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* (16030186) -2- T980(E)(M31)T DEPARTMENT OF HIGHER EDUCATION AND T RAINING REPUBLIC OF SOUTH AFRICA NATIONAL CERTIFICATE MATHEMATICS N6 TIME: 3 HOURS MARKS: 100 INSTRUCTIONS AND INFORMATION 1. Answer ALL the questions. 2. Read ALL the questions carefully. 3. Number the answers according to the numberi 4. Questions may be answered in any orde| together. 5. Show ALL the intermediate es 6. ALL the formulae used must writ a Downloaded from hlayiso.com Copyright reserved Please’ turn over

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(16030186) 3- , T980(E)(M31)T QUESTION 1 2 Lt If z* arc cot (=) calculate the following: x). Ll a ox @) 112 oe oy 13) 1.2 A rectangular container with a length of 3,5 m, breadth of 1,6 m and Ade is expanding along its length at 0,04 m/s and along its container is contracting along its depth at 0,05 m/s. Calculate the approximate change in the volume o ov ov ov HINT: AV =—— Al +——Ab+—Ah al ab dh @) {6} QUESTION 2 Determine J y a if: 2.1 (4) 2.2 6) 23 6) (4) 2. [18] QUESTIONS Use partial fractions to calculate the following integrals: 33x? -7x+6 sf Sate to (3x ~1)* (2x + 3) (6) x? +3x-3 32 J Q-x)e * e : (6) Downloaded from hlayiso.com {12} Copyright reserved " Please tum over

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(16030186) . , AF T980@) (M3 1)T QUESTION 4 Al Solve the following differential equation: x sine dey de =x dy 4.2 Calculate the particular solution oft 2 22 r4Daay = 20e', if p=0 when ¢=0 and at) when tf QUESTION 5 5.4 5.1 Calculate the points of intersection of y d y = 4-42’, Make a neat sketch of the two opr¥e ; thérrepresentative strip/element that you will use to call enerated when the area bounded by the two curves,istugtat i 5.1.2 Calculate the magnitude means of integration. 5.2 Calculate the points of intersection of x+y—7=0 and xy=6. Make a Mheat sketch of the two graphs and show the representative strip/element that you will use to calculate the volume generated when the area bounded by the two graphs is rotated about the x-axis. - 5.3.2 Calculate the volume described in QUESTION 5.3.1. 5.3.3 Calculate the moment of inertia of the solid generated when the area described in QUESTION 5.3.1 is rotated about the x -axis. Express the answer in terms of the mass. Downloaded from hlayiso.com 6) ” [12] @) Q) (2) _@) (6) @) (3) 4) © Copyright reserved Please turn over

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(16030186) 5.4 34.1 “5 : T980E\M31)T A vertical retaining wall in a canal is 8 m wide at the top, 4m wide at the bottom and 4 m high. The top of the retaining wall lies in the water level. Make a neat sketch of the retaining wall and show the representative strip/element that you will use to calculate the aréa moment of the wall about the water level. : Calculate the relation between the two variables x and y. (3) 5.4.2 Calculate the area moment of the wall about the water level. oN (3) 5.4.3 Calculate the second moment of area of the wall about thegwater ley; well as the depth of the centre of pressure on the integration. (4) {40} QUESTION 6 6.1 Calculate the length of the curve represepith.by 6) 6.2 1) [12] 100 Copyright reserved Downloaded from hlayiso.com

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higher education & training Department: Higher Education and Training REPUBLIC OF SOUTH AFRICA MARKING GUIDELINE NATIONAL CERTIFICATE APRIL EXAMINATION MATHEMATICS N6 31 MARCH 2015 This marking guideline consists of 13 pages. Downloaded from hlayiso.com Copyright reserved Please turn over

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T980(E)(M31)T MARKING GUIDELINE 2 MATHEMATICS N6 Copyright reserved Please turn over

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MARKING GUIDELINE -3- . TEBO(E)(M31)T MATHEMATICS N6 . _ as 068. am, ++ ee SW eo a. ian Se mle Di cd J aba oe "3 ffSs. - 56 41)" ie ay [ex wi)? . 4 [8 Ox at ae 4 [* waite Ge, = & 4 ere “a 7 ‘ wes ar ws Bee) Coun) oo ae vas ogo? re.

T980(E)(M31)T MARKING GUIDELINE -4- - : MATHEMATICS N6 7 4-3 afe ee Bd YY of es nee le S 2x ele ‘ fi Cie bu tn cue Bie dm — ante ha feet Dive 3 Bae) obs: ~- feve Pare ae a 2 in : ‘ le ee ae feck? Pe ork x de 07 “ee ae cat 2a mf Came doe fy cn am. de = as ly cok? 2% - “Ceeiee 22 2x jax a fees 2. oe : ; E & La, Cain Te ara gai ome ge eee? 2x. ae t 226 tuts tx) tO ann 00) : ae gh = sv Be HE ‘éxjs le™ goo - eee. { mer ay ~ ax + “fe ax foxy eee” gio = cos 2 | Eltal= ae’* yee? = ‘ o oe nara cx. | a a an > ‘ a os phen 22k, ERS wo ‘ - es 2m eB ee ad 2x T soot ee Downles eaded fronrhlayiso. com = 2 4 Le

T980(E)(M31)T MARKING GUIDELINE -5- MATHEMATICS N&

MARKING GUIDELINE -6- TS80(E}(M31)T MATHEMATICS N6

T980(E)(Mat)T MARKING GUIDELINE 7. MATHEMATICS N6 ys e? Cheat + Bat) op Le vw Os a? Chros 50 » Bawa) + wa? er as v ae = ev’ CAs bob Boab) we (hen ¥ Bent) ee Oe ee t-Asio eaojp-e? fA 948 9 ee? o= B-~ A+? O= Br (r2pr2 Bena y= OP Caco e-waste) + deh i) inner aS Downloaded from hlayiso.com

Teso(E)(M31)T MARKING GUIDELINE -8- MATHEMATICS N6

MARKING GUIDELINE -9- T980(E)(M31)T MATHEMATICS N6

TIBO(E)(M31)T MARKING GUIDELINE -10- MATHEMATICS N6

MARKING GUIDELINE -11- Tes0(E\(M31)T MATHEMATICS N6

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T980(E)(M31)T MARKING GUIDELINE -13- MATHEMATICS NG

(16030186) -2- TOTACEION 14 DEPARTMENT OF HIGHER EDUCATION AND TRAINING REPUBLIC OF SOUTH AFRICA NATIONAL CERTIFICATE MATHEMATICS N6 ‘TIME: 3 HOURS MARKS: 100 INSTRUCTIONS AND INFORMATION I. Answer ALL the questions. 2. Read ALL the questions carefully. 3. 4. Questions may be answered in together. 5 6. 7. Downloaded from hlayiso.com

(16030186) bo TO7O(E}(N14)T QUESTION 1 ll If z= e*In (xy) determine the values of: oz LL Be Oz L12 ay : az 1.1.3 axdy @Gx1) (3) L2 In measuring the length of the perpendicular is of airigl jangle, errors of 0,4% are made with each length. fe] Calculate the maximum possible err HINT: (3) [6] QUESTION 2 Determine j ydk (4) A)

(16030186) oe TOTO(E\ONIA)T _ QUESTION 3 Use partial fractions to calculate the following integrals: 3 x +1 2 dx . 3 | x(x —-1y 333 — 6x? ~ 2x +12 x(x" ~4x +6) QUESTION 4 4.1 Calculate the particular solution of: tan —ys ese at (230). dx cotx 4.2 Calculate the general solution Dicliniddevaidieehtheaytt ihlayiso.c com integration. Ealculate the volume described in QUESTION 5.1.1 by means of 6) (6) (12 (6) 0)

MARKING GUIDELINE -2. MATHEMATICS N6 QUESTION 1 be pe tacty) LL 9 L 1.1.2 “ 2 S. ws ee vin 1.1.3 2. 5) 3: S38, = Sx OG) % ~ Se (4) v x ~§F i ae Ve A= ab Sa-4b v Sheba ig “A Dé vinlosdeefrom hlayiso.com a 4 ab (&) w ah Mok A u if TO6OCE)(NI4)T Q

MARKING GUIDELINE QUESTION 2 MATHEMATICS NG glen o tn? ap a & = fo? 4 ae t% fur we Sot * 6 Di med f= a 2 a ook. . = yS ain * Ds wi | Ao | UM= dees Pre obs 5 a Wo je 2 a pleaded from hlayiso.com eae we die de Pan - ae TIGOCRYON1AyT (8)

MARKING GUIDELINE 2.3 24 2.5 -4. MATHEMATICS N6 ~ ep xe j . 2 + ux = du Pex} = x. 3 (x) se Ese) ae ” . FlQxcde 2oc toc) 2 ae ~ fax a odu. 4 Heactet "fe ** du | Peale 2e gto! =e" pr eben ae 2 ae 2 os = gules we 4 tem - ux ! ue fete ~4& fx - 1 ae | 4 we ee * + xte So Fee eRe ‘Fe % Ab 2 Z we Ltet*- beet +55 eee ° i / Vie- 20 ee OX Seen Sn al ~ jv Cnr 1s oh, a” footie Sin Vy ote a Cocty sav Sr ee = ye aa [ect + 2ee to} —_ [ese #1) ~~ 1] [eee OF aye Voo= €xe4t)% L7 tt t fC owee® Boe NS cee C cose. ® ax, nels PF 3x) Dae = Jace * ase ce Let. QL. G3 Bx ro cae BS coy oun %w - Downloade [from hlayiso.. ‘com T960(R)(NI4)T (6 [36

TOGOCIYCN L4)T MARKING GUIDELINE MATHEMATICS in QUESTION 3 St + teone F GOTHe te ag $5 = AfseetP} Bact Ove Ces) + Docgse te? Lele wor Go 4 : 1 te } - © mo ly 2 a ‘ ; = fe ‘ Peco DAa? + cou stud -2Reb aoe Vv pe AO at = LA De det . : weer = ine = lA C2 3,2 ro = -RB+D qm hlayiso,com

MARKING GU{DELINE -6- ; MATHEMATICS N6 QUESTION 4 4d dl eae e BV bun ae “ds ~ Yos “Seb ay S™, six % be Saba hak, a gy ax ™ ye Coxe, = a, 1 ye eek oe atx, ~ © Ree ae & =< eee we) we bev (site! = Ba “| ecee x y cfhe do be - ; Pt. oe Bin, = | eS 2 O Swe de -ee ye are = f er ma ok et a . sn = et te we 2. a - sae = EO +e * (aya) G =e? MW (~ 1,389) a rr a 4 % Base FF EB "phy = @* 4384 ener 4.2. - ay nore dy 2 dF +g su. By = fox ra dy. u . te a ee ay = Bat moe dew boy Dawinloadiad from hiayiso. com - Mery we . ae a ee * . : en . * : * T960(E)(N14)T (12,

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-8- ‘T96OCE)(N14)T MARKING GUIDELINE MATHEMATICS N6 5.1.3 amy = 7 {ye yr) arc KO ‘My = Fes a yy }oe tac - = Fes Pomta? -Gaetst] 2c de es Ce *- 2x4 “ab? Ryusxp dx 2ex4 2h We a ea [ee- Bt ue" oe, “ i. an bh, 22, 2 4. hS = Vit B~8 + 2) [bea Frat- alae s-sy]t va 1 as = wit -g-8+ £]- [Gaal ~(s)-tl-an +E ayy “ose asa TO 7, = -abe,O88 | , ~ -~ Bes ,og3 pe RS Sih ¥v = TO,TI4 nits a (8 5.2 5.2.4 aa _— y mace} moe | ose daeek Bu =O Lay) 7 we . © . r , hs

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MARKING GUIDELINE -10- T96O(E)(N14)T MATHEMATICS N6 5.3.2 : , BV= yh Bx i Ee v =r ye ce “Wet (Baw 223% dy ne ait P Qo ha, ” — LF we” sides £ = € us ¢ $ < My — sin UCR] L-- : e : fe Z - & ag 7 2 unis? | w or = F103 vials 3 ¢ oe eee : on be io AT, = pays C#& yp be ie . ft Lg 4 hx - /% (3am te Vt ebe ry Me aM Ooe de AT ft Cain 22% dre = See “ (4 “Fen axd(@ “2 b soto dahe as gt a Ct + cone bcm ty) de ca aig! dog hs DR ae) 1 ae i a ms = Beta 4(4)- + se ad) +e + ord aw 6C4) = 14 Gai a Le fe. OR Downloaded yee payiso- com & TF Ar “(4 Sell ue See e

MARKING GUIDELINE -ll- MATHEMATICS N6 54 54] 4 aR Sq Gh x Sek Ue cise (iS 53) 3 = atuys)% . Yo otue Be os woo aie y Downloaded fron hlayiso.com POBOERYONIA)T

MARKING GUIDELINE -12- MATHEMATICS N6 5.4.3 3 “es Wary G = sa? (By 2 -y oa 3, a Sh, aye fe = 2 oa Ss [3 ty |? Ey = s,a [3° 303)" ~26)%*)" = 2), ib units? we ae gy om Bos oh Bi bie = 1 Tdwnis meen OR a Str 3, Ly y te 78 Ty rDeya7) Ray Wb, WHT ye AO ty a 26197 _ ot ay = Cae? _ 4-20,27 4 bE 20,27 "GBF tb 3 = a[ co masa 3 )u, eas “Go t 2 ao, 3, 487 Fog ey S7 yy 0108 8 3u 7 )du “9 Re Ye, Aaa e au? bau a = O,LqU [ °" S, ad Downtordiiehtreitin Fieyivt]:com = = 2 ble wars 3 Le SaeaeeEEEENRD T960(E)(N14)T

MARKING GUIDELINE -13- TICOEMNIAT MATHEMATICS N6 QUESTION 6 Gi i. ¢ . S < ia = F [on qt = Cy sca) |] be = bp eS Tune ae ia nena Ch 62 Bk 2 & sank =O ont) + ae" enk ¥ > Bel < = tos & + ase) b Coast Bsnk)* Pos Pelco ct ask ye + cout 2 DP, Ae - ey _Rémionsc from hlayiso.com = am vie @ tae ele = i fe ~ fee" * sn wee we [steve oC) = ee 5

MARKING GUIDE) JE conte TAB. re T96O(E)J24)T MATHEMATICS NB . a {3° fe= 2s Sry JETER of [' any J EY? de — Vee =a ft oe EH FEE dx coe “s V2 Fx ao = ae Bee on t+ F ence]! = ‘st pie aac! toa. S33 sue] Le = 89,395 Tan? } gq o> L23,818 waite “ (a) (24) TOTAL: 100 Downloaded from hlayiso.com

ze (Key higher education Za «le & training ——— we SS fe ° — === hy Department: = ‘ ©) ip Higher Education and Training ———— aw REPUBLIC OF SOUTH AFRICA TLIOO(E)(I24)P AUGUST 2012 NATIONAL CERTIFICATE MATHEMATICS N6 (16030186) 24 Suly (X-Paper) 09:60 — 12:00 Calculators may be used, Downloaded from hlayiso.com This question paper consists of 5 pages and a 7-page formula sheet,

MARKING GUIDELINE -14- TIOO(E)(N IAT MATHEMATICS N6 ag tO [fe rtec te ate U% eo =p ef she +he cat ~ ab anee bbe . be . ~ bk. Den @ "sak + 60° can aa Tt Loe bee te fe cae ot bu BIDe eet tr geePame 3 (e be Sink + be” co; b) H a fA ania | 35 @ gat +be*cay| “4 = amie fee Cent + Geen) [Me a a7 = BTV peck) ie) wo Le Caw Babe d) ~ 2° (sho + besd)] oe wa avis 30 pen [e - 6 = R127 unik™ a RQ = 66s 1,193 uribs % (4) os * [24 200 + 2 TOTAL: — 100. Downloaded from hlayiso.com

a DHET TIOSOGRYEM PIT DEPARTMENT OF HIGHER EDUCATION AND TRAINING REPUBLIC OF SOUTH APRICA NATIONAL CERTIFICATE MATHEMATICS N6 TIME: 3 HOURS MARKS: 100 INSTRUCTIONS AND INFORMATION lL. Answer ALL the questions. 2. Read ALL the questions carefully. 3. Number the answers according to the numbering systen used in this question paper. 4. Questions may be answered in any order, but subsections of questions must be kept together. 5. Show ALL the intermediate steps 6. ALL the formulae used must be written down 7. Questions must be answered in BLUE or BLACK ink g. Marks indicated are percentages. 9. Write neatly and legibly. Downloaded from hlayiso.com

DHET 3 T1050(8)(M22)T QUESTION | 2x ; az i Given; z = cos (22), Determine the value of sane y Ox 3) 1.2 The dimensions of a cone are r= 25 mm and h = 70mm. Calculate the approximate increase in the volume of the cone when the radius increases by 5 nm and the height decreases by 10,5 mim. c S HINT: The volume of the cone = Lath . @G) 3 6 [6] QUESTION 2 Determine [y alx if: 21 y = V6x 3x? (4) 2.2 ye" (x? ~ 2x43) re) _ ] 28 oan? 5x 1) Gea] 2.4 y = sin’| = |.cos"| — 3 3) (4) 2.5 y = cos! x H @) 253 Downloaded from hlayiso.com 118] OTR OWT ARE ®

DHE? -4- PEOSOC Ey IAT QUESTION 4 ad Calculate the particular solution of ay Ree <2 + (i-x)y =e", if x= 0 when yal, ax (63 4.2 Calculate the general solution of: dry, 5b — tSy = ee2n-l dx? aoe (6} [12] QUESTION 5 5.1 S.A Make a neat sketch of the curve y = 3sin 2. and show the area bounded : a . by the curve and ihe lines x = Oand x = a Show the representative strip/element that you will use lo calculate the volume generated if the area bounded rotates about the y-axts 2) 5.1.2 Calculate the volume described in QUESTION SAb by means of ) integration. . . . . > | Catculate the points of intersection of y * 2x and x” = y. Sketch the é 5.2 5.2.1 graphs and show the repr resentative slrip/element (PERPENDICULAR to the x-axis) that you will use to calculate the area bounded by the graphs OS 5.2.2 Calc! Ni ii ee UEST mh 2A GQ) ed from hlayiso.com 5.2.3 Calculate the second moment of area described in van JESTION 5.2.1 with respect to the p -axis and express (he answer in terms of the area,

“5. T1OSO(EYM22)T DHET S.4 5.4.1 A vertical retaining wall in a water canal is 6 m wide at the top, 5 m wide at the bettom and 2 m high, The top of the retaining wa'l is 2 m below the surface of the water. Sketch the retaining wall and show the representative strip/element that you will use to calculate the depth of the centre of pressure on the retaining wall, Calculate the relation between the two variables x and y. 5.4.2 Calculate the area moment of the retaining wall about ‘he water level by means of integration, 5.4.3 Calculate the second moment of area of the retaining wall about the water level, as well as the depth of the centre of pressure on the retaining wall by means of integration. QUESTION 6 6.) Calculate the are length of the curve represented by the parametric equations, : a a y = rsiP20 and x = rcos’ 20, over the interval 7s Os rf. 6.2 Calculate the surface area generated when the are of x? +y?=16, between py = 4 and y =2, is rotated about the y-axis, TOTAL: Downloaded from hlayiso.com (4) (40] (6) [12] 100

ligher educatio g trainin Depart ne at: Higher Education and Traming REPUBLIC OF SOUTH AFRICA a NATIONAL CERTIFICATE APRIL EXAMINATION MATHEMATICS N6 22 MARCH 2013 Downloaded from hlayiso.com

TIOSOGE)(M. 2. MARKING GUIDELINE MATHEMATICS N6 QUESTION 1 ov 12 AN ar ar 4 ov Ph oh pehfar) + #1? Cah) ie a. yee XS = resaay + dt f2s*t-ids) = Pao 3,723 Yam? [3 bus eas oo? (e) £24 2 8 (eY = mnt pe oV3 21,732 = VB [ aa SP + * 3) 22)

3. TORO EYRE 2 MATHEMATICS N6 MARKING GUIDELINE Ete) ghd = oS caesar j «| = Roya SPATE ih ee law Oe a cae 1g! Pownlaaded from hlayiso.com

MARKING GUIDELINE “4. TLOSOCE (M227 MATHEMATICS N6 co Sata oo EE) ne OS ae — Jad “yf 03 (E) wb a OS) ay caelE en, cate]. 4G = S. c qo =] q ode 6 2)

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MARKING GUIDELINE “8. TIHOSOCE)M22YP MATHEMATICS N6 a. { (0,0) cna (ayn = Ux? a bb) ux Dade ia tee? Rye = fb $e an = 0, ORB wanes * P . aw ye rd a “5 (by veh Cs wad. "Doinntodida ay fim hlayiso.com o = “a ( 20! TH) ola ae wv

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MARKING GUIDELINE, -1)- MATHEMATICS N6 S34 — Pa (tn) 2.2 ~~ fe ady ry) ay v . . ‘yi Le 7 ae “| - = amfeeny+ Seay?) v = By oo Aes) 890 unin? &) soa = 2, say aa 3 =a sa) ly) * Flu) a Bog) «7 = Ett, BHM 368 aa haya we), (ay an, OH) on Downloaded from hlayiso.cony> acid aga/m U- AGT ea seid, 242 ™ ue TiOSOCE)(M22)T

MARKING GUIDELINE ~HL- *PUSACE M227 MATHEMATICS N6 ecu RG = Be Cea 7o) G2 ma faery? 2H Ewe MEO bea San) ly an + peg® “ Downloaded from hlayiso.com 7 \ we

T1050¢2)(M22)T DHET 3 QUESTION 1 7 . 2x : a2 Ll Given: z = cos} —— |. Determine the value of —-. ¥ ax” GB) 12 The dimensions of a cone are r= 25 mm and h= 70 mm, Calculate the approximate increase in the volume of the cone when the radius increases by 5 mm and the height decreases by 10,5 mm. < ioe HINT: The volume.ofthe.cone = bm2h. ee eee ee Seer BPE ; (6) QUESTION 2 Determine fy ax if: 2.1 y = V6x~3x? (4) 22 ye Oo 2x43) (4) ve 1 23 tan? 5x 1) 2.4 yo s'[Z) cos" (;} ; ; (4) os cos! Downloaded from hlayiso.com 2), 25. y= cos” x fag] : = L

“de £10508 )(M22)T DHET QUEST TON 4 4 Calculate the partoular solution of: a _ yt +(-x)y =e if x=Owhen y =1 dx : (6) 4.2 Calculate the general solution of: VY Vas y ax? 42n-1 ay p= x 42x - dx? dx ; ©) os [42]. QUESTION 3 5.1 S.A Make a neat sketch of the curve y = 3sin2x and show the area bounded by the curve and the lines x = 0 and x = a, Show the -epresentative strip/element that you will use to calculate the volume generated if the area bounded rotates about the y-axis. Q) 5.4.2 Calculate the volume described in QUESTION 5.1.1 by means of integration. (5) “ : : : : 5 2! 5 5.2 5.2.4 Calculate the points of intersection of y = 2x and x? mar Sketch the d graphs and show the representative strip/element (PERPENDICULAR to the x-axis) that you will use fo calculate the area bounded by the graphs. GO) ° 522 caciwaianndeade d dimsmvhiayiso.com @) SSTION 5.7 | vil 5.2.3 Calculate the second moment of area deseribed in-QU Te and aunsana the nt ta he se

(16030186) 5.3 5.3.1 5.3.2 5.4 5.4.1 $4.2 5.43 QUESTION 6 “5. T970CEKN14)T Make a neat sketch of the graph y = 3sin2x. Show the representative strip/element that you will use to caleulate ihe volume generated when the area bounded by the graph y= 0 and x= ; is rotated about the x-axis, Calculate the volume described in QUESTION 5.3.1 by means of integration, Calculate the moment of inertia of the solid generat described in QUESTION 5.3.1 is rotated about the x -ax The top of the sluice gate is 9 m wide and Ttinthe, Wa ‘er ae The sluice gate is 3 m high. Make a neat sect Gg ice Bat&2nd show the representative strip/element that you wwill.use4 ey ate the area moment of the sluice gate about the water < Be S, hee A vertical sluice gate, in the form ofa parabo, be is . ime Ned any wall, He the'sly Calculate the relation betwee x and the area moment of the s[uiée gate.ab at you will need to calculate ¢ water level, of integration if@f€’ second midinent#SF area of the sluice gate about the water level is,givétkas 342056 units ae aS : % Nee Calculate the ey cetitre of pregsure on the sluice gate by means 6.1 Caleulatéthe length ofthe ciitve represented by x? = y* between y =1 and p =3, Se e surface*area generated when the curve with * parametic equations @) 4) (6) Q) 03) 6) [40] 6) 7) [12]

Departrient: Higher Education and Training REPUBLIC OF SOUTH AFRICA be NATIONAL CERTIFICATE NOVEMBER EXAMINATION MATHEMATICS N6 14 NOVEMBER 2014 Downloaded from hlayiso.com

DEBT “5. T1050(6)(M22)T 5.4 5.4.1 A vertical retaining wall in a water canal is 6 m wide at the top, 5 m wide tert At the bettom and 2m high. The top of the retaining wall is 2m below the surface of the water. Sketch the retaining wall and show the representative strip/element (hat you will use to calculate the depth of the centre of pressure on the retaining wall, Calculate the relation between the two variables x and y. (4) 5.4.2 Calculate the area moment of the retaining wall about the water level by means of integration. @) 5.4.3... Calculate the second moment of area of the retaining wall about the water... ...... level, as well as the depth of the centre of pressure on the retaining wall by means of integration. (4) [40) i QUESTION 6 6.1 Calculate the are length of the curve represented by the parametric equations, . a nz ye rsiP20 and x =r cos’ 20, over the interval —< 8 <—, 8 4 (6) 6.2 Calculate the surface avea generated when the arc of x? + y? =16, belween y = 4 and y = 2, is rotated about the y -axis. aa) TOTAL: 100 Downloaded from hlayiso.com

higher educatio & tra aining ena riment: M ARK ING ¢ GUTD NATIONAL CHRTIFICATE APRIL EXAMINATION 22 MARCH 2013 Downloaded from hlayiso.com i. Higher Ectuce ation @ and Tratning REPUBLIC OF SOUTH AFRICA

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(16030186) “2. TOQOCE EIA? DEPARTMENT OF HIGHER EDUCATION AND TRAINING REPUBLIC OF SOUTH AFRICA NATIONAL CERTIFICATE MATHEMATICS N6 TIME: 3 HOURS MARKS: 100 INSTRUCTIONS AND INFORMATION 1. Answer ALL the questions. ne Read ALL the questions carefully. 3, Number the answers according to the numbering system used‘in this question paper. 4. Questions may be answered in any order, but subsections of questions must be kept together. & 5. Show ALL the intermediate steps.

(16030186) 3. T960(E)(J24)T QUESTIO" 1 ; lt If w= tan3x.cot determine a Q) 12 The parametric equations of a function are given as: and x= —., calculate the values of the following: a t+a? dy L2 ee dx 1.2.2 a’y dx (4) [6] QUESTION 2 Determine [y ax if 21 vy = tan® 2x (4) 22 ee @) 2.3 (4) GAN

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(16030186) 5.3 5.3.1 Calculate the points of intersection of the graphs y=3x and y=32x°, Make a neat sketch of the two graphs and show the tepreseitative strip/element (Parallel to the y -axis) that you will use to calculate the volume if the 1: -a bounded by the two graphs is rotated about the ; -axis. (3) 5.3.2 Calculate the volume described in QUESTION 5.3.1. (4 5.3.3 Calculate the moment of inertia of the solid obtained when the area in QUESTION: 5.3.1 is rotated about the y -axis and express ‘hogangwer in terms of the mass. ‘ay. (5) S4 5.4.1 A vertical weir in a rectangular canat is 6 m wide andeg.m hip 5 of the weir is 3 m below the water surface. Make a ne weir and show the representative strip/eleme: hi calculate the depth of the centre of pressure ong (2) 5.4.2 Calculate the area moment of the wei of integration. (4) 5.4.3 Calculate the second moment of'area of “8 cir about the water surface, as well as the depth of the éghire presstite on the weir by means of integration. “Ss : (4) (40) QUESTION 6 el Calculate the are lenge X= 3(cosO+ Osing) an feos0 Jan) ©) . : ° 2 2 i 62 Detolie wineeaalepl feo ta LABS CAM X21, | ‘s a aboutthe x-axi (6) y bth 21 -5- . TI6O(E)(J24)T

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THOotE {a QUESTION 4 yp 4] Calculate the general solution of: dy +45 ~~ ~ Gian) y = sind y he ( x)y * sind » 42 Calculate the particular solution of: ay 6b, 13 yt = 39 if x= 0 when ¥=2and x=0 when co =], ads ay ad QUESTION § a SL] Calculate the points of intersection of 4y =x) and y? =4y, Make a neat sketch of the two curves and show the area bounded by the curves, Show the representative strip/element that you will use to calculate the volume (use ihe SHELL METHOD only) generated if the area bounded by the curves rotates about the r-axis, 5.1.2 Use the SHELL METHOD to calculate the volume described in QUESTION 5.1.1 by means of integration. in 3.2 5.2.1 Calculate the points of intersection of 5 16 and y+2x-12=0, , x Sketch'the graphs and show the representative strip/element that you will use to calculate the area in the lirst quadrant, bounded by the graphs. 3.2.2 Calculate the area described in QUESTION 5.2.1. e < o.com 5.2.3 [Draw loaded. from.blay; a Brea described QUESTION 5.2.1. Sai ow us G) (@) 12} G) (4) Q) (3) (3)

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