Mathematics Grade 12 NSC June 2013 Question Paper

basic education Department: Basic Education REPUBLIC OF SOUTH AFRICA SENIOR CERTIFICATE EXAMINATION ADDITIONAL MATHEMATICS HIGHER GRADE 2013 MARKS: 400 ADDITIONAL MATHEMATICS HG TIME: 3 hours 3021E 30210E This question paper consists of 11 pages, a 1-page formula sheet and a normal-distribution sheet. X@5 VII GAUTENG Please turn over Copyright reserved

Downloaded from hlayiso.com Additional Mathematics/HG 4 DBE/2013 SCE INSTRUCTIONS AND INFORMATION i, A This question paper consists of FIVE sections. SECTION A is COMPULSORY. Answer any TWO sections from SECTIONS B, C, D and E. Only TWO of these sections will be marked. Answer each section in a SEPARATE ANSWER BOOK. The relevant section must be clearly indicated on the cover. Place all ANSWER BOOKS inside the ANSWER BOOK for SECTION A before handing them in. Unless otherwise indicated, non-programmable non-graphic calculators may be used. Round off answers to TWO decimal places or as stated otherwise. A formula sheet and statistical tables for SECTION E are attached. All essential calculations must be clearly shown, and writing must be legible. All angles must be expressed in radians. OT, PUT AC TCU IL Please nn over

Downloaded from hlayiso.com Additional Mathematics/HG 3 DBE/2013 SCE SECTION A: CALCULUS (COMPULSORY) QUESTION 1 1.1 Determine x if arcsin2x = a. (6) 1.2 Draw the graph of f(x) = arcsin2x and show the above solution. (6) [12] QUESTION 2 2.1 Given the function: x+1l if x<2 a(x)=} |x-4| ff 2<x<6 -(x-6)742 if x26 21 Sketch g(x). (8) 212 State, with reasons, the type of discontinuity that exists at x = 2. (3) 2.1.3 Use the definition of differentiability at a point to determine whether g(x) is differentiable at x = 6. (7) 2.1.4 Determine the value of k if: x+1 if x<2 g(x)=1 Ax-4| if 2<x<6 -(x-6)* +2 if x26 is continuous at x = 2. (7) 2.2 If f(x) =x and A(x) = (2x-3)”, find: 2.2.1 Cf ehy(x) 4) 2.2.2 D,Lf@).A()] (8) [37] Cori raed 00 gg Ps nv G4UTENG

Downloaded from hlayiso.com Additional Mathematics/HG 4 DBE/2013 SCE QUESTION 3 3.1 Determine the following limit: . cos2k—1 lim k>0 k 3.2 Determine, from first principles, the derivative of f(x) =~V2x-1. 3.3 Hence or otherwise, determine the equation of the normal to f(x) at x =2. QUESTION 4 4.1 By using the rules of differentiation, determine the following derivatives: 4.11 2 if y= cosx+Vx Ix nae d_sin2x _ dx sin(z + x) 1 4.1.3 D? ] x 42 Evaluate the following integrals (without using a calculator): -x 4.2.1 dx TS 4.2.2 Jsin30cos 6d0 0 il 4.2.3 dx J V1-2x? _—<——' (aretan4x)? 4.24 1+16x? Copyright reserved Please turn over GAUTENG (5) (10) (8) [23] (8) (6) (12) (4) (12) (6) (10) [58]

Downloaded from hlayiso.com Additional Mathematics/HG é} DBE/2013 SCE QUESTION 5 5.1 The sketch below represents functions f(x) =(x+2)?-3 and g(x) = 2x+4. 5.1.1 Show that the coordinates of A are (-3;-— 2) and that of B are (1; 6). (8) 5.1.2 (a) Show that the shaded area between A and B can be calculated by h(x) =—x? —2x+3. (6) (b) Calculate the shaded area between A and B, using a Riemann sum. (16) 5.1.3 Determine the maximum distance between f and g in the interval x €[-3;1]. (8) 5.1.4 Hence, determine the volume of the solid of revolution formed by rotating h about the x-axis between the points x = 0 and x=1. (14) [52] QUESTION 6 6.1 Show that, if f (8x? —7)dx =10 then 2a* —7a-28 =0. (6) 6.2 Use Newton's method to calculate the value of a, closest to 2,2, correct to 4 decimal places if [" (8x* — 7)dx = 10. (12) , [18] TOTAL SECTION A: 200 Co ne ugg Pamir

Downloaded from hlayiso.com Additional Mathematics/HG 6 DBE/2013 SCE SECTION B: MATHEMATICS OF FINANCE QUESTION 7 A man starts saving for his retirement in 40 years' time. He saves R500 a month in a bank account. He eventually retires after 45 years and not 40 years, but his contributions cease after 40 years. Calculate the value of his annuity after 45 years if: 7A The account is paying 8% p.a. interest, compounded monthly. (8) 7.2 The account is paying 9.5% p.a. interest. (8) [16] QUESTION 8 A contractor wants to buy trucks for his company to the value of R780 000. He takes the loan for the full amount and agrees to amortise it in 5 years' time. The bank offers 9,3% p.a. compounded monthly. 8.1 If he makes his first payment 8 months after the loan was granted, determine the value of the monthly payment if the last payment is made at the end of 5 years. (8) 8.2 The contractor experiences problems and is unable to make the 17" and 18” payments. Determine the value of the new payments if he is to amortise the loan over 5 years as agreed with the bank. (18) 8.3 What would his monthly payment be if he started paying back the loan immediately? (8) [34] QUESTION 9 Patrick takes a loan to pay for a bachelor flat to the value of R300 000. The interest rate is 8,5% p.a., compounded monthly. He can afford to pay R5 000 every month. He will make ” payments of R5 000, and then a final payment of less than R5 000. 9.1 How many payments of R5 000 will he make? (10) 9.2 Determine the value of the final payment. (10) [20] eae TL ATLU LIL Please nn over

Downloaded from hlayiso.com Additional Mathematics/HG ay DBE/2013 SCE QUESTION 10 The total revenue from weekly sales of motorcars is given by R(q) = 2q° + 39” +7996. 10.1 Find the expression for the marginal revenue. (6) 10.2 Find the total and marginal revenue when qg = 144. (8) [14] QUESTION 11 A loan of x rand from Be Bank, that offers 11% p.a compounded quarterly, is repaid in the following manner: e R2 100 after two years e R1 700 one year later ° RS four years after the loan was granted ° A final payment of R3 200 that amortises the loan 6 years after the loan was granted. Determine x. [16] TOTAL SECTION B: 100 CDS) FUTON AON A Please urn over

Downloaded from hlayiso.com Additional Mathematics/HG 8 SCE SECTION C: ANALYTICAL GEOMETRY QUESTION 11 DBE/2013 Determine the equation of the locus of a point that is equidistant from points A(2;-3) and B(-231). QUESTION 12 Given the equation of the circle x? + y? -6x-—4y+5=0 and the line y =-2x—1. Determine the shortest distance between the line and the circle correct to TWO decimal places. QUESTION 13 2 7 y Given the ellipse +2 =1. Determine: 32 13.1 The coordinates of the vertices of the ellipse 13.2 The coordinates of the focii QUESTION 14 14,1 Determine the eccentricity of the ellipse given by 8x7 + 9y? =18. 14.2 Hence, sketch the ellipse. QUESTION 15 Given two circles x? + y? +4x-2y-11=0 and x? +y?+4y—-46=0. 15.1 Determine the coordinates of the centre and the radius of the circle defined by x+y? +4x-2y-11=0. TOTAL SECTION C: 15.2 Hence or otherwise: 15.2.1 Show that the circles intersect each other. 15.2.2 Determine the value of m if x’? +mx+y? +4y—46=0 touches the circle x? + y? +4x-2y-11=0. Copyright reserved PU AL TL IL Please turn over [12] [22] (12) (8) [20] (12) (6) [18] (6) (12) (10) [28] 100

Downloaded from hlayiso.com Additional Mathematics/HG 9 DBE/2013 SCE SECTION D: ALGEBRA QUESTION 16 If A, B and C are the roots of the equation 2x° — 5x? +8x—10=0, find a polynomial whose aia 4 zeros are +, 4 and 4. QUESTION 17 By using mathematical induction, prove that 9”+3 is divisible by 4 for any natural value of n. QUESTION 18 18.1 State, without proof, the rational root theorem. 18.2 Factorise the following polynomials fully, where possible. Show all explanations and calculations. 18.2.1 x‘-9 in R[x] 18.2.2 3x4414x? +28x? -63x—42 in Q[x] ee x'4x°4+x-1 in R[x] QUESTION 19 2 = Given f(x) =2—2* 4! x+l 7 Determine the intercepts with the axes and all possible asymptotes of f. 19.2 Determine the coordinates of all turning points. 19.3 Sketch the graph of f, clearly indicating all intercepts with axes, asymptotes and turning points. TOTAL SECTION D: Cori rer A Pee nn re GAUTENG [16] [12] (6) (8) (8) (14) [36] (16) (12) (8) [36] 100

Downloaded from hlayiso.com Additional Mathematics/HG 10 DBE/2013 SCE SECTION E: STATISTICS QUESTION 20 20.1 What is the probability that a word arranged at random from the letters in the word STATISTICS will start and end with the same letter? 20.2 Tom has 3 favourite items of clothing: e A brown Due South sweater (S) e =A pair of Levi jeans (J) e A pair of white Fresh Produce running shoes (R) In 1 month (30 days) he recorded the following: On 5 of the days he wore all 3 items. On 18 of the days he wore his sweater. On 22 of the days he wore his jeans. On 17 of the days he wore his running shoes. On 8 of the days he wore his jeans and his sweater, but not his running shoes. On 6 of the days he wore his jeans and running shoes, but not his sweater. On 4 of the days he wore running shoes and sweater, but not his jeans. 20.2.1 Draw a Venn diagram to represent the above information. 20.2.2 Use the diagram to answer the following questions: (i) What is the probability that Tom wears none of his 3 favourite items? (ii) What is the probability that Tom wears at least 2 of his favourite items? QUESTION 21 A dealer in PSPs receives a delivery of 30 PSPs, some that may have defects. The dealer decides that he will accept the delivery if, in a random sample of 5 PSPs (taken with replacement), less than 3 are defective. What is the probability that he will accept the delivery if there are actually three defective PSPs? aaa TL Please nen over GAUTENG (12) (8) (4) (4) [28] [16]

Downloaded from hlayiso.com Additional Mathematics/HG I] DBE/2013 SCE QUESTION 22 Solty Chips are sold in packets with a mass normally distributed with a mean of 170 g and a standard deviation of 8 g. 22.1 Calculate the probability that a randomly selected packet will have a mass between 171 gand 177 g. (12) 22.2 The probability that a packet has more than ¢ grams is 0,67. Calculate 1. (12) [24] QUESTION 23 A probability density function is given by: Pp 1 ~(*-~) for -5<x<5 x f(x) = * 0 elsewhere Calculate: 23.1 The value of p (12) 23.2 The probability that 1<x <5 (8) [20] QUESTION 24 A sample of 100 chocolates has a mean mass of 150 grams and a standard deviation of 10 grams. Calculate the 98% confidence interval for the mean mass of the chocolate. [12] TOTAL SECTION E: 100 TOTAL: 400 Conroe NAN 00 0 A

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Downloaded from hlayiso.com Additional Mathematics/HG DBE/2013 SCE FORMULA SHEET/FORMULEBLAD Differential and Integral Calculus/ Finance/ Differensiaal- en Integraalrekene Finansies sinks 5 —cos2x) F=P(1+i)" F=P(1-i)" costi=5 (1400825) F=P(1+ni) F=P(\-ni) 1-(+i)” sinA.cosB= Slsina +B) +sin(A-B)| P=R. oe 1+i)"-1 sinA sinB=5 [cos(4 ~ B)—cos(A +BY] FER. est cosA.cosB=feos(A — B)+cos(A+B)] Analytical Geometry/ 2 Analitiese Meetkunde Sis “ee y =4ar’ yy, = 2a(x+x,) el i won-eiyont eva ae i a zZtyt 1 eam ry =1 7 6 a b a 5 aa, -L(@) ey MIM y n+] n t'(a,) a’ Bb a Bb? 3 2 V= 2) Uf @Par Statistics/Statistiek FQ) FG) ax" nax"" = C= os sinx cos x “Taant "(Ur @ ur cos x —sin x ner tan x sec’ x (x+y) >> (t eo sec x sec x.tan x " N-p cosec x —cosec x.cot x (? cotx —cosec” x P(R=r)= Aner } arcsin x 1 N bgsin x hx? n arccos x 1 - PR +1.96-F << 1.96 )=0.95 bgcos x 1-x? An saa l P(P- “fH PY cn < P+1.96,,-6— a) = 0.95 bgtan x x41 Se).g@) | f')-8() + 8'@)-f@) f@) | #'@).8@)-8'@).f@) g(x) [gr S[g@)] Stg@)13'@) Copyright reserved GAUTENG

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Downloaded from hlayiso.com Additional Mathematics/HG SCE DBE/2013 AREA UNDER THE NORMAL CURVE Entries in the table are values of H(z) for positive values of z where H(z) = P(0<Z<z) OPPERVLAKTE ONDER NORMAALKROMME Die tabel bevat die waardes van H(z) vir positiewe waardes van z waarby H(z) = P(0<Z<z) H(z) z 0.00 0,07 C,0z 0.03 0.04 0,05 0.08 6,07 C.08 C.cs 0.0 | o.ccooo 0.c0389 0,00798 0.01197 0.01585 0.01994 0.02392 0,02790 0.03188 0,1 | 0.03983 0.04380 0,04776 . 0.03172 0,05567 0.05962 0.06356 0.06743 0,07142 0,2 | 0,07926 0.08317 0,08706 0,09085 0,094830,09871 0,10257 0,106420,11026 0,3 | 0.11791 0.12172 0,12552 0.12930 0,13307 0.13683 0.14058 0,14431 0.14803 0.4 | 0.15542 0.18910 0,16276 0.16840 0,17003 0,17364 0,17724 0,18082 018439 0.18799 05 0.19146 0,19497 0,19847 0,20194 0,20540 0,20884 9 0,21226 0,21566 0.21904 0.22240 0.6 | 0.22575 0,22907 0,23237 0.23565 0.23891 0,24215 024537 0.24857 —0,25175 0.25490 0,7 | 0.25804 0.26115 0,26424 0,26730 027035 0.27337 0.27637 «0.2793 0.28230 0.28524 0.8 | 0.28814 0.29103 0.29389 0,29873 029955 0,30234 0,30511 0.30785 0.31057 0,31327 0.9 | 0.91596 031859 0,32121 0.92381 0.92634 0,92894 0.33147 0,33998 «033646 0.33891 4,0 | 034134 0.34375 0.34614 0.34842 0,35083 0,35314 0,35543 0.35769 © 0,28993 0,36214 a1 | 036433 0,36650 0,36864 0,37076 0,372860,374890,37698 —0,37900 0.98100 0.38298 4.2 | 038493 0.38686 0,38877 0,39065 0,39251 0,39435 0,395170,39798 0.39973 0,40147 13 0.40320 0,40490 0,40658 0.40824 0.40988 0.41149 0,41309 0.41466 = 0.41621 0.41774 a 0,41924 0,42073 0,42220 0,42364 0,42507 .0,42647 0,42785 0.42522 0,43056 0.43189 15 0,43319 0,43448 043574 0,43699 0,43822 0,43943 044062 0.44179 0,44295 044408 1.6 0.44520 0.44630 0,44738 0,44845 0.44950 0.45053 0.45154 0,45254 0.45352 0,45449 1,7 | 045543 0.45627 0.45728 + 0,45818 «0.45907 0,45994 0.48080 0.46164 0.46246 0.46327 18 0,46407 0,46485 0,46562 0.46538 0,46712 0,46784 = 0,46856 0.46926 0.46995 0.47062 19 0,47328 0.47193 0,47257 0,47320 0.47381 0,47441 = 0,47500 0,47558 0.47615 0,47670 2.0 0.47725 «0.47778 ~=—-0,47831 0,47882 0,47932 0,47982 0,48030 0.48077 0.48124 = 0.40169 21 0,48234 0,48257, 0,48300 0,48341 0.48382 0.48422 0,48461 0.48500 0,48537 0,48574 2.2 0.48510 048645 0,48679 0.48713 0.48745 0.48778 0,48809. 0,48840 0.48870 0.48899 23 0.48928 0,48956 0.48983 0,49010 0,49036 0.49061 0.49086 0,49111 0.49134 = 0,49158 2.4 0.49180 0.49202 0,49224 0,49245, 0.49266 0,49286 0,4930S 0,49324 0,49343 0,49361 25 0.49379 0,49396 +=—-0,49413. 0,49430 0.49446 0.49461 0.49477 0.49492 0,49506 0,49520 2.6 | 0.49534 0.49547 0,49560 0.49573 0.49585 0.49598 0.49601 0,49621 0.49632 0,49643 2,7 0.49653 0,49654 —-0,49574 0.49683 0.49693 0.49702 0,49711 0.49720 0.49728 0,49736 2.8 0,49744 0.49752 -0,49760 0.49787 0.49774 0.49781 049788 0.49795 0.49801 0,49807 2.9 | 049813 0.49819 0.49825 0.49831 (0.49836 0,49841 0.49846 0.49851 0.49856 0.49951 3.0 | 0.49865 0.49869 0,49874 0.49878 0.49882 0.49886 0.49889 0.49893. 0.49896 0.49900 34 0.49903 0,48906 0,49910 0.43313 0,49916 0,49918 0.49921 0.49924 0.49926 0,49929. 3.2 0.49931 0.49934 0.49936 0.49938 0.49955 049942 0,49944 0.48946 0.49948 0.49950 33 0.49952 0.49953 -0,49955 0.49957 0.49958 0.49960 0.4896) 0.49962 0.49964 0,49965 3.4 0.49966 0.49968 0.49969 0.49970 0,49971 0.49972 0,49973 0.49974 0.49975 0.49976 35 0.49977 0.49978 0.49978 0,49979 0.4998 0.49981 0,49981 0.49982 0.49983 0.49983 36 0,49984 0.49985 0.49985 0.49986 0.49986 0.49987 0/49987 0,49983 0.49988 0,49989 3,7 0,499aS }=— 0.49990 :0,48990 0,49990 0,49991 0.49991 0.49992 0.49992 0,49992 0,49992 3.8 0.49993 0.49993 0,49993 0,48994 0.49994 0.49994 0,49994 0.49995 0.49995 0,49995 39 0.49995 0.49995 0.49996 0.49995 0.49996 0.49996 0.49995 0.49996 0.49897 0.49997 Copsright reserved CTU ANA OM UT TH GAUTENG