r, p + q N F m L + C fm I Q 00 Q 0 σ npq z x µ σ TRIGONOMETRY Arc length, s rθ Are of sector, A sin A + cos A sec A + tn A cosec A + cot A sin A sina cosa r θ cos A cos A sin A cos A sin A tn A tn A tn A sin (A ± B) sina cosb ± cosa sinb 0 cos (A ± B) cosa cosb sina sinb tn (A ± B) sin A sin B tn A ± tn B tn Atn B c sin C + c c cos A Are of tringle sinc / ZON A KUCHING 00 [Liht seelh SULIT

4 SULIT / THE UPPER TAIL PROBABILITY Q(z) FOR THE NORMAL DISTRIBUTION N(0, ) KEBARANGKALIAN HUJUNG ATAS Q(z) BAGI TABURAN NORMAL N(0, ) z Minus / Tolk f ( z) exp z π f Q(z) / ZON A KUCHING 00 SULIT Exmple / Contoh: If X ~ N(0, ), then Jik X ~ N(0, ), mk y Q( z) f ( z) dz P(X > k) Q(k) x k z P(X >.) Q(.) O k

5 SULIT / SECTION A [0 mrks] Answer ll questions in this section. Solve the simultneous equtions x y nd x xy + y. [ mrks] f ( x ) 0 is qudrtic eqution which hs the roots nd p tht re distinct from ech other. () Write f ( x ) 0 in the form x + x + c. [ mrks] () The curve y kf ( x) cuts the y-xis t the point (0, ). Given tht the vlue of p, clculte (i) the vlue of k. [ mrks] (ii) the coordintes of the mximum point of the curve. [ mrks] A numer of wires re rrnged s shown in the Digrm. DIAGRAM The longest wire is 0 cm long. The length of ech susequent wire is 0% shorter thn the previous wire. Find () the length of the th wire [ mrks] () the totl length of the first wires. [ mrks] (c) the sum to infinity of the length of the wires. [ mrks] / ZON A KUCHING 00 [Liht seelh SULIT

6 SULIT / () Prove tht sin x tn x ( + cos x). [ mrks] () (i) Sketch the grph of y sin x for 0 x π. (ii) Hence, determine the vlue of p such tht sin x p hs only two rel solutions for 0 x π. Tle shows the distriution of mrks otined y 00 students in test. Mrks Numer of students 0 TABLE [ mrks] () Using scle of cm to 0 mrks on the horizontl xis nd cm to students on the verticl xis, construct histogrm for the given dt nd from the histogrm, estimte the vlue of mode of the mrks of the students. [ mrks] () Without drwing n ogive, clculte the medin mrk. [ mrks] Digrm shows pentgon PQTRS. P x Q y T () Express in terms of x nd y. S x DIAGRAM R (i) (ii) PR, QS. [ mrks] () Given PT m PR nd QT nqs, express PT in terms of (i) m, x nd y, (ii) n, x nd y. Hence, find the vlues of m nd of n. [ mrks] / ZON A KUCHING 00 SULIT

7 SULIT / SECTION B [0 mrks] Answer four questions from this section. () A closed cylindricl wter tnk, with rdius of r cm is to e constructed using luminium sheets to hold 0π m of wter. If the totl surfce re of the wter tnk, A m 00π, is given y A π r +. Clculte the minimum totl surfce re. r [ mrks] () Digrm shows prt of the curve y x( x) intersecting the stright line y x t point P(, ). y y x( x) P(, ) x O DIAGRAM y x (i) Find the re of the shded region. [ mrks] (ii) The region enclosed y the curve nd the x-xis is rotted through 0 o out the x-xis. Find the volume generted, in terms of π. [ mrks] / ZON A KUCHING 00 [Liht seelh SULIT

8 SULIT / Use grph pper to nswer this question. Tle shows the vlues of two vriles, x nd y, otined from n experiment. The vriles x nd y re relted y the eqution constnts. x 0 y TABLE x y where nd re () Plot log y ginst (x ), y using scle of cm to unit on the ( x )-xis nd cm to 0. unit on the log y -xis. Hence, drw the line of est fit. [ mrks] () Use your grph from () to find the vlue of (i), (ii). [ mrks] Digrm shows sector of circle with centre O. ABC is segment with the height, MC of m. C m A DIAGRAM AB is chord of the segment with the length of m nd M is the mid point of AB. Given tht r is the rdius of the sector, find () (i) the length of OM in terms of r. (ii) the vlue of r, M O [ mrks] () AOB in rdin, [ mrks] (c) the re of the shded region. [ mrks] r B / ZON A KUCHING 00 SULIT

9 SULIT / 0 Solution y scle drwing is not ccepted. Digrm 0 shows trpezium ABCD. y A. O C(m, 0). D(, ).. B(, ) DIAGRAM 0 x Given tht the eqution of the line AB is x y + 0, find () the vlue of m, [ mrks] () the eqution of AD nd hence, the coordintes of point A, [ mrks] (c) the eqution of the locus of P if P moves in such wy tht BPD is lwys right ngle. [ mrks] () The school hockey tem held trining session on penlty shooting. Ech plyer ws given trils. After the session, it ws found tht on the verge, the men for numer of gols scored y plyer is. If plyer is chosen t rndom, find the proility tht plyer (i) fils to score ny gol, (ii) scores t lest gols. [ mrks] () The ody mss of pupils in certin school follows norml distriution with men of kg nd stndrd devition of 0 kg. (i) (ii) If pupil is chosen t rndom, find the proility tht his ody mss is etween 0 kg nd 0 kg. A pupil will e plced under oesity list if his ody mss exceeds 0 kg. Estimte the numer of pupils whose nmes will pper in the list. [ mrks] / ZON A KUCHING 00 [Liht seelh SULIT

10 SULIT 0 / SECTION C [0 mrks] Answer two questions from this section. A prticle moves in stright line nd psses through fixed point O. Its velocity, v ms, is given y V t t +, where t is the time, in seconds, fter leving O. [Assume motion to the right is positive.] Find () the initil velocity, in ms, () the minimum velocity, in ms, (c) the rnge of vlues of t t which the prticles moves to the right, (d) the distnce, in m, trvelled y the prticle in the third second. [ mrk] [ mrks] [ mrks] [ mrks] The tle shows the monthly expenditure of fmily on different items in the yers 00 nd 00 nd their respective weightges, with totl weightge of. Item Expenditure (RM) Yer 00 Yer 00 Weightge Electricity 0 0 Wter 0 0 Telephone nd internet 0 No chnge x Others 0 0 () TABLE Using 00 s the se yer, clculte the price index for the (i) (ii) expenditure of electricity, expenditure of telephone nd internet in the yer 00. [ mrks] () Clculte the composite index for the expenditure on these items in the yer 00 sed on the yer 00. [ mrks] (c) The rte of increse for the expenditure on electricity nd others from the yer 00 to 0 is expected to e 0% while tht of wter nd telephone nd internet remin unchnged. Clculte (i) the expected expenditure on electricity in the yer 0, (ii) the composite index in the yer 0 sed on the yer 00. [ mrks] / ZON A KUCHING 00 SULIT

11 SULIT / The digrm shows qudrilterl PQRS. P. cm Q cm 0 o DIAGRAM S 0 cm R Given tht the re of QRS is cm nd QRS is cute. Clculte () QRS, [ mrks] () the length of QS in cm, [ mrks] (c) PQS [ mrks] (d) the re of qudrilterl PQRS. [ mrks] / ZON A KUCHING 00 [Liht seelh SULIT

12 SULIT / Use the grph pper provided to nswer this question. A fctory produces two types of commodities, A nd B. The commodities produced y the fctory stisfy the following constrints. I : The rtio of the numer of commodities A nd B produced in dy must not e less thn :. II : The totl numer of commodities A nd B produced in dy must e t lest units. III : The numer of commodity A must exceed the numer of commodity B produced in dy y not more thn. () () (c) Tking x s the numer of units of commodity A nd y s the numer of units of commodity B produced in dy, write down three inequlities, other thn x 0 nd y 0, which stisfy the ove conditions. [ mrks] By using scle of cm to unit of commodity on oth xes, construct nd shde the region R tht stisfies ll the ove conditions. [ mrks] By using your grph in (), find (i) (ii) the minimum numer of units of commodity A produced if the numer of units of commodity B produced is units, the minimum cost of production of these commodities ech dy, if the cost of production for one units of commodity A nd one unit of commodity B in dy re RM00 nd RM0 respectively. [ mrks] END OF QUESTION PAPER / ZON A KUCHING 00 SULIT

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