St Stithians College



67818019177000Mathematics Paper 1Grade 12 Preliminary Examination 2018Duration:3 hoursExaminer:R. ObermeyerMarks: 150Moderator:A. JanischDate:31 August 2018External Moderator: I. L. AtteridgeINSTRUCTIONS: See overleaf for Instructions.This paper consists of 9 pages (including cover) and an information sheet.NAME: ____________________________________ASSESSMENTQuestionLevel TestedTopicSuggested Time AllocationPossible markMark ObtainedSECTION A11 – 4 Equations20 mins1721 – 4 Sequence & Series11 mins931 – 4 Financial Mathematics 10 mins841 – 4 Probability 10 mins851 – 4 Sequence & Series16 mins1361 – 4 Functions6 mins571 – 4 Functions14 mins1281 – 4 Calculus10 mins8SECTION B91 – 4 Equations13 mins11101 – 4 Financial Mathematics10 mins8111 – 4 Sequence & SeriesCalculus11 mins9121 – 4 Functions18 mins15131 – 4 Calculus16 mins13141 – 4 Probability8 mins7151 – 4 EquationsCalculus8 mins7TOTAL:150PERCENTAGE:Teacher’s Signature: ________________________ Controller’s Signature: _______________________Moderator’s Signature: _______________________ InstructionsPLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLYThis question paper consists of 9 pages and an Information Sheet. Please check that your paper is complete.Read the questions carefully.Answer all the questions.Number your answers exactly as the questions are numbered.You may use an approved non-programmable and non-graphical calculator, unless otherwise stated.Round your answers off to one decimal digit where necessary unless otherwise stated.All the necessary working details must be clearly shown. Answers only will be merited with one mark only.It is in your own interest to write legibly and to present your work neatly.Please hand in this question paper.Answer all questions underneath each other.Start each new question on a new page.SECTION AQuestion 1Solve for x:3x2-5x-1=0 (correct to TWO decimal places)(3)2x2+x-6≤0(3)11x+20=2-x(5)Solve for x and y:2x+y=5 and 2x2-xy-4y2=8(6)[17]Question 2Given the quadratic sequence:34; 42; 48; 52; 54; …Determine an expression for the general term Tn, of the quadratic sequence.(4)Jack calculates the 20th term to be -155. Without doing any calculations, explain why this answer has to be incorrect.(1)The general term of the first row of differences is Tn=-2n+10. For a certain value of p, the pth term of the quadratic sequence is equal to the pth term in the first row of the differences. Determine the value of p.(4)[9]Question 3The school buys a Toyota Hilux Single Cab for R 242?000 and set up a sinking fund to replace it in 6 years’ time.If the rate of depreciation is 14,3% p.a. compounded annually, calculate how much the truck will be worth after 6 years.(2)If the rate of inflation is 11,2% p.a. compounded annually, calculate how much a new truck will cost at the end of 6 years.(2)What is the total required value for the sinking fund if the current truck will be traded in?(1)Determine the monthly payments into the sinking fund if interest is calculated at 9,8% p.a. compounded monthly and payments start in one months’ time.(3)[8]Question 4A main course and dessert are chosen from the following menu:What is the probability that the choice will contain:Lamb Shank and Peppermint Crisp Tart?(2)Lamb Shank or Peppermint Crisp Tart?(2)The following Venn diagram refers to probabilities. If X and Y are independent events, find the values of a and b. Show all calculations. (4) [8]1333500129540r=1∞2.71-r00r=1∞2.71-rQuestion 5Evaluate: (3)CALCULATORS MAY NOT BE USED IN THIS QUESTIONlog 2 and 2 log 2 are the first two terms of an arithmetic as well as a geometric sequence.Find the nth term of the arithmetic sequence.(2)The nth term of the geometric sequence is Tn=2n-1×log 2Calculate the difference between the fifth term of the geometric sequence and the fifth term of the arithmetic sequence.(3)If S is the sum of the first six terms of the geometric sequence and Q is the sum of the first six terms of the arithmetic sequence, calculate the ratio S:Q.(5)[13]Question 6Given:f(x)=9x+3+5Calculate the y-intercept of f.(1)Determine the equation of the positive axis of symmetry of f.(2)Write down the equations of the asymptotes of f-1.(2)[5]Question 7The graph of k is defined by the equation k(x)=ax. The point 32;4 lies on k.Calculate the value of a.(2)For what values of x will k be defined?(1)Write down the equation of k-1, the inverse of k, in the form y=….(2)If p(x)=x-32, determine algebraically the point(s) of intersection of k and p.(5)Hence, or otherwise, determine the values of x for which k(x)>p(x).(2)[12]Question 8Given:f(x)=-3xFind f'(x) by first principles.(5)ANSWER THIS QUESTION ON THE DIAGRAM SHEETThe graph of y=g(x) is given below.Use the diagram sheet to draw a possible graph of y=g'(x) on the same set of axes.(2)Indicate the point A, on the x-axis, where g"(x)=0(1)[8]SECTION BQuestion 9Given:3ax2+bx-3a=0 where a; b ∈R.Prove that the roots of the equation are real.(3)If b=8a, determine, with a reason, whether the roots will be rational or irrational.(4)You are given four statements. Write down if they are always true, sometimes true or never true. In each example, the variables are defined.xx=1where x ∈ R(1)If x<y<0 then x2<y2<0 where x ; y ∈ R(1)2xy≤x2+y2where x ; y ∈ R(1)a+xa+2x>aa+xwhere a ; x ∈ N(1)[11]Question 10John retired at the end of 2016 and received his pension of R 1?400?000.How many full months can he live off the fund if he withdraws R 13?500 every month? Interest is calculated at 10,7% p.a. compounded monthly.(4)How much can he withdraw in the 292nd month?(4)[8]Question 11In an arithmetic progression, T2016+T2017=p. Find, in terms of p, the value of: T2013+T2014+T2015+T2016+T2017+T2018+T2019+T2020(2)Evaluate:f'(x) if f(x)=3x+4x(4)Dt2a29t3 where a is a constant.(3)[9]Question 12Given:f(x)=-x2-2x+3 and g(x)=-2.2x-1+1Draw neat graphs of f(x) and g(x) on the same set of axes.Clearly label ALL important information.(8) Write down the equation of the axis of symmetry of f.(1)Determine the value(s) of k for which -x2-2x+k=0 will have no real roots.(2)Given:h(x+y)=h(x)+h(y)+xy and h(1)=3. Calculate h(4).(4)[15]Question 13The turning points of g(x)=-2x3+ax2+bx+c are at x=2 and x=5. One turning point is 2;-9. Determine the value of a, b and c.(6)If h(x)=x3+3x2-9x, determine the value(s) of k for which y=-9x+k will be a tangent to the curve of h.(7)[13]Question 14The letters of the word PROBABILITY are arranged to form different words. Assume that all the words have meaning and repeated letters are treated as identical.How many different words can be formed?(2)What is the probability that the word will start and end with the same letter?(2)There are 23 students in a class. What is the probability (correct to 3 decimal places) that at least two will have the same birthday? Assume there are exactly 365 days in a year.(3)[7]Question 15Town A is 30 km west of town B. Two athletes start walking simultaneously from the two towns.The athlete who starts from town A, walks due east in the direction of B at a constant speed of 6 km/h. He reaches point P after x hours.The athlete who starts at town B, walks due north in the direction of another town C at a constant speed of 8 km/h. he reaches point Q after x hours.Find the distance in terms of x, i.e. PQ, between the two athletes after x hours.(3)Given: PQ2=100x2-360x+900How many hours will it take for the athletes to be a minimum distance from each other?(2)What was this minimum distance between them?(2)[7] ................
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