St Stithians College



04445MATHEMATICS PAPER 2JULY PRELIMINARY EXAMINATIONJULY 2019Time: 3 hours150 MarksExaminer and Moderators: St Mary’s DSG Mathematics DepartmentName:Instructions:1.This question paper consists of 27 pages and a separate double-sided Information Sheet. Please check that your paper is complete.2.Read the questions carefully.3.Answer ALL the questions on the question paper (with reasons where needed).4.Diagrams are not necessarily drawn to scale.5.You may use an approved non-programmable and non-graphical calculator, unless otherwise stated.6.All necessary working details must be clearly shown.7.Round off your answers correct to two decimal digits where necessary, unlessotherwise stated.8.It is in your own interest to write legibly and to present your work neatly.9.Write only in black or blue ink.SECTIONAQ1Q2Q3Q4Q5Q6111214161110SECTION BQ7Q8Q9Q10Q 11Q 12TOTAL%14211113107150SECTION AQUESTION 1The distances (in cm) of the best attempts of 11 long jump athletes during this year’s Inter-High event are given:287328374486492501522583601619685Determine:a)The mean distance.(2)b)The standard deviation. (2)c)How many distances lie within one standard deviation from the mean. Give evidence for your answer. (3)Hence, what conclusion can you make regarding the distances jumped? (1)d)The official that measured the distances, mistakenly measured each of themx cm shorter than what they were supposed to be. As a result, all distances measured were x cm shorter than they were supposed to be. When the mistake was corrected, the sum of the athletes’ jumps is 5555.1)Calculate the value of x. (2)2)Write down the standard deviation of the new correct distances.(1)[11]QUESTION 2a)A-3;4, B-4;2 and C3;p are points in the Cartesian plane.Determine the value of p if AB⊥AC.(3)b)?PQR is sketched with vertices P-2;5, R(-6;3) and Q(x;y).N(-2;-1) is the midpoint of RQ.1135380113030v<>^yx(-6;3) NQ(x ; y)RP(-2;5)(-2;-1)00v<>^yx(-6;3) NQ(x ; y)RP(-2;5)(-2;-1)Determine:1)The equation of PN. (2)2)Determine the equation of the perpendicular bisector of RQ.(3)3)Determine the size of RPQ, rounded off correct to two decimal places. (4)[12]QUESTION 3a)Simplify the following completely: (5)2tan1800-x. 2cosx-1800.sin(900+x)2sinx+1800.cos?(3600-x)b)If tan360=t, express the following in terms of t, with the aid of a diagram (and without a calculator):Diagram1)cos540(2)2)1sin3060 (3)3)sin720 (4)[14]QUESTION 4* Circle A has the centre (2;5).* Circle B is given as: x2+y2+8x-2y=-1* Circle A and Circle B have the same length radius.* Tangent TP intersects Circle B where Circle B touches the positive y – axis, at point T.a)Determine the equation of Circle B in the form x-a2+y-b2=r2(3)b)Write down the equation of Circle A.(2)c)Draw a rough sketch of the circles given.(2)d)(1) Determine the coordinates of point T.(2)(2) Hence, determine the equation of tangent TP.(2)e)Show, by calculation, whether Circle A and Circle B intersect each other?(5)[16]QUESTION 5a)In the diagram A, B, C, D and E are points on a circle with centre O. AC produced meets DF at F and DF is a tangent to the circle at D. EC and AD are straight lines and A2=600.12877809525001)Show that AE // DC.(3)2)Find the value of:i)C3(2)ii)D2(1)iii)B(1)iv)C4(1)3)Is OCFD con-cyclic? Provide a reason for your answer.(3)[11]QUESTION 6Information about profits made on two different days are given in the table below.* The correlation between expenses (x) and profit made in rand (y) at the Colour Run Fun Day is recorded in the first row.* The correlation between expenses (x) and profit made in rand (y) at the Winter Dance is recorded in the second row of the table.Line of best fit (including outliers)Correlation CoefficientThe number of outliers with high expenditure and LOW profitThe number of outliers with low expenditure and HIGH profitColour Fun Dayy=4000+0,9x0,826Winter Dancey=2000-0,5x- 0,740a)Which line of best fit would you choose to maximize the profit and why? (2)b)If the outliers of the Colour Fun Day were removed, how would this affect the:1)Correlation coefficient? (Give a reason).(2)2)Gradient for the line of best fit? (Give a reason).(2)c)Using the information in the table above, explain which event will be thebest for the Matric Dance fundraising? Explain your answer.(2)d)Considering the event you choose in c), predict how much profit/loss will be made if the expected expenses for that event are going to be R5?000 in 2020.(2)[10]1644968-696278KLMO00KLMOSECTION B:QUESTION 7a)The sides of an equilateral triangle KLM are 18 cm each. Calculate the length of the radius OM of the circumcircle of the triangle, leaving your answer in simplest surd form.(6)637540138157CADBE85660°40°00CADBE85660°40°b)A design for a sail is shown. Two sails, ΔACE and ΔADB, are joined to a centre mast AB.. Show that the total height of the mast AB is 9 metres.(3)Calculate the area of the larger sail ABD, correct to 1 decimal place.(5)[14]QUESTION 8a)The graph of fx=sin?(x+a) and gx=cosbx x∈[-1800;1800] is sketched:1)Determine the values of a and b respectively.(2)2)For which values of x will:i)f(x)≥0 for x∈[-1800;1800]?(2)ii)fx.gx>0 for x∈[-600;600]?(2)3)Give the new equation if f is shifted 200 to the left.(1)b)If cos550=A and sin900-k.sin350-sin2150.cos350=B, find the values of k if A=B and 00≤k≤900.(5)c)(1) Prove that 4sinα .cos3α-4 cosα.sin3α=sin 4 α(4)(2)Hence, solve for α, if 1+4sinα.cos3α=4cosα.sin3α if ∝∈[-900;900]Give answer correct to one decimal place.(5)[21]BLANK PAGEQUESTION 9In the diagram, QVT and VS are tangents to the circle at Q and S respectively.PQ//RT as RT is a straight line. Let Q1=x and R2=y.1996440165735Prove that:a)V1=2R1(4)b)QVSW is a cyclic quadrilateral.(4)c)P1+T=R1+R2(3)[11]QUESTION 10In the diagram below, two circles intersect at A and B. 215646041021000Points A, T, S and B lie on the larger circle and points A, D, C and B lie on the smaller circle.Let B=x and show that:a)T1+T2=D3(3)b)?DTS∕∕∕?TAS(3)c)TS2=BS.CS(7)[13]QUESTION 11In the diagram below, a circle with centre P and tangent RQT has been given.Point S is where line PR intersects the circle. RS = 2 units, RQ = 4 units and T (k;-k) has been given.143256072390a)Determine the length of PQ.(3)b)If the equation of the circle is given as (x-1)2+(y-4)2=9 1)Show that QT2=2k2+6k+8(3)2)Hence, determine the length of the shortest possible tangent that can be drawn from point T to the circle.(4)[10]QUESTION 12In the diagram below, QRS is an equilateral triangle. RP ⊥ QS and QR = p.211074050801)Write down the lengths of QP and PR in terms of p. Leave your answerin simplified surd form, where necessary.(2)2)In the diagram below ABCD represents a square section of a steep ramp which makes an angle of 600 with the horizontal plane ABEF, and 300 with the vertical plane DCEF.The side of the square is p metres. A man climbing the ramp, starting at point A, chooses path AC rather than AD as it is not quite so steep. The angle made by this path with the horizontal plane is shown in the diagram as x.6400801911350043167309080500Calculate the size of x, rounded off to one decimal digit.(5)[7]TOTAL: 150 ................
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