# Afterschool Spm Add Maths Trial p12 2013

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• 8/11/2019 Afterschool Spm Add Maths Trial p12 2013

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59SOALAN ULANGJI SPM 2013

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1 e Cartesian graph above represents a relation of set A = {1, 3, 5, 6} and set B = {1, 2, 3, 4}. State Graf pada rajah di atas menunjukkan hubungan set A ={1, 3, 5, 6} dan set B ={1, 2, 3, 4}.Nyatakan (a) the image of 1, imej bagi 1,

(b) whether the relation is a function or not. samada hubungan ini merupakan fungsi atau tidak. [2 marks]

(b) ______________________________

2 Given that , find .

Diberi fungsi , cari . [2 marks]

3 Find the value ofpif the quadratic equationp(x2 + 1) = 6x has two equal roots.

[2 marks]

4 Given that , find the range of values of xwhich satisfy the inequality

Diberi , cari julat nilai x yang memuaskan ketaksamaan [2 marks]

5 Given that , find the value of kand of m.

Diberi , cari nilai k dan nilai m. [3 marks]

(m) _____________________________

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6 Solve the equation .

Selesaikan persamaan . [2 marks]

7 Given that 4 logx 2 + 2 logx7 logx98 = 3, find the value of x.

Diberi 4 logx 2 + 2 log

x

x [3 marks]

8 Given that 11, p+ 1, 19 are three consecutive terms of an arithmetic progression and(p+ 1) is the sixth term, find the value of

dan (p + 1) ialah sebutan keenam, cari nilai

(a) p,

(b) the first term. sebutan pertama. [3 marks]

Answer : (a) p = __________________________

(b) _____________________________

9 Express the recurring decimal 2133333. as a fraction in its simplest form.

Ungkapkan perpuluhan jadi semula 2133333 dalam bentuk pecahan yang termudah. [3 marks]

Answer : (a) m = __________________________

(b) _____________________________

10 e common ratio of a geometric progression is and the sum to infinity is 21. State the first three terms of the progression.

Nisbah sepunya suatu janjang geometri ialah dan hasil tambah sehingga ketakterhinggaannya ialah 21.Nyatakan tiga sebutan pertama janjang ini. [3 marks]

11 Given the straight line 4x+ 3y6 = 0 is parallel to the straight liney= ax+ bwhich passes through the point (12, 5),find the value of aand of b.

cari nilai a dan nilai b. [3 marks]

(b) _____________________________

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12 e diagram above shows the straight line graph obtained by ploing against x.

Rajah di atas menunjukkan graf garis lurus yang didapati dengan memplotkan melawan x.

(a) Calculate the value ofp. Hitungkan nilai p.

(b) Expressyin terms of x.

Ungkapkan y dalam sebutan x. [4 marks]

(b) _____________________________

13 Given that Ois an origin andM(3, 4). If MP= 2i3j,

Diberi O ialah titik asalan dan M(3, 4). Jika MP = 2i3j,

(a) Express OMin terms of iand j.

Ungkapkan OM dalam sebutan idan j.

(b) Find | OP|.

Cari | OP |. [3 marks]

(b) _____________________________

14 Given that a= 2i+ (p+ 1)j and b= 3i+ 6j, find the value ofpin each of the following cases :

Diberi a= 2i+ (p + 1)j dan b i j, cari nilai p bagi setiap kes yang berikut :

(a) a+ bis parallel to x-axis. a+ bselari dengan paksi-x

(b) ais parallel to b. aselari dengan b. [4 marks]

(b)p= __________________________

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15 Find the equation of the curve which has the gradient function x2(x+ 3) and passes through the point (2, 14).

Cari persamaan lengkung yang mempunyai fungsi kecerunan x2(x + 3) dan melalui titik (2, 14). [3 marks]

16 Given , find the value

Diberi , cari nilai

(a)

(b) k if

k jika [4 marks]

(b) k= ___________________________

17 Given , find the value off(1).

Diberi , cari nilai bagi f (1). [3 marks]

18 e volume of the liquid in a container,Vcm3is given by V= 2x3+ 4x2+ 5, where xcm is the depth of the liquid in the container.Given that Vincreases at a rate of 32 cm3s1, find the rate of increase of xwhen x= 2.

Isipadu cecair dalam sebuah bekas V cm3 diberi oleh V = 2x3+ 4x2 Diberi bahawa V bertambah dengan kadar 32 cm3s1, cari kadar pertambahan x pada ketika x = 2. [4 marks]

19 Find the equation of the normal to the curve at the point (2, 8).

20 e right diagram shows a semi circle with centre Oand diameterPOR= 8 cm. Given that the arc length ofPQis 387 cm, calculate

Rajah kanan menunjukkan sebuah semi bulatan berpusat O dengan diameterPOR = 8 cm. Diberi panjang lengkok PQ ialah 387 cm, hitungkan

(a) the value of in radian, nilai dalam radian, (b) the area of sector OQR, luas sektor OQR. (Use/Gunakan= 3142) [4 marks]

(b) _____________________________

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21 Given that cosA= and sinB= whereAandBare obtuse angles, find the value of

Diberi kos A = dan sin B = dengan sudut A dan B ialah sudut cakah, cari nilai

(a) sin 2A,

(b) cos (BA). [4 marks]

(b) _____________________________

22 Five leers from the word T S U N A M Iis to be arranged in a row begins with a vowel.Find the probability that the arrangement begins with the leer .

Lima daripada huruf-huruf dalam perkataan disusun sebaris bermula dengan suatu vokal.Cari kebarangkalian jika susunan itu bermula dengan huruf . [3 marks]

23 e mean of a set of 25 numbers is 24.

(a) If every number in the set is added by 2, determine the new mean of the set. Jika setiap nombor dalam set itu ditambah dengan 2, tentukan min baru bagi set itu.

(b) If two numbers kand k+ 2 are taken out from the set, the new mean is 22, find the value of k. Jika dua nombor k dan k + 2 dikeluarkan daripada set itu, min baru bagi set itu ialah 22, cari nilai k. [4 marks]

(b) k= ___________________________

24 In a chess competition, Peter plays five games.e probability that Peter wins one of the games is . Calculate the probability that Peterwon

Dalam satu pertandingan catur, Peter bermain lima permainan. Kebarangkalian bahawa Peter memenangi mana-mana satu permainanialah . Hitungkan kebarangkalian bahawa Peter memenangi

(a) three games aer playing four games,

tiga permainan selepas empat permainan,

(b) the first game and the last game. permainan pertama dan permainan terakhir. [4 marks]

(b) _____________________________

25 Xis a continuous random variable withX~N(, 25).IfP(X< 9) = 07257, find the value of .

[4 marks]

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Section A/ [40 marks / ]

Answer allquestions from this sectionJawab semuasoalan dalam bahagian ini.

1 Solve the equation 2p+ k = p2 + k2 5 = 5 [6 marks]

2 A function is defined by , for all values of xexcept x= h, and m is a constant.

Satu fungsi ditakrian oleh , bagi semua nilai x kecuali x = h, dan m ialah pemalar.

(a) Determine the value of h. Tentukan nilai h.

(b) Given that 2 maps onto itself under the functiong, find

Diberi nilai 2 dipetakan kepada dirinya sendiri di bawah fungsi g, cari

(i) the value of m, nilai m,

(ii) the value of

nilai bagi

(iii) the value of pifg(2p) = 5 [7 marks]

3 e straight line x+ 4y=pis normal to the curvey= (2x 1)2+ 1 at the point Q.

Find / Cari

(a) the coordinates of Q, [4 marks] koordinat-koordinat Q,

(b) the value ofp, [1 mark] nilai p,

(c) the equation of the tangent at the point Q. [2 marks] persamaan tangen pada titik Q.

4 In Diagram 1, the equation of the straight linePQR is 2x y+ 4 = 0. If PQ: QR= 2 : 3, find

Dalam Rajah 1, persamaan garis lurus PQR ialah

(a) the coordinates of R, [3 marks] koordinat-koordinat R,

(b) the equation of perpendicular bisectorPR, [4 marks]persamaan pembahagi dua sama serenjang PR.

DIAGRAM 1

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5 Solve each of the following equations :Selesaikan setiap persamaan berikut :

(a) [3 marks]

(b) log5 3x log

5(x 2) = 1 [3 marks]

6 (a) e sum of the first nterms of an arithmetic progression is given by Sn

Hasil tambah n sebutan pertama suatu janjang aritmetik diberi olehSn

Find / Cari

(i) the first term and common difference, sebutan pertama dan beza sepunya,

(ii) the sum of the all the terms from the 5th term to the 10thterm. [5 marks]

(b) Given that the 5thand the 8thterms of a geometric progression are 2 and respectively, find the common ratio of thisprogression.

Diberi sebutan kelima dan sebutan kelapan suatu janjang geometri masing-masing ialah 2 dan , carikan nisbahsepunya janjang tersebut.

[2 marks]

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Section B/ [40 marks / ]

Answer fourquestions from this sectionJawab empatsoalan dalam bahagian ini.

7 Diagram 2 shows two sectors of a circle, sector OCBAwith centre Oand sectorPCOAwith centreP.OAPandOPC are two congruent equilateral triangle with sides 8 cm. Find

Rajah 2 menunjukkan dua sektor bulatan iaitu sektor OCBA yang berpusat di O dan sektor PCOAyang berpusat di P. OAP dan OPC adalah dua segi tiga sama sisi yang kongruen dan mempunyai sisi8 cm. Cari

(a) the reflex angle AOC in radian, [1 mark]

(b) the perimeter of the shaded region,

perimeter kawasan berlorek, [3 marks]

(c) the area of the major sector OCBA,luas sektor major OCBA, [2 marks]

(d) the area of the shaded region.luas kawasan berlorek. [4 marks]

[Use/ Gunakan= 3.142]

DIAGRAM 2

8 Table 1 shows the values of two variables, x and y which are related by the equationy = ax(x + b) , where aand bare constants.

Jadual 1 menunjukkan nilai bagi dua pembolehubah x dan y yang dihubungkan oleh persamaan y = ax (x + b) , dengan keadaan adan b ialah pemalar.

(a) Reduce the equation y = ax(x + b) to the linear form.

Tukarkan persamaan y = ax (x + b) kepada persamaan bentuk linear. [1 mark]

(b) Plot the graph of against x. /Lukiskan graf melawan x . [4 marks]

(c) Use the graph from (a) to find Gunakan graf anda dari (a) untuk mencari

(i) the value of aand of b

nilai a dan nilai b,

(ii) the value of y when x= 6 [5 marks]

9 (a) A set of game score x1, x

2, x

3, x

4and x

5has the mean 6 and standard deviation 12.

Satu set skor bagi suatu permainan x1, x

2, x

3, x

4dan x

(i) Find the sum of the squares of the score, x2Cari hasil tambah kuasa dua skor itu, x2 [2 marks]

(ii) If each score is multiplied by 3 and 2 is added to it, find the mean and variance of the new score.Jika setiap skor itu didarabkan dengan 3 dan ditambah dengan 2, cari min dan varians

bagi set skor yang baru. [3 marks]

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(b) Table 2 shows the marks obtained by a group of students in a particular test. Jadual 2 menunjukkan markah yang diperoleh sekumpulan pelajar dalam satu ujian.

(i) Without drawing an ogive, find the median marks. [3 marks]Tanpa melukis ogif, cari median bagi markah-markah itu.

(ii) Calculate the mean marks. [2 marks]Hitungkan markah min.

10 Given the quadratic functionf(x) = 4x2 + 8x1. Diberi fungsi kuadratik f(x) =4x2 + 8x 1.

(a) Express the quadratic function f(x) in the form p(x+ q)2+ r, where p, qand r are constants.Determine whether the functionf(x) has a minimum or maximum value and state its value.

Ungkapkan fungsi kuadratik f(x) dalam bentuk p(x + q)2+ r dengan keadaan p, q dan r ialah pemalar.Tentukan sama ada fungsi f(x) mempunyai nilai maksimum atau nilai minimum dan nyatakan nilainya. [3 marks]

(b) Sketch the graph of the functionf(x) for the domain 1 x 3. State the corresponding range for this domain.Lakarkan graf fungsi f(x)untuk domain 1 x 3. Nyatakan julat yang sepadan dengan domain ini. [3 marks]

(c) Find the range of values of ksuch that f(x) = kx has no real roots.

Cari julat nilai k supaya f(x)= kx tidak mempunyai punca nyata. [4 marks]

11 (a) Given thatABCDis a parallelogram whereBC= 2i+ 3jand CD= 2i 2j, find

Diberi ABCD ialah sebuah segi empat selari dengan BC = 2i+ 3jdan CD = 2i j, cari

(i) AC,

(ii) the unit vector in the direction of AB. / vektor unit pada arah AB. [4 marks]

(b) Diagram 3 shows a triangleABCwhere CP=CB.

Given thatAB= a and AC= b.

Rajah 3 menunjukkan sebuah segi tiga ABC dengan CP=CB

Diberi AB = adan AC = b.

(i) Show thatAP= a+ (1 )b

Tunjukkan bahawa AP =a+ (1 )b

(ii) IfBP= 3a+ 3b, find the value of .

Jika BP = 3a + 3b, cari nilai . [6 mark]

DIAGRAM 3

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Section C / Bahagian C[40 marks / 40 markah]

12 A particle moves along a straight line from afixed point O. Its displacement, Sm is given by S= 9t2 t3, where tis the time in secondsaer leaving the point Q.

(Assume that motion to the right is positive)

Suatu zarah bergerak di sepanjang suatu garis lurus bermula dari satu titik tetap O. Sesarannya, S m, diberi oleh S= 9t2 t3, dengankeadaan t ialah masa, dalam saat, selepas meninggalkan titik Q.

(Anggapkan gerakan ke arah kanan sebagai positif)

Find / Cari

(a) the distance travelled the 2ndsecond, [2 marks] jarak yang dilalui dalam saat kedua,

(b) the time at which the velocity of the particle is uniform,masa ketika zarah mencapai halaju seragam, [3 marks]

(c) the value of twhen the particle passes the point Qagain,nilai t apabila zarah itu melalui titik Q semula, [2 marks]

(d) the range of values of tduring which the particle is moving towards the point Qaer instantaneous rest. julat nilai t apabila zarah menuju ke titik Q selepas rehat seketika. [3 marks]

13 Table 3 shows the price of a set of football jersey which has been worn by a particular team at Lion Championship tournament in the year2003 and 2004.

(a) Calculate the values of x,yand z.Hitung nilai x, y dan z. [4 marks]

(b) Given that the index number of Adibas jersey is 120 in the year 2003 based on the year 2002,find the price in the year 2002.

[2 marks]

(c) If the price of Nikas jersey increases at a constant rate every year, determine the price in the year 2005 to the nearest RM. terdekat. [4 marks]

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14 Given that x and y are positive integers with the following conditions : Diberi x dan y ialah dua integer positif dengan keadaan berikut :

I : e value of xis more than the value ofyby 5 or more

II : e minimum value of 2x+ 3yis 60

III : e maximum value of 4x + 3yis 5 times the minimum value of 2x+ 3y

(a) Write down three inequalities which satisfy the above conditions.Tulis tiga ketaksamaan bagi setiap keadaan di atas. [3 marks]

(b) Construct and shade the region Rthat satisfies all of the above conditions.Bina dan lorekkan rantau R yang memuaskan syarat-syarat di atas. [3 marks]

(c) By using your graph from (b), answer the following questions : Dengan menggunakan graf anda dari (b), jawab soalan-soalan berikut :

(i) Find the minimum value of the sum of the integers.Cari nilai minimum bagi hasil tambah integer tersebut.

[1 mark]

(ii) Given that xandyrepresents the number of itemMand itemNrespectively which was sold by the company.Find the total maximum sale obtained if the price of one unit itemMandNare RM15 and RM30 respectively.

Diberi bahawa x dan y masing-masing mewakili bilangan barang M dan barang N yang dijual oleh sebuah syarikat.

[3 marks]

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15 (a) Diagram 4 shows a triangle PQR. Calculate QPR.Rajah 4 menunjukkan sebuah segi tiga PQR. Hitungkan QPR. [3 marks]

DIAGRAM 4 / RAJAH 4

DIAGRAM 5 / RAJAH 5

(b) In Diagram 5,ABCandBCDare two triangles which aached atBC.Calculate

(i) the length ofBC, /panjang BC,

(ii) BCD,

(iii) the area of whole diagram, / luas seluruh rajah.[7 marks]