Mathematics P1 (HG) (English) June 2012 QP hlayiso.com

Downloaded from hlayiso.com basic education Department: Basic Education REPUBLIC OF SOUTH AFRICA SENIOR CERTIFICATE EXAMINATION MATHEMATICS P1 HIGHER GRADE 2012 MARKS: 200 TIME: 3 hours MATHEMATICS HG: Paper 1 3011E 30111E This question paper consists of 9 pages, 1 sheet of graph paper and 1 formula sheet. X@5 MMT GAUTENG Please turn over Copyright reserved

Downloaded from hlayiso.com Mathematics/HG/P 1 2 DBE/2012 SCE INSTRUCTIONS AND INFORMATION Read the instructions below carefully before answering the questions. iI, This question paper consists of EIGHT questions. Answer ALL the questions. 2. Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in determining the answers. eh You may use an approved calculator (non-programmable and non-graphical), unless stated otherwise. 4. If necessary, round answers off to TWO decimal places, unless stated otherwise. 5. Use the attached graph paper only for QUESTION 8. Detach it and place it inside the front cover of the ANSWER BOOK. 6. Number the answers correctly according to the numbering system used in this question paper. Th Diagrams are not necessarily drawn to scale. 8. A formula sheet is included at the end of the question paper. 9. Write neatly and legibly. Copia reserved FHT AOUA A A Pease urn over GAUTENG

Downloaded from hlayiso.com Mathematics/HG/P 1 3 DBE/2012 SCE QUESTION 1 1 Solve for x: LL (2x +1)3-x)=-4 (4) 1.1.2 3x? 4x = 5 (Correct to TWO decimal places.) ©) 1.13 |3-2x|<7 (4) 7 Given: f(x)= oe Determine the values of x for which: 1.21 f(x)=0 (1) 1.2.2 ui (x) will be a real number (4) 1.3 Solve for x and y if they satisfy the following equations simultaneously: 3x2 +4xy +2y? =9 x+2y=3 (7) 1.4 The roots of a quadratic equation h(x) = 0, are given by: —~5+.,/25— x= ast where k is a constant. 141 Write down ALL the integers k, k > 0, for which the roots will be rational. Q) 1.4.2 For which value of & will one of the roots be 0? Now write down the value of the other root. (2) 1.4.3 If k= 8, what is h(x)? (3) 1.5 Prove that the roots of 3x” +2kx—k =2x are never equal, for any real number k. (5) [37] iaiiiead FAUT AAO EA A Pease urn over GAUTENG

Downloaded from hlayiso.com Mathematics/HG/P 1 4 DBE/2012 SCE QUESTION 2 Le 2.1 Given: S@e)=3% —4x and g(x)=3-x 2.1.1 Sketch the graphs of f and g on the same system of axes, clearly indicating the intercepts with the axes as well as the turning point of f on your graph. (7) 2.1.2 If the roots of 4x” -12x=9-3x are 3 and 3 respectively, use your graph to determine: (a) The solution of 3—x>3 qd) (b) The values of x for which | (x) < g(x) (2) (c) The values of k for which the roots of sx? -4x=k are non-real (2) 2.2 The point A(zit] lies on the graph of f(x)=6", b > 0. 2.2.1 Prove that b= > qd) 2.2.2 If f and g are symmetrical about the line y =x, write down an equation of g. (2) 2.2.3 Sketch the graphs of f and g on the same set of axes. Clearly show the intercepts with the axes as well as point A. (4) 2.2.4 Point B is the mirror image of A as seen in the line y=x. Clearly show on the curve of g, in QUESTION 2.2.3, where B will be situated. Write down the coordinates of B on the sketch. (2) Copyright reserved FUE AO ATO TA Please nurs ver GAUTENG

Downloaded from hlayiso.com Mathematics/HG/P I 3 DBE/2012 SCE 2.3 The diagram below shows graphs of the following: e The hyperbola having equation xy =-8 e The semicircle centre O that touches the hyperbola at B ( 22:22 ) e The absolute value graph of y =| xt+a , which passes through the intercepts C and D of the semicircle with the axes, as shown below y A (2 23202 @ D ad _ Determine: 231 The equation of the semicircle (3) 22 The values of a (3) 2.3.3 The x-coordinate of A, the intersection of the hyperbola and the absolute value graph (4) [31] QUESTION 3 3.1 When f(x)=8x* - x? +5ke-6 is divided by (x—1), the remainder is k? —5. Determine the values of k. G) 3.2 Given: g(x)= ax? +bx-2 g(x) is divisible by (2x—1) and has a remainder of 50 when it is divided by (x + 2). Determine the values of a and b. (6) [11] Corina mg Pete GAUTENG

Downloaded from hlayiso.com Mathematics/HG/P 1 6 DBE/2012 SCE QUESTION 4 4.1 Simplify the following without using a calculator: 9* .10*? 4.1.1 rere 4 Ce ee ® log, 2 4.1.2 3 * +log,,8 (5) 42 Solve for x, without using a calculator: 4.2.1 2.37 437.271 =7L (3) 4.2.2 log, x +log, 125=4 (5) 4.2.3 log(x + 4) +log(x +2) < log8 (6) 43 If 2*=12*—', prove that x = loge 12. GB) [26] QUESTION 5 5.1 In an arithmetic sequence the 7" term is 17. The sum of the 1 and 2™ terms is 12. Determine: 5.1.1 The first three terms of the sequence (5) 5.1.2 The sum of the first 40 terms of the sequence (4) 5.2 Determine the value of n if »y (5r — 4) = 235. (5) r=] a3 The first term of a geometric sequence is 2 and the sum of the first 4 terms is 10 times the sum of the first 2 terms. The common ratio is greater than 1. Determine the first three terms of the sequence. (6) 5.4 A geometric sequence has first term 3 and common ratio 7 1 Show that f alue of n, S,. - S, =——> w that for any value of n, S., n= 5am 6) [26] Connright reserved UIUC AON A A A Pease nn ver GAUTENG

Downloaded from hlayiso.com Mathematics/HG/P 1 7 DBE/2012 SCE QUESTION 6 6.1 Given: (x)= ax? —3x 6.1.1 Determine the derivative, f' (x), from first principles. (5) 6.1.2 Calculate the values of a if f’(a)=50. (3) 6.2 Differentiate the following with respect to x: 1 a 6.2.1 h(x) = [« + J (4) x Vx +3 622 y= (3) 3x 6.3 The line y =4x—3 isa tangent to the curve g(x)= 2ax? —bx? at x = 1. Determine the values of a and b. (8) [23] Coir A Powe ne GAUTENG

Downloaded from hlayiso.com Mathematics/HG/P1 8 DBE/2012 SCE QUESTION 7 7A The graph represents the functions f and g with f(x)= ax? +bx4+2 and g(x) _ y a A 3 f A C x -1 O - & v qld Determine the co-ordinates of C, the x-intercept of f and g. (1) dal Show, by calculation, that a=-1 and b=3. (5) 7.13 Determine the co-ordinates of B, a turning point of f (4) 714 Show that AB is parallel to the x-axis. (5) TAS For which values of x is T(x) > g(x)? (3) 7.1.6 For which values of x is f’(x)<0? (2) 72 The cost (c in rand) for a boat trip for one hour is given by the formula c= =" +80, where v is the average speed in kilometres per hour. eon Show that the total cost for a boat trip of 500 kilometres, at an average speed of v kilometres per hour, is equal to 20v? + mt rand. 6) v Ue Calculate the most economical speed for the 500 kilometre boat trip. (5) 7.2.3 Determine the total cost of the 500 kilometre trip at the most economical speed. (2) [30] Copyright reserved UAC EA demniiniel GAUTENG

Downloaded from hlayiso.com Mathematics/HG/P1 9 DBE/2012 SCE QUESTION 8 Keep-it-cool is a small company that produces two types of cooler bags, Cool-6 and Cool-12. The shaded area of the graph represents the feasible region if x units of Cool-6 and y units of Cool-12 are produced, subject to FOUR constraints. TWO of the constraints are: 15<x<40 S<y<20 y 50 40 30 an 7 4 ~ ° 6 20 o 10 ~ x 0 10 20 30 40 50 60 70 COOL-6 8.1 Write down the remaining TWO constraint inequalities. 8.2 If Keep-it-cool makes a profit of R10 on each Cool-6 cooler bag and R25 on each Cool-12 cooler bag, write down an equation in terms of x and y which will represent the profit (P) that is made. 8.3 The company wishes to make maximum profit. 8.3.1 What is the slope of the search line? 8.3.2 Show, on the attached graph paper, the optimal position of the search line which will ensure maximum profit. 8.3.3 How many units of each cooler bag should be produced to ensure maximum profit? 8.3.4 What is the maximum profit? 8.4 Determine the minimum profit that can be made subject to the same constraints. TOTAL: Conpright reserved CATAL A IA A GAUTENG (4) (2) Q) (2) 2) (2) @) [16] 200

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Downloaded from hlayiso.com Mathematics/HG/P1 DBE/2012 SCE CENTRE NUMBER: EXAMINATION NUMBER: GRAPH PAPER FOR QUESTION 8: TOBE HANDED IN y 50 40 30 an 7 =| SPS lela © 2 ° o 10 x 0 10 20 30 40 50 60 70 COOL6 Conrigiereserved FETT AAO A A GAUTENG

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Downloaded from hlayiso.com Mathematics/HG/P1 DBE/2012 SCE Mathematics Formula Sheet (HG and SG) Wiskunde Formuleblad (HG en SG) _ —b+ Vb? —4ac 2a T,=a+(n-1)d Sy=5 (a+T) or / of Sy =Fla+0) Sy = fea + (n -1)d] PQ) = tim LEAM= FO) ho h >! d =V (x - 4) +02 -1)" y=mxt+e Y-Yy = m(x- x4) peal x2- xX m=tan0 xy+x + (33.93) = (A) vayar (x- p)? +(y-g)" =r? In AABC: a__b _-¢ sinA sinB sinC a? =b? +c” —2bc.cosA area AABC = 4ab.sin c ae NANO 0 GAUTENG | |