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1 7/ Additioal Mathematics Paper September 00 Jam NAMA :. TINGKATAN :. PEPERIKSAAN PERCUBAAN BERSAMA SIJIL PELAJARAN MALAYSIA 00 ANJURAN BERSAMA PERSIDANGAN KEBANGSAAN PENGETUA-PENGETUA SEKOLAH MENENGAH CAWANGAN NEGERI PERLIS DAN JABATAN PELAJARAN PERLIS ADDITIONAL MATHEMATICS Paper Kertas Two hours Dua jam JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU. Tulis ama da tigkata ada pada ruaga yag disediaka.. Kertas soala ii adalah dalam dwibahasa.. Soala dalam bahasa Iggeris medahului soala yag sepada dalam bahasa Melayu.. Calo dibearka mejawab keseluruha atau sebahagia soala sama ada dalam bahasa Iggeris atau bahasa Melayu. 5. Calo dikehedaki membaca maklumat di halama belakag kertas soala ii. Kertas soala ii megadugi halama bercetak Utuk Keguaa Pemeriksa Soala Markah Peuh Markah Diperoleh Jumlah 0 7/ MOZ@C

2 The followig formulae may be helpful i aswerig the questios. The symbols give are the oes commoly used. Rumus-rumus berikut boleh membatu ada mejawab soala. Simbol-simbol yag diberi adalah yag biasa diguaka. ALGEBRA b b ac a a a m m ( a a a m ) a a a m m m 5 log m log m log a m 6 loga loga m loga 7 log m log m a a a a logc b loga log a 9 T a ( ) d 0 S [a 9 ) d] T ar c a( r ) r a r r S, r a S, r r y uv, u v CALCULUS dy d dy d dv u d du d v u y, dy d dy du du d Area uder a curve Luas di bawah legkug = b a y d = b dy a v or (atau) v du d dv d 5 Volume geerated/isipadu jaaa b a = y d or (atau) = b a dy Distace/Jarak = GEOMETRY Midpoit/Titik tegah y (, y) =, y ( y ) ( y ) A poit dividig a segmet of a lie/titik yag Membahagi suatu tembereg garis m y my (, y) =, m m Area of triagle/ Luas segitiga = ( y y y) ( y y y) 5 6 r rˆ y i y j y 7/ MOZ@C

3 STATISTICS/STATISTIK N TRIGONOMETRY/ TRIGONOMETRI Arc legth, s r Pajag legkuk, s j f f ( ) = N f ( ) = f N F 5 m L C fm Q 6 I 00 Q 7. I P r 0 W I i W! ( )! i i N f 9.! C r ( )! r! 0 P( A B) P( A) P( B) P( A B) r r P( X r) C p q, p q Mea/ Mi, p pq Z X r f Area of sector, A r Luas sector, L j si A cos A sec A ta A 5 cos ec A cot A 6 si A si Acos A 7 cos A cos A si A = cos A = si A sia B si Acos B cos Asi B 9 cos A B cos Acos B si Asi B 0 taa B ta a si A ta A ta B ta A ta B ta ta A b si B A A c si C a b c bc cos A Area of triagle/ Luas segitiga = absi C 7/ MOZ@C

4 Diagram shows the fuctio f ( ). Rajah meujukka fugsi f ( ). k 6 f() 6 6 (a) State the value of k, Nyataka ilai k, Diagram /Rajah (b) By usig the value of k i (a), rewrite the above relatio i ordered pairs. Dega megguaka ilai k dalam (a), tulis semula hubuga di atas dalam pasaga bertertib. [ marks /markah] Aswer/Jawapa :(a) k = (b) Give the fuctio f :. Diberi fugsi f :. Fid Cari (a) f (), (b) the value of which maps oto itself. ilai bagi yag memetaka kepada diriya sediri. [ marks /markah] Aswer/Jawapa :(a) f () = (b) = 7/ MOZ@C

5 . Give the fuctio f : ad g :, 0. Diberi fugsi f : da g :, 0. Fid gf (5). Cari gf (5). [ marks/markah] Aswer/Jawapa : (a) Solve the followig quadratic equatio : Selesaika persamaa kuadratik berikut : 5 0 (b) The quadratic equatio p 0, where p is a costat, has two differet roots. Persamaa kuadratik p 0, di maa p ialah pemalar, mempuyai dua puca berbeza. Fid the rage of values of p. Cari julat ilai p. [ marks/ markah] Aswer / Jawapa : (a) (b) 7/ MOZ@C 5

6 5 Fid the rage of values of for which ( )( ) ( ) Cari julat ilai bagi ( )( ) ( ) [ marks/markah] Aswer / Jawapa : 6 The quadratic equatio f ) h k miimum poit (, ). (, where h ad k are costats, has a Persamaa kuadratik f ( ) h k, di maa h da k adalah pemalar, mempuyai titik miimum (, ) Fid Cari (a) the value of h ad of k ilai h da k (b) the equatio of the ais of symmetry persamaa bagi paksi simetri [ marks/markah] Aswer/ jawapa : (a) h = k = (b) 7/ MOZ@C 6

7 7 Solve Selesaika = = [ marks/markah] Aswer / Jawapa : Solve + log ( ) log Selesaika + log ( ) log [ marks/markah] Aswer / Jawapa : 7/ MOZ@C 7

8 m 9 Give log m r ad log t, epress log i terms of r ad / or t. 6 Diberi log m r da log t, ugkapka log m 6 dalam sebuta r da/ ataut [ marks/markah] Aswer / Jawapa : 0 The fourth terms of a arithmetic progressio is 5 ad the sum of the first eight terms of the progressio is. Sebuta keempat bagi suatu jajag aritmetik ialah 5 da hasil tambah lapa sebuta pertama jajag itu ialah. Calculate Hitug (a) the first term. sebuta pertama (b) the commo differece of the progressio. beza sepuya jajag itu [ marks/markah] Aswer / Jawapa : (a) (b) 7/ MOZ@C

9 . Give a geometric progressio,,,., 56. Calculate the umber of terms of the progressio. Diberi suatu jajag geometri,,,.., 56. Hitugka bilaga sebuta bagi jajag itu. [ marks/markah] Aswer / Jawapa : The first three terms of a geometric progressio are m, m, m.fid Tiga sebuta pertama suatu jajag geometri ialah m, m, m. Carika (a) the value of m ilai m (b) The commo ratio of the progressio isbah sepuya jajag itu. [ marks/markah] Aswer / Jawapa : (a) m = (b) 7/ MOZ@C 9

10 Diagram shows a straight lie graph of y agaist. Rajah meujukka graf garis lurus y melawa. y (6, 6) (, ) O Diagram/ Rajah Epress y i terms of. Ugkapka y dalam sebuta. [ marks/markah] Aswer / Jawapa : 7/ MOZ@C 0

11 Diagram shows a semicircle ABC with cetre O. Rajah meujukka satu semibulata ABC berpusat di O B C O 0.95rad Diagram/Rajah A The legth of arc BC is cm ad the agle AOB is 0.95 radias. Pajag legkok B ialah cm da sudut AOB ialah 0.95 radia. Fid Cari (a) the legth of OB pajag OB (b) the area of sector OAB. luas sektor OAB [use / guaka =.] [ marks/markah] 7/ MOZ@C Aswer / Jawapa : (a) (b)

12 5 I Diagram, the poit O is the origi ad the straight lie AB is perpedicular to the straight lie BC. Rajah meujukka titik o ialah titik asala, da garis lurus AB berserejag dega garis lurus BC y B (, 5) C (-, ) A O Diagram/ Rajah Fid the equatio of the straight lie AB. Cari persamaa garis lurus AB [ marks/markah] Aswer / Jawapa : 7/ MOZ@C

13 6 Diagram 5 shows two vectors, PO ad QO. Rajah 5 meujukka dua vektor PO da QO y Q(6, ) 0 P(-, -5) Diagram / Rajah 5 It is give that QP mi j. Fid the values of m ad of. Diberi bahawa QP mi j. Cari ilai m da ilai. [ marks/markah] Aswer / Jawapa : m = = 7. Give that O(0, 0), A(, ) ad B (, ). Diberi O(0, 0), A(, ) da B (, ). Fid i terms of uit vectors i ad j, Cari dalam sebuta vektor uit i da j (a) AB (b) the uit vector i the directio of AB vektor uit dalam arah AB [ marks/markah] 7/ MOZ@C Aswer / Jawapa : (a) (b)

14 Solve the equatio cos si for 0 o 60 o. Selesaika persamaa cos si bagi 0 o 60 o. [ marks/markah] Aswer / Jawapa : ( ) 9 Differetiate with respect to. ( ) Bezaka terhadap. [ marks/markah] Aswer / Jawapa : 0 The surface area of a sphere icreases at the rate of 5cm s. Luas permukaa sebuah sfera bertambah dega kadar 5cm s. Fid the rate of chage of the radius, i cm s at the istat whe its radius is 0 cm Cari kadar perubaha jejari sfera itu dalam cm s pada ketika jejariya ialah 0 cm. [Surface area of sphere / Luas permukaa sfera = r, use / gua =.] [ marks/markah] 7/ MOZ@C Aswer / Jawapa :

15 Give that g ( ) d 6. Fid Diberi g ( ) d 6. Cari (a) g( ) d (b) ( g( ) 9) d [ marks/markah] Aswer / Jawapa : (a) (b) A set of positive itegers cosists of 7,,, m, 5,, ad. The mea for this set of itegers is 5. Satu set iteger positif terdiri daripada 7,,, m, 5,, da. Mi bagi set iteger itu adalah 5. (a) State a possible value of m ad of. Nyataka satu ilai yag mugki bagi m da. (b) Fid the mea for m ad. Cari mi utuk m da. [ marks/markah] 7/ MOZ@C 5 Aswer / Jawapa : (a) m = = (b)

16 A team of 5 studets are to be chose from a group of boys ad 6 girls. Satu pasuka yag terdiri daripada 5 orag pelajar dipilih daripada sekumpula orag pelajar lelaki da 6 orag pelajar perempua Fid Cari (a) the umber of ways to form the team, bilaga cara pasuka itu dibetuk, (b) the probability of formig a team of all boys. kebaragkalia pasuka yag dibetuk itu adalah terdiri daripada semua pelajar lelaki [ marks/markah] Aswer / Jawapa : (a) (b) Diagram 6 shows letters ad 5 digits. Rajah 6 meujukka huruf da 5 digit. A B C D 5 6 Diagram 6 / Rajah 6 A code is to be formed usig the letters ad the digits i diagram 6. The code must cosist of letters followed by digits. Satu kod dibetuk megguaka huruf da digit dalam rajah 6. Kod itu mesti mempuyai huruf diikuti dega digit. Fid the umber of ways the code ca be formed if o letter or digit is repeated i each code. Cari bilaga cara kod itu dapat dibetuk jika tiada huruf da digit boleh diulag dalam setiap kod. [ marks/ markah] Aswer / Jawapa : 7/ MOZ@C 6

17 5 The height of studets i a school has a ormal distributio with a mea of cm ad a stadard deviatio of 5 cm. Ketiggia pelajar-pelajar di sebuah sekolah mempuyai tabura ormal dega mi cm da sisiha piawai 5cm. Fid Cari (a) the Z score for the height of 56 cm, skor Z bagi ketiggia 56cm, (b) the value of t whe P( t < z < t) = 0.. ilai bagi t bila P( t < z < t ) = 0.. [ marks/ markah] Aswer / Jawapa: (a) (b) t = END OF QUESTION PAPER 7/ MOZ@C 7

18 INFORMATION FOR CANDIDATES MAKLUMAT UNTUK CALON. This questio paper cosists of 5 questios. Kertas soala ii megadugi 5 soala.. Aswer all questios. Jawab semua soala.. Give oly oe aswer for each questio. Bagi setiap soala beri satu jawapa sahaja.. Write your aswer i the spaces provided i the questio paper. Jawapa ada hedaklah ditulis pada ruag yag disediaka dalam kertas soala ii. 5. Show your workig. It may help you to get marks. Tujukka lagkah-lagkah petig dalam kerja megira ada. Ii boleh membatu ada utuk medapatka markah. 6. If you wish to chage your aswer, cross out the aswer that you have doe. The write dow the ew aswer. Jika ada hedak meukar jawapa, batalka dega kemas jawapa yag telah dibuat. Kemudia tulis jawapa yag baru. 7. The diagrams i the questios provided are ot draw to scale uless stated. Rajah yag megirigi soala tidak dilukis megikut skala kecuali diyataka.. The marks allocated for each questio are show i brackets. Markah yag diperutukka bagi setiap soala ditujukka dalam kuruga. 9. A list of formulae is provided o pages ad. Satu searai rumus disediaka di halama da. 0. The ormal distributio table is provided. Sifir tabura ormal disediaka.. You may use a o-programmable scietific calculator. Ada dibearka megguaka kalkulator saitifik yag tidak boleh diprogram. 0. Had i this questio paper to the ivigilator at the ed of the eamiatio. Serahka kertas soala ii kepada pegawas peperiksaa pada akhir peperiksaa. 7/ MOZ@C

19 Mark Scheme For Additioal Mathematics Paper Trial Eamiatio (SPM) 00 Questio Distributio Of Marks Sub Marks Total Marks. (a) (b) (,) (,) (,6) (6,6). (a) (b) B : =. B : B : ( ) + f = or f (5) =. (a) 5, (b) p > B : ( ) ()( p) > 0 5., 5 B : or equivalet 5 B : ( + )( 5) 0 6. (a) h =, k = (b) =, 7. B : = 9 or equivalet B : or ( + ) MOZ@C

20 . 7 B : log ( ) = log or = log B : log or 9. r + t r t log B : log + log - log log m log log 6 B : log + log log B : log m + log log 6 0. (a) 6 (b) 7 B : 5 = a + d or = (a + 7d). 9 B : = or,,,,6,,6,,56.. (a) (b) B : + or equivalet (substitute m to fid r). y = + B : y = + B : gradiet m = or y-itercept =. (a) 5.06 cm B : π or π or (b).5 cm B : * * follow his (a) aswer MOZ@C

21 5. y = + 6 or equivalet B : y 5 = ( ) or equivalet usig y = m + c B : gradiet = 6. m =, = 9, 7. (a) i + 6 j B : i 5 j 6i j i + j (b) or equivalet 0 B : i+ 6 j (a) 90,.,.9 or ' 0 ', 9, ' B : 90,. or 90, 9 B : ( si + )(si ) = 0 or si =, B : ( si ) + si = 9. ( + ) ( B : + )(6) (+ ( + ) )() B : 6 for differetiatig + or for differetiatig + 0. or cm s - 6π dr B : dt = π (0) 5 da B : = 5 dt. (a) 6 (b) 6 or 6.75 ( 6) 9 + B : [ ] MOZ@C

22 B : 9 g ( ) + d. (a) m =, =, (b). (a) 00 B : C 5 56 (b) or 00 B : C 5. 0 B : 60 or 5 or 5 P P B : P or 5 P or.. or. 5! 5. (a) 0.5 (b) 0.5 B : P( z > t) = 0.05 or P ( z < t) = t t B : 0.95 MOZ@C

23 ANSWER SCHEME TRIAL 0 PAPER ADD MATHS SECTION A 5. = 5 y or y = M (5 y) + y y(5 y) = 5 + ( 0) ± y = ( 7)( ) ( 0) ± = or (y )( y ) = 0 or or or 5 = ( 0) = 0 () ( 0) () ()() ()(7) y =, y = M 7 =, = M. (a) T6 = 00 + (6 )(50) M = 50 M M M dy. (a) = p + d p() + () = p = (b) M M M m = M y 5 = ( ) M y = + M or equivalet.(a) (i) r ( π θ ) = 55 ad r ( π θ ) = M Solve simultaeous eq. M r = 5 M (ii) θ =. or. or. or 07.9 o or 07.9 o or 07.7 o M (b) AP = 0 or OP = 5 M (5) (.) or (5 )(.) M (5) (.) (5 )(.) M =. or. or M 0 M = M (b) S = [ (00) + (0 )(50) ] 0 (c) [ (00) + ( )(50) ] = 50 = 9 Depth = 9 = 57 M M M MOZ@C

SULIT /. This questio paper cosists of questios. Kertas soala ii megadugi soala.. Aswer all questios. Jawab semua soala.. Give ol oe aswer for each qu

SULIT /. This questio paper cosists of questios. Kertas soala ii megadugi soala.. Aswer all questios. Jawab semua soala.. Give ol oe aswer for each qu SULIT / NAMA ANGKA GILIRAN PEPERIKSAAN PERCUBAAN SPM TAHUN 9 / ADDITIONAL MATHEMATICS Kertas September 9 jam Dua jam JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU. Kertas soala ii adalah dalam dwibahasa..