St Stithians College



GRADE 12:JUNE EXAMINATIONS 2015MATHEMATICS: PAPER 1Time: 3 hoursTotal: 150Examiner: A. BrowneModerator: G. ClaassenRead the following instructions carefully:This question paper consists of 9 pages and 12 questions. Please check that your question paper is complete.Read the questions carefully.Number your answers exactly as the questions are numbered.All the necessary working details must be clearly shown.Approved non-programmable calculators may be used unless otherwise stated. Answers should be rounded off to two decimal digits where necessary, unless otherwise stated.It is in your own interest to write legibly and to present your work neatly.Question 1Solve for x: (Leave answers in surd form where necessary)2x2-3=3x+12(3)x3+5x2-8x-48=0(4)2x2-3x≥5(4)x+5<2 and 2x2-3x≥5(2)[13]Question 2Consider the arithmetic sequence: 4; 1; – 2; – 5; – 8; … Write down the next term in the sequence.(1) If the nth term of the sequence is -461, determine the value of n.(3)Given the geometric series: 3+2+x+89+1627+… State the value of x.(1) Write this infinite geometric series in sigma notation.(2) Calculate S∞.(2) If S∞-Sn=256729, determine the value of n.(5)Show that there is no real value of x for which the sequence1; x + 3; x – 1 will be a geometric sequence.(4)The first two terms of a geometric sequence and an arithmetic sequence are7 and 7r. The sum of the first three terms of the geometric sequence is 28more than the sum of the first three terms of the arithmetic sequence.State the third term of each sequence in terms of r.(2)Determine TWO possible values of r.(4)[24]Question 3Consider the graph of f.3457575490220f00fIs the graph of f a one-to-one function? Explain.(2)Write down the range of f.(2)On the graph provided on the answer sheet, draw the inverse of f.(3)Explain why the inverse of f is not a function.(2)1137920548005Write down a possible restriction for the domain of f so that the inverse of the graph of f will now be a function.(2)[11]Question 4If f(x)=log5xState the domain of f.(1)State the equation of f-1 in the form y = …(2)State the equation of k(x), the line of symmetry of f and f-1 (1)Sketch k, f and f-1 on the same system of axes and label all intercepts with the axes. Indicate the coordinates of two points on each graph. (7)If h is the reflection of f-1 in the y-axis, state the equation of h(x). (1) [12]QUESTION 5 R12?000 is invested at an interest rate of 13% per annum, compounded monthly. Calculate the value of the investment after 5 years.(3) Money is invested at an interest rate of 10,5% per annum, compounded quarterly.After 6 years the value of the investment has increased by R8?624,14. What was the initial amount?(5) R10?000 is invested at 12% per annum, compounded monthly.Calculate the effective interest rate.(3)A man invests R200?000 at 12% per annum compounded monthly. After 2 years he withdraws R50?000 and after another 3 years he withdraws another R30?000.Calculate the value of his investment after 8 years.(6) A firm buys computers. After 5 years the value of the computers will be R12?000. Calculate the initial value of the computers if the computers decreased at a rate of12% on a reducing basis.(3) [20]QUESTION 6The graphs of fx=-12x2+x+c, gx=-2x+q and h(x) = q are sketched below.Q and R are symmetrical to each other about the symmetry line of f.P is the turning point of f and f-1=g-1=6.Show that c=152 and q=4.(4)Determine the range of f.(4)One of the axes of symmetry of g is a decreasing function.State the equation of this axis of symmetry.(2)If f-3=f5=g12=0, state the values of x for which f(x).g(x) 0.(4)Calculate the average gradient of f between x= –1 and x=5.(2)Show that f and g share a common tangent at Q(–1; 6)(3)[19]QUESTION 7Determine f'(x) from first principles if fx=9-x2. (5)Evaluate dydx ify=1-6x(3)y = 8 - 3x48x3(4)In the diagram ΔPQR is equilateral with sides 12 cm. F and G are points on PQ and PR respectively. FGHJ is a rectangle such that QJ=HR=x.1621331173456 P FG Q J H R Prove that the area of the rectangle FGHJ is 3 x (12-2x)(3)For which value(s) of x is the area of FGHJ a maximum.(3)[18]QUESTION 82229774215966In the figure A(0;0) and B(2;-1) are the turning points of the graph of f. Determine the values of x in each of the following: 3269392919754A(0;0)00A(0;0)f(x)<0(2)f'(x)≤0(2)x×f'(x)≥0(3)4134879140009B(2;-1)00B(2;-1)[7]QUESTION 9The graph of y=f'(x), where f is a cubic function, is sketched below.f(-1)=f(5)=0, f0=-5, f1=-16 and f3=-32.Use the graph and the information above to answer the following questions:For which values of x is y=f(x) a decreasing function?(2)State the x-coordinate of the point of inflection of f.(1)Sketch y=f(x), indicating the intercepts with the axes, thestationary points and the point of inflection.(4) [7]QUESTION 10The flight of a cricket ball, hit by a batsman, is modelled by the equation s(t)=-13t2+2t+1 where st is the height of the ball in metres, t seconds afterit was hit. A(0,3; 1,57) and B(5,7; 1,57) are two points which satisfy the equation. What was the height of the ball when the batsman hit it?(1)What was the average speed of the ball during the first 3 seconds?(2)The ball was caught on its way down, at a height of 1,57 m.After how many seconds was the ball caught?(1)What was the speed of the ball at the moment it was caught?(3)[7]QUESTION 11Given: f(x)= x3+qx-px2-pqx-pDetermine:fp(1)f'(p)(4)limx→pf(x)(3)[8]QUESTION 12The roots of the equation: 4x2-24x+p=0 are rational and have a ratio of 5:7. Calculate p and the roots.[4]TOTAL 150DIAGRAM SHEETNAME:______________________Question 3.30337185 ................
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