Mathematics Grade 11 Paper 2 KZN November 2016 Paper & Memo

\ basic education Department: Basic Education REPUBLIC OF SOUTH AFRICA Ao ee eS ; MATHEMATICS P2 1 iy au i al i NOVEMBER 2016 " OT TTTTTTTETITrrrerrerrrerreeecrereerreeceeeee ere sd NATIONAL SENIOR CERTIFICATE GRADE II MARKS: 150 TIME: 3 hours This question paper consists of 13 pages and a 22-page answer book. Copyright reserved UIE AVE A Please turn over

Mathematies/P2 2 DBE/November 2016 NSC ~ Grade 11 INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. 2. 3. This question paper consists of 12 questions. Answer ALL the questions in the SPECIAL ANSWER BOOK provided. Clearly show ALL calculations, diagrams, graphs et cetera that you used to determine the answers, Answers only will NOT necessarily be awarded full marks, If necessary, round off answers to TWO decimal places, unless stated otherwise. Diagrams are NOT necessarily drawn to scale. ¢ ) You may use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise. Write neatly and legibly. Copyright reserved HRCA CAA A Please tum over

() Mathematics/P2 3 DBE/November 2016 NSC - Grade 11 QUESTION 1 The table below shows the number of cans of food collected by 9 classes during a charity drive. 5 8 15 20 25 27 31 36 75 Ld Calculate the range of the data. 1.2 Calculate the standard deviation of the data. 13 Determine the median of the data. 14 Determine the interquartile range of the data. 1.5 Use the number line provided in the ANSWER BOOK to draw a box and whisker diagram for the data above. 1.6 Describe the skewness of the data. L7 Identify outliers, if any exist, for the above data. QUESTION 2 The table below shows the time (in minutes) that 200 learners spent on their cellphones during a school day. TIME SPENT (IN MINUTES) FREQUENCY 95 <x <105 15 105 <x <115 27 115 <x <125 43 125 <x < 135 52 135 <x <145 28. 145 <x <155 21 155 <x < 165 10 165 <x $175 4 2.1 Complete the cumulative frequency column in the table provided in the ANSWER BOOK. 2.2 Draw a cumulative frequency graph (ogive) of the data on the grid provided. 2.3 Use the cumulative frequency graph to determine the value of the lower quartile. 24 Determine, from the cumulative frequency graph, the number of learners who used their cellphones for more than 140 minutes. Comodo 0) (2) @) (3) (3) @) ) [12] (2) (3) (2) 2) 9}

Mathematics/P2 4 DBE/November 2016 NSC ~ Grade 11 QUESTION 3 In the diagram, A(6;~—2), B(2; 15) and C(—4; 3) are the vertices of AABC. M is the midpoint of AB. N isa point on CA such that MN || BC. 3.1 3.2 3.3 3.4 3.5 a B(2; 15) C4; 3) wt [e] | A(6 ; -2) Determine the coordinates of M, the midpoint of AB. Determine the gradient of line MN, Hence, or otherwise, determine the equation of line MN, inthe form y = mx-+e. Calculate, with reasons, the coordinates of point N. If ABCD (in that order) is a parallelogram, determine the coordinates of point D. Covprigh reserved UME A AAA A Please turn over @) @) OC @) (4) [15]

Mathematics/P2 QUESTION 4 5 NSC - Grade 11 DBE/November 2016 In the diagram, Rand A are the x- and y-intercepts respectively of the straight line AR. The equation of ARis y= ~ e+ 4. Another straight line cuts the y-axis at P(O ; 2) and passes through the points M/k; 0) and NG ; 4). a and £ are the angles of inclination of the lines MN and AR respectively. ty C) Mc; 0) a () 44 Given that M, P and N are collinear points, calculate the value of &. 4.2 Determine the size of 0, the obtuse angle between the two lines. 4.3 Calculate the length of MR. 4.4 Calculate the area of AMNR. Copyright reserved {MAUI AAA Please turn over 3) ) (3) G) (13)

Mathematics/P2 DBE/November 2016 6 NSC ~ Grade 11 QUESTION 5 51 In the diagram below, P(-8 ; 4) is a point in the Cartesian plane such that OP = 17 units and reflex XOP=0. (21 17 PC-8;1) 3.41 Calculate the value of f, 5.1.2 Determine the value of each of the following WITHOUT using a calculator: (a) cos(-) (b) 1-sind 3.2 If sini7°= a, WITHOUT using a calculator, express the following in terms of a; 5.2.1 tan17° 5.2.2 sin 107° 5.2.3 cos? 253° + sin? 557° 53 Simplify fully, WITHOUT the use ofa calculator: cos(—225*), sin 135° + sin 330° a 2 6 7 1 ~] (cosx-+1)(cosx-1) tan?x.cos?x 5.4 Prove the identity: 5.5 Determine the general solution for 2sin x.cos x =cos x, Copyright reserved URNA Plegee ram aver 6) @ (2) (2) @) (2) (4) (4) (6) 1]

Mathematics/P2 7 DBE/November 2016 NSC - Grade 11 QUESTION 6 In the diagram the graphs of f(x)=cosx and g(x)=sin(x+5) are drawn for the interval -180° <x < 90°. () 6.1 6.2 63 6.4 () 65 AS Write down the value of 0, Write down the period of g. Write down the value(s) of x in the interval -180°< x <90° for which S(s) - g(x)=0. For which values of x in the interval ~180°< x < 90° is sin(90°-x) > g(x)? The graph of A is obtained by shifting f 3 units upwards. Determine the range of h. comet YM LUUNUIIN 8 (1) (1) Q) GB) (2) [9]

Mathematics/P2 8 DBE/November 2016 NSC - Grade 11 QUESTION 7 VA In the figure below, acute-angled AABC is drawn having C at the origin. y B x i A Wd Prove that c? =a? +b? ~ 2abcosC. (6) - 4 7A2 Hence, deduce that 1-+cosC = enter. @) a 72 Quadrilateral ABCD is drawn with BC=235m and AB= 90,52 im. It is also given that ADB =3 1,23°; DAB = 109,16° and CBD = 48,88°, Determine the length of: 7.21 BD (3) 7.2.2 cD (3) ee ee

Mathematics/P2 DBE/November 2016 9 NSC — Grade II QUESTION 8 The diagram below shows a water tank which is made up of a cylinder and cone having equal radii. The height of the tank is 1,8m and the radius is 0,5m. The angle between the perpendicular height, AB, and the slant height, AC, of the conical section is 35,5°, Volume of cone “5 ah Total surface area of cone=71’ +75 4 Volume of cylinder = 77°h t ! Total surface area of cylinder = 2a r?+2arh H ' 1 H SC ' 1 18m 1 1 ' t t ' ' ' ' t v 8.1 Calculate the perpendicular height, AB, of the cone. (2) 8.2 When the tank is full, an electric pump switches on and pumps the water from the tank into an irrigation system at a rate of 0,52 m’/h, The pump automatically switches off when the tank is full. Calculate how long, in hours, the pump feeds water into the irrigation system. (4) [6] Comvaaenes NAMEN oe

Mathematics/P2 10 DBE/November 2016 NSC - Grade 11 Give reasons for your statements and calculations in QUESTIONS 9, 10, 11 and 12, QUESTION 9 91 Complete the statement so that it is TRUE: The angle subtended by an are at the centre of a circle is ... 9.2 O is the centre of circle TNSPR. POS=60° and PS=NT. T N R | P ' s Calculate the size of: 9.2.1 PRS 9.2.2 NST Conyright rserved UAT AA A Please turn over (2) | (2) [6]

() Mathematics/P2 at NSC - Grade It QUESTION 10 D, B, F, G and Hi are points on the circumference of the circle. G,=x+20° and H=2x+10°. DEI|FG. 10.1 Determine the size of DEG in terms of x. 10.2 Calculate the size of DHG. copies 00 00 0 A DBE/November 2016 Please turn over (2) i) [6]

Mathematies/P2 12 DBE/November 2016 NSC. Grade If - QUESTION 11 O isthe centre of the circle PTR. N isa point on chord RP such that ON LPR. RS and PS are tangents to the circle at R and P respectively. RS = 15 units; TS=9 units; RPS = 42,83°, R 11.1 Calculate the size of NOR. (5) 11,2 Calculate the length of the radius of the circle, (4) [9] Conia reserved UAE AAA Please tum over

} Mathematics/P2 QUESTION 12 13 NSC ~ Grade 11 DBE/November 2016 12.1 Use the diagram below to prove the theorem which states that EFG = EDF. D. F G (5) 12.2 In the diagram below, BOC is a diameter of the circle. AP is a tangent to the circle at A and AE = EC. Prove that: 12.2.1 12.2.2 12.2.3 Copyright reserved A D P ; a/\i 2 2 3 4 EBiX 7% Y 3 C a7 1 3 0 BAI|OD (4) AOCD isa cyclic quadrilateral (5) DC isa tangent to the circle at C (4) [18] ANH TOTAL: 150

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Mathematies/P2/Wiskunde V2 3 DBBNovember 2016 CAPS/KABV ~ Grade/Graad 11 —-Memorandum QUESTION/VRAAG 2 ud CUMULATIVE Tipepanpeng | FREQUENCY | FREQUENCY! FREKWENSIE | KUMULATIEWE (IN MINUTES/ FREKWENSIE MINUTE) f of Lif 95 <x <105 15 15 “first 4 comect/ 105 <x<115 27 a2 eorste 4 korrek 15 <x 125 8 35. CF values 125 <x<135 52 137 KF waardes 135 <x<145 28 165, “last 4 correct CF 145 <x< 155 21 186 values/ 155 <x< 165 10 196 daaste 4 korrekte KF 165 <x <175 4 200 waardes 2 22 Grounding! OGIVE/OGIEF Aner iow CCCP Ce TTT] — 8 points coreci/ i Le a 4 8 punte korrek 20 Peet a: £ T 200 a i: 5 OR/OF 380 +} a - ¥ Grounding! 2 160 x i : i gs Ankerpunt = £ i fh . eae Sr 1 @ a0 cE v ge et 4-7 points correct. z 100 = crt “t 4-7 punte karrek cot 60 + i oF Fe cr £ i i = EEB AREER Ce ee a ee [ oO 20 40 60 80 100 1206 140 160 180 200 i Time spent on cellphone @) Copyright reserved/Kopiereg voorbehou mi -11- 11 |Please turn over/Biaai om. NO Mathematics/P2/Wiskunde V2 4 DBE/November 2016 CAPS/KABV — Grade/Graad 11 —~Memorandum 23 Qi= 118 ] CA | ¥ CA answer/antw Accept any answer between (115 and 120) : Q 24 | Number of learners / Getal leerders=200-150 i YEA 150 =50 YCA 50 te) Accept 150 or any other reading between (145 and 155) 9] EoucATION DEPARTMENT OF BASIC 206 -i- 11 PRIVATE BAG 2896, PRETORIA BOT APPROVED MARKING GUIDELINE Copyright reserved/Kopiereg voorbehou PUBLIC EXAMINATION | | Please turn over/Blaai om asseblief

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| | | | | Mathematics/B%/Wiskande V2 Tt DBE/Noveniber 2016 Mathematics! 2/Wiskunde V2 8 ' DBE/November 2016 CAPVRABY. ~tiaite/Gioail 1 Meher asta ‘CAPS/KABV — Grade/Graad 11 Memorandum ; — -QUESTION/VRAAG 4 ! | OR/OF ai o Why =Mpy From Bto A Vor 9-0 4-2 | ¥ hg =Wpg G5y) @t4sy- IT) Yy-17 O-k 3-0 | © subst in gradient D-444;3-17) 7 2 62 vom DO ;-14) Ysubst * °, | ¥ answer/antw = \ GB) ¥v 3-14 - DO ;-14) 42 | tana = mpg i) tna=2 fist : 3 / 0533, 69° ! ¥ 0=33,69° as | — v tanB=—— i tap 2 | B 2 i B=-26,57° +180° ! | =153,43° | | i i ¥ 153,43° ORVOF tanB = 745 : = | tan =—> ‘G) | KA=ten"} — : 2 i =2657° I B =180°-26.57° | i =153,43° | o @=153,43°— 33,69° ¥CA 119,741 ® =119,74° \ v y= a ~prHde0 1 yo x=8 ' ¥x=8 R(8;0) i CA answer/antw MR=8-(-3) ORIOF UR = (3-37 -0° ; 8 =llunits / eenhede i DEPATDDUSATION 1 ! area PRRATE BAG Ob, PRETORIA OIOI 44 | dreaof AMNR =—(MR). height | fornula/formute 2016 -1I- 44 Lan vaieoe 9) | APPROVED MARKING GUIDELINE 1 1 i PUBLIC EXAMINATION =D) | 2 ¥ subst y-value of N =22 sq utlits/vk eenh j subst y- waarde van | \ z : N i ! YCA answer/antw 7 ‘Hlease tum over/Blaai om asseblick ¥ | Copyright reserved/Kopiereg voorbehou Please tum over/Blaai om asseblief ‘Copyright reserved/Koptereg voorbehou 1

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Mathematics/P2/Wiskunde V2 11 DBE/November 2016 CAPS/KABV —Grade/Grand 11 —Memoraadam =t : S34 RHS = tan’ x.cos’ x . _ 1 Y tango St =a cosx Se xcos* x cos?x v -1 simplification’ vereenv. © sin?x ¥ identity/ identiteit -1 ~ Tocos? x _ 1 cos*x-1 -—_1 Y factors /faktore (cosx +1)(cosx—1) = LHS OR/OF Lag=——__! {cos x +1)(cosx—1) ¥ cos’ x-1 = 2 “cos? x1 vositt = i v = “Tas sin’ x =} vy sivtx cos’ x sin? x Py cosx _ -1 sin’? x | cos? x cos*x 1 _ -1 tan’ x.cos? x EDUGATION DEPARTMENT OF BASIC) 206 tt 14 PRIVATE GAG X056, PRETORIA 0004 APPROVED MARKING GUIDELINE) PUBLIC EXANINATION Copyright reserved/Kopiereg voorbehow Please turn over/Blaai om assebligh, Ms Mathematics/P2/Wiskunde V2 2° DBE/November 2016 CAPS/K ABV ~ Grade/Graad 11 —- Memorandum i OR/OF RES = — 508 | Gin? xcos? x | yl 2 tx 2 | tan?x sin? x sin? x | vai | in? LHS = a 5 om cos’ x—. 1 ¥ cos*x-1 Ll sin® x | ¥~sit x RES = | @ | | \ | | | | | | | 55 2sinx cos x—cosx=0 cosx(2sinx—I) <0 | “factors / faktore cosx=0 or sinz=t 2 | both equations/ ' beide verg. == 90°4360 keZ 2=30°+360°K REZ or x= 270°+ 360°. & keZ #1504360" hkeZ VBOTH general or solutions for 4=90°H18PEkeZ | cosx= O/ | Altwee algemene or i oplossings vir £=190°436PkeZ | cosx=O i vv general solutions i el | for sinx= = cerAtpecmen ‘ algemene PRIVATE BAG xt00, PRETORIADOE| 4 splossings or M6 -Het sing= APPROVED MARKING GUIDELINE | V kez 2 PUBLIC EXAMINATION | © © l (st ] Copyright reserved/Kopiereg voorbehou | Please tum over/Blaai om asset

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Mathematics/P2/Wiskunde V2 8 . DBE/November 2016 CAPS/KABY — Grade/Graad 11— Memorandum ORIOF etna? +?—2aboosE 2 25? i . 2 gt e=a +8 -2abeosC costed tne a_@ +b -? 2ab 2.cos€ = ¥ Making cos C the 2 2ab subject of the formula/ arb) ~¢ Maak cos C die 2ab onderwerp van die formule _@ +2ab+B? — 0? _ 2ab V writing asa 2a p2_ 2 difference of 2 ath -e ae 2ab Steyfas die verskil . tussen twee vierkante =cosC +1 Y expansion/ontwikkel =LHS > splitting up the OR/OF fraction / Deel die ¢? =a" +b? —2abcosC breuk in twee 2abcosC =a? +b? —c? 2ab+2abcosC =a" +2ab+b? —c? 2ab(L+-cosC)=(a+5)*-c? (a+by?—<? 1+cosC = 2ab “making 2zbeosC the subject of the formula/ Maak 2abces C die onderwerp vn die formule ¥ adding 2ab on both sides of equation/ tel 2ab aan beide ante van die vergelyking “common factor! gemene foktor ins "PRIVATE BAG X03, PRETORIA 0001 ¥ factorise the trinomial/ faktoriseer Copyright reserved/Kopiereg voorbehou 2016 -it- 11 die drieterm APPROVED MARKING GUIDELINE z = = re Try Please tun over/Blaai om eti\ Mathematics/P2/Wiskunde V2 DBE/November 2016 i i | | 16 i CAPS/KABV — Grade/Graad 11 — Memorandum. 724 in AABD | BD __ 90,52 Y sine sule/sinusreél A Tay 530 | sinl09,16° sin31,23' | Y subst | | | _ 90,52xsin109,16° | | BD= ans 23 Y answer/antw. =164,92 m i @ 72.2 | CD? =164,92? +235" —2%164,92%235xc0s48,88° | 'Y cos nile/cosinusredl CD? =31448,4874 : Y subst CD=17734m : CA answer/antw. | | @) [16] | QUESTION/VRAAG 8 | : 7 ¥ subst ' B81] tn35,5°= 2% | | AB | | _ 0s | | AB aaa i ¥ answev/aniw | =0,7 m fe | 82 aden (0,5)? x07 | CA V of coneftedl Volume of cone 3 7 =0,180* | Volume of a cylinder = (0,5)? x1,1 | = 3 i YCA V of cylinder! 0,86m i silinder | 3 3 of volume==-x (0,18 + 0,86) i 4 4 i =34¢,04)m° | 4 1 =0,78m* | 3 YCA 0,78 m 0,78? j Timetakenby pump = 05am =1,5hours | | ¥CA answerlantw 4) DEPARTMENT OF 5, EDUCATION 206 -H- 41 Convricht reserved/Kopieree voorbehou PRIVATE BAG X605, PRETORIA GOGT dl RN case turn over/Blaai ont nai} |

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Mathematics/P2/Wiskunde V2 19 CAPS/KABV — Grade/Graad 11—Memorandum QUESTION/VRAAG 12 A D PEA 5 DBE/November 2016 12.1 | Construction: Draw diameter AOF. Join A to D. FoF =90° [tan 1 diameter] B, =F [Z'sinthe same segment] Bb, +B, =90° [Linasemicircle] wh =B, EPG = FBE ORIOF Construction :Draw diameter AOF Join Ato E. Fo4F, =90° [tan 1 diameter] AEF =90°[Z'sinthesemi circle] C+ PF =90°[sumofZ'sin AT oR+h=C+k but C =D [Z's inthe same segment] nf=5, EPG = FDE ~Constt/Kons vs vSYR ©) aay Bre EDUCATION 206 -fi- 41 PRIVATE BAG 1096, PRETORIA Gout APPROVED MARIGNG GUIDELINE Coovrishtreserved/Kopieres voorbehou ASAE ACD A RAEN ECON Y Mathématics/P2/Wiskende V2 20 | DBE/November 2016 CAPS/KABV — Grade/Graad 11 —-Memorandum OR/OF | Construction: Drawradii OF and OE Let BOF =2x | o Box [Zatcentre=2x Lat circumference] OFE =90°—x [sumof int. Z's of A] ! wv. EPG=x frad Ltan] | < EPG= FDE | i} 12.2.1 | BAC=90° [/ina semicircle] YS/R £, =90° [line drawn from centre to midpont of chord] VSR -. BAC=E, ! . | BAL{OD [ comesp. 2s are equal] | YR if oR/or | YSIR BAC=90° [Lin asemicircle] | YSVR £,=90°[Line fromcentreto midpoint of chord| | VR BAC=E, | i @ = BA| [OD [Ait2’s are equal] i 222 [Zax B=x [ten—chord theorem] | YSYR O, = x [corresp Z's equal, 4B || OD] VSVR 4=6, ; | +. AOCDisacyelicquadritateral [conv.Z'sinthe same segment] |“ ® OR/OF | Let 0,=a | €,=90°~a [int. Zs of A] ' vs 2.4, =90°—a [2 sopp =sides| | * | cd=a — ftan trad] | YSVR +. AOCDis acyclic quadrilateral. (Converse 2'sinthe same segment] | ¥ R ey | DEPARTMENT GF BASIC EDUCATION. PRRATESAG Made, BRETORA OOF 2016 -1i- 41 | Copyrightreserved/Kopiereg voorbehou APPROVED MARKEE Rees over/Blaat om assed)

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