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center16954500center8890JUNE 2015Mathematics P200JUNE 2015Mathematics P2left149860Grade:12Time:3 hoursDate:19 June 2015Marks: 150Examiner:ExternalModerator:G. Claassen00Grade:12Time:3 hoursDate:19 June 2015Marks: 150Examiner:ExternalModerator:G. Claassen-16891081280Name:00Name:INSTRUCTIONS:1. This examination consists of 14 pages, including the front cover. 2. Read the questions carefully. 3. Answer all the questions in your answer book.4. You may use a non-programmable, non-graphical calculator, unless otherwise stated. 5. Round off your answers to ONE decimal digit unless otherwise stated.6. All necessary working details must be clearly shown.7. Write legibly and present your work neatly.8. Figures are NOT drawn to scale.9. Write your name on the question paper.QUESTION1234567891011TOTALMAXIMUM15812152319131410147150MARK49174409525 %00 %QUESTION 1The straight line passing through A(-2;-3) and B(7;2) is parallel to the straight line with equation rx-3y+5 = 0. Calculate the value of r.(5)_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________1.2The circle with equation 3x2+3y2-12x-18y+m=0 touches the x-axis at point H. Determine:1.2.1the coordinates of the centre of the circle(5)____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________1.2.2the coordinates of H.(2)___________________________________________________________________1.2.3the value of m.(3)_________________________________________________________________________________________________________________________________________________________________________________________________________[15 marks]QUESTION 2114300058102500A driver, Chris, of a courier motorcycle, recorded the distance (in km) he had travelled on 15 trips. The 5-number summary of his data is: (12;21;25;32;34) and the box-and-whisker diagram for his travel appears below.Another driver in the same company, Colin, also recorded the distances (in kilometres) he travelled on 15 trips. The data appear alongside: 2419212717203222 261813233010132.1Determine the median for the data.(2)___________________________________________________________________2.2Determine the 5-number summary for the data of Colin’s travels and draw a box-and-whisker diagram in the space provided below the box-and-whiskerdiagram above.(4)_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________2.3Carefully analyse the box and whisker diagrams for the two drivers, and comment on the differences or similarities, if any, between the distances covered by each on the 15 trips.(2)____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________[8 marks]QUESTION 33259896335332AFE1 23DO1019845B6C700AFE1 23DO1019845B6C73.1In the figure, O is the centre of the circle AEBCD, with line BOF//EA, with F on AD. It is given that BOD = 100° and D9=20°. The sizes of the angles are as indicated. In each case, supply a valid reason.3.1.1A2=50° _________________________3.1.2O1=80° _________________________3.1.3F10=80° 43995009525___________________________3.1.4A1=30° ___________________________3.1.5B5=30° right219075PSOQMR132121PSOQMR132121___________________________(5)3.2In the sketch, O is the centre of the circle PQRS. (3) SP is produced to M. O1=66° and R2=41°. Calculate the size of P1, giving reasons.__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________405511012700DCBA1212DCBA12123.3In the diagram, AB is the diameter of the circle ABCD. (4)It is given that A2=57°. Calculate the size of D, giving reasons.__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________[12 marks]QUESTION 44.1 SEQ CHAPTER \h \r 1It is given that cos 27°=k. Use a sketch to express each of the following in terms of k, clearly showing all working detail:4.1.1cos?(-27°) ____________________________________________________________________________________________________________(2)4.1.2sin 423° ____________________________________________________________________________________________________________(3)4.1.3sin 27° ___________________________________________(2) 4.1.4sin 54° ____________________________________________________________________________________________________________(3)4.2 Prove the following identity: SEQ CHAPTER \h \r 1cosA - cos2A + 23sinA-sin2A = 1 + cosAsinA(5)__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________[15 marks]33362908255ABDOM ●yx0ABDOM ●yxQUESTION 5In the figure, the straight line with equation2y-x-10=0 cuts the x-axis at A 46869353200400463105527315900and the y-axis at B. M is the midpoint of AB and OD ⊥ AB. SEQ CHAPTER \h \r 1Determine:5.1the coordinates of A, B and M.______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(4)5.2the coordinates of D.(6)______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________5.3the equation of a circle with centre O, such that AB is a tangent to the circle.(4)__________________________________________________________________________________________________________________________________________________________________________________________________________________5.4the equation of the circle with AB as diameter. Give your answer in the form Ax2+By2+Cx +Dy +E = 0(6)__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________5.5the size of ABO, correct to one decimal place.(3)________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________[23 marks]QUESTION 66.1Determine the general solution of the equation: sin2x +sin2x =1 and cosx ≠0(7)________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________4241605489816ABCDMzyPxABCDMzyPx6.2P is on the same horizontal plane as the vertices B and C of a vertical, rectangular picture ABCD, attached to a wall. The perpendicular distance PM from P to BC is n metres, BPM=x°, CPM =y° and the angle of elevation DPC=z°.6.2.1Show that BC can be expressed as:BC=n(tan x+tan y)______________________________________________5147378764470________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(5)6.2.2Hence, prove that the area of ABCD = n2tanz (tanx+tany)cosy(7)________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________________________________________________________________________ [19 marks]411480010985500QUESTION 77.1 SEQ CHAPTER \h \r 1The following table shows the heartbeat per minute of 100 adults between the ages 20 to 60.7.1.1 Complete the cumulative frequency table.(2)Heartbeats/minute intervalsNumber of individualsCumulative Frequency50 < x ≤ 60460 < x ≤ 701870 < x ≤ 802680 < x ≤ 903290 < x ≤ 1006100 < x ≤ 1107110 < x ≤ 1202120 < x ≤ 13057.1.2Draw an ogive curve to represent the data. Label the axes and use an appropriate scale.(4)31114143510007.1.3Show on your graph, using the letter B, where you would read off the 75th percentile.(1)7.2Given, the following set of values: {a; a+d; a+2d; a+3d; a+4d; a+5d; a+6d.7.2.1Determine, in terms of a and d, the variance of this set of values.(4)______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________7.2.2Hence, without further calculation, write down the variance of the following set of numbers -7; -4; -1; 2; 5; 8; 11(2)__________________________________________________________________________________________________________________________________________________________________________________________________________________[13 marks]3850640-173355PSRQM12PSRQM12QUESTION 88.1In the figure, P, Q, R and S are pointson a circle with centre M. It is given that Q=2x+20°and S=5x+20°,Calculate, stating reasons, the size of M1.___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(6)2893060342900FASBCDE12341112xy00FASBCDE12341112xy8.2In the figure, ABCD is a cyclic quadrilateral and FAS is a tangent, meeting CB produced at F. AD is produced to E. CDE=y and ACB = x.8.2.1Give, stating reasons, ONEother angle equal to x.________________________________ (2)8.2.2Give, stating reasons, TWO other angles equal to y. (4)____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________8.2.3Write down the size of F in terms of x and y.(2)____________________________________________________________________________________[14 marks]3574415-411480AHEBGCD0AHEBGCDQUESTION 9In ΔABC, HG//AC, ED//BC, ADDC= 32 , and BGGC= 21.If AB=15 units, determine, with reasons, 451739061595K0Kthe values of the following:9.1the length of AE(4)_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 9.2the length of AH(3)____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________9.3the value of GKKH(3)____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________[10 marks]QUESTION 101164590384175SR11232TP ●123Q00SR11232TP ●123QRefer to the sketch. The radius of the larger circle, centre P, is twice that of the smaller circle, with centre Q. SR is a tangent to both circles, ST, SQ and QR are drawn. Reasons need to be supplied.10.1Prove that Δ TSQ///ΔSRQ.(4)_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________10.2If the radius of the smaller circle is r, find the length of SQ in terms of r.(4)____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________10.3Calculate the size of TSR.(6)_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________[14 marks]QUESTION 1111.1Determine the general solution for θ and α if cos 2θ . cos α = 1.(4)_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________11.2Determine the maximum value of 42-cosx . Show relevant working.(3)________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________[7 marks]TOTAL 150 ................
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