GAUTENG DEPARTMENT OF EDUCATION PREPARATORY EXAMINATION
GAUTENG DEPARTMENT OF EDUCATIONPREPARATORY EXAMINATION201810612MATHEMATICSPAPER 2TIME:3 hoursMARKS: 15015 pages, 1 information sheet and a 21 page answer bookP.T.O.
MATHEMATICS(Paper 2)10612/182GAUTENG DEPARTMENT OF EDUCATIONPREPARATORY EXAMINATIONMATHEMATICS(Paper 2)TIME: 3 hoursMARKS: 150INSTRUCTIONS AND INFORMATIONRead the following instructions carefully before answering the questions.1.This question paper consists of 11 questions.2.Answer ALL the questions in the ANSWER BOOK provided.3.Clearly show ALL calculations, diagrams, graphs et cetera that you used to determinethe answers.4.Answers only will NOT necessarily be awarded full marks.5.You may use an approved scientific calculator (non-programmable andnon-graphical), unless stated otherwise.6.If necessary, round-off answers to TWO decimal places, unless stated otherwise.7.Diagrams are NOT necessarily drawn to scale.8.An INFORMATION SHEET with formulae is included at the end of the questionpaper.9.Write neatly and legibly.P.T.O.
MATHEMATICS(Paper 2)310612/18QUESTION 1In a Mathematics competition learners were expected to answer a multiple choice questionpaper. The time taken by the learners to the nearest minute to complete the paper, wasrecorded and data was obtained. The cumulative frequency graph representing the timetaken to complete the paper is given below.100Cumulative frequency908070605040302010001020304050Time taken to complete the paper in minutes60An incomplete frequency table for the data is given below.Time taken tocomplete the paper inminutesFrequency10 x 2020 x 3030 x 4040 x 5050 x 60a6828341.1Determine the value of a in the frequency table.(2)1.2How many learners wrote the paper?(1)1.3Identify the modal class of the data.(1)1.4Calculate:1.4.1The estimated mean time, in minutes, taken to complete the paper1.4.2The number of learners that took longer than 35 minutes to completethe paper(3)(2)[9]P.T.O.
MATHEMATICS(Paper 2)10612/184QUESTION 2A group of students did some part-time work for a company. The number of hours that thestudents worked and the payment (in rand) received for the work done is shown in the tablebelow. The scatter plot is drawn for the data.Number of hoursworked67810131518202325Payment (in AYMENT (in rand)40003500300025002000150010005000024681012 14 16 18TIME IN HOURS20222426282.1Calculate the standard deviation of the number of hours worked.(1)2.2Determine the number of hours that a student needed to work in order to receivea payment that was more than one standard deviation above the mean.(3)2.3Determine the equation of the least squares regression line of the data.(3)2.4Mapula who worked for 11,5 hours was omitted from the original data. Calculatethe possible amount that the company has to pay Mapula.(2)2.5Use the scatter plot to identify an outlier and give a possible reason for this pointto be an outlier.(2)[11]P.T.O.
MATHEMATICS(Paper 2)10612/185QUESTION 33.1In the diagram below, points A(–2 ; –3), B(3 ; –4), C(4 ; r ) and D(2 ; 1) are thevertices of quadrilateral ABCD. P is the midpoint of line AD.yD(2 ; 1)xOPA(–2 ; –3)C(4 ;)B(3 ; –4)3.1.1Calculate the value of r if AD BC.(4)3.1.2What type of quadrilateral is ABCD?(1)3.1.3Determine the coordinates of P.(2)3.1.4Prove that BP AD.(2)3.1.5Determine the equation of the circle passing through PBA in the form3.1.6( x a ) 2 ( y b) 2 r 2 .(5)Calculate the maximum radius of the circle having equationx 2 y 2 2 x cos 4 y cos 2 for any value of .(5)P.T.O.
MATHEMATICS(Paper 2)3.2610612/18In the diagram below, points P(–2 ; 1) and Q(3 ; –2) are given and R is a point inthe third quadrant. PQ and PR cut the x-axis at S and T respectively.QP̂R 77,47 º.yP (–2 ; 1)77,47ºxTSOQ (3 ; –2)R3.2.1Determine the equation of line PQ in the form ax by c 0(3)3.2.2Determine the equation of PR in the form y mx c .(6)[28]P.T.O.
MATHEMATICS(Paper 2)710612/18QUESTION 4In the diagram below, AB is a chord of the circle with centre C. D(–1 ; –2) is the midpointof AB. DC AB. The equation of the circle is x y 6 y 4 x 12 .22yBxD(–1 ; –2)OAC4.1Determine the coordinates of C.(3)4.2Determine the radius of the circle.(1)4.3Calculate the length of AB.(5)4.4Calculate the area of ABC.(3)[12]P.T.O.
MATHEMATICS(Paper 2)10612/188QUESTION 55.1Simplify the following expression to a single trigonometric function.sin x.sin 90 y cos x.sin 180 y cos x.cos y 360 sin x sin y5.2(6)Given: cos(A B) cosAcosB sinAsinB5.2.1 Prove that cos(A B) cosAcosB sinAsinB(2)5.2.2 In the diagram, T is a point such that HÔT P and sin P a . T isreflected about the x-axis to R such that HÔR QT(x ; a)QPOH R(a) Determine the coordinates of T in terms of a.(b) Write down the coordinates of R in terms of a.(c) Calculate cos(P Q).(d) Hence, show that P Q 360º.5.3(2)(2)(2)(1)Given: cos d5.3.1 Write down the values of d such that cos is defined.5.3.2 Determine the general solution for if :15cos cos 6(2)(6)[23]P.T.O.
MATHEMATICS(Paper 2)10612/189QUESTION 6The functions f ( x ) tan 2 x and g ( x) 1 sin 2 x are sketched for x 135 ; 135 .f g-135 -90 -45 45 90 6.1Write down the equation of the asymptote in the interval x 135 ; 0 .6.2If h( x ) 6.3Determine the equation of p in its simplest form, if graph g is translated bymoving the y -axis 45 to the right.6.4sin x 2 sin 3 x, determine h in terms of f .2 sin 2 x. cos xDetermine the values of x for which tan 2 x .( 1 sin 2 x) 0 forx [ 135 ; 0 ) . 135 (1)(4)(3)(3)[11]P.T.O.
MATHEMATICS(Paper 2)10612/1810QUESTION 7The given figure represents a roof in the form of a triangular prism. The beams EG and EDhave length p metres. EF GD and GÊD 30 .AEB30 GFCDWithout using a calculator:7.1Prove that GD 2 p 2 ( 2 3 ) .(3)7.2Hence, determine the value of CD in terms of p, if CĜD 60 .(3)[6]P.T.O.
MATHEMATICS(Paper 2)10612/1811GIVE REASONS FOR ALL STATEMENTS AND CALCULATIONS IN QUESTIONS8, 9, 10 AND 11.QUESTION 8In the diagram below, TAP is a tangent to circle ABCDE at A. AE BC and DC DE.TÂE 40 º and AÊB 60 º .TE60º1240º1A2DP12C12B8.1Identify TWO cyclic quadrilaterals.8.2Determine, with reasons, the size of the following Ê1(3)P.T.O.
MATHEMATICS(Paper 2)8.310612/1812In the diagram below, radius CO is produced to bisect chord AB at D.CA 34 mm and AB 40 mm34CA.ODBCalculate the size of Ĉ .(4)[15]P.T.O.
MATHEMATICS(Paper 2)1310612/18QUESTION 9In the diagram below, O is the centre of the circle. ABCD is a cyclic quadrilateral. BA andCD are produced to intersect at E such that AB AE AC.B12A1 2 3O 4xE2 1D213CDetermine in terms of x:9.1B̂ 2(2)9.2Ê(5)9.3Ĉ 2(3)9.4If Ê Ĉ 2 x, prove that ED is a diameter of circle AED.(4)[14]P.T.O.
MATHEMATICS(Paper 2)10612/1814QUESTION 1010.1In ABC below, D and E are points on AB and AC respectively such thatDE BC.AD AEProve the theorem that states that .DB EC(6)ADEB10.2CIn DXZ below, AC XZ and BP DZ. DY is drawn to intersect AC at B.DAXProve that:BYCPZBC DA YZ DX(5)[11]P.T.O.
MATHEMATICS(Paper 2)10612/1815QUESTION 11In the diagram below, NPQR is a cyclic quadrilateral with S a point on chord PR. N and Sare joined and RN̂S PN̂Q x .Nxx1P1221S21Q1 2RProve that:11.1ΔNSR ΔNPQ(3)11.2ΔNQR ΔNPS(3)11.3NR.PQ NP.QR NQ.PR(4)[10]TOTAL:END150
MATHEMATICS(Paper 2)10612/18INFORMATION SHEETx b b 2 4ac2aA P(1 ni)A P(1 i ) nA P(1 ni)n 2a (n 1)d 2Tn a (n 1)dSn Tn ar n 1a r n 1Sn r 1 x 1 i 1if ( x h) f ( x )f ' ( x) limhh 0F nA P(1 i ) nP ; r 1S a; 1 r 11 rx[1 (1 i ) n ]i x x y y2 M 1 2 ; 12 2y y1y y1 m( x x1 )m 2x 2 x1d ( x 2 x1 ) 2 ( y 2 y1 ) 2y mx c x a 2 y b 2In ABC:m tan r2abc sin A sin B sin Ca 2 b 2 c 2 2bc. cos Aarea ΔABC 1ab. sin C2sin sin . cos cos .sin cos cos . cos sin . sin sin sin . cos cos .sin cos cos . cos sin . sin cos2 sin 2 cos 2 1 2 sin 2 2 cos2 1 sin 2 2 sin . cos 2nx xnP(A) n( A )n S yˆ a bx 2 xi x i 1nP(A of B) P(A) P(B) – P(A en B)b x x ( y y ) (x x)216
P.T.O. GAUTENG DEPARTMENT OF EDUCATION PREPARATORY EXAMINATION 2018 10612 MATHEMATICS PAPER 2 TIME: 3 hours MARKS: 150 15 pages, 1 information sheet and a 21 page answer book
ANNEXURE O PROVINCIAL ADMINISTRATION: GAUTENG DEPARTMENT OF COMMUNITY SAFETY Gauteng DCS is an equal opportunity employer and gender sensitive employer and it is its intention to promote representivity in the Public Service through the filling of these posts. The Department
PROVINCIAL ADMINISTRATION: GAUTENG DEPARTMENT OF COMMUNITY SAFETY Gauteng DCS is an equal opportunity employer and gender sensitive employer and it is its intention to promote representivity in the Public Service through the filling of these posts. The Department committed to the achiev
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