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2357975-51194800MathematicsPaper 2FORM 414 June 2018Session 1TIME: 2 hours TOTAL: 100 marksExaminer: Miss M. Eastes Moderator: Mrs.D. AlgieName and Surname:Teacher:PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY BEFORE ANSWERING THE QUESTIONS.This question paper consists of 17 pages. Formulae are given on page 15.Please check that your question paper is complete.Answer all questions on this question paper.Read and answer all questions carefully.It is in your own interest to write legibly and to present your work neatly.All necessary working which you have used in determining your answers must be clearly shown.Approved non-programmable calculators may be used except where otherwise stated. Where necessary give answers correct to 2 decimal places unless otherwise stated.Ensure that your calculator is in DEGREE mode.Diagrams have not necessarily been drawn to scale.Give reasons for all statements used in geometry.Questions123456Out of911121769MarkQuestion7891011TOTALOut of67995100MarkSECTION AQUESTION 1 [9]The equations of the following lines are given:ABy = 1/2 xCDy = 2EF3y – 2x - 3 = 0GH3y + 2x – 6 = 0JKy = - 2x + 4LMx = 2Choose from the above list a line which will satisfy the conditions in each of the following cases. Each line may only be used once. Show all calculations used to arrive at your answer.1.1 The line cuts the x-axis at (3; 0). (2)________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________1.2 The line is parallel to the line y = 2/3 x + 3 (2)________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________1.3 The line has a gradient of zero. (1)________________________________________________________________________________________________________________________________________________________________________________________________________________________1.4 The line represents all points 2 units from the y-axis. (1)________________________________________________________________________________________________________________________________________________________________________________________________________________________1.5 The line has an angle of inclination of 116,6 (2)________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________1.6 The line is perpendicular to line JK. (1)________________________________________________________________________________________________________________________________________________________________________________________________________________________ QUESTION 2 [11]In the diagram below, the coordinates of A(4; 4), B(12; -4) and C(-1; -4) are given.25268277233000AC // DE and CEB is a straight line.D is the midpoint of AB.Determine:2.1the length of AB. Give answer correct to one decimal digit. (2)________________________________________________________________________________________________________________________________________________________________________________________________________________________2.2the coordinates of D, the midpoint of AB. (2)________________________________________________________________________________________________________________________________________________________________________________________________________________________2.3the equation of the line DE. (4)________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________2.4the coordinates of E. (3)________________________________________________________________________________________________________________________________________________________________________________________________________________________ QUESTION 3 [12]3.1 is the midpoint of and . Find the value of a. (3)_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________3.2 Given the points , and : Determine the equation of the straight line through R perpendicular to PQ. (4)________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________3.3 Consider the points: : Show that these 3 points are collinear. (5)________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ QUESTION 4 [17]A(1;5), B(4.3) and C(0;-3) are the vertices of the triangle.10337805715004.1Determine, using analytical methods, the coordinates of D if ABCD is a parallelogram. (5)________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________4.2If the distance between C and F(8;p) is 12 units, find the value(s) of p (to the nearest integer). (5)________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________4.3Find the size of ACB in ? ABC (7)________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________QUESTION 5 [6]The diagonals of rectangle STVW are equal in length and bisect each other at P.Calculate the co-ordinates of T and W.220386315727y0y608330158750V (3 ;2)S (-1 ;-1)TPW0V (3 ;2)S (-1 ;-1)TPW1150620527050________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ QUESTION 6 [9]A, B, C, D, and E, are points on the circumference of a circle with centre O. TBS is a tangent to the circle at B. DB and AC intersect at X. 2327910139700400DBTEAC12222111OX34521231S00400DBTEAC12222111OX34521231SGive the reasons to the following statements made.6.1 ______________________________________________________________ (1)6.2______________________________________________________________ (1)6.3 ______________________________________________________________ (1)6.4______________________________________________________________ (1)6.5 ______________________________________________________________ (1)6.6 ______________________________________________________________ (1)6.7_______________________________________________________________ (1)6.8 _______________________________________________________________ (1)6.9_______________________________________________________________ (1)QUESTION 7 [6]left267335O is the centre of the circle, and COT XY.If OC = 10 and XY = 12,Find the length of CT.00O is the centre of the circle, and COT XY.If OC = 10 and XY = 12,Find the length of CT.2773104141266 C X ? O T Y00 C X ? O T Y31883064694500 (6)________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________QUESTION 8 [7]205740164836 F C B 1 1 E A 00 F C B 1 1 E A 11809225056373 3 3003 3 326687723324593 2 3003 2 32664460387985006112761511550011526021511550089656915115500 893050972750091851593548008965698623203851910862320105219588903 2 3003 2 389004865123 1003 12817628568403 2 3003 2 3216941462730 28176282802423 4 3003 4 32902334153330 3 D 3 D282533018866700 BE is the diameter. BD = DF. BCF and ADF are straight lines. = 10.Calculate:8.1 (3)___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________8.2 (1)____________________________________________________________________________________________________________________________________8.3 (3)______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________QUESTION 9 [9]14992356286599441070485 F 2 2 2 3 D 1 E0 F 2 2 2 3 D 1 E2823210107315 O0 O391372443991409211720320 G G417066569203111110011111366823279995111110011111FE is a tangent to the circle with centre O. D and F are joined so that EG = GF. 9.1 If = p, name, with reasons, two other angles which are equal to p.(2)________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 9.2 Express in terms of p.(3)________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________9.3 Express in terms of p.(4)______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________QUESTION 10 [9]26155651397000In the diagram alongside,?ABC is drawn in thecircle. TA and TB are tangentsto the circle.The straight line THK is4671060180975parallel to AC with H onBA and K on BC.AK is drawn. Let A3=x10.1Prove that K3=x. (2)________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________10.2Prove that AKBT is a cylic quadrilateral. (2)________________________________________________________________________________________________________________________________________________________________________________________________________________________10.3Prove that TK bisects AKB. (3)________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________10.4Prove that TA is a tangent to the circle passing through the points A, K and H. (2)________________________________________________________________________________________________________________________________________________________________________________________________________________________QUESTION 11 [5]83312025400 D A O B C0 D A O B C1122045252285500 CD is a tangent at D to the circle with centre O. CD = 10 units. AC = 25 units. If r is the radius, show that r = 10,5 units.(5)______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________FORMULAE:175260245173500-342909906000M46710001312425ore space for working if needed: ................
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