l I l trjfp,i l I I a\ 3l rrr I lltl!l 5 i 5I iluiibeb OF OUESflOHS ;,.Eos$,,,,','"{psS'. ffii Z. Mean _ sum of (frequency x midpoint) Min

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1 FORM 4 Statistics lll KERTAS I il*l LlI f l I I : l -J l I l trjfp,i l I I a\ 3l rrr I lltl!l i I 6.1 Class lntervals iluiibeb OF OUESflOHS ;,.Eos$,,,,','"{psS'. ffii t.,,.,ffi.' &0S,", "-ffii,.2&11',1 6.2 Mode and Mean of Grouped Data Histograms 6.4 Frequency Polygons 6. Cumulative Frequency Measures of Dispersion sum of data 1. Mean = number of data hasil tambah nilai data Min = bilangan data Z. Mean _ sum of (frequency x midpoint) sum of frequencies Min hasil tambah (kekerapan x titik tengah) = hasil tambah kekerapan ffi Mode and Mean of Grouped Data SPM ffi Question fr 1 Table 1 is a frequency table which shows the mass of fish caught by 20 fishermen. Jadual I ialah jadual kekerapan yang menunjukkan jisim tangkapan ikan oleh20 orang nelayan. 'trv[flss,,,of, sh-*nu tftg) lishn tangkapan ikan {kg} ll - 13 t4-t6 t Table I Jadual I FFeQuenc}-,Kcfurcfr*n, Calculate the mean mass, in kg, of fish caught by a fisherman Hitungkdn min jisim, dalam kg, tangkapan ikan bagi seorang nelayan itu. A 16.1 c 18.1 B t7.l D GEJ Gumulative Frequency SPM mg Question N 2 Table I is a cumulative frequency table which shows the scores obtained in a competition. Jadual I ialah jadual kekerapan longgokan yang menunjukkan skor yang diperoleh dalam suatu pertandingan.,.s(ore,,,,,.:.s&or, I CumulatiY 'frfquencf Kekerupan longgolun, t6 Table I Jadual I If the mode of the scores is 1, state one possible value of x. Jika skor mod ialah l, nyatakan satu nilai yang mungkin bagi x. A2 C B3 D6 x 9

2 KERTAS 2 ill' :, tfl. I I t i I r I I =Ir{ IEIr,I I I I L't'J r1 I lj, I :l I : l: 6.1 Class lntervals 6.2 Mode and Mean of Grouped Data 6.3 Histograms 6.4 Frequency Polygons ffis,i.,,',l. m:l$i:ri,ffi?4 ltio"',.${t. i*,..$'. S...,ir.. B I E.,,B.*,,,,"8i :t E 1 1 1i_ i,.4iit :. 1 T -l_i_ 6. Cumulative Frequency 2i 6.6 Measures of Dispersion t_l_ 4i T 1 Z 1i_ 2t tr 2 3. I Mathematical Formulae and Facts ', Class interval - (upper.tus ti-it - lowe, class limitj + I Selang kelas = (had atas kelas - had bawah kela,s) + 1 Modal class = class interval with the highest frequency Kelas mod = selang kelas dengan kekerapan tertinggi r,^^- sum of (frequency x midpoint) _ sum of (classmark x frequency) sum of frequencies Min = hasil tambah (kekerapan x titik tengah) oro, hasil tambah kekerapan Interquartile range = third quartile - first quartile Julat antara kuartil = kuartil ketiga - kuartil pertama -E?.hag,?n ffi Sil Class lntervals ftej Mode and Mean of Grouped Data ffi Histograms sum of frequencies hasil tambah (nilai titik tengah kelas x kekerapan) hasil tambah kekerapan dhw Frequency Polygons IEil Gumutative Frequency #tr Measures of Dispersion SPM M Question 14 1 The data in Diagram 8 shows the marks for an English Language monthly test for 42 pupils. Data dalam Rajah 8 menunjukkan markah ujian bulanan Bahasa Inggeris bagi 42 orang murid Dtagram I (a) Using data in Diagram 8 and a class interval of marks, complete Table 2 in the answer 6l space. Menggunakan data dalam Rajah 8 dan dengan saiz selang kelas markah, lengkapkan Jadual 2 pada ruang jawapan. 14 marksl 14 markah) 96

3 (b) Based on your table in (a), Gil Berdasarkan jadual anda di (a), (i) state the modal class, nyatakan kelas mod, (ii) calculate the mean mark for the English Language monthly test and give your answer correct to 2 decimal Places. hitungkan min markah-ujian bulanan Bahasa Inggeris dan berikan iawapan anda betul kepada 2 tempat perpuluhan' [4 marks) 14 markahl (c) For this part of the question, use graph paper. O gv using a scale of 2 cm to marks on the horizontal axis and} cm to 1 pupil on the vertical axis, draw a histogram for the data. Untuk ceraian soalan ini, gunakan kertas graf. Dengan ntenggltnskun skilcr 2 cm kepada nrurkah padu poksi mengufuk dan 2 cm kepada I orang muritl pucln ltaksi nrcnconcung, lukiskan satu histogram bagi data itu. [4 marksl 14 markahl Answer/,/awupltt: (a) Marks Markah Midpoint Titik Tengah Frequoncy Kekerapan (b) (i) Table 2.Iadual2 (ii) (c) Refer to Your graph. Rujuk graf anda. 97

4 SPM mg Question 14 2 The data in Diagram 7 show the donations, in RM, collected by 40 pupils. Data dalam Rajah 7 menunjukkan derma, dalam RM, yang dikutip oleh 40 orang murid l l 38 3s (a) Based on the data in Diagram 7 and by using a class interval of, complete Table 2 in the Gtl answer space. [3 marksl Berdasarkan data dalam Rajah 7 dan dengan menggunakan saiz selang kelas, lengkapkan Jadual2 pada ruang jawapan. [3 markah] (b) Based on Table 2 in (a), calculate the estimated mean of the donation collected by a 13 marksl Berdasarkan Jadual2 di (a), hitungkan min anggaran kutipan derma bagi seorang murid. 13 markahl (c) For this part of the question, use graph By using a scale of 2 cm to RM on the horizontal axis and 2 cm to 1 pupil on the vertical axis, draw a frequency polygon for the data. [ marks] Untuk ceraian soalan ini, gunakan kertas graf. Dengan menggunakan skala 2 cm kepada RM pada paksi mengufuk dan 2 cm kepada I orang murid pada paksi mencancang, lukis satu poligon kekerapan bagi data itu. 1 markah) (d) Based on the frequency polygon in (c), state one piece of information about the Diagram 7 Rajah 7 markl Berdasarkan poligon kekerapan di (c), nyatakan satu maklumat berkaitan dengan kutipan derma tersebut. U markaht Answer/"Iawapan: (a) Class interval Selang kelas Midpoiht Titik tengah Frequency Kekerapan (c) Refer to your graph. Rujuk graf anda. 2t (d) Table 2 Jadual 2 (b) 98

5 SPM m7 Question 16 3 Table 2 shows the frequency distribution of the masses, in kg, of a group of 80 students. Jadual 2 menunjukkan taburan kekerapan jisim, dalamkg, bagi sekumpulan 80 murid. Mass (kg) lisim (kg) Frequency Kekerapan l s Table 2 Jadual 2 (a) (i) State the modal Nyatakan kelas mod. (ii) Calculate the estimated mean of the mass of the group of students. Hitung min anggaran jisim bagi kumpulan murid itu. [4 marks] 14 markah) (b) Based on Table 2, complete Table 3 in the space to show the cumulative frequency distribution of the masses. For this part of the question, use the graph paper. Untuk ceraian soalan ini, gunakan kertas graf. 13 marksf Berdasarkan Jadual 2, lengkapkan Jadual 3 pada ruang jawapan untuk menunjukkan kekerapan longgokan jisim itu. 13 markahl By using a scale of 2 cm to kg on the horizontal axis and2 cm to 10 students on the vertical axis, draw an ogive for the data. [4 marks) Dengan menggunakan skala 2 cm kepada kg pada paksi mengufuk dan 2 cm kepada l0 murid pada paksi mencancang, lukis satu ogif bagi data tersebut. [4 markah] 27o of all the students in the group have a mass of less than p kg. These students will be supplied with nutritional food. Using the ogive you had drawn in (c), find the value of p. ll markl 2Vo daripada murid-murid dalam kumpulan tersebut mempunyai jisim kurang daripada p kg. Murid-murid ini akan dibekalkan dengan makanan berkhasiat. Dengan menggunakan ogif yang telah anda lukis di (c), cari nilai bagi p. 17 markah) Answer/Jawapan: (a) (i) (ii) (b) Upper Boundary Sempadan Atas (kg) Cumulative Frequency Kakerapan l*nggokan (c) Refer to your graph. Rujuk graf anda. (d) 34. Table 3 Jadual 3 99

6 SPM m Question 14 4 The data below shows the payment, in RM, by 40 drivers at a toll booth. Data di bawah menunjukkan bayaran, dalam RM, oleh 40 pemandu di sebuah pondok tol T t Based on the data, complete Table 14 in the answer space. Berdasarkan data itu, lengkapkan Jadual 14 di ruang jawapan t marksl 13 markahl Based on Table 14 in (a), calculate the estimated mean of the toll paid by a driver. [3 marks] Berdasarkan ladual 14 di (a), hitung min anggaran bayaran tol bagi seorang pemandu. 13 markahj For this part of the question, use graph paper. Untuk ceraian soalan ini, guna kertas graf. By using a scale of 2 cm to RM on the horizontal axis andz cm to I driver on the vertical axis, draw a frequency polygon for the data. Dengan menggunakan skala 2 cm kepada RM pada paksi mengufuk dan 2 cm kepada 7 pemandu pada paksi mencancang, lukis satu poligon kekerapan bagi data tersebut. [ marks) [ markah] Based on the frequency polygon in (c), state the number of drivers who paid more than RM34 for the toll. $ mark) Berdasarkan poligon kekerapan di (c), nltatakan bilangan pemandu yang telah membuat baltaran tol lebih daripada RM34. ll markahl Answer/"/awapan: (a) Class interval Setang kelas Midpoi{t Tirik tungfr,h 1-19 t7 Frequency Kekerapan (b) (c) Refer to your graph. Rujuk graf anda. Table 14 Jadual L4 (d) 100

7 SPM mg Question 14 A group of 30 students are attending a motivation camp. The data in Diagram 14 shows the distances, in km, between their houses and the camp. Sekumpulan 30 orang murid sedang menghadiri kem motivasi. Data di Rajah 14 menunjukkan jarak, dalam km, antara rumah mereka dengan kem itu t t t Diagram 14 Rajah 14 (a) Based on the data in Diagram 14, complete Table 14 in the answer space. ($fi1 Berdasarkan data di Rajah 14, lengkapkan Jadual 14 di ruang jawapan. (b) State the size of the class interval used in Table Nyotakan saiz selang kelas yang digunakan dalam Jadual 14. I 13 markst 13 markahl ll markl markah) (c) Based on Table 14, calculate the estimated mean of the distance between the students' houses t3 marksl Berdasarkan Jadual 14, hitung min anggaran jarak antara rumah murid-murid itu dengan kem Gl and the camp. tersebut. (d) For this part of the question, use graph paper. Untuk ceraian soalan ini, guna kertas 13 markahl Using a scale of 2 cm to 10 km on the horizontal axis and 2 cm to I student on the vertical axis, draw a histogram for the data. 14 marks) Menggunakan skala 2 cm kepada 10 km pada paksi mengufuk dan 2 cm kepada I murid pada paksi mencancang, lukis satu histogram bagi data tersebut. (e) State one information based on the histogram in (d). l@ Nyotakan satu maklumat berdasarkan histogram di (d). 14 markah) U markl U markahl Answer//awapan: (a) Distance (km) Jarak 1fm) Frequency Kekerapan Midpoint Titik tengah (b) 2t -30 3t-40 (c) 41-0 l r (d) Refer to your graph. Rujuk graf anda. (e) Table 14 Jadual

8 SPM nlo Question 14 6 Table 14.1 shows the frequency distribution of the ages of 88 workers in a factory. Jadual 14-1 menuniukkan taburan kekerapan umur bagi 88 orang pekerja di sebuah kilaig. Age (Years) Umur (Tahxn\ Frequenfcy Idekerqphn t l Table 14.1 (a) Calculate the estimated mean of the ages a worker. t3 marksl Hitung min anggaran umur bagi seorang pekerja itu. [3 markah] (b) Based on Table 14.1, complete Table l4.2inthe GEI answer space by writing down the values of the upper boundary and the cumulative frequency. [3 marks] Berdasarkan Jadual 14.1, lengkapkan Jadual 14.2 di ruang jawapan dengan menulis nilai-nilai sempadan atas dan kekerapan longgokan. 13 markahl (c) For this,:#:rtl"'or"r,ion, the graph paper. you may use a flexible curve By using a scale of 2 cm to years on the horizontal axis and 2 cm to 10 workers on the vertical axis, draw an ogive for the data. 14 marksl Untuk ceraian soalan ini, gunakan kertas graf. Anda boleh menggunakan pembaris fleksibel. Dengan menggunakan skala 2 cm kepada tahun pada paksi mengufuk dan 2 cm kepada lo pekerja pada paksi mencancang, lukis satu ogif bagi data tersebut. [4 markah] (d) The factory plans to organise a motivational camp for the workers whose age are 47 years old. Using the ogive in (c), find the number of workers who will attend the motivational camp. 12 marksl Kilang itu bercadang untuk menganjurkan satu kem motivasi kepada pekerja-pekerja yang berumur lebih daripada 47 tahun. Menggunakan ogif di (c), cari bilangan pekerja yang akan menghadiri kem motivasi tersebut. L2 markah) Answer/Jawapan: (a) (c) Refer to your graph. Rujuk graf anda. (b) Upper Boundary Sempadan Atas Cumulative Frequency Kekerapan Longgokan (d) Table 14.2 Jadual 14.2 toz

9 SPM 2011 Question 14 7 Diagram 14 shows the number of books read by a group of 24 in the year Rajah 14 menunjukkan bilangan buku yang dibaca oleh sekumpulan membaca pada tahun L students in a reading prograflrme 24 orang murid dalam satu program ",;:;:f,; (a) Based on the data in Diagram 14, complete Table 14 in the answer space. l@ Berdasarkan data di Rajah 14, lengkapkan Jadual 14 di ruang jawapan. (b) State the modal class. (@ Nyotakan kelas mod. (c) Calculate the estimated mean for the number of books read by a student. Gl Hitung min anggaran bagi bilangan buku yang dibaca oleh seorang murid. [4 marks] [4 markah] $ markl fi markahl t3 marks) 13 markahf (d) For this part of the question, use graph By using a scale of 2 cm to books on the horizontal axis and 2 cm to 1 student on the vertical axis, draw a histogram for the data. 13 marksl Untuk ceraian soalan ini, gunakan kertas graf. Dengan menggunakan skala 2 cm kepada buku pada paksi mengufuk dan 2 cm kepada I murid pada paksi mencancang, lukis satu histogram bagi data tersebut. 13 markah) (e) Based on the histogram drawn in (d), state the number of students who read less 32 books in that programme. li markt Berdasarkan histogram yang dilukis di (d), nyatakan bilangan murid yang membaca kurang daripada 32 buku dalam program tersebut. ll markahl Answer/,/cwapan: (a) Class interval Sel*ng kelas Midpoint Titik tengah (b) (c) Table 14 Jadual 14 (d) Refer to your graph. Rujuk graf anda. (e) r03

10 REVIEW AND ANSWERS KERTAS 1 ry. Mean _ sum of (frequency x midpoint) sum of frequencies i Mean _ 7 (t2) + 4(1) + 2(18) + 2(21) + (24) 20 _ = 17.l kg Answer: B %. Construct a frequenry table based on the, cumulative frequency table given.,. Mode : score with the highest frequency Answer: D KERTAS 2 Score I Frequency r = =2 16-1=l Since the mode of the scores is l, its frequency must be greater than (frequency of score 3). Thus, a possible value of "r is 6. 1 (a) 'Mar,ks lvlidpoint Fre{uenc; s (b) (i) Modal class = 3-39 (ii) Mean s(22) + 7(27) + 8(32) + 10(37) _ +6(42)+4(47)+2(2) t marks (2 d.p.) x t' to4 2 (a) (c) Frequency t J 2 I 0 t (b) Estimated mean s(23) + 6(28) + 8(33) + _ 10(38) + 7(43) + 4(48) +6+8+lO+7+4 _t = RM3.0 (c) F "qu"n"y l J 2 1 Class interval Midpoint Frequency; 2t l - 3 aa JJ t t (d) The modal class is Donation (RM)

11 3 (a) (i) Modal class = 0-4 (ii) Estimated mean s(32) + 8(37) + Lr(42) + 2t(47) _ + 22(2) + 10(7) + 3(62) t s 80 = 47.6 kg (b) (c),,fj.ppetr,,,. Boundary 29.s (b) (c) Estimated mean 3(17) + (22) +7(27) +9(32) + 7(37) + 6(42) + 3(47\ _ t = RM32.2 hrequency 12 t Payment (RM) (d) Number of drivers who paid more than RM34 =7+6+3 =16 (a) Distance (km) Frre{uGnc} Midpoin=t 2t -30 I 2. 3t t-0 4. (d) 4 (a) )\ '" x 80 = 20 students 100 From the ogive, p = 43. Mass (kg) Class.,, interyal' ',Midpoint F,r,eqiienry l t9 L] 1 J a -1 (b) (c) t Size of class interval =(30-2L)+l =10 Estimated mean 1(2s.s) + 2(3.) + (4.) + 6(s.) _ + 8(6.) + (7.) + 3(8.) l = 60. km 10s

12 l (d) Frequency ) 2 1 :::' r rr_. I i:' :: :i::...,t-;- -.-:i:.-_:: _:a-.':...) r..::..''.: r-':,]: :... rlr:iil -r rr.ti: l.::l:i:r:-rt!l-:--; I.1 i _- i lili:: rj. :l ir r :t' : il '::ii :lr i-.::laiil'rt- :ll : :.: I l -s->j-.-:1::. -+ j :lir:; ::.:.'.'1 -t ::,:" : :i I :. :.'l:i, : : r., J --. i :...) - -i.{.,::. :1.:.:a.l 0 20,s s Distance (km) (e) The modal class is 6l (a) Estimated mean 3(28) +9(33) +27(38) +31(43) + l1(48) + (3) + 2(s8) (b) = years Upper Boundary 88 Cumulativte Frequenci s 1 J 3. l l s 88 (d) 7 (a) (b) (c) (d) Number of workers who will attend the motivational camp =88-14 =74 Class interval Freqnency Midpoint r l I 49 Modal class = 27-3l Estimated mean 4(24) + 7(29) + 6(34) + 4(3e) + 2(44) + t(49) = books Frequency (c) Cumulative frequency I.i ;ii'l -,f ll 'i lr,."i, - [,,.,,r, s,sl.s Number of books :,I t: ',, ";'',,l, l i,-'r'i ' - J,''', "t, i'l:; '-'j t' ' '!,'t.',,, :;l'lr'l:ii":1: ' l"':t ' ',1'::,r. tj', Age (years) (e) Number of students who read less than 32books -4+7 _ tt - ll r06

13 FORM 4 KERTAS 1 Mathematical Formula and Fact number of times event A occurs Probability of event A - number of trials Kebarangkalian peristiwa A - bilanga! 'kali berlaku Peristiwa A bilangan Percubaan P(A) = n(a) n(s) ffi ProbabilitY of an Event SPM M Question U 1 Table 2 shows the distribution of a group of 90 pupils PlaYing a game. Jadial 2 menunjukkan taburan sekumpulan 90 orang muridyang mengambil bahagian dalam satu permainan. Girls Perempuan Boys I*lnki Form Four Ti.nglutan Empat Form Five Tingkatan Lilna 33 1 t8 24 Table 2 Jaduol 2 A pupil is chosen at random from the group to start the game. What is the probability that a girl from Form Four will be chosen? Seorang murid dipilih secara rawak daripada kumpulan itu untuk memulakan permainan' Apakah kebarangkalian seorang murid perempuan daripada Tingkatan Empat akan dipilih? A# c+ SPM m Question il 2 It is given that set G is { 1,2,3,, 6,7,8,9, 13, 14, 1). A number is chosen at random from the elements of set G. Find the probability that the number chosen is a prime number. Dibeii bahawa set G ialah {1, 2, 3,, 6, 7, 8, 9, 13, 14, 1). Satu nombor dipilih secara rawak daripada unsur-unsur set G. Cari kebarangkalian bahawa nombor yang dipilih itu ialah nombor Perdana' 4 BTI ATi t D il ll n 107

14 SPM Question?4 3 A box contains red marbles and Zl green marbles. Lim puts another 4 redmarbles and 1 green marble inside the box. A marble is chosen at random from the box. What is the probability that a red marble is chosen? Sebuah kotak mengandungi biji guli merah dan 2l biji guli hijau. Lim memasukkan lagi 4 biji guli merah dan I biji guli hijau ke dalam kotak itu. Sebiji guli dipilih secara rawak daripada kotak itu. Apakah kebarangkalian sebiji guli merah akan dipilih? A+ B c* D 9 x 9 % SPM 2@9 Question & 4 Table 2 shows how a group of 400 students travel to school. Jadual 2 menunjukkan bagaimana sekumpulan 400 orang murid ke sekolah. SPM nlo Question 36 In a group of 80 students, 2 are girls. Then 10 boys leave the group. If a student is chosen at random from the group, state the probability that the student chosen is a boy. Dalam satu kumpulan 80 orang murid, 2 orang daripadanya adalah murid perempuan. Kemudian seramai 10 orang murid lelaki meninggalkan kumpulan itu. Jika seorang murid dipilih secara rawak daripada kumpulan itu, nyatakan kebarangkalian bahawa murid yang dipilih itu adalah lelaki. AA o B1z 7 ct6 o Di6 Type of transport lenis pengangkutan Bicycle Basikal Motorcycle Motosikal Car Kereta Bus Bas Number o{ students Bilangan murid Table 2 Jadual 2 A student is chosen at random from the group. Find the probability that the student travels to school by car. Seorang murid dipilih secara rawak daripada kumpulan itu. Cari kebarangkalian bahawa murid itu pergi ke sekolah dengan kereta. B# A# C1 D+ 108

15 REVIEW AND ANSWERS KERTAS 1 'l. P(a girl from Form Four), number of girls from Form Four.:. Cive the answer in the simplest form. P(a girl from Form Four) aa JJ =- 90 l1 30 Answer: A ''Y. ldentify all the prime numbers from the set C. '. P(prime number) number of prime numbers in set C number of elements in set G n(g) = 11 Prime number - {2,3,,7,l3l n(prime number) = J.'. P(prime number) = + Answer: B Number of students who travel to school by car =80 Number of students in the group = 400.'. P(student travels to school by car) Answer: C Number of boys =80-2 = After l0 boys left. number of boys =-10 /l< -aj- Total number of students =80-10 =JO L\.'. P(boy, = fr 9 3 Total number of red marbles = Total number of green marbles = =22 Total number of marbles = =31.'. P(red marble) = + Answer: B Answer: B

16 Circles III KERTAS 1 1. Angle subtended at the centre of a circle Uy un arc - 2 x angle subtended at the circumference by the same arc sudut yang dicangkum pada pusat sebuah bulatan oleh suatu lengkok = 2 x sudut yang dicangkum pada lilitan oleh lengkok yang sama 2. sum of interior opposite angles of a cyclic quadrilateral = l80o Hasil tambah sudut pedalaman bertentangan dalam sisi empat kitaran = l80o 3. The exterior angle and the coffesponding interior opposite angle of a cyclic quadrilateral are equal. Sudut peluaran dan sudut pedalaman bertentangan ))ang sepadan dalam sisi empat kitaran adalah. samo. 4. Angle between a tangent and a chord = angle in the alternate segment subtended by the chord Strdut di antara tangen dengan perentas = sudut dalam tembereng selang seli yang clicangkum oleh perenta,s itu 110

17 ffiffi Tangent to a Circle SPM ffi Questron 8 1 In Diagram 2, QRf is a tangent to the circle centre O, at R. PUS, OPQ and OUR are straight lines. Dalam Rajah 2, QRT ialah tangen kepada bulatan berpusat O, di R. PUS, OPQ dan OUR ittlah garis lurus. SPM m Question 9 3 In Diagram 3, PQR is a tangent to the circle centre O, at point Q. PTOS is a straight line. Dalam Rajah3, PQR ialah tangen kepada bulatan berpusat O, di titik Q. PTOS ialah garis lttrus. Diagram 3 Rajah 3 Find the value of x. Carikan nilai x. A B C D Diagram 2 Raiah 2 ffi Find the value of x. Cari nilai x. A8 842 c3t Dzt Angle between Tangent and Ghord SPM nos Question 8 4 In Diagram 3, IPS is a tangent to the circle POR at P. Dalam Rajah 3, TPS ialah tangen kepada bulatan PQR di P. SPM n@ Question 8 2 In Diagram 3, JKL is a tangent to the circle centre O at K and JNM is a straight line. Dalam Rajah 3, JKL ialah tangen kepada bulatan berpusctt O di K dan JNM ialah garis lurus. Diagram 3 Rajah 3 Find the value of x. Cari nilai x. A4 c1 Diagram 3 Raiah 3 82 D The length of arc PR is equal to the length of arc QR. Find the value of x. Panjang lengkok PR adalah sama paniang dengan panjang lengkok QR. Carikan nilai x. A c44 D68 111

18 SPM Question 8 Diagram 3 shows a circle, PFQ, centre O. EFG is a tangent to the circle at F. Rajah 3 menunjukkan sebuah bulatan, PFQ, berpusat O. EFG ialah tangen kepada bulatan itu di F. SPM 2011 Question I 7 In Diagram 2, PQR is the tangent to rhe circle TQS with centre O, at Q. Dalam Rajah 2, PQR ialah tangen kepada bulatan TQS berpusat O, di Q. Find the value of x. Cari nilai x. A32 c61 Diagram 3 Rajah D74 SPM 2O1O Question 9 6 In Diagram3, PQRS is a tangent to the circle RTU at point R. TVR is a straight line. Dalam Rajah 3, PQRS ialah tangen kepada bulatan RTU di titik R. TVR ialah garis lurus. Find the value of x. Cari nilai x. A10 81 c20 D2 Diagram 2 Rajah 2 Diagram 3 Rajah 3 Find the value of x. Cari nilai x. A 100 c 110 B 10 D t2 ttz

19 REVIEW AND ANSWERS KERTAS 1 Y tt:w$,. Angle at centre :2 x angle at circumference '. OR is perpendicular to QRf where IORQ:90o. '. Sum of interior angles of a triangle : I BO',i,: ZPOR=2xIPSR =2x2o = 0o I.ORQ = 90o ln A,OQR, -ro + 0o + 90o = 180o x+140=180 x=40 Answer: A. ZNOK:2 x ZKMN. Sum of interior angles of a quadrilateral : 360o ;Sum of angles on a straight line : lb0o lvlethod ft ZOSQ = ZOQS = 90" - 74" = 16o ZPSQ = ZOSQ = 16o tspq+ Z-PSQ= /.RQS xo+16o=74" x=74-16 =8,Method 2; ZOSQ = /.OQS = 90o - 74o = 16o x"+16o+76o+90o=180o x+122=180 x=8 Answer: A Exterior angle of a triangle = sum of interior opposite angles ry. Angle between tangent IPS and chord PR ', : angle in the alternate segment subtended. by the chord,. The angles at circumference subtended by equal. arc lengths are equal. Since lengths of arcs PR. and QR are equal, 0Qn: IQPR. IJKO = 90o ZKON = 2 x 0o = 100o In quadrilateral JKON, ZJNO = 360o - 3o - 90'- 100" = 13o xo+13o=180o x= =4 Answer: A ZPQR = Z.RPS = 68o ZQPR = ZPQR = 68o x= =44 Answer: C 113

20 ZFPQ = IGFQ = 8" ZFOQ=2x IFPQ =2x8o = 116o lfeo- l8o'- 116' =32o 2 x" - ZPQF =16+32" = 48" Answer: B $u'j.4rtlj:z.ljrs.. Sum of angles at point f : 360". Sum of interior opposite angles quadrilateral : I B0o.IPQV + /.PTV: l80o of a cyclic ZQOS=2xLQTS =2x40o = 80o ZOQS = IOSQ 180' - 80" =z = 0o ZTSQ = /.PQT = 6o x"+/-osq=ztsq xo+0o=6o ro = 1o Answer: B ZRTU = ZURS = 70o /.PTV +70o +210o =360o /.PTV + 280" = 360o ZPTV = 80o ZPQV+IPTV=180" xo+80o=180o xo = 100" Answer: A tl4

21 FORM 4 KERTAS 1.41,. i: 'rri_":rr:1:l:!_l':. r:i::r-''.:tl.:jr1 : r.':i. : Mathematical Formulae and Facts it" I. (t) \'/ stn 0 = -:- opposite side hypotenuse.sisi bertentangan sin e = ---iq*rrw ^ adjacent side \ " ' hypotenuse sisi bersebelahan ktrs d = ri,puterus opposite side (ru) tan A-- adjacent side 3. Quadrant II Sukuan ll sin (+) Quadrant III Sukuan lll tan (+) Quadrant I Sukuan I All (+) Semua (+) Quadrant IV Sukuan IY cos (+) kos (+) 00, 3600 tan0= sisi bertentangan sisi bersebelahan 4. (i) ) v I -1 o ^/ff' (i) (ii ) (iii) -l sin 0 - _y-coordinate of P sin 0 = koordinat-y bagi P cos 0 = x-coordinate of P kos 0 = koordinat-x bagi P ^ v-coordinate of P LattA=x-coordinate of P. A koordinat-1' bagi P t?n t, = koordinat-x bagi P 11

22 ffi Values of Sin 0, Gos 0 and Tan g SPM m Question I It is given that e < 36A'. Find the value Diberi bahawa 0 < Carikan nilai 0. A ztg" 27, c 309" 27' SPM rcs Question 12 2 In Diagram 6, QRS Dalam Rajah 6, QRS is a straight line. ialah garis lurus. _I 4cm Diagram 6 Rajah 6 What is the value of cos 0o? Apakah nilai kos 0"? A+ c-1 ln 3cm 1^ s 11 cos 0 = and 180o < of 0. kos 0- B D B1 D-I dan 180" < 230" 33', 32A" 33', D SPM m Question 12 4 In Diagram 6, MPQ is a right-angled triangle. Dalam Rajah 6, MPQ ialah sebuah segi tiga bersudut tegak. Diagram 6 Rajah 6 It is given that 8N - 13 cm, MP = 24 cm and N is the midpoint of MNP. Find the value of tan yo. Diberi bahawa QN - 13 cm, MP = 24 cm dan N ialah titik tengah MNP. Carikan nilai tan y". A _ 13 \ v D SPM m7 Question '11 In Diagram 6, USR t2 -t3 t3 -n and VQTS are straight lines. Dalam Rajah 6, USR dan VQTS ialah garis lurus. SPM 2M Question 11 3 In Diagram, S hne QST. Dalam Rajah, S lurus QST. is the midpoint ialah titik tengah of straight bagi garis 1' 6cm I, The value of cos Nilai kos xo ialah A A+ J c3 4 Diagram Rajah x" is BT D1 Dra9ram 6 It is given that fs - 29 cm, PQ = 13 cm, QR = 16 cm and sin *' = #. Find the value of tan yo. Diberi bahawa 7S = 29 cffi, PQ = 13 c0, QR = 16 cm dan siar' = #. Cari nilai tan yo. A9 ) c-* B* D-+ ) Lt6

23 SPM no7 Question 12 6 In Diagram 7,, O is the origin and JOK is a straight line on a Cartesian plane. Dalam Rajah 7, O ialah asalan dan JOK ialah garis lurus pada suatu satah Cartesan. The value of cos 0 is Nilai bagi kos 0 ialah v Diagram 7 Rajah 7 A-+ B-1 c1 )) SPM W Question 11 7 Diagram 6 shows a right-angled trian gle PQR. Rajah 6 menunjukkan sebuah segi tiga bersudut tegak PQR. PKQ DL Diagram 6 Rajah 6 I Given sin xo - ;, find the value of h. 1 Diberi sin xo - ;, cari nilai h. R SPM m Question '12 8 Given that tan *" = Twhere 180" < x ( 360o, 4 find the value of cos xo. Diberi bahawa tan xo = ] arngon keadaan 4 180' < x S 360o, cari nilai kos xo. A-+ c1 D+ )) SPM mg Question 12 9 In Diagram 6, PQST is a parallelogram and PQR is a straight line. Dalam Rajah 6, PQST ialah segi empat selari dan PQR ialah garis lurus. TS POR Diagram 6 Rajah 6 Given sin ZSQO = *, cos /.PTS. find the value of Diberi sin ZSQR= +, cari nipti bagikos ZPTS. IJ '.r13 t Bi c -n D SPM nlo Question 13 L0 In Diagram 6, point P and point Q lie on the arc of a unit circle with centre O. Dalam Rajah 6, titik P dan titik Q terletak di atas lengkok suatu bulatan unit berpusat O. v B-1 P(0.60,0.80) B C k tr, 30' k tan 30o k."s 60' k k.. 60' k cos 60" k kos 60" Find the Cari nilai A r.az c 1.1 Diagram 6 Rajah 6 Q(0.91,0.42) value of cos xo + sin yo kos x" + sin y". B r.22 D T,7I 117

24 SPM 2oll Question Diagram 4 shows a right-angled triangle PQR.PRS is a straight line. Rajah 4 menunjukkan sebuah segi tiga bersudut tegak PQR. PRS ialah garis lurus. O reffij Graphs of Sine, Cosine and Tangent SPM m Question Which of the following graphs represents Y = cos 2x? Antara berikut, yang manakah mewakili graf Y = kos 2x? Av Diagram 4 Rajah 4 'o ; -1. calculate the length, in cm, of QR. Diberisin x" = hiturtg paniang, dalam cm, 13, blgi QR. At2 824 c36 D39 SPM 2oll Question Diagram shows a sailing boat. The sail TPN has the shape of a right-angled triangle and MNPQ is a straight line. Rajah menunjukkan sebuah kapal layar. Layar TPN berbentuk segi tiga bersudut tegak dan MNPQ ialah garis lurus. B C Given that NP = 4 m find the height, in m, DiberiNP=4mdan tinggi, dalam m, TP. A 1. B 8. c 1 D16 Diagram Rajah andcos ZMNT=-8-17' Of TP. kos ZMNT cari t1 D 118

25 SPM mg Question Which of the following represents the graph of y = cos -r for 0o 1 x 1 180'? Antara yang berikut, yang manakah mewakili graf y = kos x bagi0o ( x < 180"? AY BY SPM m Question Which of the following represents the graph of y= tanx for0o <.x<-360"? Antara graf berikut, yang manakah mewakili graf )=tanxbagi0o(x<360"? Av 1 SPM no7 Question 13 1 Which of the following graphs represents )=sinxfor0"<xs180'? Antara yang berikut, Yang manakah graf y = sin xbagi 0o (x< 180"? AY mewakili 1L9

26 SPM m Question t3 17 Which graph represents y = sin xo for 0o < x<270o? Graf manakah yang mewakili )r = sin x" bagi 0o(x<270"? Ay SPM 2oll Question t3 19 Which graph represents part of y - sin xo? Graf manakah yang mewakili sebahagian daripada J,, = sin-ro? AY B 0-1 C -l 0 D SPM NlO Question Which graph represents part of!=tanx? Graf manakah yang mewakili daripada gral'y = tan x? AY BY the graph sebahagian

27 REVIEWAND ANSWERS KERTAS 1 t. cos 0 is negative in quadrant lll. '. Use a scientific calculator to find its corresponding acute angle in quadrant l. cos 0 = and 180' < A < 360" Corresponding acute angle in quadrant I = cos-l 0.772r = 39" 27' 0 = 780" + 39" 27' - 219" 27', Answer: A. Use Pythagoras'theorem to find the length of PR.. 0o is an obtuse angle in quadrant ll. 'cos lprq : ldiacent side hypotenuse. I PR =,tof;td - \t 3r;O =cm cos 0o = --cos ZPRQ Answer: C OR --- PR a J =_- rff.qs:qt-2. Use Pythagoras'theorem to find the length of QR. QS=6cm+2=3cm i_---;-- QR = r/rs2 - QS', ="'[iz s =4cm cos x" ^or =;S Answer: B 4 :ir4'i. NP: MP + 2. Use Pythagoras'theorem to find the length of PQ. op'losite sjde 'tan IQNPadjacent side. Since y" is an obtuse angle, tan f NP=24cm+2 =12cm PQ = "'{OF NA = \[BT:1r' =cm tan y" = -tan LQNP PQ NP t2 Answer: C is negative. ',...sinxo- 8 : QR 17 QS. Use Pythagoras'theorem to find the lengh of PT.. Since y" is an obtuse angle, tan yo is negatve. sln x- 8 t7 &=a QS t7 QS=rcx{ 8 =34cm QT=QS-TS =34-29 =cm Pr =^[Fe-@ = tf1p -P =12cm tan Yo = -tan ZPQT = -PT QT =-12 Answer: D r21

28 ffi. Since 0 lies in quadrant lll, cos iitffi OK=\tTP coso=-3 ',! Answer: B =J 0 is negative. a cos.tro = 0.91 sin vo "l = 0.80 ' 0.9r 0.60 o.o, ry t0 1 sln -r" 3 -J- 2 xo=90o+60o = 10" IRPQ=180o-10"=30o tan 3oo = L k h = k tan 30o Answer: B. Since x' lies in quadrant lll, cos xo is negative. tan xo = +, ^4 COS X- = -- Answer: A 180'<x<360o er=1[1i2-y =12 cos ZPTS = cos IPQS = {os ISQR Answer: D Method trl = _12 13 P(0.60, 0.80) and Q(0.91,0.42) cos,o = r-coordtnate of Q = 0.91 sin y" - y-coordinate of P = 0.80.'. cos xo + sin )lo = = 1..7I -f;tt.'. cos ro + sin )o = = 1.7I Answer: D. sin IPRO -PR : PQ. Use Pythagoras'theorem to find the lengh of QR. sin xo = sin QR - ^'[tr:t* = r[l 296 =36cm Answer: C IPRO= l3 1= PR T3 ị NP ' ' cos ZPNT: ---::- NT "=+ =39.. Use Pythagoras'theorem to find lh:. length of IPcosIMNT=-8 t7 cos IPNT = =8 NT I7 NT=Lxq 8 =8.m Tp =.'lg.sz - +z = t[fr2 =7.m Answer: A x1 cm 122

29 13!=cos2x When x = 90o,, = :::?[?9", I - -l When x = 180o, ), = cos 2(180') = cos 360" _1 -l L7 sin 0o = 0, sin 90o = 1, sin v o"l 90' r8h" 270" l80o = 0, sin 270" = -l Graph in B represents )l = sin xo for 0o S x <270". Answer: B l8 The graph of y - tan x for 0o 1 x 1360o is The graph in C represents j = cos 2x. Answer: C t4 cos 0o = 1, cos 90o = 0, cos 180" = -l The graph J=tanxv in B represents part of the graph Graph in B represents ) = cos x for 0o (x < 180". Answer: B 1 sin 0o = 0, sin 90" = 1, sin 180o = 0 v t9 Answer: B The graph of y = sin r" for 0o ( -ro ( 360" is 0 Graph in C represents -I = sin x for 0" ( x ( 180". t6 Answer: C tan 0o = 0, tan 180o = 0, tan 360" = 0 v The graph in A represents part of ) = sin xo v Graph in D represents )l = tan x for 0o ( x < 360". Answer: D Answer: A r23

30 Angles of Elevation and Depress ion KERTAS 1 ffi Angles of Etevation and Depression SPM 2o0 Question 1 1 Diagram 8 shows two vertical poles, PS and QR, on a horizontal plane. Rajah 8 menunjukkan dua batang tiang tegak, PS dan QR, yang terletak pada permukaan mengffik. P s<- l0 m - +4 Diagram 8 Rajah 8 The angle of depression of vertex Q from vertex P is 42". Calculate the angle of elevation of vertex Q from ^S. Sudut tunduk puncak Q dari puncak P ialah 42". Hitungkan sudut dongakan puncak Q dari S. A32 B 41" 9' c J" 9', D 680 ll', SPM m Question 1 2 In Diagram 8, PQ and TR are two vertical poles on a horizontal plane. S is a point on 7R such that PQ = SR. Dalam Rajah 8, PQ dan TR ialah dua batang tiang tegak pada satah mengufuk. S adalah satu titik di atas TR dengan keadaan PQ = SR. OR Diagram 8 Rajah 8 The angle of depression of P from Z is Sudut tunduk P dari T ialah A ZTPS C ZPTS B ZRPT D ZSTS SPM mg Question 16 3 In Diagram 9,,IrK is a vertical flagpole. KL is horizontal. Dalam Rajah 9, JK ialah sebatang tiang bendera tegak. KL adalah mengffik. Kl<- 3m l, Diagram 9 Rajah 9 The angle of elevation of.,i from L is 42 The height, in m, of the flagpole is Sudut dongakan J dari L ialah 42". Tinggi, dalam m, tiang bendera itu ialah A B 26.0t c 31.1 D

31 SPM 2OO7 Question 1 4 In Diagram 9, PQ and RS are two vertical poles on a horizontal plane. The height of RS is twice the height of PQ. The angle of depression of P from vertex R is 3". Dalam Rajah 9, PQ dan RS ialah dua batang tiang tegak yang terletak pada suttr permukaan mengufuk. Tinggi RS ialah dua kali tinggi PQ. Sudut tunduk P dari puncak R ialah 3". SPM m Question 16 6 Diagram 9 shows a tower PQ. The points P, R and S lie on a horizontal plane. Rajah 9 menunjukkan sebuah menara PQ. Titik P, R dan S terletak di atas satah mengufuk. o Diagram 9 Rajah 9 Diagram 9 Rajah 9 Calculate the height, in m, of RS. Hitung tinggi, dalam m, bagi RS. A 1.07 B 26.4 c D 3.08 SPM m Question 1 Diagram 8 shows two vertical flagpoles on a horizontal plane. P, Q, R. S and I are five points on the poles such that QR = S7. Rajah 8 ntenunjukkan dua tiang bendera tegak pada satah mengufuk. P, Q, R, S dan T ialah lima titik pada tiang-tiang itu dengan keadaan QR - ST. Q m ----+S The angle of depression of R trom Q is 4" Find the angle of elevation of Q from S. Strtlut tunduk R dori Q iolah 4". Curi suclut tlongakcut Q cluri S. A 22 3J', B 30 13', c 4" 47', D 6J" 23', SPM m Question /6 7 Diagram 8 shows a vertical tower,/rk a horizontal plane. The points L and K on the plane. Rajah 8 menunjukkan sebuah menara tegak JK di atas satah mengufuk. Titik L dan K terletak di atas satah itu. J on lie lom K Diagram 8 Rajah 8 Diagram 8 Rujah 8 Name the angle of elevation of point P from point S. Namakan sudut dongakan titik P dari titik S. A C tqsp IRSP B Z.QPS D IRTP Find the angle of elevation of point -/ from point L. Cari sudut dongakan titik J dari titik L. A 30" 28', B 36" 02', c 30 8', D 9" 32', 12

32 SPM m Question 1 8 T, P and Q are three points on a vertical pole. S is a point on the horizontal ground. The angle of depression of S from Q is x" and the angle of depression of S from P is y'. Which diagram represents the situation? T, P dan Q ialah tiga titik pada sebatang tiang tegak. S ialah satu titik yang terletak pada tanah mengufuk. Sudut tunduk S dari Q ialah xo dan sudut tunduk S dari P ialah y". Rajah manakah yang mewakili situasi tersebut? A B C e<+p 20m Diagram 8 Rajah 8 The angle of elevation of point R from point P is 44". Calculate the height, in m, of the pole. Sudut dongakan titik R dari titik P ialah 44'. Hitung tinggi, dalam m, tiang itu. A c B D 20.7 t SPM nlo Question Diagram 9 shows a boy standing on a horizontal stage. Points P, Q and R are on a same vertical plane. The angles of depression of point R from his eye level when he is at P and Q are 6lo and 29" respectively. Rajah 9 menunjukkan seorang budak lelaki berdiri di atas sebuah pentas mengufuk. Titik P, Q dan R terletak pada satah mencancang yang sama. Sudut tunduk titik R dari aras matanya bila dia berada di P dan Q adalah masing-masing 6lo dan29". Eye level D o R-rTl+ Diagram 9 Rajah 9 Stage Pentas SPM nlo Question 1 9 Diagram 8 shows a vertical pole RQ. P rs a point which is 20 m from Q on a horizontal plane. The point R is vertically above point Q. Rajah 8 menunjukkan sebatang tiang tegak RQ. P ialah satu titik yang berjarak 20 m dari Q pada satah mengufuk. Titik R berada tegak di atas titik Q. It is given that the horizontal distance of point R from the base of the stage is m. Calculate the distance, in m, of PQ. Diberi jarak mengufuk titik R dari tapak pentas adalah m. Hitung jarak, dalam m, bagi PQ. A 2.77 c t6.21 B D

33 SPM 2oll Question 1 11 Diagram 7 shows two vertical towers on a horizontal plane. P and Q we two points on the two towers. Rajah 7 menunjukkan dua menara tegak di atas satah mengufuk. P dan Q adalah dua titik pada dua menara itu. SPM frll Question 16 t2 Dragram 8 shows a man in a lookout station. The angle of depression of a tent from his eye is 40'. The tent is y m from the base of the lookout station. Rajah 8 menunjukkan seorang lelaki berada di stesen peninjau. Sudut tunduk sebuah khemah dari matqnya ialah 40". Khemah itu adalah y m dari tapak stesen peninjau itu. Lookout station Stesen peninjau Diagram 7 Rajah 7 Point R is vertically above P, at the same level as S.Point S is vertically below Q, at the same level as P. Which diagram shows the angle of elevation, 0, of point Q from point P? Titik R berada tegak di atas P, pada paras yang sama dengan Q. Titik S berada tegak di bawah Q, pada paras yang sema dengan P. Rajah yang manakah menunjukkan sudut dongakan, 0, titik Q dari titik P? AB RO RO )'m Diagram 8 Rajah 8 It is given that his eye level is 6 m above the horizontal ground. Calculate the value of y, in m. Diberi bahawa aras matanya adalah 6 m dari tanah mengufuk. Hitung nilai y, dalam m. A 4.96 B.03 c 7.11 D VP C I o A PS D PVo APS 127

34 REVIEW AND ANSWERS KERTAS 1 r.'"i+. Draw a horizontal line through P and a horizontal line through Q to touch PS at L. Draw lines joining P and Q, and Q and S.. Since the angle of depression of Q from P is 42", ZPQT is also 42'.. With tan 42o : +,the length of PTcan be found. Hence, the lengt[s of IS and QR can be found. *]B''. Since the angle of elevation of / from I is 42o, Z.KLJ : 42o.. Use tangent to find the height of the flagpole JK. Height of the flagpole = JK JK +^^ Ano KL - LatL al JK? = tan 42" JK = 3 tan 42 = 31.1 m Answer: C s+_10 PT = 10 tan 42" = m 7S = = m QR = 7S = m Angle of elevation of Q from S = ZQSR tan /QSR= # =ro = L.gg6 ZQSR = tan-r t.996 = 7" 9' Answer: C m -- -+R :;xtft. The angle of depression of P from f is the angle between line Pf and the horizontal line through L. Since the horizontal line through f rs parallel to PS, the angle of depression of P from f is equal to ITPS. 4. Draw two horizontal lines, IR and PN. PrV is parallel to fr. N is automatically the midpoint of RS and RS: 2 x RN. ran 3o = # RN = 20 tan 3o = 26.4 m RS=2xRN = 2 x 26.4 = 3.08 m Answer: D Q+20 m-+s Angle of depression of P from 7 = angle of elevation of Z from P = ITPS Answer: A Angle of elevation of point P from point S = ZQSP Answer: A 128

35 tan 4o = 12 PR.12 I l- -- PR PR=12cm PS=12+=17cm tan zpsq = X t2 t7 LPSQ = 3o 13', Answer: B P lzm R m S Angle of elevation of point J from point L = IJLK tan LJLK = # 17 l0 ZJLK = Answer: D o- _11 - t.t 9o 32' 10,? Eye level l1 [, P o R- m* Let the vertical distance of point R from level be h. tan6lo- h h=tan61o = 9.02 m tan 2eo =;#d +PO_ 9.02 tan 29" Po= 9'02 - ran 29" = m Answer: B the eye The diagram which shows the angle of elevation ofqfrompis, --\ The diagram in B represents the situation given. Answer: B. Angle of elevation of point R from point P : ZQPR : 44" t2 Answer: D Height of the pole = RQ RO tan 44" = fi RO -&-20 RQ = 20 tan 44" = m Answer: C tan 40o =! v 6 / - tan 40o Answer: C t29

36 Lines and Planes in 3-Drnenslbns KERTAS 1 ffi Angles between Lines and Planes SPM ffi Question 14 1 Diagram 7 shows a right pyramid a quadrilateral base EFGH. Rajah 7 menunjukkan sebuah piramid dengan tapak sisi empat EFGH. EF V Diagram 7 Rajah 7 What is the angle between the base EFGH? Apakah sudut di antara garis EFGH? A ZVFE C ZFVH B ZVFH D /.FVE &ffi Angles between Two Planes with tegak the line VF and VF dengan tapak SPM mo Question 14 2 Diagram 7 shows a cuboid with a horizontal base TUWV. Rajah 7 menunjukkan sebuah kuboid dengan tapak mengffik TUWV. o T Diagram 7 Raiah 7 Name the angle between the and the plane SRWT. Namakan sudut di antara satah satah SRWT. A ZQTR B IQTS C ZQWR D ZQWS plane PQVW PQV{T dengan SPM m7 Question 14 3 Diagram 8 shows a pyramid with its rectangle base QRST. Rajah 8 menunjukkan sebuah piramid dengan tapak segi empat tepat QRST. Diagram 8 Rajah 8 Vertex P is vertically above 7. Name the angle between the plane PIS and the plane PfQ. Puncak P berada tegak di atas T. Namakan sudut di antara satah PTS dengan satah PTQ. A ZPQT B ZPST C ZSPQ D ZSTQ 130

37 SPM m Question 14 4 Diagram 7 shows a right-angled triangular prism with the horizontal base $STV. Rajah 7 menunjukkan sebuah prisma segi tiga tegak dengan tapak mengufuk QSTV SPM nlo Question 14 6 Diagram 7 shows a right prism PQRSTU with a rectangular base, PQRS. X and Y are midpoints of the lines TP and UQ respectively. Rajah 7 menunjukkan sebuah prisma tegak PQRSTU dengan tapak segi empat tepat, PQRS. X dan Y adalah masing-masing titik tengah bagi garis TP dan UQ. Diagram 7 Rajah 7 What is the angle between the plane SZU and the base QSTV? Apakah sudut di antara satah STU dengan tapak QSrv? A ITUV C ZUSV B ZUTV D ISUV SPM mg Question 14 Diagram 7 shows a pyramid with an equilateral triangle PQR as the horizontal base. M is the midpoint of QR and N is the midpoint of SP. Rajah 7 menuniukkan sebuah piramid dengan segi tiga sama sisi PQR sebagai tapak mengffik. M ialah titik tengah QR dan N ialah titik tengah SP. Name the the plane Namakan satah TPY. A ZSYT C ZSXY Diagram 7 Raiah 7 angle between the plane ISP and TPY. sudut di antara satah TSP dengan B ZSTY D ZSYX SPM frll Question 14 7 Diagram 6 shows a cuboid with a horizontal base PQRS. Rajah 6 menuniukkan sebuah kuboid dengan tapak mengufuk PQRS. Name the angle the base PQR. Namakan sudut tapak PQR. A IPQN C IPMN Diagram 7 Rajah 7 between the plane NQR and di antara satah NQR dengan B ZPNM D ZPRN Dragram 6 Name the angle between the plane TWQ and the plane ISRW Namakan sudut di antara satah fwq dengan satah TSRW. A ZWQS B zwqr D tqws c tqwr 131

38 KERTAS 2 Bahagian,ffi ffiw Angles between Lines and Planes #ffi Angles between Two Planes SPM M Question 4 1 Diagram I shows a right prism. Right-angled triangle PQR is the uniform cross section of Rajah I menunjukkan sebuah prisma tegak. Tapak segi tign ber.sudut tegak PQR ialah kertttan renta.\ seragam prisma itu. U T tzcm t8 cm Diagram I Rajah 1 Calculate the angle between the plane RTU and the plane PQTU. Hitungkan sudut di antara satah RTU dengan satah PQTU. 13 marksl [3 markah) Answer/"/awapan: 132

39 SPM N6 Question 2 2 Diagram 1 shows a right prism. The base PQRS is a horizontal rectangle. The right-angled triangle llfrlurg is the uniform cross section of the prism. Rajah L menunjukkan sebuah prisma tegak. Tapak PQRS ialah sebuah segi empat tepat yang mengufuk. Segi tiga bersudut tegak UPQ ialah keratan rentas seragam prisma itu. U 9cm P cm O Diagram I Raiah 1 Identify and calculate the angle between the line RU and the base PQRS. Kenal pasti dan hitungkan sudut di antara garis RU dengan tapak PQRS. [4 marks) 14 markah) Answer/"/cwapan: SPM no7 Question 4 3 Diagram 1 shows a right prism. The base PQRS is a horizontal rectangle. Right-angled triangle Gtrl QnU is the uniform cross section of the prism. V is the midpoint of PS. Rajah I menunjukkan sebuah prisma tegak. Tapak segi empat tepat PQRS adalah mengufuk. Segi tiga bersudut tegak QRU adalah keratan rentas seragam prisma itu. V ialah titik tengah PS. Diagram 1 Rajah I Identify and calculate the angle between the line UV and the plane RSTU. Kenal pasti dan hitung sudut di antara garis UV dengan satah RSTU. L3 marksl 13 markahl Answer/"/awapan: 133

40 SPM m Question 11 4 Diagram 11 shows a cuboid. M rs the midpoint of the side EH and AM = 1 cm. Rajah 11 menunjukkan sebuah kuboid. M ialah titik tengah bagi sisi EH dan AM = 1 cm. Diagram I I Rajah ll (a) Name the angle between the line AM and the plane ADEF. 3I Namakan sudut di antara garis AM dengon satah ADEF. (b) Calculate the angle between the line AM and the plane ADEF. llll Hitung sudut di antara garis AM dengan satah ADEF. Answerfiawapan: (a) (b) 13 morksl 13 markahl SPM m Question 4 Diagram 4 shows a cuboid with horizontal base ABCD.,I is the midpoint of the side AF. Rajah 4 menunjukkan sebuah kuboid dengan tapak mengcrfuk ABCD. J ialah titik tengah bagi sisi AF. Diagram 4 Rajah 4 (a) Name the angle between the plane BCJ and the base ABCD. (fip Namakan sudut di antara satah BCJ dengan tapak ABCD. (b) Calculate the angle between the plane BCJ and the base ABCD. ildl Hitung sudut di antara satah BCJ dengan tapak ABCD. Answer/"Iawapan: (a) (b) [3 marks] [3 markah] 134

41 SPM nlo Question 2 6 Diagram 2 in the answer space shows a right prism. The base ABCD is a horizontal rectangle. Trapezium ABGF is the uniform cross section of the prism. Raiah 2 di ruang jawapan menunjukkan sebuah prisma tegak. Tapak ABCD ialah segi empat tepat yang mengufuk. Trapezium ABGF adalah keratan rentas seragam prisma itu. (a) On Diagram 2 in the answer space, mark the angle between the plane BCHG and the base ABCD. ll Pada RajahT di ruang jawapan, tandakan sudut di antara satah BCHG dengan tapak ABCD. (b) Hence, calculate the angle between the plane BCHG and the base ABCD. SlDl Seterusnya, hitung sudut di antara satah BCHG dengan tapak ABCD. [3 marksl 13 markahl Answer/,/awapnn; (a) E cm H (b) Diagram 2 Rajah 2 SPM frll Question 4 7 Diagram 4 shows a right prism with a rectangular base EFGH on a horizontal plane. Trapezium FGML is the uniform cross section of the prism. Rajah 4 menunjukkan sebuah prisma tegak dengan tapak segi empat tepat EFGH di atas satah mengufuk. Trapezium FGML adalah keratan rentas seragam prisma itu. Effi Diagram 4 Rajah 4 (a) Name the angle between the plane JEM and the plane Namakan sudut di antara satah JEM dengan satah JHGM. (b) Calculate the angle between the plane JEM and the plane JHGM. ildl Hitung sudut di antara satah JEM dengan satah JHGM. Answer/,/awapan: (a) (b) [3 marks) [3 markah) 13

42 REVIEWAND ANSWERS KERTAS 1 ry. fh. angle between a line and a plane is the angle between the line and its orthogonal projection 6n, the plane.. ldentify the normal (perpendicular line) to the base and the orthogonal projection of the line t/f, onto the base EFCH.. Draw a right-angled triangle with the line t/f, the normal and the orthogonal projection as its sides. tu /B, /." \ OR Angle between the plane PIS and the plane pte = ZSTQ Answer: D EF Angle between the line VF and rhe base EFGH = ZVFH Answer: B Angle between the plane STU and the base esfv = ZUTV Answer: B %. The angle between two planes is the angle between the lines perpendicular to the line of intersection of the two planes.. The line of intersection of the plane pewt and the plane SRWT is line T\IL. Thus, the angle between the plane pew and the plane SRWT is the angle between the lines QW and RtzV which are perpendicular to the line of intersection W. t'68 a Angle between the plane NQR and the base PQR = ZPMN Answer: C. The angle between the plane ISP and the plane TPY is the angle between the lines SX and Xy which are perpendicular to the line of intersection TP of the two planes. Angle between the plane PQWT and the plane SRWT = ZQWR Answer: C 3. The line of intersection between the plane PIS and the plane PTQ is line PL Angle between the plane ISP and the plane TPY = ZSXY Answer: C 136

43 U 9cm Angle between the plane fwq and the plane ISRW = ZQWR Answer: C KERTAS 2 ry. The angle between the plane RTIJ and the plane, PQTU is the angle between the lines RT and QT which are perpendicular to the line of intersection TU,. Use the tangent of an angle in triangle QIR to find Z.QTR.. Use a scientific calculator to find the angle. 18 cm I Qr)- - - zj ''-'- 'T'r P Scm O Angle between the line RU and the base PQRS = ZPRU PR = \t7.,'* = 13 cm tan ZPRU - PU PR _9 13 IPRU = tan-t a 13 = 34.7o or 34" 42'. Since t/s is a perpendicular line from t/ to the plane RSTU, US is the orthogonal projection of the line UIl onto the plane RSIU.. Thus, the angle between the line UV and the line US is the angle between the line Utl and the plane RSIU, that is ISUV.. Use the tangent of an angle in triangle SUt/ to find ZSUV. u1,.'". Angle between the plane RTU and the plane PQTU = ZQTR OT ZRTQ tan IOTR = QR Qr =12 l8 lqfr = tan-t 9 18 =33.69" or 33o 4l'. Since UP is a perpendicular line from U to the base PQRS, RP is the orthogonal projection of the line RU onto the base PQRS.. Thus, the angle between the line RU and the line RP is the angle between the line RU and the base PQRS, that is ZPRU,. Use the tangent of an angle in triangle PRU to find zpbu. Angle between the line UV and the plane RSTU _ ISUV SV=16cm+2=8cm SU = \t7;@ = 13 cm tan ZSUV = sy SU _8 13 ZSUV = tan-r A 13 = 31.6" or 31" 36' 137