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1993900-58420000Mathematics Paper 2 June 2017FORM 4Examiner:A GunningModerator:P DenissenTime:2? hoursMarks:125NAME: Ques No12345678Out of1111685191410MarkQues No910111213TOTAL%Out of589811125100MarkAll questions are to be answered in this booklet.This question paper consists of 20 pages. Included in this, is a list of useful formulae. Please check that your question paper is complete.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.Diagrams have not necessarily been drawn to scale.Question 1Given points R(1;3), S(3;7) and T (-1; -1)Determine the gradient of RS. (2)____________________________________________________________________________________________________________________________________________________________The length of ST is w5. Find the value of w. Show all relevant working detail.(3)________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________Determine the equation of the line through T parallel to RS, giving your answer in standard form..(2)__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________Determine the equation of a line which is perpendicular to RS AND which bisects RS. (4)____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ [11]Question 2This diagram has not been drawn to scale. In the diagram, K(-1; -2) is the midpoint of LN with N(3;6).3270139250328y00y348170516764000PK LN with P on the x-axis.The angle of inclination of NL is θ.263652014668500382397069850N (3 ; 6)00N (3 ; 6)3662680107315003360420201930θ00θ2482215201295P00P4490720321945x00x270510010033000216535092075002483485-1270K (-1 ; -2)00K (-1 ; -2)31585068889900242062058420L00LDeterminethe gradient of LN (2)__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________the size of θ, correct to 1 decimal place(2)__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________the co-ordinates of L(4)__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________the length of NK, leaving your answer in the simplest surd form.(3)______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________[11]Question 3The points A(3;1), B(2; -2) and C(2;3) are given.If CD is perpendicular to AB with D-3;t+1, determine the value of t. (3)________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________If A, B and E are collinear, given E(r;4), evaluate r.(3)______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________[6]Question 4In the given sketch (not drawn to scale) A(1; 1) is a point on the line AB. And ABC=45°The line BC makes an angle α with the x-axis and the line AB makes an angle θ with the x-axis.1468120187961002690037193306Points B and C lie on the line y=3x+22460625-635C020000C212407523749045°02000045°2607310165735A(1; 1)020000A(1; 1)161925165735001872615-2540B020000B1871980217170α020000α1494155-2540θ020000θ-53164214113Determine the inclination of the line BC.(2)__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________Determine the value of θ, the inclination of the line AB.(2)__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________Determine the equation of the line AB.(4)__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________[8]Question 5267940595693y020000y2987749224923In the diagram below, reflex TOP= α and P has coordinates (-5; -12)3789045211455T020000T244470661595α020000α2968477171450O020000O4103370177505x020000x357218543387 020000 21265122155311041990215531172085091381P(-5; -12)020000P(-5; -12)Determine the value of each of the following, WITHOUT USING A CALCULATOR.cosα(3)__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________tan?(180°-α)(2)__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________[5]Question 6Without the use of a calculator, simplify each of the following. sin100°cos?(350°)(2)__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________sin2-20°+cos2(200°)(4)______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________cos130°-sin90°-θsin400°+cos-θ(5)______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________cos140°.tan?(-320°)sin220°(4)________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(e) Evaluate 4sin150°. 23tan135° without the use of a calculator.(4)__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ [19]Question 7If cos62°=k, determine the value of each of the folling in terms of k.sin28°(2)__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________cos242°(2)__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________Simplify to a single ratio tan360°-x.sin(90°+x)sin(-x)(5)________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________Consider 4sin2x-3=0 for 0°≤x≤270° , In how many quadrants will there be solutions for this equation?(1)____________________________________________________________________________________________________________________________________________________________Without the use of a calculator, solve for x, giving all possible values of x(4).______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________[14]Question 8On the same set of axes below, draw sketch graphs of each of fx=-2sin x and gx=cos 2x for the domain x∈[0°;360°]. You must show clearly any intercepts with the axes as well as the coordinates of the maximum and minimum values for each graph.(6)1428752227580142875141605(b) Without doing any working, write down the period of the graph kx=sin3x(1)______________________________________________________________________________State the amplitude of the graph fx+1(1)______________________________________________________________________________What is the range of gx=cos2x?(1)___________________________________________________________________________If f(x) has been shifted 30° to the right, write down the equation of the new graph. (1)______________________________________________________________________________ [10]Question 9260032536195T020000T267713025271623°02000023°2753833135572743200135571668780135422703195166370102000012472690263525O020000O3266440127192102000012699385111760202000022743200114166370522533020S020000S5052503383865R020000R2540000164170P020000P27538331676250O is the center of the is the diameter and PR is a tangent.If T =23°, to determine the size of R, the following steps have been given. For each of the statements below, fill in the relevant reason. (Remember, you are building a logically sequenced proof.)STATEMENTSREASONSTO = OSS1=23°O2=46°P=90°R=44°[5]Question 10In the given diagram, O is the centre of the circle. C1=30°Name with reasons, three other angles each equal to 30°(4)_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________Calculate, with reasons, the following angles.B(2)______________________________________________________________________________O1(2)____________________________________________________________________________________________________________________________________________________________[8]Question 11Complete the following statement:(1)A line drawn from the center of a circle perpendicular to the chord ____________________________________________________________________________________________________In the circle with centre O, OT⊥SQ, OT⊥PR, OP=10 units and PU=8 units. Determine TU.(8) ________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________2764007-52572102000013124023-55245302000032936328-147320202000022945234-382772A020000A3104707-148855001807535-148856003104205-148855001658679-1488561552205-149225Question 124020820197485102000014391025117475C020000C40829029897120200002143129020955B020000B29452192454300180753524543001658546194664003033395189865O020000O180264724716320200002395986021336030°02000030°1805940135255102000013946392154778102000011551940190500F020000F4354830187325D020000D29452192756490180721027495500308959927238528°02000028°2755738115378102000012842260281940E020000EIn the diagram, O is the centerof the circle. Chords AB = AC.CED=28° and ADB=30°.Calculate with reasons, the sizes of each of the following angles.E1(2)__________________________________________________________________________________________________________________________________________________________F2(2)_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________A2(4) _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________[8]Question 131868170392430In the diagram, AB is the diameter of the circle, center O. AB is produced to P. PC is a tangent to the circle at C. OE ⊥ BC at D.3465360178462A020000A2498725236855O020000O9939131240460027590751079500026683801437032020000224580852133601020000121548041951110198884818685610200001172504774295B020000B1828800101602020000223069558255020200002219202023771102000012605405230505102000012091690131224D020000D197167526670002157095285752020000224593552984520200002197548557981020000176697093345P020000P1968500110490E020000E2675255109247C020000C9939132072310Prove with reasons that EO ∥CA.(4)_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________If C2=x, name with reasons, two other angles also equal to x. (4)____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________Calculate, with reasons, the size of P in terms of x(3)_________________________________________________________________________________________________________________________________________________________________________________________________________________________________ [11]Useful formulae ................
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