N5 MATHS SPOT ON R80 .00

a SPOT ON MATHEMATICS NS QUESTIONS AND SOLUTIONS R80.00 Downloaded from hlayiso.com

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higher education & training Department: Higher Education and Training REPUBLIC OF SOUTH AFRICA Downloaded from hlayiso.com Please turn over Copyright reserved

(16030175) -2- T10S0(E)(M30)T DEPARTMENT OF HIGHER EDUCATION AND TRAINING REPUBLIC OF SOUTH AFRICA NATIONAL CERTIFICATE MATHEMATICS N5 TIME: 3 HOURS MARKS: 100 INSTRUCTIONS AND INFORMATION 1. Answer ALL the questions. 2. Read ALL the questions car 3. Number the answers accordii Downloaded from hlayiso.com Copyright reserved Please turn over

(16030175) -3- TLOSO¢E)(M30)T QUESTION 1 11 11.1 dx-3 Determine In y if In y=lim ny if iny=lm— (2) 1.1.2 Now, determine the value of y qd) : 1x)? 1.2 Determine lim (=*) xt x* —x—xInx 4) [7] QUESTION 2 2.1 Given: y -_! 4x? : so. ay Determine the derivative —, (4) 2.2 2) (4) (2) | 2 2.3 Determine ed with the aid lic differentiation if y= yx ~ltandx dx Inx (4) 2.4 Given: y = arctan x 24.1 Make a neai, rough sketch of the graph of y for the range i 5 4 @) 2.4.2 Derive a formula to determine » if y = arctan x dx Q) 2.5 Given: the implicit function e**” -— tL 2y? x Calculate the derivative a F dx (4) [25} Downloaded from hlayiso.com Copyright reserved Please turn over

(16030175) ~4- T1050(E})(M30)T QUESTION 3 3.1 Given: f(x)=x° +x? -2x-1 3.1.1 Determine the coordinates of the point of inflection of f(x). (2) 3.1.2... Draw up a table of values for x and f(x) with x ranging from x = 1,1 to x = 1,4 using intervals of 0,1. Round answers off to ONE decimal. Draw a neat graph of ffx) between these values. (4) x=1,23. Use Taylor/Newtdii's method to determine a better approximation of this (4) 3.2 4 should be as 6) 3.3 from a fixed Obtain an exp ‘or the velocity of the o! after ¢ seconds and then calculate the values of ¢ Scbject is at rest, 4) [19] Downloaded from hlayiso.com Copyright reserved Please turn over

(16030175) -5- T1050(E)(M30)T QUESTION 4 4.1 Determine [Se by resolving the integral into partial fractions. (5) x° -2x-3 42____ De Determine [ _ydx in each of the following cases: 42.1 See Ses, (3) 4.2.2 howe dx (3) 42.3. [sin 3nx.cos 8m ) (5) 43 (4) [22] QUESTION 5 5-1 Given: 5.1.1 Za enclosed between the two curves, the show the representative strip and the (3) 5.1.2 itlide of the enclosed area. A) 5.1.3 Calculate the volume generated when this area rotates about the X-axis. (4) 1 1 5.2 Calculate the value of J V6 —9x? (4) Downloaded from hlayiso.com Copyright reserved Please turn over

(16030175) -6- T1050(E)(M30)T 5.3 Determine the second moment of mass of a rectangular lamina of mass (7m) about the axis parallel to one side of the lamina. The distance from the representative strip to the axis is x. The lamina’s length and breadth is a and 6 respectively. Y a (a;b) QUESTION 6 6.1 TOTAL: Downloaded from hlayiso.com Copyright reserved (4) [19] GB) (3) [8] 100

(16030175) “1 MATHEMATICS NS5 FORMULA SHEET Any applicable formula may also be used. TRIGONOMETRY cos*x = 1 — sin’x sin? x + cos? x = << sin’x = 1 —cos’x 1 + tan’ x = sec” x ————>» tan?x = sec?x - 1 1 + cot” x = cosec” x ——-» cot?x = ¢ sin 24 =2 sin A cos A cos 24 = cos’ - sin’A cos? A cos (4 + B) } tan A+ tanB + B) = ———_—. tan (42 B)= Tan Atan B dv cos 4 cos B=% [cos (A + B) + cos (4 - BY] sin A sin B= % [cos (4 - B) - cos (A + B)] sinx , 1 tanx = 3 Sina = : 5; COSx = cosx COSEC X secx cos x 1 cot x =— 5, cos ecx = — ;secx = sin x sin x cosSx Downloaded from hlayiso.com Copyright reserved T10S0(E)(M30)T Please turn over

(16030175) -2- T1050(EXM30)T BINOMIAL THEOREM non - -) n(n-N(n—-2)_ n3,3 eth)" =x" txt 14 2 Wt + (eth) = 2" +m a Taxa A DIFFERENTIATION ___ F(@) vrate PRODUCT RULE y = ux). v(x) dy dy du S=u,—t+v.— dx dx dk =u-v +eeH CHAIN RULE y =fue)) wv _ ey du de du dk Downloaded from hlayiso.com Copyright reserved Please turn over

(16030175) 3+ T1050(E)(M30)T d Se) Zl [ f@ac x” nx”? a +e (n#-1) n+1 axtb dete a In(ax) Se) al? In fix) sin ax cos ax tan ax cot ax Sec ax cosec ax Copyright reserved a sec’ ax _intseo(axy}+C a datsin(ax)]+C a - a cosec” ax «tn [sec ax + tan ax) +C a e(s) Downloaded from hlayiso.com Please turn over a sec ax:tan ax = a cosec ax:cot ax

(16030175) 4. T1050(E)(M30)T fe) ai) fo sin f(x) cos fixdef"(x) _ cos f(x) - sin fx)of Gx) _ tan fx) sec” fix)of'" (x) _ cot f(x) - cosec” f(a sec fix) cosec f(x) sin” f(x) tan" (G0) cot? f(x) _ sec” f(x) _ cosec? f(x) _ fOWVLF@P -1 sin’(ax) a x BnGer) + ¢ cos*(ax) _ 7 + see +c tan’(ax) __ 5 tan (ax) — x +€ cot?(ax) __ - cot{ax}—x+C Downloaded from hlayiso.com Copyright reserved Please turn over

(16030175) 5 INTEGRATION J 00) g'(@) dx = fo) - J £1) g(a) dx [U/@O"* p(a)dx = Yor". C (#-1) J Fey Ba BIE+C APPLICATIONS OF INTEGRATION AREAS 4A,=[) yar; 4,=[) 0, -y2)de SECOND MOMEN =f? 2a-7 =f? 2 =f dA: 1y=[) rdA MOMENTS OF INERTIA Mass = density x volume M= pv DEFINITION: [=m GENERAL: J=f r’dm= pf’ ray Downloaded from hlayiso.com Copyright reserved T1050(E)(M30)T

higher education & training Department: Higher Education and Training REPUBLIC OF SOUTH AFRICA... . MARKING GUIDELINE [ NATIONAL CERTIFICATE MATHEMATICS N5 APRIL 2017 This marking guideline consists of 10 pages. Downloaded from hlayiso.com Copyright reserved Please turn over

MARKING GUIDELINE -2- T1050(E)(A)T MATHEMATICS NS INSTRUCTIONS AND INFORMATION 1. ¥ equals 1 mark. 2. Half marks are not allocated, unless indicated otherwise. 3. Where formula is required, using the wrong formula is a principle error and NO marks are allocated. 4. Candidates should show ALL formula, intermediate steps and simplify where possible, 5, ALL final answers must be rounded off to THREE decimal places (unless indicated otherwise). 6. Questions may be answered in any order, but subsections of questions must be kept together. If subsections are separated, the candidate can be penalised by ONE mark. 7. Where a candidate copied incorrectly from the question paper, and the standard of the question is still the same, the candidate will be penalised with ONE mark. If the copying error simplifies the question and makes it easier, the candidate forfeits the marks, 8. Questions must be answered in blue or black ink. Answers in PENCIL are not marked as it is regarded as rough work. Downloaded from hlayiso.com Copyright reserved Please turn over

MARKING GUIDELINE 3 3. MATHEMATICS N5 QUESTION 1 1.1 111 - iny=timY 3 x9 x—9 = jiim2¥2 9] aly 6 LL2 yy =1,181 1.2 —x)? tin = xl x —x-xInx : —2+2x 0 = lim —————_. — x1 2x-1-Inx-1 vv ry “aT x =2 v QUESTION 2 21 ye 1 V4 x? 1 1 Worthy drt P0)= fi +H 2x vx -2xh-W 1 yy = lim .,— b0 Dx" (x+h)? h h(-2x-h) 1 slim ty 40 2x"(xt+hy kh x Downloaded from hlayiso.com Copyright reserved T1OSOCEMA)T Q) qd) (4) [7] 4) Please turn over

T1050(E)(A)T MARKING GUIDELINE 4. MATHEMATICS N35 2.2 2.2.1 1, y=arcsin(e? +3) 1d 2 v wy 3 dx loo . 1-2? y x 2a (2) 22 2.2.2 3x ° = a tan’ (--5 G ) Vo 4 4-4 ay 3.tan’(—5)—3x3tan’(—5).— > de (tan(A-5)? ¥ : x (4) 2.2.3 ys gieos’ x) y=Cos' x y De 2cosx.-sinx ~ dx 2) 23 _ vx? -1tan4x » Inx Iny = | In(x? 1) + Intan 4x—In(n x) ld 2& 4 Asec? 4x _ 1 ydz 2G?) tan dx a v¥v¥ ONE mark per term Wy *_,Asectde 1) dx x1 tandx xinx (4) Downloaded from hlayiso.com Copyright reserved Please turn over

MARKING GUIDELINE -5- T1050(E)(A)T MATHEMATICS N5 2.4 2.4.1 ¥- coordinates ~~ asymtotes v- shape. (3) 2.4.2 y =arctan x tan y=x sec? yal v dx dy 1 dx 1+tan? y v dy 1 de 1+x (2) 2:5 sec y e ~1uny x v v v sec dl dy e” sec y tan y+ = 4 — y Vox 2 Ve dy _& secytany—4y ax 1 x (4) [25] Downloaded from hlayiso.com Copyright reserved Please turn over

MARKING GUIDELINE 6: TIOSO(EYAYT MATHEMATICS N5 QUESTION 3 310 BA ptgyex? ex? 2x] yl=3x? +2x-2 6x+2=0 % x=-0,333 ~ for both x and y =-0,26 Q) ¥- values v- shape (4) 3.1.3 xi +x7-2x-1=0 -— 0,086 =-——_ = 0,0172 vvv . 4,9987 for approximate value v r=1,23 + 0,0172 =1,247 4 3.2 Vex2h na? vy x A=x’+4xh V A=x? +4(205)x7 27-820 _ 9 v x x= 410 x=7,429 ¥ h=3,714 ¥ G) Downloaded from hlayiso.com Copyright reserved Please turn over

MARKING GUIDELINE -7- MATHEMATICS N5 33 xeit a1 8-1 vals? —24-8 ¥ (@-4)(t+2)=0 v t=4¢=-2 V NA ¥ QUESTION 4 4.1 j x-8 , x? -2x-3 x-8 _ A + B (x-3)(@+1) x-3 x41 x-8= A(x+1)+ B(x-3) x=-tn2=B v 4 eg ed ¥ 4 oe 4,4 4 x-3 x41 v ov 5 9 =——In(x~3)+—In(x + D+e 4 (x ~ 3) 4 (x+) 42 42.1 joa us 8 2 3 dea dy V 1 footu.3du v =3In(sinvx)+e ¥ Downloaded from hlayiso.com Copyright reserved T10S0(E)(A)T (4) [19] 6) @) Please turn over

MARKING GUIDELINE -8- MATHEMATICS NS 4.2.2 fe 62" dx u=-2x' de =: di —8x je ilies ae] ” = Jota v -1 6" 8° Iné 4.2.3 f sin3zx.cos8ax dx +¢ V : f sin] 1+ sin(—Sax)dx v v 1,—cosllax cos(—Sax) = —[——___ - te =e 5x 4.2.4 2 vv quotient/remainder 4 foe 424 9 — 2) yov ~ ONE mark per term wo2 4 =—+=x+—InGx-2)+e 3.2 9 27 430 f£@8' = f@a@ - [fed x" Ina’dx v v roe —f 78 ee =5x fasta v 6 35x cv Downloaded from hlayiso.com Copyright reserved T1050(E)(A)T 3) Q) 6) (4) [22] Please turn over

MARKING GUIDELINE -9- MATHEMATICS N5 QUESTION 5 5 5A 1 T10S0(E)(A)T v - shading and strip (3) 5.1.2 1s Je —x'dx Vv-ONE mark for limits/ONE mark for equations 0 4 = le" a .s v =3,216-1 =2,216 ¥ (4) 513 Ls V=nfe*—x%dx ¥ 0 e x1, v =a =v al > 7 a V = a{7,602—-0,5] ¥ = 12v V = 22,312 (4) 5.2 1 1 9 ¥16 - 9x? If 1 =f dk v 34 oy _ 3 al +t 3X. 1 vo = 3 sin ¢ 4 Mo =; [0.848 -0]) ¥ = v = 0,283 (4) Downloaded from hlayiso.com Copyright reserved Please tum over

~-10- MARKING GUIDELINE MATHEMATICS N5 4.3 QUESTION 6 6.1 a _ 2x? -1 de 10” 10” dy = 2x? ~ 62 oa —2e* + Downloaded from hlayiso.com Copyright reserved TOTAL: T1050(E)(A)T (4) [19] QB) qd ee 4 o£ "2 7p tansx—L2x+c¢ v~ penalise with ONE mark for every wrong term 2 YY penalise with ONE mark for every wrong term 3) 18} 100

7 her education & training Department: Higher Education and Training REPUBLIC OF SOUTH AFRICA pana 22, November 2016 (X-Papéx) ys 09:00—12:00 ay / Scientific caleutators may be used. This question: paper consists Gh6 pages und a formula sheet of 5 pages. ane Downloaded from hlayiso.com Piease turn over Copyright reserved

(16030175) -2- T890(E)(N22)T DEPARTMENT OF HIGHER EDUCATION AND TRAINING REPUBLIC OF SOUTH AFRICA NATIONAL CERTIFICATE MATHEMATICS NS TIME: 3 HOURS MARKS: 100 INSTRUCTIONS AND roy 108 5 1. Answer ALL the question 2. Read ALL the questions Psa 3, Number.thSainswers accohigg to the num "Re, 4. Show ed in this question paper. 5. ot final answers a rounded off to THREE Seong laces on 6. Questibas may beslperel in any a but subset of questions must be kept together> .N y al % a “s at 7. Questions _* answered in blu or black ink. i) 8. Write neatly and ieeibly. “Sh. | Downloaded from hlayiso.com Copyright reserved Please turn over

(16030175) -3- QUESTION 1 Ll so Inx Inx GB) 112 jim * 72"-5 4300 x2 Oy (2) ree =, 1.2 Determine whether f(x) =4% rs is continuous at x =— y rl « QUESTION 2 | 2.1 Deteffine the derivative of Ke (x) =¥/2x ° fiom first rl (6) é - 2.2 (4) (3) (3) 2.3 Given: A particle moves along a line such that displacement (in meters) in time (t seconds) is given by Determine the velocity at ¢ = * seconds with the aid of logarithmic differentiation. ‘6 2.4 Given: cos(x+ y)+sin x+sin y =2 Make use of implicit differentiation to calculate —— dx (4) [24] Downloaded from hlayiso.com Please turn over Determine the following limits: 1.11 lim-—-- + Copyright reserved T890(E)(N22)T

(16030175) -4- T890(E)(N22)T QUESTION 3 3.1 Given: f(x) =3x° —4x? —2x+2 3.1.1 Determine the coordinates of the turning points of f(x). QB) 3.1.2 ..... Verify. by.using.a table that the equation 0 = 3x3 = 4x? ~2x+.2. has a root between the points x =1 and x =2. Use values on the table: ~1Sx52 @) 3.1.3 Hence, make an sketch of the graph of the function f(x). (3) -_- 3.1.4 If the i of f(x)is estimated as 1,7, use Taylor's/Newton's t method to detégiine a better approximation of this root. (2) 3.2 A farmer Wnts to enclége a field rinse mat x meter and y meter Th esst of fencing is R36,00/m for the length and R45, ni for the breadth. He has ait atfount of R56 000,gppevailable. ‘ f os BET Give the fons of the area and the cost i ’ of x and y. (2) 3.2.2 “tiga (5) 3.3 HINT: Y=ar'h (4) [21] Downloaded from hlayiso.com Copyright reserved Please turn over

(16030175) -5- T890(E)(N22)T QUESTION 4 41 Determi ef 1= cos2x ~2x+sin 2x 4.2 Determine J ydx in each of the following cases: 42.1 J 2x. dx 4.2.2 P= 4 4.23 424 \ a yA dott” a? 43 Delerfnine f ydx by resolving the integrand into partial fractfons: 8 ‘ x roy 44 Determine QUESTION 5 5.1 Given: The curves X=—y’ +4 and x=4-2y S11 Sketch the TWO graphs. Shade the area enclosed by the two graphs and show the representative strip with dimensions x and dy. 5.1.2 Calculate the magnitude of the enclosed area towards the Y-axis. 5.1.3 Calculate the volume generated when this area rotates about the Y-axis. Downloaded from hlayiso.com Copyright reserved Piease turn over @) 3) G) G) (3) 6) (2) [23] (3) 4) (4)

(16030175) -6- T890(E)(N22)T 5.2 Determine the second moment of mass of a rectangular lamina of mass (m) about the axis parallel to one side of the lamina, which bisects the lamina. The distance from the representative strip to the axis is x, The lamina’s length and breadth is a and b respectively. AY Pe a »X PY) (4) Wy [15] QUESTION 6 oA (: i= a& 6.1 Solvg the differential equation e“*” a _ Po ly (4) “ard ee 6.2 Detetmine the genefal Solution of the differential ¢ Se , = 4 Do, d’y \ “s ae % (6) i [10] ¢ id TOTAL: 100 Downloaded from hlayiso.com Copyright reserved

(16030175) -L MATHEMATICS N5 FORMULA SHEET Any applicable formula may also be used. TRIGONOMETRY sin? x + cos’?x=1 1 + tan” x= sec" x 1+ cot’ x = cosec” x sin 2A =2 sin A cos A cos 24 = cos"A - sind na iy é J on ad” « i (3 . sin (A+ B)=sin Aviad Zin Bcos A ~~ cos (4+ B)=cos A cosBsinAsinB | tanA+tanB : tan (4B) = 7 tan AtanB sin A cos B= 4 [sin (4 + B) + sin (4 - BD] cos A sin B= % {sin (A + B) - sin (4 - BY] cos 4 cos B= [cos (A + B) + cos (4 - B)] sin A sin B= % [cos (4 - B) - cos (A + B)] sinx 1 1 = 3 . cos x = cosec x secx tan x = 3 sin cosx Downloaded from hlayiso.com Copyright reserved T890(E)(N22)T Please turn over

(16030175) 20 T890(E)(N22)T BINOMIAL THEOREM (xt hy)" =x" + nx"! i MD We... DIFFERENTIATION S@ r=ate PRODUCT RULE y=ulx). v(x) a Downloaded from hlayiso.com Copyright reserved Please tum over

(16030175) -3- T890(E)(N22)T fs) =f) f(ayax n+ n+l 0 axte & & e+e sinx cos x tanx cotx 4 +tanx] +¢ In (cosecx - cotx) +c sec x oN cosec x sin” x cos" x tan™ x cot? x sec’ x —— _ xy? -! cosec" x —_ _ ax? -l Downloaded from hlayiso.com Copyright reserved Please turn over

(16030175) » P 4 INTEGRATIO@ P 4 4 % mn J f(x) 2) de = ABs | £°G) gta) dx | FO ~~ Fray =n fate 4 Lor f) e- OT +6 f(x) __A4 B (ax+byext+d) axtb cxt+d fi) A B c \ Z (@+ay” Gta) Gta Gray Gray APPLICATIONS OF INTEGRATION AREAS b b 4A,= fxd A= for — y)dx 4=f. xdy A, =f (4 — 4)dy Downloaded from hlayiso.com Copyright reserved T890(E)(N22)T Please turn over

(16030175) -5- T890(E)(N22)T VOLUMES Ve=a[ Pde :Ve=nf? bp -yF)ae Vy =a xdy iVy =a (2 - x3 ay SECOND MOMENT OF AREA t.=[) Pda; t,=f" Pad MOMENTS OF INERTIA Mass = density x volume M= pv DEFINITION#// = m 7? GENERAL” Downloaded from hlayiso.com Copyright reserved

higher education & training Department: Higher Education and Training ‘REPUBLIC OF SOUTH AFRICA MARKING GUIDELINE NATIONAL CERTIFICATE NOVEMBER EXAMINATION MATHEMATICS N5 22 NOVEMBER 2016 This marking guideline consists of 9 pages. Downloaded from hlayiso.com Copyright reserved Please fur over

MARKING GUIDELINE -2- T890(E)(N22)T MATHEMATICS NS INSTRUCTIONS AND INFORMATION 1. equals mark. 2. Haif marks are not allocated, unless indicated otherwise. 3. Using the wrong formula when one.is required is a principle error and NO marks should be allocated. 4. Candidates should show ALL formula and intermediate steps and simplify where possible. 5. ALL final answers must be rounded off to THREE decimal places (unless indicated otherwise). 6. Questions may be answered in any order, but subsections of questions must be kept together otherwise candidates can be penalised with ONE mark. 7, When something has been copied incorrectly from the question paper and the standard of the question is still the same, the candidate will be penalised by ONE mark. If the copying error simplifies the question the mark will be forfeited. 8. Questions must be answered in blue or black ink. Answers in PENCIL will not be matked as it is regarded as rough work. Downloaded from hlayiso.com Copyright reserved Please turn over

MARKING GUIDELINE -3- MATHEMATICS N5 QUESTION 1 1.1 1.1 li x 1 xe Inx Inx 1.2 _ [+9 rey= 2 v undefined discontinuous ¥ QUESTION 2 21 F(x) =V2x “¢ binomial theorem £42 9 S37 1 : 203 sina A-px" Ft) Aa f'@)=lim - = ss WaW2x 3 ~2y9¢3 Ay v =lim—23 9 2 20 h i 3x3 Downloaded from hlayiso.com Copyright reserved T890(E)(N22)T (3) (2) 2) | 6) Please turn over

MARKING GUIDELINE -A- MATHEMATICS N5 = 2.2.1 y =n? 32 1332 ev v v v dy _ 3x. 1 3e*-3xe* 6x a an 3x° ge 3x” e 2.2.2 y = Sarccos ec(x sin 2x) ._ S0sin2x+2xcos2x) “~~ product rule xsin2x(xsin2x)?-1 ¥ 2.2.3 y =4ycos'(x?—x-1) v v v I y= 40s! (x? ~x—1).-sin(x? —x-DQx-1) 2.3 s(t) = pos! Ins=cost.Int ~ cost 148 | snp ne + ¥ s dt 2 = s(-sintInt + 2088) at t v for t=4:5=08%3 velocity = 0,843(0,17140,9) “ =0903 2.4 cos(x+y)+sin x+sin y=2 v v —sin(x+y).(1 +2) +cosx+cos ye =0 & (sin(x-+y)-+e08y) =-cosyt+sinixt+y) vv dy _~—cosy+sin(x+ y) dx —sin(xt+ y)+cosx Diiv d& Downloaded from hlayiso.com Copyright reserved T890(E)(N22)T (4) (3) (3) (5) A) [24] Please turn over

MARKING GUIDELINE 5- MATHEMATICS NS QUESTION 3 3.1 3.1.10 f(x) =3x?-4x? -2x42 3.1.2 3.1.4 3.2 3.2.1 3.2.2 Copyright reserved 9x?-8x-2=0 VY x=1,092 of x=-0,203 ¥ for both x-values y=-L047 of -y=2;216 for both y-values 0 =3x°—4x7-2x+2 : -lSx<2 x-l| 0 1 2v y f 2-1 «Ov a Y y-intercept Y turning points v shape -1 fy XC) 2 e=— 7? _ i709 v 10,41 r=1,7—0,709=1,5291 ¥ A=xy v C= 26x) +2(45y) C=72x+90yp ¥ 56000 = 72x+90y 56000—72x _ 9 A=622,22x-0,8x" y 622,22~16x=0 ¥ 388,89 =x B1Lll=y y Downloaded from hlayiso.com v TR90(E)(N22)T @) 2) (3) 2) (2) 6) Please turn over

MARKING GUIDELINE -6- T890(E)(N22)T MATHEMATICS N5 3.3 V=arh r=1,5m dV > dh = = ary. Y derivati dt (7) dt derivative 3% mrs) ~ substitution dt 0,424m = “ v (4) [21] QUESTION 4 4] { 1-cos2x d a —2x+4sin 2x apg oak sin' 2s) ==24 2 cos 2x v ~20- cos 2x) & — | -2x+sin2x -2 . v = Sineax +sin 2x)+¢ ; @) 42 421 faxe™de 20% v 2x —— [26 dx 2 =xe*-e* 40 0 B) 4.2.2 3 [= 4 de xtl Long division: ¥¥ quotient remainderY J 2x? -2x4.24— de x+l 2x, = 3% +2x-6In(x+1)+e¢ ~~ ONE mark for two correct terms (6) Downloaded from hlayiso.com Copyright reserved Please turn over

MARKING GUIDELINE 7. T890(E)(N22)T MATHEMATICS NS5 4.2.3 sin’ 3x Je a sin3x fsiv’ 3xdx 11” = J5=s00s 6xdx 2.2 _ lv sin 6x 2 12 (G3) 4.2.4 for [58 + 1d u=2x' +1 dx= ae v 6x? [oxta de 6x? ov i} Alu vos v 5 3 ==(2x7 +1) +e 9 (G3) 43 pe ae 4x? -1 4° -3x+l _ A +2 (2x-1(2xt1) 2x-1 2x41 4x? -3x4+1= A(2x+1) + B(2x-1) ¥ x=-05:-L28 4 : Z J --he (2x-1) 2x41 = diag) -2 = inex )) g inax+ +e (5) Downloaded from hlayiso.com Copyright reserved Please turn over

MARKING GUIDELINE -8- T890(E)(N22)T MATHEMATICS NS 44 [ sin(aresin x de J xd ¥ slaxtev Q) [23] QUESTION 5 5.1 SAL x=-y'+4 and x=4-2y v - parabola graph Fy Y - straight line Vv - shading and strip | ——~ | (3) 512 2, Jo’ +4)-(4-2y)dy - equations 0 -y =F3-+9"lo v ¥ - integration =1,333 ¥ (4) 5.13 2 V =nf-" +4) —(4-2y)'dy V - equations 0 2 =a] y‘-12y?+16ydy_- simplification QO 5 =a —4y° +8y’2 “> integration = 7(6,4) =20,106 (4) copyright reserved Downloaded from hlayiso.com ___,.cmumover

MARKING GUIDELINE 5.2 4 2fx be Y ! ab z =27 fxtde 4% a 24° QUESTION 6 6.1 2xt3y a =e S8y é€ dy 62 2 =210%4 v v dy 2.10 cot2x =—— -—— -4x4+ dx 41n10 2 v v 10 In(sin2x) 2.4In?10 4 Downloaded from hlayiso.com Copyright reserved -9- MATHEMATICS NS v e v 2x? +ex+d T890(E)(N22)T @) [15] (4) (6) [10] TOTAL: 100

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higher education & training Department: Higher Education and Training REPUBLIC OF SOUTH AFRICA TS90(EA2)T AUGUSEEXAMINATION NAGPIONAL CERTIFICATE is MATHEMAT CSNS J 2,August 2016 (X-Paper} This question paper.gonsists 6f'6 pages and 1 formula sheet of 5 pages. “OP a a Downloaded from hlayiso.com Copyright reserved Please turn over

(16030175) -2- T890(E)(A2)T DEPARTMENT OF HIGHER EDUCATION AND TRAINING REPUBLIC OF SOUTH AFRICA NATIONAL CERTIFICATE MATHEMATICS NS TIME: 3 HOURS MARKS: 100 INSTRUCTIONS AND INFORMAT OR, 1 Answer ALL the questiong. 2. Read ALL the questions 3 Number thegnswers 4. Show/AKL intermediate steps \ 5. final answers must be rounded off to THREE decimals. 6. ions may be wife in any order, but subse: togethe. hd } 7. answered in uy or biaeietne A Fe 8. Am copyright reserved Downloaded from hlayiso.com ,____._....

(16030175) 3. QUESTION 1 1.1 Determine In y if Ll Iny=lim x3 Insin 3x 1.1.2 Now, determine the value of y 1.2 Determine limx’.e* a QUESTION 2 | 2-; @ 2.1 Given: = et ee Desiegpine the derivative f'(x) using first principles. j op ee) T890(E)(A2)T 2.2 Determine 2 in eagif6f the following cases (simplificatijin is NOT required): aN y- 4 — ‘ PS 4 ~ A - S ‘ . A “ 221 aie \ a 222 yy =[In(Cdsec2x)) 2.2.3 y=2et 3 23 Determine 2 with the aid of logarithmic differentiation if y = (9—.x*)"* 2.4 Given: y = arcsin x dy 2.4.1 Derive a formula to determine G, if y = arcsin x 2.4.2 Hence, determine 2 if y =aresin (5.107) Downloaded from hlayiso.com Copyright reserved Please turn over GB) (1) (3) 7} (4) (4) (3) @) (4) (2) (2)

(16030175) -4- 2.5 Given: the implicit function x? +2= y?.cot3x Calculate the derivative 2 QUESTION 3 3.1 Given: f(x) = x(x? -3x-9) 3.1.1 Determine the coordinates of the point of inflection of f(x) T890(EMA2)T (4) [25] Q) Q) 3.1.3 Draw a neat gtaph of JSG) between the values in QUESTION 3.1.2 if the turing points bal given as (3;-27) and (-1;5) 3.2 HINT: V=mh 3.3 The sum of the diameter and height of a cylinder is 6 cm. Calculate the dimensions of the cylinder to ensure a maximum volume. QUESTION 4 4.1 Given: _ A + B i x? ee) (10+ x) 4.1.1 Determine the numerical values of A and B. 3 4.1.2 Hence, calculate In 4x 1 ~“x Downloaded from hlayiso.com Copyright reserved @) @) @ 6) 6) [21] (2) (3) Please turn over

(16030175) -5- T890(E)(A2)T 4.2 Determine f ydx in each of the following cases: 4.2.1 yex? sine x 4.2.2 y=cos? 5x 4.2.3 y =cos12x.sin3x 4.2.4 . ay (4x3) (12) 43 \ (4) [21] ‘ 4 a, 4 fe mca 9 y ad « > XxX 4= y vy 3.1.1 Prove that the points of intersection are: (0;0), (2,828;1,414) and (—2,828;-1,414) (3) 5.1.2 Calculate the area of the bounded region in the first quadrant. (4) Downloaded from hlayiso.com Copyright reserved Please tum over

(16030175) -6- T890(E)(A2)T 5.1.3 Calculate the volume generated when the area in QUESTION 5.1.1 rotates about the X-axis. 52 $2x? : Calculate the value of Jie x 1 5.3 Show that the volume of the figure formed when the line y = Gor rrotates about : h the X-axis equals 7 = sry x QUESTION 6 Te ——— | 6.1 Solve the differential equation ede—y'dy =e" dx 6.2 Determine the general solution of the following differential equation: (1+ x)sin y.dy = cos y.dx TOTAL: Downloaded from hlayiso.com Copyright reserved (4) (3) (4) [18] @) (4) [8] 100

(16030175) 3. T890(E)(A2)T fa) = 70) {player sin x cos x tan x cot x sec x cosec x sin? x cot” x sec’ x cosec' x Copyright reserved n+] nax”! n+1 axtc & ete Ina i aafx? -t - Downloaded from hlayiso.com Please turn over

(16030175) 4. T890(E)(A2)T fo = 40) Sf (dx INTEGRATION™.) # [ f() 8) dx = £9) eo £1) g(x) dx | oN SO “a Fag @ fe) te Des, atl Ler se ar-YET fe) A (axtb\ex+d) axtb oxtd f@) A B c Z (xta)" (eta) (xt+a)? Gra) Otay Downloaded from hlayiso.com Copyright reserved Please turn over

(16030175) De BINOMIAL THEOREM (xt hy =x" 4 nx" h MaDe Reo. DIFFERENTIATION -_£@ £@ roate PRODUCT RULE =u) . vO) dy dv CHAIN RULE y=fu) - w_w ha dx du ax Downloaded from hlayiso.com Copyright reserved T890(E)(A2)T Please turn over

(16030175) -1- T890(E)(A2)T MATHEMATICS N5 FORMULA SHEET Any other applicable formula may also be used. TRIGONOMETRY sin’ x + cos’ x= 1 1 + tan? x = sec? x 1 + cot” x =cosec” x , sin 24 =2 sin A cos A cos 24 = cos" - sin Wa cos" A= 4+ A coe2d i , A y sin (A+ B)=sin A coi tan A+ tanB 1+ tan Atan B tan (A + B) = sin A cos B = ¥4 [sin (A + B) + sin (4 - BY] cos A sin B= [sin (4 + B) - sin (4 - B)] cos A cos B = 4 [cos (4 + B) + cos (A - BY] sin A sin B= [cos (4 - B) - cos (4 + B)] sinx | 1 1 tanx = 5 sinx = 5 COSX = cosx cosec x secx Downloaded from hlayiso.com ajcose tun over Copyright reserved

(16030173) 5 APPLICATIONS OF INTEGRATION AREAS b b A, = foe ae J Oirydde 4=f. xdy ; A, =f. (% — x,)dy VOLUMES ~ OF AREA haf r thy iis AM Mass = density x volumé™) wee M= pv DEFINITION: J=m?? b GENERAL: 1=[” rdm= pf” Par Downloaded from hlayiso.com Copyright reserved T890(E)(A2)T

higher education & training Department: Higher Education and Training REPUBLIC OF SOUTH AFRICA MARKING GUIDELINE NATIONAL CERTIFICATE AUGUST EXAMINATION MATHEMATICS N5 2 AUGUST 2016 This marking guideline consists of 10 pages. Downloaded from hlayiso.com _,, lease turn over Copyright reserved

MARKING GUIDELINE 2 T890(E)(A2)T MATHEMATICS NS INSTRUCTIONS AND INFORMATION 1, Y equals 1 mark. 2. Half marks are not allocated, unless indicated otherwise. 3. Using the wrong formula is a principle error and NO marks will be allocated. 4, Candidates should show ALL formulae and intermediate steps and simplify where possible. 5. ALL final answers must be rounded off to THREE decimals unless indicated otherwise. 6. Questions may be answered in any order, but subsections of questions must be kept together. If subsections are separated, the candidate can be penalised with ONE mark. 7. Where something has been copied incorrectly from the question paper, and the standard of the question is still the same, the candidate will be penalised with ONE mark. If the copying error simplifies the question and makes it easier, the student forfeits the marks, 8. Questions must be answered in blue or black ink. Answers in PENCIL are not marked as it is regarded as rough work. Downloaded from hlayiso.com Copyright reserved Please turn over

T890(E)(A2)P MARKING GUIDELINE 3- MATHEMATICS N5 QUESTION 1 Ll Ld Iny <li v ay ti osx sin3x 1 Iny = lim tan 3x Iny=0 ¥ (3) 112 y=l¥ (1) : x? 120 Ne ree =lim v 2 =0 v @) [7] QUESTION 2 21 2-x IQ) = 3 flet+hy 222th _ 2-4-h , 3-2(x+h) 3—2x—2h . , 2-x-h 2-x_1 ‘(Od = lim( A" — LO) nae 2h 32x Fe) = Him 222 AMG = 20) B= 2x-2wYO—z) 1 ; am (3—2x—2h)(3—22) h h 1 \(x) = lim ————2____. = PO)=|ia Gy -tHG—2In hk 1 I'(x) = v (3—2x)" (4) Downloaded from hlayiso.com Copyright reserved Please tum over

MARKING GUIDELINE 4. T890(E)(A2)T MATHEMATICS N5 _ l-x 2.2 2.2.1 aa an x y YY a dy ~1(vtanx)— (2), (tanx)? sec’ x a (vianxy ” (4) 2.2.2 y =[In(cosec2x)]° & ¥ - cot 2x & = S[In(cosec2x)]* = OOS ee x 00k 2x cosec2x ” @) 2.2.3 ys Demsinsx Me v v . 5 ~ 1 2 y= dees acme Ma + 20S? 1-(Sx) (2) 2.3 y=(9-x°)™* w In y = (In x) In(9-x°) —3x? vw 1h 21 nox) +Inx. 3x ydx x (9-x’) . 3 4,2 & _ Om) ine. 3x v & (9-x') (4) 2.4 2.4.1 y = aresin x sin y=x ay cos y—=1 y ax dy 1 dx cosy wt v dx -sin? y w__s v dy Vi-x (2) Downloaded from hlayiso.com Copyright reserved Please turn over

MARKING GUIDELINE 5- MATHEMATICS N5 24.2 y = aresin(5.10*) dy _ 5.10% Inl0 v dx ft-(5.10")? Y 2.5 x? +2=y? cot3x v ¥ 2x =3y? D cbt3xty’.—Beosee?3x 2x+3y’ cosee?3x =3y" cote ¥ 2x+3y* cosec?3x _y 3y’ cot3x dx Copyright reserved Ce Downloaded from hlayiso.com T890(E)(A2)T Please turn over (2) (4) [25]

MARKING GUIDELINE -6- MATHEMATICS NS QUESTION 3 3.1 3.1.1 f@=xe —3x-9) 3.1.4 3.2 3.2.1 3.2.2 Copyright reserved S(@) =x -3x? -9x S'(x) =3x? -6x-9 f'"(®) =6x-6 6x-6=0 | x=l y=-ll v X}-1 0 1 2 3 Y|5 0 -11 -22 -27 v — y-intercept : 2 4 ¥ — shape a) ex - LOO) _ 0869 _ ying vy FCO) | -837 a=-0,1+0,104 = 0,004 T890(E)(A2)T v¥¥ (Any 2 correct=1 mark) All these units indicate rates of change“ of the volume, radius and height tespectively. “ a ra DY age Ky? th dt dt dt aVv ” A a 27(11)(0,5)(30) + #11)? (-1) cla = 656,593 Downloaded from hlayiso.com Please turn over 2) @) Q) G) Q) 6)

MARKING GUIDELINE -7- T890(E)(A2)T MATHEMATICS NS 33 D+h=6 D=6-h 2 wv V=mh yan3—")h ¥ 3 3 ¥ =9nh—3ah + 2 V'= 90-642 0=2(36—24h+3h7) 0=(4-6)(h-2) ¥ h=6;h=2 r=0;r=2 [21] QUESTION 4 41 4.1.1 j a 4,38 100-—x? (10-x) (10+x). 4x= A(10+x)+B(10-x) ¥ x=-10:-2=B -iIn-4— w x=10:A=2 (2) 4.1.2 3 ay S07 + 2 -2 aos v 4(l0-x) (0+x) =[-2InQ0-x)-2n0+x)] |, = (-9,022)-(—9,19) =0,168 » (3) Downloaded from hlayiso.com _,, lease tum over Copyright reserved

MARKING GUIDELINE 4.2 4.2.1 4.2.2 4.2.3 4.2.4 Copyright reserved -8- MATHEMATICS N5 2. 1 y=x? sin— x 1 u=—sde=—x' duw x Jez sin ux? du ¥ =~ fsin udu =cosut+ec I =cos—+cr x y =C0s" 5x 11 Jz + zoosl0xdx v 1” sinlOx ¥ =ix+ + 2 20 y =cosl2y.sin3x J + (sin 5x) sin 9x) v = 1 eoslsx + hs 9x 2 15 J+e 3 fox? Vv tL. =csin'xt+e Vv 5 Downloaded from hlayiso.com T890(E)(A2)T (4*3) 12) Please tum over

MARKING GUIDELINE -9- MATHEMATICS N5 43 fx” In2x dx F(x) = In 2x gia? , 1 xt x)=— =— f'@) x 80) = 14 14 in2x2—— [1 ge vy 14 4x 14 =In2x 2 — cae ‘14-414 =In2x 2 = 4¢ “14 196° ¥% QUESTION 5 5.1 5.1.1 Vea 2 x -8x=0 x=0;x? =8 ve P ; y =0;x = 42,828 one mark per x-value y=tl4l4 - 5.1.2 2,828 1 J * ary 0 2 =? 5 x 2908 v = 4 re =0)-@ y =l y 5.1.3 288 2 02 ¥ au J x a ry 4 32 x =x} — pes oy Be ygh = A{(1,508)-(0)]_ v =4,738 y Copyright reserved T890(E)(A2)T (4) [21] @G) 4 (4) Downloaded from hlayiso.com aise ium over

MARKING GUIDELINE -10- MATHEMATICS N5 5.3 QUESTION 6 6.1 e*dx— y dy =e"dx ¥ w dxfe* —e*) = y'dy v sv e+e* +c0=— 6.2 (i+ x)sin y.dy = cos y.dx v 1 i tan =— dy 1+x Inst y = Ind #%) +c Downloaded from hlayiso.com Copyright reserved T890(E)(A2)T GB) (4) [18] (4) (4) [8] TOTAL: 100

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higher education & training Department: Higher Education and Training REPUBLIC OF SOUTH AFRICA (ForoGiyadyr APRIL EXAMINATION CERTIFICATE (16030175) 4 April 2p 16 (X-Pap 09:00-12:00 This question paper cons! f 6 pages and 1 formula sheet of 5 pages. Downloaded from hlayiso.com Copyright reserved Please turn over

(16030175) 2- T970(E(A4)T DEPARTMENT OF HIGHER EDUCATION AND TRAINING REPUBLIC OF SOUTH AFRICA NATIONAL CERTIFICATE MATHEMATICS NS TIME: 3 HOURS MARKS: 100 INSTRUCTIONS AND INFORMATION 1. Answer ALL the questions. Read ALL the questions care: js question paper. Downloaded from hlayiso.com Copyright reserved Please turn over

(16030175) 3 QUESTION 1 1.1 Determine the following limits: 111 lim—* x30 xe* ~ 2x lim (sec x — tan x) xr 2 1.2 Determine whether f(x) = QUESTION 2 2.1 Determine th i first principles. 2.2 Determine —in & i is not required): 2.2.2 ~ 2Incosec4x 2.2.3 y =arcsec(2.4™) 2.3 Determine ke with the aid of logarithmic differentiation if: aresin(sin y) =(x* ) 2.4 Given: 3x‘ — xy =2** dj 24.1 Determine the slope = of the tangent at the point: (1;-5). 2.4.2 Hence, determine the equation of the tangent al this point. Copyright reserved Downloaded from hlayiso.com TOTO(E\(A4)T (2) 3) 2) 7] (4) (4) 6) 2) (4) @) Q) [24] Please turn over

(16030175) -4- T9T0(EMA4)T QUESTION 3 3.1 Given: f(x) = x(x’ -5)-4 3.1.1 Determine the co-ordinates of the turning points of f(x). @) 3.1.2 Verify, using a table, that the equation 0=x(x°-5)-4 has a root between the points x=2 and x=3. Use values on the table; O<x<4 (4) 3.1.3 Hence, make a neuit sketch of the graph of the function f(x). (2) 3.1.4 If the positive raokof svi estimated as 2,7, use Taylor's/Newton's method to determitie Qhetter approximation of this root. (4) 3.2 Two sides of alivctingle are lengthened at a sate vi'S cm/s wiitethe other two sides are being shotiened in)such a way that the figure remains a rectangle with a constant area of 50 ci’. 3.2.1 Calculate the rate of change of the perimeter of tie rectanjle when the length of an increasing side is 7 cm. (5) 3.2.2 Prove that when the rate of change of the perimeter is zero, the figure must be a square. > (2) 3.3 A particle moves in 4 )§iraight line accofting to the distance formula s(t) =V#G-3t-2). 3.3.1 Calculate the velocity of th2\particle after,2,7 seconds. 4) 3.3.2 Calculate the acceleration after 2 seqiuids. (3) [27] Copyrishtreseved DOWwNnloaded from hlayiso.com Pease tmover

(16030175) 5- T9TO(E\(A4)T QUESTION 4 44 Determine: fe +e)‘ (e? -e")dx Q) 4.2 Determine j ydkx in each of the following cases: 42.1 y= ~~ 1+ cosx (3) 4.2.2 G3) 4.2.3 (2) 4.2.4 (3) 43 Determine = 9) 44 Determine: J 4) [22] Downloaded from hlayiso.com Copyright reserved Please turn over

(16030175) -6- ‘1970(EW AAT QUESTION 5 5.1 Given: The curves f(x)=VI16—x° and g(x)=4-x No es about the x-axis. 5.2 Prove that f-se" 0 QUESTION 6 6.1 Solve the differential equation: dy _ tan’ y dx cosec?x 2 6.2 Determine the particular solution of the differential equation = aes + ; x40 for which y=2 and y'=—3 when x=1. TOTAL: Downloaded from hlayiso.com Copyright reserved GB) 4) (4) [11 (4) (5) [9] 100

T970(E)(A4)T (16030175) DIFFERENTIATION __ f@ f (a) r=ate PRODUCT RULE y= ulx).v@) A dx ax dx =u +vu' QUOTIENT RULE - u(x) v(x) yt _ & Dds ds dx y _ vu! —av' y CHAIN RULE ¥ =flu@)) ay _ dy du dx du dx Downloaded from hlayiso.com Copyright reserved Please turn over

(16030175) -1- TS70(E)(A4)T FORMULA SHEET Any other applicable formulas may also be used. TRIGONOMETRY sin’ x + cos’x =1 I+ tan’ x =sec’ x 1+ cot’ x =cosec’ x sin 2A = 2 sin A.cos A cos 2A = cos’ A - sin? A 2 tan A tan 2A = Z 1-tan* A sin? A at = De 2 cos” azi + —cos2 2 2 sin (A + B) =sin A.cos B + sin cos (A+ B) = cos A.cos B F sin A.sin tan (4+ B)= tan A+ tan B 1-tan A.tan B sin A.cos B = % [sin(4 + B) + sin (4 - B)} cos A.sin B = 4 [sin(4 +.B) —sin(A - BY] cos A.cos B= [cos(A + B) + cos(A - B)] sin A.sin B= % [cos(4 - B) - cos(4 + B)] sinx . 1 1 tanx = jsinx = 5 cosx = cosx cosec x secx BINOMIAL THEOREM (xt A)" =x" 4+ nx" th + MOD ray? +... Downloaded from hlayiso.com Copyright reserved Please turn over

(16030175) -3- T T970(E)(A4)T d fe) af [fea atl ax’ nax™! = n+1 a 0 ax te sin x cos x tan x cot x sec x cosec x sin’ x cot” x 1+x’ sec” x —— _ ax? -1 cosec™ x > _ xa)x -I Downloaded from hlayiso.com Copyright reserved Please turn over

(16030175) 4 T97O(E)(A4)T (axtbext+d) axtb cxt+d fe) _ 4, 8B Cc Zz (xt+a)’ (x+a) (x+y “Ga+ap "Gray APPLICATIONS OF INTEGRATION AREAS 4, =f) yar; A, =) (v2 -¥,)4e A, =" xdy; A, =. (a —a)ey Copyright reserved Down loaded from hlayiso.com Please turn over

(16030175) -5- T970(E)(A4)T VOLUMES V,=af yids s¥,= Af! (vy -») dx Vv, =n xdy iV, =a (x, x, )ay SECOND MOMENT OF AREA I= { dA Ty = { dd MOMENTS OF INERTIA Mass = density x volume M=pV DEFINITION: J= GENERAL: Downloaded from hlayiso.com Copyright reserved

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higher education & training Department: Higher Education and Training REPUBLIC OF SOUTH AFRICA MARKING GUIDELINE | N NATIONAL CERTIFICATE APRIL EXAMINATION MATHEMATICS N5 4 APRIL 2016 This marking guideline consists of 10 pages. Downloaded from hlayiso.com Copyright reserved Please tum over

MARKING GUIDELINE -2- T970(E)(A4)T MATHEMATICS N5 INSTRUCTIONS AND INFORMATION 1. Half marks are not allocated, unless indicated otherwise. 2. Where formula is required, using the wrong formula is a principle error and NO marks are allocated. 3. Students should show ALL formulae and intermediate steps and simplify where possible. 4. ALL final answers must be rounded off to THREE decimal places (unless indicated otherwise). 3. Questions may be answered in any order, but subsections of questions must be kept together. If subsections are separated, the student can be penalised by ONE mark. 6. Where a student copied wrong from the question paper, and the standard of the question is still the same, the student will be penalised by ONE mark. If the copying error simplifies the question and makes it easier, the student forfeits the marks. 7. Questions must be answered in blue or black ink. Answers in PENCIL will not be marked as it is regarded as rough work. Copyright reseved Downloaded from hlayiso.com Pissetumover

MARKING GUIDELINE 3 T970(E)(A4)T MATHEMATICS NS QUESTION 1 Ll 1.1.1 e lim. x90 xe" — 2x ll 8 ol- Q) 1.1.2 lim (sec x — tan x) BOD 2 = tim =8inx Ov ee cosx 0 _ COS x = lim — v sof —sinx =0 ¥ @) 1.2 7 _ 3-27 _ -54_ F(@)is continuous aix=-3 Vv (2) 7) QUESTION 2 24 - WENK: [imeem tig stim SM ay io 430 ft lim cos(x+h)-—cosx 40 .__ cos x.cosh—sin x.sinh—cos x =lim h0 h v sh-l . — sinh —sin a v . co! =limcos x. oat) =cos x(0)—sin x(1) =-sinx y (4) 2.2 2.2.1 y =cos* (x? —4)+cos(x? —4)* = 4cos* (x? —4).—sin(x? —4).2x — sin(x? -—4)4.4(x? -4)°.2x ¢ D (x 74) Ae ) @) Downloaded from hlayiso.com Copyright reserved Please turn over

T970(E)(A4)T MARKING GUIDELINE 4. MATHEMATICS N5 v v v v 2.2.2 _ - 6tan¥1—x.sec” V1-x. | aIncosecdx —3tan? y1— x speed cot Ax _ 2v1-x cosec4x (2incosec4xYy 6) 2.2.3 y=arcsec(2.4**) 3x dy _ 6.47 In4 Vv de 2.4/2.4")? -1 Q) 2.3 yas Iny=e*.Inx 1® eine yas x ® oye (ef Inx+) ax x (4) 2.4 2.4.1 d 12x° -1y-x =3.2* In2 x —ly-x hk v 12x7-y-3.2"In2_ dy : x dx at(,-5) ay _ 12 -—(-5)-3.81n2 dx 1 » = 0,364 v a 8) 2.4.2 y=mete -—5=0,364(l) +c v -5,364=¢ v equation : y = 0,364x — 5,364 (2) [24| Copyightreseveas DOWnloaded from hlayiso.com pris uno

MARKING GUIDELINE “5 T970(E)(A4)T MATHEMATICS N5 QUESTION 3 3.1 3.1.1 f(a=x? -5x-4 3x? -5=0 ¥ x=1,291 or x»=-1,291 mark for both x-values yp =-8,303 y= 0,303 » mark for both y-values (3) 3.1.2 Use values on the table: OS x <4 xX | 0 1 2 3 4 we'v marks for any 4 correct values y | 4 8 6 8 40 y-value changes from negative to positive indicating a root” (4) 3.1.3 40 8 # shape of the graph y-intercept < ZN > -2 / -1 9 1 2/3 4 -6) 8 ’ (2) 3.1.4 w we a=2,7—0,129= 2,571 » (4) Downloaded from hlayiso.com Copyright reserved Please turn over

MARKING GUIDELINE -6- MATHEMATICS N5 32 32 Azay 50=14y + 50x7 =y P=2x+100x"! ¥ dP 9% 100 dx ada ea ld o =-0522 v 32.2 &=0 only if <.% v Therefore: x=y Meaning the figure is a square 3.3 3.3.1 i 3 5 s(t) =32? -31? -2? 1 3 yop 27 vw wi 2 2 V= 23,987 “we 3.3.2 a=3- 4B we 4t? 4¢? a= -7,159 v Downloaded from hlayiso.com Please turn over Copyright reserved T970(E)(A4)T (5) (2) 4 (3) 27

MARKING GUIDELINE PV MATHEMATICS N5 QUESTION 4 4.1 eo cass fete Le 5 ww 4.2 4.2.1 sinx ys 7 vV1+cosx u=l+cosx du -sinx ¥ sinx du Saas v dx= a = fu? du i —2u? +e —2Vl+cosxt+e ” 4.2.2 y=x.sec’ x S(@=x g'(x) =sec’ x f'@)=l a(x) =tanx x.tanx— f tan xde x.tan x—In(secx)+e we a 4.2.3 y =cos6x.cos2x Joos(8x) +cos(4x)dx sin8x sin4x = 3 + ” ¥ 4.2.4 2 Copyright reservea 4 4DOWwnloaded from hlayiso.com T9T0(E)(A4)T 2 (3) (3) (2) (3) Please turn over

MARKING GUIDELINE -8- 43 44 Copyright reserved MATHEMATICS NS xe-2 OP ae x-l _A B (Oirt+2 x x42 xP -1=A(x+2)+ Bx v 9 x=2:—=B 3 v x=0:-L=4 a 2 1 9 x (x+2) al iny-2ines 2 +e 2 2 we “ ~, 25 = |x+5+——d&k fx x—5 ad x? ay t5xt25lnG@—5) +e a w T970(E)(A4)T @) 4) [22] Downloaded from hlayiso.com =?! amove

MARKING GUIDELINE QUESTION 5 5.1 5.1.1 §.1.2 5.2 : en =-5 a —-s 5.1... =-Lylo ¥ e =0-1) ” Ss _75 =— Downloaded from hlayiso.com Copyright reserved -9- MATHEMATICS NS [vies -(4-xde ov 0 2 =(ssin“ 3 +5 16—x? ~4r4]} =( 4,566-0) = 4,566 ¥ 4 n[Vi6—x? - (4-2) ax v 0 ; 3 8 x = n{16x-7 16x + - = = A[(21,333) - (0)] =67,02 ¥ 4” T970CE)(A4)T (3) 4) 4) a) Please turn over

MARKING GUIDELINE -10- MATHEMATICS NS QUESTION 6 6.1 a _ tan? y dx cosec*x 1 dy= ! dx tan? y Cosecx cot? ydy = sin? xdx ca 2 liv (cos ec“ y—1)dy = G08 Hae “1 sin 2x Toby Ysa 4 Cw 6.2 w 13 2 7 ede ee “ 6 4 -x* 3x3 am? ++ 24 12 w ax +ext+d y= wv -~3=3,725+¢ -6,725=¢ 2=-4,946 +d 6946=d v Downloaded from hlayiso.com Copyright reserved T970(E)(A4)T 4) (5) 19] TOTAL: 100

higher education & training Department: Higher Education and Training REPUBLIC OF SOUTH AFRICA T960(E)(N25)T NOVEMBER EXAMINATION NATIONAL CERTIFICATE MATHEMATICS N5 (16030175) 25 November 2015 (X-Paper) 09:00-12:00 Scientific calculators may be used. This question paper consists of 6 pages and 1 formula sheet of 5 pages. Downloaded from hlayiso.com Copyright reserved Please turn over

(16030175) 2- T960(E)(N25)T DEPARTMENT OF HIGHER EDUCATION AND TRAINING REPUBLIC OF SOUTH AFRICA NATIONAL CERTIFICATE MATHEMATICS N5 TIME: 3 HOURS MARKS: 100 INSTRUCTIONS AND INFORMATION 1, 2. bed Answer ALL the questions. Read ALL the questions carefully. Number the answers according io the numbering system used in this question paper. Show ALL intermediate steps and simplify where possible. ALL final answers must be rounded off to THREE decimal places Questions may be answered in any order, but subsections of questions must be kept together. Use ONLY blue or black ink. Write neatly and legibly Copyright reserved Downloaded from hlayiso.com pics: in over

(16030175) 3 T960(E)(N25)T QUESTION 1 1.1 Determine: 2 LL iny = lim x90 tanx 1.1.2 The value of y 1.2 Determine lim Insecx x90 xe * QUESTION 2 2.1 Given: —- = S08 ecx Determine the derivative oy making use of the derivatives of sinx and cos x 2.2 Determine o in each of the following cases (simplification is NOT required): se 2.2.1 y <0 2.2.2 y=e™ In(ax’ — bx +c) 2.2.3 y =aresin(10**) 23 Determine ”. with the aid of logarithmic differentiation if y= (cos3x)™" 2.4 Given: y = arecos x 2.4.1 Make a neat, rough sketch of the graph of y for the range [0; 7] dy 2.4.2 Derive a formula to determine 7, if: y= arecos x 2.5 Given the implicit function: sin(5x—2y)=4 Copyright reserved Calculate the derivative 2 Ie Downloaded from hlayiso.com 3) ¢3) GB) [7] 3) (4) (3) 2) (4) G) (2) (4) [25] Please tum over

4 T960(E)(N25)T (16030175) QUESTION 3 3.1 Given: f(x) =2x° —4x? —8x+8 3.1.1 Determine the coordinates of the turning points of f(x) 3.1.2 Draw up a table of values for x and /fx) with x ranging from x = -2 to x= 4. 3.1.3 Draw a neat graph of f(x) between the values determined in QUESTION 3.1.2 3.1.4 An estimated root of the graph is 0,7. Use Taylor's/N ewton's method to determine a better approximation of this root. 3.2 An open cardboard box with a square base is required wo hold 163 m. Determine the dimensions that require the least amount of material, and neglect the thickness of the material and waste in construction. Veh / A=4xh+x? | A Po x 3.3 An electric current is varying through a resistor of 1692. The rate of change of the current is 0,6 A/s when the current is 12A. Find the rate of change of power developed with the current at 12A. HINT: P=I?R Copyright reserved Downloaded from hlayiso.com Q) (2) (2) (4) (3) (4) [19] Please turn over

(16030175) -5- T960(E)(N25)T QUESTION 4 4 3 by 2 4.1 Determine f 3x0 43x" Sx tanh, by resolving the integral into partial fractions. x°+x-2 (5) 42 Determine j ydx in each of the following cases: . 3 421 _ sin® x secx (3) x 422 ya 2x? +3 (3) arctanx 42.3 yoo — l+x (2) 42.4 yo 3x-2 (4) 43 Given f /@)8'@) = S@).8(2) - ff C.g(dde Determine. f2xe*dx (4) [24] QUESTION 5 1 5.1 The sketch shows the area bounded by the curve x= ae x=y’ and theline y= ry y ay a v < ‘ Downloaded from hlayiso.com Copyright reserved Please tum over

(16030175) -6- T960(E)(N25)T 5.1.1 Prove that the point of intersection of the two curves equals (2,52; 1,59). (3) 5.1.2 Calculate the area of the bounded region. (4) 5.1.3 Calculate the volume generated when the area in QUESTION 5.1.1 rotates about the Y-axis. (4) é 1 5.2 Calculate the value of : |——« 1vi-x (4) 5.3 Determine the second moment of area of a rectangular lamina with coordinates (a;5) about the axis parallel to one side of the lamina. The distance from the representative strip to the axis is ‘x’. The lamina's length is 'a’ and breadth is 'b’. | - (a;b) X » xX_y, “y (4) {19] QUESTION 6 6.1 6.1.1 Solve the differential equation Wz e™ dx (3) 6.1.2 Determine the equation of the curve through the point (1;-2). (2) 6.2 Determine the general solution of the differential equation (1 — y*)dx = —2xydy (4) 19} TOTAL: 100 Downloaded from hlayiso.com Copyright reserved

(16030175) -1- T96O(E)(N25)T FORMULA SHEET Any other applicable formulas may also be used. TRIGONOMETRY sin’ x +cos’x =1 1 + tan’ x = sec’ x 1 + cot’ x = cosec’ x sin 2A = 2 sin A.cos A cos 2A = coy’ A-sin’ A 2 tan A tan 2A = — 72, t—tan* A sin? 421A cos24 2 3 cos? A= i, | cos2d 2 2 sin (A + B) = sin A.cos B+ sin B cos A cos (A+ B) = cos A.cos B F sin Asin B tan A+tanB 1-tan A.tan B tan (4 + B)= sin 4.cos B= % [sin(A + B) + sin (4 - B)] cos 4.sin B= % [sin(4 + B)— sin(A - B)] cos A.cos B= % [cos(A + B) + cos(A - BY] sin A.sin B= % [cos(A - B) - cos(4 + B)] sinx , 1 l tanx = 3 sinx = 3 cosx = cos.x cosec x secx BINOMIAL THEOREM (x thy ax" tnx h nln 7 1) vn? ownloaded from hlayiso.com Copyright reserved Please turn over

(16030175) 2 DIFFERENTIATION f(a) f@ r=ate PRODUCT RULE y =ula).v@) ak dx dk =uv't+vu' QUOTIENT RULE = UO) v(x) CHAIN RULE v=flue)) dy _ ay du dx du dx Downloaded from hlayiso.com Copyright reserved T960(E)(N25)T Please turn over

T960(E)(N25)T (16030175) f@) £ Fx [¢@)ar dx 1 n+1 ax" nax” ——+e ntl a 0 arte & é ete ao a a lna te Ina log. x = x 1 1 OB x Ba xlna — sin x cos X -COSX re cos x -sin x sinx +¢ tan x sec” ¥ In (sec x) +€ cot x -cosec” x In (sin x) + secx sec x.tan In [seo x + tan x] + cosec x -cosec x.cot x In [cosee x - cot x] + ¢ ca 1 sin” x a cara _ yl-x# ot 21 cos" x _ 1x? 4 1 tan” x z _ 1l+x -l cot” x - _ d+x* sec’ x ! xy eal a a ~i cosec” x = Copyright reserved Downloaded from hlayiso.com Please turn over

T96O(E)(N25)T (16030175) 4. d fe) gf [7a | _ sin” 6 te ax a i 1 af Xx pan _ zi [2+ : ‘() —see”|—J+e xfer -a - a a eax? @ on! (2)+28 -x +e a ! 1 {x-a)\ it 1 atx ax ~~ puss] [ INTEGRATION J/@)e'@)= SO) e@) - J HO) ee jE a= in sles c (LrCol F ear = £0" 5. ntti fe) AB (ax+b)extd) axt+b oxt+d f(x) __ A B c Zz (x+a)" 7. @+a)- (x+a)’ * (x+a) Feet («+ a)’ APPLICATIONS OF INTEGRATION AREAS A.=f yar; A= [ . — yee A, = [ xdy; A, = [ (x, — x, Downloaded from hlayiso.com Copyright reserved Please turn cver

(16030175) -5- VOLUMES v, sap ytd 5V,=2f'(y.? — 9?) ax V, =a x*dy; vy, = af x," = x" )dy SECOND MOMENT OF AREA t=! Pda st, =f? Pa MOMENTS OF INERTIA Mass = density x volume M=pV DEFINITION: J=mr Ab GENERAL: J ai dm = rl, rdV Downloaded from hlayiso.com Copyright reserved T960(E)(N25)T

higher education & training clarks Depariment: ey Higher Education and Training ae” REPUBLIC OF SOUTH AFRICA — MARKING GUIDELINE an NATIONAL CERTIFICATE NOVEMBER EXAMINATION MATHEMATICS N5 25 NOVEMBER 2015 This marking guideline consists of 9 pages. Downloaded from hlayiso.com Copyright reserved Please turn over

MARKING GUIDELINE -2- T960(E)(N25)T MATHEMATICS N5 INSTRUCTIONS AND INFORMATION 1. equals lmark. 2. Half marks are not allocated, unless indicated otherwise. 3. Where a formula is required, using the wrong formula is a principle error and NO marks are allocated. 4. Students should show ALL formulae and intermediate steps, and simplify where possible. 5. ALL final answers must be rounded off to THREE decimal places (unless indicated otherwise). 6. Questions may be auswered in any order, but subsections of quesiions must be kepl together. If subsections are separated, the student can be penalised by ONE mark. 7. Where a studeitt copied incorrectly from the question paper and the standard of the question is still the same, the student will be penalised by ONE mark. If the copying error simplifies the question and makes it easier, the student forfeits the marks. 8. Questions must be answered in blue or black ink. Answers in PENCIL are not marked as it is regarded as rough work. Downloaded from hlayiso.com Copyright reserved Please turn over

MARKING GUIDELINE -3- T96O(E)(N25)T MATHEMATICS NS QUESTION 1 11 1.1.1 e lim— x90 xe* — 2x u 8 ole AON 2) lim (sec x — tan x) xs 2 =lim Insinx (0. a x95 COSK 0 _ ~COSX = lim ¥ xot —Sinx 2 0 a =0 @) L2 ay _ (3) -27_ -54_ f= Ty 6 f(x) is continuous at x =3 “ ow (2) [7] QUESTION 2 24 _ F pint: tim 20S! 9 stim SIMA Ly td i0 ft ._ cos(x+h)-cosx tim ———+__+>——— hoo h cos x.cosh— sin x.sinh—cos x =lm Oa 4290 h : cosh-l . — sinh y =limcos x. — sin x.— 00 h h = cos .x(0)-sin x(1) =~sinx 4) 2.2 2.2.1 y = cos* (x? ~4)+cos(x? —4)* = 4cos*(x* — 4).-sin(a°—4).2x - sin(x? —4)*.4(x? -4)'.2 ( 5 ( ) LD ( ) & y 2x Downloaded from hlayiso.com Please turn over ¥ (4) Copyright reserved

MARKING GUIDELINE 4 T960(E)(N25)T MATHEMATICS NS 2.2.2 _3tan? VIl-x * 2Incosec4x ee ae “ 6tan-¥1—x.sec’ V1—x. =) a Incos ec4x —3tan? v1- x. Boosecde. cot 4x 2Vl-x cosec4x (2Incos ec4x)? w (5) 2.2.3 y =aresec(2.4*") as wy 6A ns dx 2.4" (2.4) -1 (2) 2.3 arcsin(sin y) = x") pax” Inyee™.Inx ! 1D _ ys nxt we ydx x dy . ew Sax (lnx+— = x ( x z ) (4) 2.4 2.4.1 ; 12x° -ly-x% 23.2" In2 yo dx 7 12x°-y-3.2¥n2_ dy x dx at -5) dy _12~-(-5)-3.8In2 dx 1 ® 0364 ¥ ax G3) 2.4.2 yrmxtc —5=0,364(1) +e —5,364=¢ ” equation: y = 0,364x -5,364 Q) [24] Downloaded from hlayiso.com Copyright reserved Please turn over

MARKING GUIDELINE -5- MATHEMATICS NS QUESTION 3 3.1 3.1.1 f()ax? -5x-4 ¥ 3x? -5=0 x=1,291 or y =-8,303 y = 0,303 “mark for both y values T960(E)(N25)T x =—1,291 » mark for both x values 4) wv’ marks for any 4 correct values 312 X,0 1 2 3 y | \4 8 6 8 y-value changes from negative to positive indicating a root ” 40 3.13 40 8 < . 6 8 NT 3.1.4 ” _ fQN 2183 _ F'Q27) 1687, 0,129 a=2,7-0,129=2,571 y (3) (4) shape of the graph y-intercept 4 (2) (4) Downloaded from hlayiso.com Copyright reserved Please turn over

MARKING GUIDELINE 3.2 3.2.1 3.2.2 3.3 3.3.1 3.3.2 Copyright reserved -6- MATHEMATICS N5 x A=xy 50=xy + 50x71 =y P=2x+100x'! ¥ aP _ dx 100 de dt dt x* dt 00 aP = 33)-4 —3 dt: T v aP __o192 dt dy —=0 only if -——=-> y ae Therefore: x=y Meaning the figure is a square “ 1 305 (0) =302 -322 -2 3 ye 9858 we 2/2 2 2 v= .23,987 vw -3 9 185 we a=—y-— yt 402 49? a= -7,159 ¥ Downloaded from hlayiso.com T960(E)(N25)T 6) (2) @ (3) [27] Please turn over

MARKING GUIDELINE QUESTION 4 4.1 -J= MATHEMATICS N5 (er +e") te OH 5 4.2 4.2.1 sinx v1+cosx —2Vl+cosx+e~ 4.2.2 y=xsec? x f(x)ax J'(@)=1 g'(x) = sec” x g(x) =tanx Y x.tan x f tan xd x. tan x — In( “ 4.2.3 y=cos6x.c secx)+c ¥ OS 2x f cos(8x) + cos(4x)dx _ sin8x 8 4.2.4 2) 2 an”! =—= tan v3 Downldéaded from hlayiso.com Copyright reserved sin4x + 4 x +e V3 T96O(E)(N25)T (2) GB) (3) (3) Please turn over

T960(E\(N25)T MARKING GUIDELINE -8- MATHEMATICS N5 43 a) x +2x v-l 4, B (x)(xt+2) x x42 xi -1=A(x+2)+ Be + 9 =-2:-=B x 2 vw x=0:-t24 “ 2 1 9 more x (+2) =—dinx-2 ine +2) +e 2 2 a v 6) 4.4 25 = |x+5+——a&k if x—5 we = +5x+25In(—5) +e “ (4) [22] QUESTION 5 5.1 5.1.1 [lie -a-ne y =[Ssin '4 +2 Vi6—3? ~ 4x42; v = (4,566-0) =4,566 5.1.2 ‘ n{Vi6—x” -(4-x)'de ¥ CO) 3 2 3 = nfl6x ~*~ 16x 4 8 Eyre 3 2 3 (G3) = a[(21,333) — (0)] =67,02 4) Downloaded from hlayiso.com Copyright reserved Please turn cver

-9- MATHEMATICS NS MARKING GUIDELINE 5.2 eo =-S—]; ¥* -s 5.1... = sigue v 5 =-(@-l) v s => ¥ S QUESTION 6 6.1 ay _ tan” y dx cosee*x 1 1 ee dx tan’ y y coseex cot? ydy = sin? xdx “ v (cosec’y —1)dy = 500s 2x)dx “1 -coty-yerx- 2 sin 2x cr 6.2 for which y =2 and y'=-3 when x=1 ad ~1x3 2 dy _-ix 43% +iaxt+e # ax 6 -x* 3x mm? y= + +— +ex4d 24 12 ~3=3,725+¢ -6725=¢ vy 2=-4946 +d 6,946=d _ Downloaded from hlayiso.com Copyright reserved Ww T960(E)(N25)T (4) 11) 6) (4) 19] TOTAL: 100

y, higher education mie & training wi, Department: 4 i; § — Higher Education and Training as aoe REPUBLIC OF SOUTH AFRICA T970(EMA7)T AUGUST EXAMINATION NATIONAL CERTIFICATE MATHEMATICS N5 (16030175) 7 August 2015 (Y-Paper) 13:00-16:00 This question paper consists of 7 pages and 1 formula sheet of 5 pages. Downloaded from hlayiso.com Please turn over Copyright reserved

(16030175) -2- T970(E)(A7)T DEPARTMENT OF HIGHER EDUCATION AND TRAINING REPUBLIC OF SOUTH AFRICA NATIONAL CERTIFICATE MATHEMATICS N5 TIME: 3 HOURS MARKS: 100 INSTRUCTIONS AND INFORMATION 1. . 2 Answer ALL the questions. Read ALL the questions carefully. Number the answers according to the numbering system used in this question paper. Show ALL intermediate steps and simplify where possible. ALL final answers must be rounded off to THREE decimal places, Questions may be answered in any order, but subsections of questions must be kept together. Questions must be answered in blue or black ink. Write neatly and legibly. Copyright reseved Downloaded from hlayiso.com Please umove

(16030175) -3- T9TOE MATE QUESTION 1 1.1 Determine the following: 2x Ll Iny=lim2*- nye tan x (3) 1.1.2 The value of y (1) 1.2 Determine lim Inseo x x30 yer * (3) 7] QUESTION 2 2.1 Given: y= cos ecx. Determine the derivative a making use of the derivatives ofsinx andcos x. GB) 2.2 Determine 2 in each of the following cases: {simplification is NOT required) xla 221 yo\gma 7 2.2.2 y=e™ In(ax? —bx +e) 3) 2.2.3 y= aresin(L0"*) 2) 2.3 Determine = with the aid of logarithmic differentiation if} y = (cos3x)"?*" (4) 2.4 Given: y = arceos x ZAt Make a neat rough sketch of the graph of y for the range [0; 7] @) dy 2.4.2 Derive a formula to determine GY if ¥ = 4reoos x (2) 25 Given the implicit function sin(5x — 2y) =4 Calculate the derivative & dx (4) [25] Downloaded from hlayiso.com Please turn over Copyright reserved

(16030175) -4- T970(EAT)T QUESTION 3 3.1 Given: f(x) =2x* — 4x? -8x +8 3.1.1 Determine the coordinates of the turning points of f(x). (2) 3.1.2 Draw up a table of values for x and f(x) with x ranging from x= -2 to x=4. (2) 3.1.3 Draw a neat graph of f(x) between these values. (2) 3.1.4 An estimated root of the graph is 0,7. Use Taylor's/Newton method to determine a better approximation of this root. (4 3.2 An open cardboard box with a square base is required to hold 108 m’. Find the dimensions that require the least amount of material Neglect the thickness of the material and waste in construction. V=x*h A=4xh+x? h x x (5) 3.3 An electric current is varying through a resistor of 16€, The rate of change of the current is 0,6 A/s when the current is 12A. Find the rate of change of power developed with the current 12A. Hint: P=1°R (4) {19] Copyighteseves DOWnloaded from hlayiso.cOmM yje6¢ 2 over

(16030175) -5- T970(E)(A7)T QUESTION 4 4 3_ ey? _ 41 Determine j 3x0 t3x0 — Sx tan1 by resolving the integral into partial fractions. xt 4x2 6) 42 Determine [vax in each of the following cases: 4.2.1 _ sin? x secx (3) x 4.2.2 y “2x? 43 (3) 4.23 ee 1+ Q) 2 424 __* o 3x2 (4) 43° Given: [ 7.2) = FO - fF @Oa@ax Determine: f 2x.e* dx (4) [21} Downloaded from hlayiso.com Copyright reserved Please turn over

(16030175) -6- T970(E)(A7)T QUESTION 5 1 3.1 The sketch shows the area bounded by the curve x= 4 ;x=y’ and theline y *5 y ay 1 ba F « »X v ~ 5.1.1 Prove that the point of intersection of the two curves equal (2,52; 1,59). (3) 5.1.2 Calculate the area of the bounded region. (4) 5.1.3 Calculate the volume generated when the area in QUESTION 5.1.1 rotates about the Y-axis. (4) 6. 5.2 Calculate the value of : — 1 vl-x7 (4) 5.3 Determine the second moment of area of a rectangular lamina with coordinate (a;sbjabout the axis parallel to one side of the lamina. The distance from the representative strip to the axis is "x". The lamina’s length and breadth is "a" and "b" tespectively. q (a;5) x 4) [19] A copmtight reserved Downloaded from hlayiso.com__ a... un ove

(16030175) 1 T970(E)(A7)T QUESTION 6 ‘ . ees 6.1 6.1.1 Solve the differential equation m =e x 6.1.2 Determine the equation of the curve through the point (1;,-2). 6.2 Determine the general solution of the differential equation (l- y? dx = -2xydy TOTAL: Downloaded from hlayiso.com Copyright reserved (3) 2) (4) 19] 100

(16030175) -1- MATHEMATICS N5 FORMULA SHEET Any applicable formula may also be used. TRIGONOMETRY sin’ x + cos’ x = 1 1 + tan? x= sec? x 1 + cot” x = cosec” x sin 24 =2 sin A cos A cos 24 = cos"A - sin? tan 24= 2 4n4 L—tan‘ A sin? A =% -% cos 24 cos’ d =%4 + ’cos 24 sin (A+ B)=sin A cos B+ sin Bcos A cos (4 + B)= cos A cos B F sin A sin B an (4S) 15 tan dtan B sin A cos B= % [sin (4 + B) + sin (4 - B)] cos A sin B= [sin (4 + B) - sin (4 - B)] cos 4 cos B='4 [cos (A + B) + cos (4 - B)] sin 4 sin B= [cos (A - B)- cos (A + B)] sinx , 1 1 tanx = ;sinx = COS x = : cosx cosec x secx Copyright reserved Downloaded from hlayiso.com T970(E(A7)T Please turn over

(16030175) 2 BINOMIAL THEOREM n(n-1) (xt hy =x" + nx" 1h aa Wt... DIFFERENTIATION _f@ f(@) r=ate PRODUCT RULE y=ulx). vx) dy dy du S=u.—+7,— dx dx dx Suey teen QUOTIENT RULE - #Q) v(x) du dy & ates y dx. dx CHAIN RULE y = fu) wb _y du dx du dz Downloaded from hlayiso.com Copyright reserved T970E)(ATIT Please turn over

(16030175) -3- T970(E)(A7)T d fe) = fe) Up (ayae atl ax" nax") +e n+1 a axte é x & + c x x x a @ Ina e Ina 1 logex = — x 1 1 lo; Bax xlna 2 sin x cos x -cosxte cos x ~sin x sinxt+e tan x sec” x In (seox) +e cot x -cosec” x In (sinx) +¢ sec x sec x‘tan x In [seex + tanx] + cosec x -cosec x'cotx In (coseex - cotx) +c oot 1 sin” x = 1— x" cos” x 7 ie _ yi-x 4 1 tan” x 5 I+x a 1 cot x Z I+x sec? x 1 ax? -1 _ 4 1 cosce” x Copyright reserved Downloaded from hlayiso.com siciss sum ove

(16030175) TITOEMADT d Se) ke nic) Sf(dx 1 al (2) _ sin” |—|+e Ja? _ x a 1 1 a(x —tan’|—jt+e : ‘() _ —sec|—|+e¢ xx? —a? a a 2 a? — x? © sin' (2 +2 P-P 4c = 2 aj} 2 t L x-a — In}: +e x’ — a? _ 2u.\xt+a f 1 at+x —In +e @—x _ 2a \a-x INTEGRATION $9) g(x) dx = £6) g(a) - J £'@) a) de £2 F(x) dx=lIn f(a) te att Leon fe) d= LET +e f@) __A B = + (axt+bjext+d) axt+b cxt+d f@) A B c (tay Gta) (tay (wta)® - (x +a)" Downloaded from hlayiso.com Copyright reserved Please turn over

(16030175) -5- APPLICATIONS OF INTEGRATION AREAS 6 b A, = J yds Ay = { (4 - ya)ax b b 4, =f xdys4,=f (-m)ay VOLUMES boo b Vy, =a] ¥ dx Ve =f (2 — y3 Jax b 2 b (2 2a Van], Vay Vy =n), ( — x3 Jdy SECOND MOMENT OF AREA Iy=[) Pda: 1y =? Pad MOMENTS OF INERTIA Mass = density volume M= pv DEFINITION: Z=m ?* A b GENERAL: 1 = [’ rdm = of. r'dv T970(E)(A7)E Downloaded from hlayiso.com Copyright reserved

higher education & training Department: Higher Education and Training REPUBLIC OF SOUTH AFRICA MARKING GUIDELINE NATIONAL CERTIFICATE AUSUST EXAMINATION MATHEMATICS N5 7 AUGUST 2015 This marking guideline consists of 10 pages. CM RAL ati. . - TS Mork4h— CM) S wetticacke AK Copyright reserved Downloaded eS filayiso.com Please turn over

MARKING GUIDELINE 2- T970E)AT)T MATHEMATICS NS INSTRUCTIONS AND INFORMATION 1. Vv equals 1 mark. 2, Half marks are not allocated, unless indicated otherwise. 3, Where formula is required, using the wrong formula is a principle error and NO marks are allocated, 4. Students should show ALL formula, intermediate steps and simplify where possible. 5. ALL final answers must be rounded off to THREE decimal places (unless indicated otherwise) 6. Questions may be answered in any order, but subsections of questions must be kept together. If subsections are separated, the student can be penalised by ONE mark. 7. Where a student copied wrong from the question paper, and the standard of the question is still the same, the student will be penalised by ONE mark. If the copying error simplifies the question and makes it easier, the student forfeits the marks. 8. Questions must be answered in blue or black ink. Answers in PENCIL are not marked as it is regarded as rough work. Copyrightreseves Downloaded from hlayiso.com Pease tum over

MARKING GUIDELINE -3. MATHEMATICS N5 QUESTION 1 11 11.1 Iny = lim x0 tan x : 2 44 nye ee sec! x 112 -y=e?=7389% 1.2 sec x. tan x ; secx ¥ tim ie ee v = lim —"* ~ x90 pF _ yg __0 2-0 =0 v QUESTION 2 21 y = cosecy = — sin x dy _0-l.cosx dx sintx ~* 8 oe dx = sinx.sinx dy 2.2 2.2.1 Copyright reserved —=-—cotx.cosecx ¥ a& xha y= 1Q Sse vv a xIna Ina.cos ax+xInaasinax = == 10cosax "19. costax Downloaded from hlayiso.com T97TO(E}{AT)T Please turn over G) () 3) [7] (3) (4)

MARKING GUIDELINE -4- | MATHEMATICS NS 222 ye e™ In(ax? ~dxt+e) v v Bw be™ In(ax? — bx +2) + of 2b de ax’-bxte y 2.2.3 y= arcsing 0") v 4x gy 4.10 InfO |, ax h — (10)? 2.3 Iny=~2x41 . Incos3x v v vw 3 LD hi ayeni F-2 incos3x-+Vrana1 2s ydx 2 Cos3x dy ~Incos3x == ~3y~2x+1.tan3x dx WS v-2x41 1” 24 24.1 Y vn eed i 2.4.2 COs y=x v —sin y =1 ax & - ; BUT:sin y=./1-cos’ y ad -siny dy _ 1 a ’ Downloaded from hlayiso.com Copyright reserved TITOCLEMKAT)T @B) 2) (4) (3) (2) Please turn over

MARKING GUIDELINE -5- MATHEMATICS NS 2.5 dy cos(Sx—Zy).[5 eae =0 v dy ¥ Scos(Sx-—2y)- Ca -2y)=0 he YL ~ Scos(Sx—2y) dx —2cos(Sx-2y) v dy 5 T970(EMATT a2” 1) [25] QUESTION 3 3. 31d 6x? ~8x-8 =0 3x? -4x-4=0 (x+2)(x-2)=0 2 Fan gine2, y=10,962;y =-8 % [One mark per coordinate x AND y} (2) 3.1.2 x | -2]-1/ 04 1 2 3 4 v foo [8 [10] 8 f 2] -8 [2 [40] |v Q) 3.1.3 K Shape “ 8 v¥ of V4 i A 2) Copyright reserved Downloaded from hlayiso.com Please turn over

MARKING GUIDELINE -6- MATHEMATICS NS 3.1.4 v _ __ 1126 = 0106" —10,66 y r=0,7 + 0,106 = 0,806 “ 3.2 x A= dap) A=432x1 4x2” dA _—432 az x ~432+2x7 =0 x = 216 x=6 4 +2x¥ h=3 33 .P a a2 RY dt dt dP vv — =2(12).0,6(16 z (12).0,6(15) aP =230,4 % at Copyright reserved TI70(EMATIT ) ©) 4) [19] Downloaded from hlayiso.com piease tum over

T970(E)(A7)T MARKING GUIDELINE -7- MATHEMATICS NS QUESTION 4 3x4 +3x> —5x? -i “fs ic 4 x? Dow 1 A B = 3x0 + Ft ——— et ee Be = + («+ 2)(x-1) @&+2)G@-1) (*+2) (x-D v = i = [3x?-+ 3 dy fore =1:B=4 and forn=-2:Aars arm a (x-D 3 3 vv Tonle lin -pe vv G) Give the candidates two marks if long division is not used for partial fractions 4.2 4.2.1 sin? x " seex if sin’ x.cos xdv u=sinx = fu’.cosx, au dy ay cosx cos x sa4 _sintx ov GB) 4.2.2 _ x © Ox? 43 alee 243% a = Fin(ax? +3)te yy OR w=2x?+3 = u 4x = In(ax? +3) @) . Copyrigntreseved Downloaded from hlayiso.com sese wm over

MARKING GUIDELINE -8- MATHEMATICS NS 4.2.3 pica acre e” 2 f 5 du(l+x) u =arotan x l+x =e" te de=(14+x")du * = ent tg 4.2.4 4 i xh 9 y _using longdivision 30 9 3x-2 2. a 24 ingr-2) +0 vvv 6 9 27 43 22 2x = 2x [2d vv 2 2 e* 2 dx vo =xe~ — +¢ TOTOCEKATIT 2) @) 4 [24] Copyrghtreseved Downloaded from hlayiso.com °2""™*

“MARKING GUIDELINE ~9- MATHEMATICS NS QUESTION 5 5.1 5.1.1 4_2 —=y y 4zy? 159=y ¥ = 1,59 x= 2,52 5.1.2 “i 4 =-y dy ¥ os ¥ 3 =[4iny-“ he y = (0,515) ~(~2,814) % #3329” 5.1.3 439 a [(6y7)-O* ay ¥ 6,5 ~16 yas ¥ = a[— 78 = n[-12,095 -(-32,006)] “ = 62,552" 5.2 : $4 |e 1 1-x? =[aresin x]¢ v = (0,551)—(1,57) VY =-1,02 5.3 a vv 2 J bx'de fone for limits; one for integral} b 3 “kv ot 3 Downloaded from hlayiso.com Copyright reserved TITOE(AT)T 2) A) 4) 4) 4) {19] Please turn over

QUESTION 6 dl @ 61 611 dy ogy e” [evay = | erax o y apr te (3) 612 ~-e"CDeetgc -e?elec c =-10,107 ¥ —ey = e* - 10,107 ¥ 2) 6.2 1 y dye -2x I-y? ay 1 1 vv Zins Find-y') +e (4) 1] TOTAL: 100 Downloaded from hlayiso.com Copyright reserved Please turn over

(16030175) -2- T970(E)(A1)T DEPARTMENT OF HIGHER EDUCATION AND TRAINING REPUBLIC OF SOUTH AFRICA NATIONAL CERTIFICATE MATHEMATICS N5 TIME: 3 HOURS MARKS: 100 INSTRUCTIONS AND INFORMATION 1. Answer ALL the questions. 2. Read ALL the questions carefully. 3. Number the answers correctly according to the numbering system used in this question paper. 4A. Show ALL intermediate steps and simplify where possible. 5. ALL final answers must be rounded off to THREE decimal places (unless indicated otherwise). 6. Questions may be answered in any order, but subsections of questions must be kept together. 7. Questions must be answered in blue or black ink. 8. Write neatly and legibly. Copyright reserved Please turn over Downloaded from hlayiso.com

Qe y higher education sie & training afo, | Department: A) Higher Education and Training Oy REPUBLIC OF SOUTH AFRICA cooooacoco T970(EA1)T APRIL EXAMINATION NATIONAL CERTIFICATE MATHEMATICS N5 (16030175) 1 April 2015 (Y-Paper) 13:00-16:00 Scientific calculators may be used. This question paper consists of 6 pages and a formula sheet of 5 pages. Copyright reserved Please turn over Downloaded from hlayiso.com

(16030175) “3. QUESTION 1 Lt 112 Prove that: lim 282° — 9 x30 Ke = Show ALL calculations. 1.2 x Consider the function f(x) = . f v2x—3 Determine whether f(x) is continuous for x =1 QUESTION 2 2.1 . toes 1-x we Determine the derivative of (x)= eae from first principles. +x 2 2 Determine 2 in each of the following cases: aaa yeeton® + tant (hy x 2.2.2 y =cos ec2x.5™ 2.2.3 _ Ssin 2x In(cos x) 2.3 Given: y=arcsinx 2.3.1 Make a neat rough sketch of the graph of y for the range [0;7] 2.3.2 : . 3 Determine s if y = aresin(3x) am Determine % if xsin x + ysin y = 76,29 Copyright reserved Determine the following limits: 11.1 : 3x lim——; xe x — +2 Downloaded from hlayiso.com T970(EMA1)T Please turn over @) GB) 1) 8] (4) 3) Q) 3) 2) Q) G)

(16030175) 4s T970(E)(AL)T 25 Determine 2 with the aid of logarithmic differentiation if: [23] QUESTION 3 3.1 Given: f(x) =x? —5x-8 3.1.1 Determine the coordinates of the turning points of f(x) - Show ALL calculations. (3) 3.1.2 Draw up a table with values 0 < x <3for the equation x° —-5x—8 =0. Use the table and turning points to make a neat sketch of the function of F(x). G) 3.1.3 Let x =3,7 be an estimated value of the rout. Use Taylor/Newton’s method ONCE to determine a better approximation of THIS root to FOUR decimal numbers. (4) 32 Prove that a rectangular field with a perimeter of 100 m will have the greatest area when it is a square. (5) 3.3 The distance of a particle is given by f(x) =10tanx. f(x) gives the distance in metres after x seconds 33.1 7 . + Determine the velocity of the particle after 7 seconds, Q) 3.3.2 Determine the acceleration of the particle at this moment. (3) 3.3.3 Determine the distance travelled by the particle in 4 seconds. (2) [22] QUESTION 4 4.1 Determine [ ydx if 4.1.1 y= (x? -2x)? @B) 4.1.2 y=x.secx.tanx (4) Copyright reserved Please turn over Downloaded from hlayiso.com

(16030175) -6- TITO(EYADT 3.2 Determine the moment of inertia of a circular lamina of radius r and mass m about an axis through its centre and perpendicular to the plane of the lamina. ry v (4) [16] QUESTION 6 6.1 : : 2. ax Solve the differential equation sin 1 +cos2yp=0 B) 6.2 a . @ y s 2 . Determine the particular solution of ars = x° +2x° — x, given that dy —=Ly=2.x=1 hk Ly d’y Change the subject to ax? ; before solving (5) (8) 100 Copyright reserved Downloaded from hlayiso.com

(16030175) -5- T9I7OCENAI)T 4.1.3 _ i > Ty 6x (3) 4.1.4. _ +inx) =" @) 45. __tanx v= Incos x (2) 4.2 Given: 5s ~11 x? —2x-8 4.2.1 Using partial fractions, determine the numerical values of 4 and B oxcll A,B x? —2x-8 (a+b) (ex+d) (3) 4.2.2 : 5x-11 Hence, determine | ————— J x? -2x-8 63) [21] QUESTION 5 3.1 The sketch shows the area bounded by the curves xy =6 and y+x=5 v \ 5.1.4 Show that the points of intersection of the graphs are (2;3) and (3;2). (3) 5.1.2 Calculate the area of the shaded region. (4) 5.1.3 If the shaded area revolves about the Y-axis, calculate the volume of the- solid generated. G) Copyright reserved Please turn over Downloaded from hlayiso.com

(16030175) 1 MATHEMATICS N5/ WISKUNDE NS FORMULA SHEET / FORMULEBLAD Any other applicable formula may also be used. Enige ander toepaslike formule kan ook gebruik word. Trigonometry / Trigonometrie sin? x+cos* x =1 1+ tan? x =sec? x 1+cot? x =cosec?x sin 2.4 =2sin A.cos.4 cos2.4=cos? A—sin? A 2tan A tan 24 =————-— 1—tan* 4 sin? 4-4 cos24 2 2 cos” A =F 450082 sin(A + B) = sin A.cos B t sin B.cos A cos(A + B) = cos 4.cos BF sin A.sin B tan. dt tan BE tan(4+ B) = 94 1-tan A.tan B sin A.cosB = WA [sin(4+.B) +sin(4—B)] cosAsinB = Molsinc +B)-sin(4- 3B) cos. d.cosB = ¥, [cos(4 +B) + cos(4~ B)] sin A.sin B= Vo[eost — B)—cos(A+ B)] cos x cosecx secx Copyright reserved Downloaded from hlayiso.com TI7O(E)(A1)T Please turn over

(16030175) -2- BINOMIAL THEOREM / BINOMIAAL STELLING (xh) =x" +nx"th ee +... DIFFERENTIATION / DIFFERENSIASIE --£@ f@ roate PRODUCT RULE / PRODUKREEL y= u(x)v(x) =uv'tyvu' QUOTIENT RULE / KWOSIENTREEL aC) v(x) du di & ak a& v CHAIN RULE / KETTINGREEL v= fu) yy _ dy du de dud Copyright reserved Downloaded from hlayiso.com T970(E)(A1)T Please turn over

(16030175) 3- T970(E)(AL)T d f) rade) | Rice a al La ax RnaX a 0 n+] ; x axt+e [4 ée ; e+e a @ Jna x a om +c nda log, x 1 x log, x a _ xIna si cOosx nx ~sinx —cOosx+e — secu sinx+¢ tan x 2 In(seex)+¢ cotx —cosec*x . In(sin x) +c secx secx, tan. cosettcat Infsee x + tan x] +e Cos eck In[cosecx —cotx} +e sin? x 1 I-x? — cos’ x = 1-x? ~ tan x a 1+x? = -1 cot" x [4x2 — sec x xv x? -1 cosec'x xx? —1 Copyright reserved Please turn over Downloaded from hlayiso.com

(16030175) “4 T970(E)\AL)T fe) < ft) (Reo a _ sin'()+e 1 1 44g = —tan™(—}+¢ a +x? a a 1 Leo (j4¢ xvx?—a’ _ a a ax? © sin'(2) +2 Va? -¥ +e _ 2 a 2 1 1 x-a xa? — 2a Mao 5 ! 7 sin At *) be a’ ~-x _ 2a -a-x INTEGRATION / INTEGRASIE [f@2'@) = f@.8@)-[f'@.g@ax LR) 4 _ Vig Baa ete #yyrt [cor fear -29— +6 n+i f@) A, B (ai+bex+d) axt+b cxid fe) 4, Bc, Z (t+a)” (eta) (eta)? (x+0)° (x+a)" Copyright reserved Please turn over Downloaded from hlayiso.com

(16030175) “5 T970(E)(A1)T Applications of integration / Toepassings van integrasie AREAS b bs A, = [yds 4, = [02-year b 4,=[,x05 4,=[@—ade VOLUMES V, = aff yd; V, =A) Oy - 9 bac bs bs V, =a] dy, V, =a] Ge a ay SECOND MOMENT OF AREA / TWEEDE AREAMOMENT I, = ['r7aa; 1, = [rad MOMENTS OF INERTIA / TRAAGHEIDSMOMENTE Mass = density « volume / Massa = digtheid x volume M=pV Definition / Definisie: J = mr? General * Algemeen: J= [Pam re pf rar la a Copyright reserved Downloaded from hlayiso.com

0006000 A higher « education & training Department: Higher Education and Training REPUBLIC OF SOUTH AFRICA _ MARKING & SUIDECINE | | i NATIONAL ‘CERTIFICATE APRIL/JU NE/AUGUST/NOV. EMBER EXAMINATION MATHEMATICS NS (EQURTH PAPER) BACKUP 2014 —T | _t ty This niarking guideline consists of 11 pages. IM: T. Molcleln <GAe- CM: RMatiener Libattlar CM: S& Tow ar Apress Copyright reserved ( [ 3) § / 20/ Ss ‘Please turn over Downloaded from hlayiso.com

MARKING GUIDELINE -2e MATHEMATICS NS QUESTION 1 Ll Determine the following limits: LL . 3x lim———-> sae x" — x42 vl im =2% OW xre2x—] LL2 Prove (show ALL calculations) that: 12 Consider the function f(x) thie Fh ‘ oN 2k : , Insecx lim ——— x0 yer LHS secx.tanx J lime a0 1e°% — xe =0 we . tan x = lim 290 @°* — xe x Determine whether 7 eS} is egntidtious for x=1 SQ= v Copyright reserved undefined — f(x} discontinuous for x=1 cont Downloaded from hlayiso.com T..(E)BU)T (2) 3) @) [8] Please turn over

MARKING GUIDELINE 3- T..AEXBU)T MATHEMATICS NS QUESTION 2 2.1 : oe ; 1-x ee Determine the derivative of f(x) = aa from first principles x l-(xt+h) 1-x,1 = limf—-——— - —-].— Vv fO)= eh Bee + _ -d-x~A3+x)-U-x)G+x+h), 1 =] _ FO)= ae (tx +hY3 +2) hy —4h 1 PO lina hasn hv 4 -4 f@s= aay Vi " “ 22 Determine 2 in each of the following cases, 2.2.1 ye giteasv=) +tan?(hy x ye cos x + tan’ ie Ye J ay | iV | 21-1 ss = + 3 tans —. a _ sin nr faa see > 3) 2.2.2 a oe (2) 2.2.3 _ Ssin2x In(cosx) v a Vv o 10cos2x.In(cos x) -—5Ssin 2. dx [Incosx?? y (3) Copyright reserved Please tum over Downloaded from hlayiso.com

MARKING GUIDELINE -4- TeufEXBU)T MATHEMATICS N5 rw te Given: y= arcsin x 2.3.1 Make a neat rough sketch of the graph of y for the range {0; 7] 1 mark for the shape 1 mark for any intercept Q) 2.3.2 _. ay, Det = if etermine 74 y =aresin(3x) (2) 24 Determine » if xsinx+tysiny = 76,29; 4 Vo gv tv “ae sinx + xcosx+—) sin y +.y COs ye 10 sin yr yoos y) =, sinx~ xtosx dy _—sing—xtoex dx = siny+ ¥cos3 a rs 6) 25 Determine 2 with the aid of logarithmic differentiation if: core yex Iny =cot3x.Inx “i ~v lay —3eosec’3x.Inx $cot3x- v yax x ay 2 = = y[-3cosec’3x.Inx+ ye x. cot 3%) J @) [25} Copyright reserved Please turn over Downloaded from hlayiso.com

MARKING GUIDELINE ~5- T...E)(BU)T MATHEMATICS NS QUESTION 3 3.1 Given: f(x) =x’ -Sx~-8 3.1.1 Determine the coordinates of the turning points of f(x) 3x’-5=0 Vv x= 4/1667 x=1291 or x=-1291 ¥ y=-12303 y=-3,697 Vv ‘ QB) 31.2 Draw up a table with values 0 < x <3 for the equation x —5x—8=0, Use the table and turning points to make a néatsketch of the function of f(x). x {0 y_ |-8 1 Mark for shape 1 Mark for any shown turning point , @) 3.1.3 Let ye 37, Eo an estimated value of the root. Use Taylor/Newton’s method ONCE to determine a better approximation of this root to FOUR decimal numbers. ; Vv 3D =53.)=8 __ 24153 _ 0.6696 3(3,7)° -5 36,07 ¥ root :3,7 — 0,696 = 3,0304 V (4) Copyright reserved Please turn over Downloaded from hlayiso.com

MARKING GUIDELINE 7 T..ACEMBUYT MATHEMATICS NS QUESTION 4 4.1 Determine f yde if 44d y=? —2xy fx = 4x3 + 4x? dy aia x' +40 B) 4.1.2 y= x.secx.tanx f=2x g =secxtanxy f=l g =Seex — Vv xseox~ [sec ke Vv ) 4.1.3 a ee V7 ce 6) Copyright reserved Please turn over Downloaded from hlayiso.com

MARKING GUIDELINE 6- T...E)QBU)T MATHEMATICS N5 3.2 Prove that a rectangular field with a perimeter of 100 m will have the greatest area, when it is a square P=2x+2y 100=2x+2y 50-x=y v A= xy A=x(50-x) A=50x-x* b 50-2x =0V jh x=25 WV os a y=50-25=25 ¥ therefore: rectan gle is a square % Sinadt (5) 3.3 The distance of a particle is given by f(x) . f(x) gives the distance in meters after x seconds a 3.3.1 ; Determine the, velocity of the particle after = 4 seconds vel0sectx ¥ v=l0see(T she vs20mts ev (2) 3.3.2 Determitéthe acceleration of the particle at this moment a=10.2secx,sec x tan x v a= 20sec’ x.tanx a= 20sec? (~).tan(= Guten a=40m/s? (3) 3.3.3 Determine the distance travelled by the particle in 4 seconds. v s=l0ian4=11,578 ¥ Q) [22] Copyright reserved Please turn over Downloaded from hlayiso.com

MARKING GUIDELINE -8- T..(E)(BUIT MATHEMATICS NS 4 N . @) 4.1.5. f tanx a Incosx derivative: * 2 ~tanx J cose -1 j —tanx x Incosx —In(Incosx)+e Vv % 42 Given 424 $x-11= aGrt2)+B0-4) jorx = 2: 2 =B V forx =4 a A a oo Q) 4.2.2 Hence, determine { 5x-11 x? -2x-8 3 7 _f.2 2 a If the values of A and B are “l@-4 + (+2) incorrect, mark with incorrect 3 7 J values = ine 4) +7 ine +2) +e 6) f24] Copyright reserved Please turn over Downloaded from hlayiso.com

MARKING GUIDELINE ~9- T...0E(BU)T MATHEMATICS NS QUESTION 5 3.1 The sketch shows the area bounded by the curves xy=6 and y+x=5 ‘ aN 4 ¥ s=5=5 x x?-$x+6=0 , (x -3)(x-2)=0 * q x=3 or xE'P ay on ya2 or ya3o ws 3) ‘e rs 5.1.2 Caliah the aréa of the shaded region =[Sy-7 lyk y =(15— 4,5 -61n3)~(10—2-61n2) =0,067 ¥ {4) Copyright reserved Please turn over Downloaded from hlayiso.com

MARKING GUIDELINE -10- T..AEXBU)T MATHEMATICS NS 5.1.3 If the shaded area revolves about the Y-axis, calculate the volume of the solid generated. ; 6 z[G-*-Cyay v 3 ¥ 3 =2[25-l0y+y?~36y%dy V 2 3 =al(25y~5y” cae E V = nl(95 ~45+9+12)—(50~20+5 +18] VB, =0,3337 ¥ , @) =1.047 ‘ :. §.2 Determine the moment of inertia of a circular lampina’@f radius r and mass m about an axis through its centre and perpendicular to the plane of the lamina (4) (16) Copyright reserved Please turn over Downloaded from hlayiso.com

MARKING GUIDELINE -H- T..{EMBU)T MATHEMATICS N35 QUESTION 6 6 Solve the differential equation sin xe +cos2y=0 sinxdx =—cos2ydy V_sin2y 4 —cosx =—-——+¢ (3) 6.2 : : : d’y 5 2 + Determine the particular solution of sar =x’ +2x° — x, given that dy ® oy ya22=1 “a ay Ms Change the subject to ds” before solving. ; 2 a ‘ 1=02+1-l+e O8sc ¥ 1 $121 . 2=—+2-= +d i 303 2 rs Y 1333=d ¥v x x? x? aq! Yay tpg POG +1933 v 6) [8] TOTAL: 100 Copyright reserved Downloaded from hlayiso.com

Qe y higher education le & training afaols Department: 4 The Higher Education and Training Se REPUBLIC OF SOUTH AFRICA cooogooo00 4 S ‘ + T960(E)(N17)T NOVEMBER EXAMINATION NATIONAL CERTIFICATE MATHEMATICS N5 (16030175) 17‘Noyember 2014 (Y-Paper) 13:00-16:00 Scientific calculators may be used. This question paper consists of 7 pages and a formula sheet of 5 pages. Copyright reserved Please turn over Downloaded from hlayiso.com

(16030175) -2- T960(E)ONI7)T DEPARTMENT OF HIGHER EDUCATION AND TRAINING REPUBLIC OF SOUTH AFRICA NATIONAL CERTIFICATE MATHEMATICS N5 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 numbering system used. in this question paper. 4. Show ALL intermediate steps and simplify where possible. 5. ALL final answers must be rounded off to THREE decimal places (unless indicated otherwise) 6. Questions may be answered in any)order, but subsections of questions must be kept together. 7. Use only blue or black ink, 8. Write neatly and legibly. Copyright reserved Please turn over Downloaded from hlayiso.com

(16030175) QUESTION | LL Determine the following limits: LL lim sin. 290 In tan x L12 lim sin(2x —12) 96 x G Fay 1.2 Determine the value(s) of x for which f(x) is continuous if f(x) = ai 7 QUESTION 2 2.1 Determine the derivative of f(x) = 43x from first principles. 2.2 Determine 2 in each of the following cases; (simplification of answers et required) 2.2.1 y=sin(cos/2@ 1) 2.2.2 y=e™ Infax? +br—c) ox a » sec3x 2.3 Determine a with the aid of logarithmic differentiation ift ¥ = (cos5x)* Copyright reserved Downloaded from hlayiso.com T96OCE}NIT)T Please turn over @) 2) @) 13] 4) @G) @ Q) (4)

(16030175) -4- T960(E)(NI7)T 2.4 Given is the sketch of y = aresin x, x €[-1;7] aA A B/ -z 2.4.1 Give the co-ordinates of A and B 42 If y = arcsin(tan 4x), give the derivative 2 of ¥ d 2.5 Determine the equation of the tangent to the curve xy? +3x? = x?y +12 at the point (231) QUESTION 3 3.1 Given: f(x) =2x°=4x*42x41 3.1.1 Determine the coordinates of the turning points of f(x) Show ALL the calculations 3.1.2 Draw up a table of values of x and f(x) for values of x from -1 to 2 and makea neat sketch of this part of the curve. 3.1.3 If the value of the negative root is estimated as -0,3 use Taylor/Newton's method ONCE to determine a better approximation of THIS root correct to FOUR decimal numbers. 3.2 A metal ball is covered by a layer of ice. The diameter of the ball is 100 mm. When the layer of ice is 5 mm thick, it melts at a rate of 10 000 mm*/s. At what rate is the thickness of the ice decreasing? vesne 3 Copyright reserved Please turn over Downloaded from hlayiso.com (2) Q) (5) [24] (3) (4) 4) 6)

(16030175) 5 T960(E\NLT 3.3 A rectangular wall is to be erected on a river bank. Building cost budgeted for is R5 600-00. The building cost of the two parallel walls are R10-00 /m and the cost of the wall parallel to the river bank is R15-00 / m (see the sketch). river x x ¥ Calculate the dimensions of the wall to enclose a maximum. area for the money available. (5) [21] QUESTION 4 41 Determine f ydx in each of the followitig cases 41.1 y = tan 3x.sec 3x (3) 41.2 y= (8-5x¥ (2) 4.1.3 a Fae 3) 4.14 v=sin(e yp (4) 415 y= x? In8x (4) 42 7 7 7 4x—2 : Use partial fractions to determine f ; ad x’ -4x Show ALL calculations. (5) (21] QUESTION 5 tx? —3x 5.1 Evaluate the definite integral [ dk yal ) Copyright reserved Please turn over Downloaded from hlayiso.com

(16030175) -6- T960CE)(NI7)T 5.2 The sketch shows the area bounded by the curves y=3x° and i =x, intersecting at (0;0) , (-2:-8) and (2:8) - > b 2 $.2.1 Calculate the area of the shaded region. 4 5.2.2 If the shaded area revolves about the X-axis, calculate the volume of the solid generated, 6) 5.3 Determine the second momentiof area of a uniform rectangular lamina of area 'ab' about an axis which bisects thédamina. Make use of the information on the given sketch. a a, 4) ‘ (4) [17] Copyright reserved Please turn over Downloaded from hlayiso.com

(16030175) -7- T960(E)(NI7)T QUESTION 6 d’y 6.1 Determine the general solution of: mo xr+ert4 1) 6.2 Determine the particular solution of (1+ x)cos’ ydy = sin? yd, given that y(1) 7 (5) 9] TOTAL: 100 Copyright reserved Please turn over Downloaded from hlayiso.com

(16030175) Rk. -1- FORMULA SHEET Any applicable formula may also be used. Enige toepaslike formule kan ook gebruik word. Trigonometry sin* x+cos? x=1 1+ tan’ x = sec? x 1+cot? x =cosec?x sin2.A = 2sin A.cos A cos24 =cos? A-sin? 4 tan2A _—— 1-tan‘ A sin? A= Ere | 2 2 cos’ A= rey | 2 2 sin(A + B) = sin A.cos B+ sin B.cos.4 cos(A + B) = cos 4.cos BF sin Asin B tan.d + tanB tan(4 + B) =F nA tan B sin A.cosE = Vy{sin(4 + B)+sin(A—B)] cos A.sin B = 4 [sin(4 +.B) ~sin(A— B)] cosid.cosB = ¥, [cos(4 + B) +cos(4— BY] sin Asin B= V5 [cos(4- B) -cos(4+ BY] sinx.. tan x = ssinx = 3cosx = cosx cosecx secx Copyright reserved Downloaded from hlayiso.com TIGOCE)(N17)T Please turn over

(16030175) - BINOMIAL THEOREM (eth) =x" + nxn MAD 7 D rays vee DIFFERENTIATION iO) F'(@) F=at+e PRODUCT RULE y¥ = u(x).v(x) wb ®t a = Me +y. hk =uy'tvu' QUOTIENT RULE = 4G) v(x) CHAIN RULE v= fu) Bh _h da ad du dk Copyright reserved Downloaded from hlayiso.com T96O(E\(N17)T Please turn over

(16030175) -10- T96O(E)(NI7)T -3- d re) — fe) | fooee dk rd ax” nL +c nox n+l a 0 ax+c e e* 2 +0 a a’ .ina +e Ina log, x 1 x _ log, x I xing sinx HOSS, —cosx+c cosx = a sinxt+c tanx = * In(secx) +¢ 2 . cotx —cosee’x In(sin x) +e secx ox. tan. In[secx + tanx]+c¢ Cosecx —cosecx.cot x Infcosecx —cotx} +¢ sin’ x a cos? x tan“ x cot x sec! x cosec 'x Copyright reserved Please turn over Downloaded from hlayiso.com

(16030175) “lle T960(E)\(N17)T “4. d f@) 510 J Hosi sin*(Sj +e a’ -x? 7 a 1 1 pat (“the a +x? a Cs 1 _ J se01(%) +¢ xvx? -a* a a 2 ax? © sin (4 Va? — x? +e “wy 2 a 2 1 1, x-@ 2 zn +e * 7a — 2a & + 2 7 : 7 I (cs = = a7x P— 2a -a-x INTEGRATION Jf@e'@=fa.8@)-| PO.s@ax [Fea-inger+e fU@orP ea -20™ 6 n+l f@ __A B (ax+bYex+d) axt+b cxt+d f@)___A B Cc 4 (xtay’ (+a) (x+a)? “Gia? fete Copyright reserved Please turn over Downloaded from hlayiso.com

(16030175) 19- 5 Applications of integration AREAS 4, = de; 4, =f, — yee A, =f xdy; A, =[@ — dy VOLUMES _ fh yt. _ 2 2 V,=2[ yds, V, =a (7 ~ 9) b ‘b Vy, =a x dy; Vv, =ai) (x,? — x, dy SECOND MOMENT OF AREA o2 2 1,=[r dA: Le[r dA MOMENTS OF INERTIA. Mass = density x volume e M=pV Definition I = mr? General 7 = [dm = pilav Copyright reserved Downloaded from hlayiso.com T960(EKN17)T

i six yY higher education == SF & eraining no Sg fee, |) DOspartment. OO Higher Education and Training SS REPUBLIC OF SOUTH AFRICA gooce000 Sa MARKING GUIDELINE NATIONAL CERTIFICATE NOVEMBER EXAMINATION MATHEMATICS NS 17 NOVEMBER 2014 This marking guideline consists of 8 pages. Copyright reserved Please turn over Downloaded from hlayiso.com

MARKING GUIDELINE 2 T960(ENKNI7)T MATHEMATICS NS QUESTION 1 1.1 11.1 li Insin x x0 In tan x cosx = lim-Sin 90 sec” x tan x = lim cos? x =1 (4) 1.1.2 i - lim sin(2x —12) me x-6 Jim 2c0s(2x -12) 296 1 =2 2) 1.2 x-1 f= (x-1)(x+1D =0 continuous : x € R;x # +1 2) [8] QUESTION 2 2.1 1 . V3(0+h)? 34x SF @) = lim 0 h 1 a 3 1 3x2 43, 2h- 13. 2h? V3x2 =lim 2 h30 h a 3 ned 43, 2 ht.) ~ lim —2. 8 0 h _ 3 2vx (4) Copyright reserved Please turn over Downloaded from hlayiso.com

MARKING GUIDELINE 3. T96O(EXNI7)T MATHEMATICS N35 2.2 2.2.1 yr sin(cos (2a? —) D_ cos(cos 208 —1).-sin (20-F)L00-2)? - 6x? @) a 2.2.2 y=e™ In(ax? + bx —e) & _ be™ In(ax? +bx—c)+ em 2d ad ax” +bx—c (2) 2.2.3 10> y= sec3x dy _ 9.10°*.1n10.se03x — 10°*.3sec3x.tan 3x dx (sec3x)? (2) 2.3 y =(cos5x)" Iny = x.Incos5x ly = Lincos5x+x,—> Si 5® cos5x & = y[Incos5x —5xtan 5x] e co) 2.4 2.4.1 A(z) B(-1-2) (2) 2.4.2 ay _ 4sec? 4x de (tan 4x)? Q) 2.5 3 2 dy 2d y +3xy et OX PV TX de ® oxy" -x") = 2xp- y? -6x dy _ 2xy -y? -6x dk 3xy?-x? Ue2Md)-1412 “3(-2)0)- (2)? = 10 ye a1, te 10 14 =i9t? -O0,4=c Equation : y = xT, -0,4 10 6) [24] Copyright reserved Please turn over Downloaded from hlayiso.com

MARKING GUIDELINE -4- T960(E)(N17)T MATHEMATICS N5 QUESTION 3 3.1 3.1.1 6x? -8x+2=0 3x? -4x41=0 (3x—-1)(x-1) =0 x=0,333 or x=] y=1296 or y=l @) @) 3.1.3 _0.3)3 — 7 = e= _ 20,3)" — 4(-0,3)* +.2(-0,3) +1 _ 0,854 = 01729 6(-0,3)? — 8(—-0,3) +2 4,94 root : —0,3 + 0,1729 = -0,1271 (4) 3.2 Ve 4 nr 3 DY nag & NB: Radius =50+5= 55mm dt dt 2dr 10000 = 42(55)° — at dr 0,263 = — , dat G) Copyright reserved Please turn over Downloaded from hlayiso.com

MARKING GUIDELINE -5- MATHEMATICS N5 2(10x) +15y = 5600 5600 20x 15 15 3.3 A = 373,333x -1,333x" Derivative :373,333 — 2,666x = 0 x =140,035m y = 186,62 m QUESTION 4 41 41.1 y= J/tan3x.sec? 3x u =tan3x _ du 3sec” 3x [vu-sec? 3x = 3sec* 3x 3 a a2 == (tan3x)? +c 4.12 y=(8-5x)3 f@-5x)° ax 4 4 = B-5x)" 5... 8-59)" 4-5 20 4,13 =. 4+9x (et 9 G2) 11, ix oom re 3 3 alia? 46 6 2 Copyright reserved Downloaded from hlayiso.com T960(E)(NI7)T (5) [21} @ @) 8) Please turn over

MARKING GUIDELINE -6- 4.2 MATHEMATICS N5 4.14 y =sin(e™)e” 2x u=e _ du _ 26" . Me fsinue™ <= t a =—-cos(e*) +e 2 (e"*) 4.1.5 y =x? In8x f=in8x g =x? = Pe f=— gry 3 3 an Je?ingxdx = In 8x tra 3 3 ~ nsx = + 3 nex 9 c 4x-2 oat 4x-2 A B 2 =—t x?-dx x (x-4) 4x—2 = A(x—4) + B(x) 1 =0:-=4 Sorx 2 7 =4:-=B Sforx : i 7 2, 2 x (x-4 al inxs Zinge—4)+e 2 2 Copyright reserved Downloaded from hlayiso.com T96OO(EMNITIE A) i) Please turn over

MARKING GUIDELINE -7- T960(ENI7)T MATHEMATICS N5S QUESTION § 5.1 aye (2 3X oe 5 xt x-4 x+1)x? -3x x tx -4x Ax-4 4 - remainder x? 3 = — 40+ Aine + OE = (32-32 +41n9) —(12,5—20+41n6) = 9,122 @) [(2e—3e?e 5,2 $.2.1 i 12x? 3x", es 4 To =(24—12)—(0) =12u? 4) 22 ° V= al [a2x)? -@x°)? fee V = af (44x? - 9x8 Je 144x° 9x’, Vent - am; 7 Io V =a[219,429 -0] V =689,355u° (5) Copyright reserved Please turn over Downloaded from hlayiso.com

MARKING GUIDELINE -8- MATHEMATICS NS 5.3 4 2 2] x? bd o 34 xf = 20,1 3 a = 2-0 A ] _a’b 12 QUESTION 6 6.1 2 Caw set ss 3 Bo® et paxse a& 3 = et 42x tered av) 6.2 cos*y, sin’ y l+x cot? ydy=— ae l+x —coty~y=Inl+x)+e a Tn —cot——— =In(l+f+e a4 (1+) 0,215=In2+e -0,478 =e equation : —cot y — y = In(1 + x) -— 0,478 Copyright reserved Downloaded from hlayiso.com T960(E)(N17)T 4) [17] (4) G) [9] TOTAL: 100

Department: Higher Education and Training REPUBLIC OF SOUTH AFRICA Tl 040(E)(A9)T APRIL EXAMINATION NATIONAL CERTIFICATE MATHEMATICS N5 (16030175) 9 April 2013 (X-Paper) 09:00~12:00 Calculators miay be used This question Paper consists of 6 Pages anda 5-page formula Sheet. Downloaded from hlayiso.com Copyright reserved

DHET “2 TLO40(E(A9)T DEPARTMENT OF HIGHER EDUCATION AND TRAINING REPUBLIC OF SOUTH AFRICA NATIONAL CERTIFICATE MATHEMATICS N5 TIME: 3 HOURS MARKS: 100 INSTRUCTIONS AND INFORMATION 1, Answer ALL the questions. 2. Read ALL the questions carefuily 3. Nuraber 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. Siniplify where possible. 6. ALL graph work must be done in the ANSWER BOOK. 7. Questions must be answered in blue or black ink. 8. Write neatly and legibly. t Downloaded from hlayiso.com Copyright reserved Please turn over

DHET 3. QUESTION 1 Determine the values of the following limits: T1040(E(A9)T 11 . Sa) lim| SUBM42= 2e? nl x-6 ) 1.2 . (see ens) lim] -— xox x (A) [7] QUESTION 2 21 Calculate 2 with the aid of logarithmic differentiation if y = e' @) 2.2 2.2.1 Make a neat sketch of the graph y = arcsin x for the range - a 2) 2.2.2 Prove that if y=arcsin x, then (3) 2.3 Determine 2 in each of the following cases: (Simplification is NOT required.) 23.1 y =sin(cosy¥x? +1 (3) 4 2.3.2 y= | (5x-4) (3) 2.33 y =(arecot x)[In(l+ x7)] (3) 24 Determine 2 if ytJytx= xf by using implicit differentiation. (5) [23] Downloaded from hlayiso.com Copyright reserved Please tum over

DHET 4 T1O40(EXAN)T QUESTION 3 3.1 A rectangular tank to hold 342 m? is to be built with a square base and open top and is to be lined with metal. If it costs R48/m? for the lining of the sides and R22/m* for the bottom, what should the dimensions for the least cost for the lining be? (5) 3.2 A balloon is in the shape of a cylinder with a hemisphere at both ends. The height of the cylinder is 4 times that of the radius. Air is pumped in the balloon at a rate of 60 cm/s. At what rate does the surface area change if the radius is 8 cm? HINT: V =i + arth and A= 4ar?42mrh (8) 33 Given: F(Q=x -2x? — x43 3.3.1 Determine the co-ordinates of the turning points of f(x) (2) 3.3.2 Draw up a table of values for x and f(x), with x ranging from x=—-2 to x=3 (2) 3.3.3 Draw a neat graph of f(x) between these values and show the turning points on it. (2) 3.3.4 Use the table and the graph to estimate a value for the root between x=-2 and x=-1 of the equation x’ -2x? -x+3=0 and then use Taylor’s/Newton’s method to determine a better approximation of this root. (Root correct to THREE decimal figures.) cy) [23] QUESTION 4 4.1 Determine [y.dx if: 4b. y=e*.sin3x (4) 4.1.2 as a 8) 4.1.3 y =sin? x.cos? x 4) 4.14 __ cosec*x > Go cotx 3) Downloaded from hlayiso.com presse unover Copyright reserved

DHET Se T1040(E)(A9)T 4.2 Determine f y.dx for the following by using partial fractions: ys x71 4x? —20x4 25 (5) [19] QUESTION 5 i 2, Hoe 1 5.1 Evaluate the definite integral Te J 1x? 2 (5) 1 3.2 The sketch below shows the area bounded by the curves y=-x?+4x and y=e?" The intersection points of these graphs are given as (0,173; 1,173) and (2,59 ; 3,652) aes -- yooxt4x , 5.2.1 Draw the above sketch in the ANSWER BOOK and show on the sketch the enclosed area, the representative strip and the intersection points. (4) 5.2.2 Calculate the magnitude of the area in QUESTION 5.2.1. (3) 5.2.3 Calculate the volume generated when the area in QUESTION 5.2.2 rotates about the y-axis. (4) 5.3 Determine, from first principles, the second moment of inertia of a rectangular lamina with respect to a reference axis parallel to one side of the lamina that bisects the lamina. (4) [20] Downloaded from hlayiso.com Copyright reserved Please turn over

DHET 6- QUESTION 6 61 Solve the differential equation: awe dx y(i+e") 6.2 Determine the particular solution of : 2 29 294-20 2 Given thar 2 =-2,y=2 and x = 1 be Downloaded from hlayiso.com Copyright reserved TLO4OCEMAI)T 16) G) [8] TOTAL: 100

DHET T1040(E)(A9)T MATHEMATICS NS FORMULA SHEET Any applicable formula may also be used. TRIGONOMETRY s*x = 1 -sin’x sin? x + cos’ x= 1 sin*x = 1 —cos?x 1 +tan? x= sec? x ————-» tan’x = sec”x - 1 1 + cot” x =cosec” x ———+eot?x = cosec’x- 1 sin 2A =2 sin A cos A cos 2A = cos"A - sin’A tan 24 = 2 A 1-tan“ A sin? A = %4- ¥2c08 2A =A(1 - cos2A) cos” A =%+ ¥cos 24 =14(1 + cos2A) sin(A+B)=sinAcosB+4sinBcosA cos (A +B)=cosAcos B+sinA sin B tan At tanB 1+ tan AtanB tan(A+B)= sin A cos B = % [sin (A+ B) + sin (A - B)] cos A sin B =% [sin (A+ B)- sin (A- B)] cos A cos B = % [cos (A + B) + cos (A - BY] sin A sin B = % {cos (A - B) - cos (A + B)] sinx , 1 tanx = ;sinx = $COsx = cos x cosec x cos.x 1 1 cotx =— 3 COS ecx = — 7 secx = sin x sin x cos.x Downloaded from hlayiso.com Copyright reserved Please turn over

DHET TLO40(E){A9)T BINOMIAL THEOREM n(n-l(a—2) 3 3 + nn) _,- PAD TT yn? py? xn A tee +h) ax" tnx"! ht Cet hy" =a! + nx 2 1x2x3 DIFFERENTIATION _£@ f(a) roate PRODUCT RULE y = ulx) vx) QUOTIENT RULE v(x) du dv —--n. yy a ae CHAIN RULE y = fue) ay _ dy da dx du dx Downloaded from hlayiso.com Copyright reserved Please turn over

DHET -10- T1040(EA9)T a Aw) 310 [fade iat F') sin” f(x) Farr ate _ . -f'() 4 f'@) an t® 1+F@P = cot” f(x) aor _ sec’ f(x) __ fo) _ FOWL @)P -1 cosec” f (x) _ =f) ~£@) _ FOYLF@)P -1 in (2 sin’(ax) _ ; - sna) +C x _ sin(2ax) cos*(ax) __ 7 +- a +C 1 tan*(ax) __ az tan(ax)—x+C 1 cot"(ax) — - = cot(ax) =x+C — sin? (2) +C a a : — 1 tan” (2) +C a+x a a ; —= 1 sec (2} +C x x -a? a a 2 a ax — Ssinr(2)+3 ax? +C * a uf x= 2) +C a? 2a \xta ! Downloaded from hlayige:e m a? — x? a a \a-x

DHET ~9- T1040(E)(A9)T fx) ce f reeke J fax 7 nl ad ( ) nx + #- n+1 cu ax” aft al *" dx dx ax+b eth ere Slax +b) > ee £ +C —(ax+b) dx dxte ate a? Ina Stas +e) — +C * Ina.—(dx+e) ax id In(ax, —— In ax—x+C (ax) nwa” adn ax —x e ef) gy OLE “dx . d a £Q) af? a™ Ina.— fx, — +C dx fo) Ina 1 od In fix) sa) — f(x) dx : cosax sin ax acos ax = 4c a . sinax COS ax -a sin ax ——— 4+ a tan ax asec? ax _ntsec(ax)] + a cot ax - acosec” ax _ntsin(ax)] +C€ a sec ax a sec ax-tan ax tin [sec ax + tan ax} +C a L ax cosec ax -acosec axcot ax —In, taal 2 +C oralso a Copyright reserved fn [cosec ax - cot ax}+C a Downloaded from hlayiso.com

DHET INTEGRATION [£02 g'(2) de = 10) ga) - | £'@) g0) de "6 ‘ _@y _ [LAO © fade Sate (n#~1) =In epee inst f(x) A, 2 (ax+bXcx+d) axtb ex+d f@) _ oA ,_3 Cc Z (xt+a)" (xta) (x+a)? "Gray "ct a)" APPLICATIONS OF INTEGRATION AREAS A= [yar A= [Ox yddr A= f adysAy= fm —ndey VOLUMES Vamp yae:ve=al (yt - y3)ax Vy =a dy Vy =f (x? ~ 23 )ay SECOND MOMENT OF AREA I= ff Paa:t,= 0 aa MOMENTS OF INERTIA Mass = density x volume Ages pv DEFINITION: J=mr GENERAL: [= [ r’dm= [ Pav Copyright reserved Downloaded from hlayiso.com T1040(E)(A9)T

a her education & training Department: Hig ther Education and Training REPUBLIC OF SOUTH AFRICA MARKING GUIDELINE | SETI RENE NATIONAL CERTIFICATE APRIL EXAMINATION MATHEMATICS N5 9 APRIL 2013 This marking guideline consists of 10 pages. Downloaded from hlayiso.com Copyright reserved Please turn over

MARKING GUIDELINE 2: MATHEMATICS N5 QUESTION 1 Ld in(2x~ in( 22==12)) 2 ow. indeterminate 196 x-6 0 =tin{ 25 we y x6 x =2c082(0) V =27¥ 12 __{ tan 4x all 0 0,4 lim| - =o N x x 0 0 =-30) ._{ tan4x—sin4x = lim| ————— |¥ 10) x 3490 mtin( te e=tesste y 40)" 40) y 4 =0 QUESTION 2 2.4 y= eta Iny=Ine*"®? =(x* -Inx)eIne Iny=x* —Inx vov 4 1 gpl eee) 4x3 -— Dou aded from hlayiso.com Copyright reserved T1040(E)(ANT (3) 4) [7] (4) Please turn over

MARKING GUIDELINE 3° TIOAEKANT MATHEMATICS NS 2.2 2.2.1 Y = y =arcsin x ¥ labels (2) 2.2.2 y = aresin x siny=x cos ye ai Vv ax cam wherecos y=1—sin? y dx cosy ¥ dy 1 : >= where sin y= x dx J1-sin? y yd a Vix? (3) 23 23.1 y =sin(cosv¥x? +1 yoy v ® —(oscosVx* +0) sinyx? +1 ! e2x ax Ix? +1 (3) 2.3.2 x y=ln| ——_; (5x—4) v v v dy _{ Gx-4) | 42? (5x4)? — x4 (2Gx-4)95 dx x [6x-477 Also yelnx* —In(5x-4)?V y=4Inx—2In(5x—-4) V¥ yiotize Vv x 5x-4 (3) Downloaded from hlayiso.com Copyright reserved Please turn over

MARKING GUIDELINE 4 T1040(E)(A9)T MATHEMATICS N5 2.3.3 y =(arecot x)[In(i + x”)] V ay dx ¥ Q) 24 ® 2.5 ytyytaca” V Vv 1o24o(y4x) Meo] 12% v Vv atari area -t(y+ay ci eee A a4 x 7 (y +3) dx 1 PA l+-(y+x) 7 gt) (5) [26] QUESTION 3 3.1 C =48(4xh) + 22(x”) A C =192xh + 22x" = sas 2 422° x x x c-(S), 22x ¥ x dC (65664 ! Vaxrh a 2 } 440-09 =32 ¥ x= 1143 mV x hee 2.62 mV 11,43 @) Downloaded from hlayiso.com Copyright reserved Please turn over

-5- MARKING GUIDELINE MATHEMATICS NS 32 V -fa +mh h=4r aim + ar (4r) veo ey _ 6 167(8)" oe 0,019 cm/s V dt 3.3 3.3.1 f'@)=3x? ~4x-1=0 A aA a T1O40E)(A9)T A=4ar’ +2mmh =4ar?+ 2m(4r) =12m7V =12”.2r ar at dr = Mm at = 24n(8)(0,019) V 7 11,25 em?/s¥ (8) _ 244 V4? -4@Cd _ 4+ V28 2(3) 6 x= 1,549 and x=-0,215 cif x= 1,549; y = 0,369 ¥ of x=—0,215; y =3,113 ¥ 3.3.2 2 -1 “tl {dl Z 3.3.3 y 120 « —90 L = 30, (1,167;1,093) 3 2 NY me x NY3 me Fal (05:-0.57 i —3 —-60 —-90 Q) Ww 2) Q) Downloaded from hlayiso.com Copyright reserved Please tum over

TLO4O(EM AQT MARKIN -9- (G GUIDELINE MATHEMATICS N5 5.2.3 OV =myi - yp ).& vot, OV = 2\ (-x7 +4x) —1e? a Nv 2,590 uVon Ie) ~ 8x) +16x? —e7).dx 0,319 590 fe +x re] 10.319 yeti 300 unit (4) 54 y * , v Aly =bx Axx x° = bx’ dx a $5) 5 (-$2] 2°) Tyg =b fx? de x a — Pam ¥ 259 a |= (£0| -as1. , 4 a 2 3 b. AA=bx Ax Y he = ts! V (4) [20] QUESTION 6 6.1 dy | e dx y(I+e*) e dy = ax V ray ae Ly stndte) + 2 Voy 3) Downloaded from hlayiso.com Convrieht reserved Please turn over

MARKING GUIDELINE -8 T1040(E)(A9)T MATHEMATICS N5 QUESTION 5 5.1 1 [eere e “iv1-x? 2 v V v 1 = |sin7 xP, =sin™ (3) ~sin7 (- | + 2 2 (2) _ (- 2) -Z 6 6) 3 Vv Vv ©) 5.2 5.2.1 ¥ t Suip v ay - (2,590; 3,652) ¥ Jah ee | x y=-xtdx 4 5.2.2 dh = (y, —y,) 0 & L atc 40) ~(0 Joa 2,590 V 1, A= f [-* +4x-e? Jee 0,319 4 L 2,590 AnD -20| 3 0,319 = 2,476 units* (3) Downloaded from hlayiso.com Copyright reserved Please turn over

MARKING GUIDELINE -7- MATHEMATICS NS 4.1.3 sin’ x.cos? xdx = [sin’ x.cos” x.cos x.dx = sin’ x(1—sin? x).cos xdx V = fosin? x—sin’ x).cosxdx = flu? =u lu 4 a4 -24¢ 4 6 . a __ sin’ x_sin’ x, 6 4 6 v v 4.14 f cosec?x (1-cot x)? = fu? edu e-lac y ui = icy (1—cot x) 42 ys x-1 4x? -20x+25 x-l oA + B (2x-5) (2x-5)’ (2x-5) “.x—1= AC) + B(2x—5) ¥ x-l 3 1 Vee po a 2x-5)P ao lang 5) 3. 1 7 Gr Gaya din(ax- 5)+C v V T1O40(EA9)T u=sinx du=cosxdx V (4) u=1-cotx du =cosec?xdx V GB) coefficient of x: B= A v constant: -2=A-5 ..A= ; V (5) [19] Downloaded from hlayiso.com Copyright reserved Please tum over

6 T1L040(E)(A9)T MARKING GUIDELINE MATHEMATICS NS 3.3.4 Let x, = -i,] 1, EX - fv fe) fCL1) = 0,349 ¥ ==-11-{2349)y 7,03 f-L1) = 7,03 ¥ x2 =-1,150N (4) [23] QUESTION 4 J at 411 Je* sin3xdx=e*.—4cos3x- fe*.— Leos 3xde 3 3 =-1e: cos3x+-4 fe*.cos3x.dx 3 =~ 1 6! cos3x4+{e7tsin3x- fe? 4 sin3xdx v 3 3 3 7 4 fet sindiude =—e" cos3x+ Ze" sins = e* sin 3x.dx 10 et.sinxdx=—e* cos3x+te? sinx 9 3 9 fe*.sindudx=-—2 e* cos3x+te* sinx+C 10 10 v Vv (4) 4.1.2 L er l edx =-— Fs u x od = fe * o—edr —du=x? edxv L =— fe" ed, 1 fet edu sae" +CV 1 a =-e "+04 2° (3) Copstight sea Downloaded from hlayiso.com Pies un ov

MARKING GUIDELINE -10- MATHEMATICS NS 6.2 d’y ~222=2- 4x-2x? dx? 2 cies +2x-1 ¥ oa tx —x4C, V =50° +O? -@M+e, aa ip aly woe Ci, 12 3 2 3 1 1 1 7 —()* +=? -=()? -= M+ rhe 3 2) 3) 2 Bee 12 s.l.5 1. 7. ,53 YE +2 KX HT K+ 12 3 2 30 «12 Downloaded from hlayiso.com Copyright reserved TLO4O(E\(AN)T 6) [8] TOTAL: 103

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ey higher education g training == Bo; Department: == UN )e es Higher Education and Training == aoe” REPUBLIC OF SOUTH AFRICA T1090(E)(N19)T NOVEMBER 2012 NATIONAL CERTIFICATE MATHEMATICS N5 (16030175) 19 November. (X-Paper) 09:00 — 12:00 yy Copyright reserved NOV 2012 Please turn over Downloaded from hlayiso.com

(16030175) -2- T1O90(E)(N19)T DEPARTMENT OF HIGHER EDUCATION AND TRAINING REPUBLIC OF SOUTH AFRICA NATIONAL CERTIFICATE MATHEMATICS N5 TIME: 3.HOURS MARKS: 100 INSTRUCTIONS AND INFORMATION 1. Answer ALL the questions. 2. Read ALL the questions carefully. 3. Number the answers correctly 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. 4. Show ALL the intermediate steps. Simplify where possible. 6. ALL graph work must be done in the ANSWER BOOK. 7. Questions must be answered only in BLUE or BLACK ink. 8. Write neatly and legibly. Copyright reserved NOV 2012 Please turn over Downloaded from hlayiso.com

¥ (16030175) 3. QUESTION 1 Determine the values of the following limits: Ll im | P+] won| TF T1090) N19) (2) 1.2 lim f° = 4 3x6 (2) 13 in | 4x2 4x =] oo i Se | 3x? + x 4] (3) (7] QUESTION 2 2.1 Prove that the derivative of » = arecot (2) + arecotx is zero, (4) x 221 Make a neat sketch of the graph y = arccos ec x for the range [-2:2 . (2) ~ ody . 2.2.2 Derive a formula to determine a if y = arccosec x, (4) 23 Determine 2 in each of the following cases, (Simplification igs NOT required.) 23.1 Y = arctan (a ~x) (3) 2.3.2 y=tan fi—x : (3) 2.3.3 Y= (X+1(2r + Gx + 2) (3) 24 Calculate x ify= e with the aid of logarithmic differentiation, (4) {23} Copyright reserved NOV 2012 Please turn over Downloaded from hlayiso.com

(16030175) 4. T1090(E)(N19)T QUESTION 3 3.1 Given: f(x) =- 4x3 + 10x? -7x+2 3.1.1 Determine the co-ordinates of the turning points of f(x). 3.1.2 Draw up a table of values for x and F(x) with x ranging from x =~-3 to x= 3, 3.1.3 Draw a neat graph of f(x) between these values and show the tuming points on it. 3.1.4 Use the table and the graph to estimate a value for the root between x =1,5 and x=2 of the equation -4x°+10x7-7x42=0 and then use Taylor’s/Newton's method to determine a better approximation of this root (root correct to THREE decimal figures). 3.2 Determine the cylinder of greatest volume. inscribed .in a given right circular cone with a height of 34 cm and a base radius of 10 cm. 3.3 Two resistances R,and R, are connected in parallel, where the total resistance R is given as aE +> If Ry and Rz are decreasing at rates of 0,02Q.5~' and 1 2 0,03 Q.s"' respectively, caleulate at what rate is R changing at the instant that R, =40Q and R, =1009. QUESTION 4 41 Determine the integrals of each of the following: 4.1.1 f 1 aV2x" 4.1.2 e* J eas” 4.1.3 feria 4.14 i ax 7+16x? Copyright reserved NOV 2012 Please turn over Downloaded from hlayiso.com (2) (2) (2) 4) (6) (5) [21] @) (3) QB) 8)

(16030175) 5: 4.2 Determine foos2x.cos.xdx by using the following: 4.2.1 Cos A. Cos B= host + B)+cos(4~ B)] 4.2.2 “= sin x 4.3 Determine! yx for the following by tesolving the integrand into partial fractions: Sx+4 “43x 10 QUESTION 5 5.1 Evaluate the definite integral 2 te f yx ve +16) } 5.2 The sketch below shows the area bounded by the curvesx=2e-” 41 and x=-y?+6y-5. The intersection points of x= 2e~’ +1 and x=-y?46~Sare given as (1,4861; 1,415) and (1,018; 4,727). x=~y?+6y-5 5.2.1 Draw the above sketch in the ANSWER BOOK and show on the sketch the enclosed area, the representative strip and intersection points. 5.2.2. Calculate the magnitude of the area in QUESTION 5.2.1 5.2.3 Calculate the volume generated when the area in QUESTION 5.2.2 rotates about the y-axis. Copyright reserved NOV 2012 Please turn over Downloaded from hlayiso.com TIOSOCE)N19)T Q) (3) (3) [23] 4) (2) GB)

(16030175) -6- TLO9O(E)X(N19)T 5.3 Determine, from first principles, the moment of inertia of a uniform circular disc of radius 2 m and mass 10 kg about am axis through its centre and perpendicular to the plane of the disc, HINT: Consider a ring of thickness Ar a distance 'r' units from the zz'-axis. The area of the ring is 27rAr and its mass is 27 rar. (6) ki [20] QUESTION 6 Solve the following differential equations: 6.1 dy __ tx da (3) 6.2 ld’y 1, 3 Qa 2” *Q**? 2) IN TOTAL: 100 Copyright reserved NOV 2012 Downloaded from hlayiso.com

(16030175) MATHEMATICS N5 FORMULA SHEET Any applicable formula may also be used, TRIGONOMETRY ; cos*x = | _ sintx Sin’ + cos? = | <= sin’x = 1 ~ cog’ i+tantxs sec’x —————>» tan’y = sec?x - | 2 2 2 2, 1 + cot*x = cosecy ———— 01" = cosec’x - | sin 24 =2 sin 4 cos A =cnetd . ot2 Cos 24 = cos" - sin?4 2 tan A tan 24 = 7 1-tan* 4 sin’d =%- 4 cos 24 = “(1 ~ cos2A) cost4 = 4 4 %cos2d=%(1+ cos2A) sin (At B) = sin A cos Bt sin B cos A cos (At B)=cos A cos B+ sin A sinB tan (4p) = lands tan B it tanAtanB sin A cos B= % [sin (4+ B) + sin (4-B)] cos A sin B= 4 [sin (A+B)-sin (4 - BJ cos 4 cos B =¥, [cos (A + B)+ cos (4 - B)] sin dA sin B = 4 [cos (A-B)-cos (4 + B)} sinx _ 1 tanx = ;Sinx = 7 COSx = —— cosx cosec x secx cosx Lo colx = —— ; cosecr = —— ; sec x = 1 sin x sin x cosx Copyright reserved NOV 20 i2 TO90(E\(Ntoyp Downloaded from hlayiso.com

(16030175) BINOMIAL THEOREM a(a-1) yr pra n(a—l)(n—2) as (a+h)" =x" +x" he 2! DIFFERENTIATION =f f (a) rr=ate PRODUCT RULE ¥ = u(x). v(x) dy dv du ered aerate a iene dx a& dx Surv tyew QUOTIENT RULE = He) ve) CHAIN RULE ¥ =fu)) dy yd dx du dx Copyright reserved NOV 2012 Downloaded from hlayiso.com 1x2x3 h+ T1O90CE)(N19)T

(16030178) TIO90(E)(N19)T d fo at”) [rear sin” f(x) fe) 1-[/()P _ cos" f(x) F'@) v!-[f@?P _ tan" f(x) f'@) 1+ @F ~ : -f'@) efi) FO 10) ier 4 f'(x) sec” f(x) AM SOWIFC)P -1 — 4 -f'@) cosec™ f(x} ———AM/ SO LFOoP =1 7 sin“(ax) #_sinltas) | —— 2 4a. cos*(ax) tan’(ax) cot*{ax) __ Va°-x 1 xg _. 1 2 — a -x Copyright reserved NOV 2012 % | sin(2ax) z ——— +e 4. gtan{ax)-—x+¢ — 2 cot(ax) -K+E sin'(Z}nc a Phan (ax) -—x4€ a2] +C a a 2 Soiw(2) 2a -24C 2 a 1 x-a = Inf —— J+ 2a x4] ma S2*) +c 2a \a-x Downloaded from hlayiso.com

(16030175) TIOSOCEXNID)T d fe) AG J tara n+] x" nx"! +e (n=-1) ntl ax" at af <"dx dx arth ete te gx +) sc dx d =~ (ax +b) dx dete ate a“*** Ing Slax +e) —~— +#C x Ina.—- (de +e ae ¢ } 1d tn(ax) —.. ax xin ax-x+C ax dx ef ef ao Si) dx _ a” a® Ina.— f(x) _ | od ln fx, Se Hy) Fo ax Si) — sin ax a COS ax —— +C a COS ax -a sin ax me +C a tan ax asec’ ax dinfsec(ax)] +C a cot ax -a cosec? gx | infsin(ax)) + a sec ax asec ax-tan ax i In [see ax + tan ax] +C a 1 ax cosec ax -@ COSEC ax-cot ax — In} tan z +C oralso a Copyright reserved NOV 2012 1 ~ In [cosee ax - cot ax} +C a Downloaded from hlayiso.com

(16030175) TIO90(E)(N19)T INTEGRATION J x) a(x) de = fx) g(x) - 1 f°) gO) dx [LPO]" « fi(xjdx = Yen". C (n#-15 (Fie -insle)ec f(x) AB (ax+bXex+d) axtb cx+d fe) __A B c Z = + Foe (Gay Gre Gea? Gay Gray APPLICATIONS OF INTEGRATION AREAS 4,= [yer de= | Oi-ydee fe a A, = [954,= [Gime VOLUMES Vv, anf yrdx ; ye =a (2 — v3 ae ra : a =a] xy 2V, anf? (2 - day SECOND MOMENT OF AREA fea P Pda: t, = [Pada MOMENTS OF INERTIA Mass = density x volume M= pv DEFINITION: [= m7 GENERAL: J = [/ rdm= of ray Cormriaht reserved “YY gwnloaded from hlayiso.com

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higher educati & training venon Department: Higher Education and Training REPUBLIC OF SOUTH AFRICA | MARKING GUIDELINE | | NATIONAL CERTIFICATE MATHEMATICS N5 | 19 NOVEMBER 2012 This marking guideline consists of 9 Pages. |_ Copyright reserved Please turn over Downloaded from hlayiso.com

MARKING GUIDELINE -2- T1O90(E)(NI9)T MATHEMATICS N5 QUESTION 1 Lt xP 1 : xt ty? . ‘F421 ye 1 1 (=:2 T =\tv/ 7 xt +7 SL Fay =1 Vv (2) " aE i). had 1 mf 4 2m oc ae v Q) 1.3 7 4x3 x8 se 3x" FOr +t k 4ytax—-1\_ oie? + et i): y (3) {7} QUESTION 2 21 xu =erecot (S}+ arccotx ; (4) Copyright reserved Please turn over Downloaded from hlayiso.com

MARKING GUIDELINE 3. MATHEMATICS NS THOSOCEY(N19)7 2.2.1 ¥ arccosec x we Ma, = aN ] d ho x a ! x (2) 2.2.2 Y= arccosecx cosecy =x —~Ccasec 3, coty. =1 ¥ dy ~1 == ax cosecysoty V where coty = ¥eosecty —] ns ax" cosec y.Jeosecty =F ’ ». ec VY where cosec you ” ax” BRTST (4) 23.1 -mmy ta ar 14 (aaa? ag F-* Vv i. =t¢@-y-? y ~i =ir@-xyF y @B) 2.3.2 Y= tanvI—X ayfdx= Usecl t2 fa — 29 ax G-wahin y = Kseck "2 (v1 -2) “U2 (1 —aye—1y2) .-1 dv sec*yT —& =Wi-z y (3) 2.3.3 y=et+ 12x + 1)(3e + 2° Z @ B= 6+ Mert Pert ase Nore aZex+14 2 Vv = (2x7 + 3x4 198) 4 Gr? +axr4 2x + 22) + (x3 + 72+ 21) = 6x? + Ov +3 +6x7 + 1Ox+44 6x2 47742 ay 2 dg 7 18x? + 26x +9 ¥ (3) +1)(8x4 Qretr ax(xt+T) Copyright reserved Please turn over Downloaded from hlayiso.com

MARKING GUIDELINE 4. TLO9O(EXN19)T MATHEMATICS NS 24 Ps Iny = lae*™* Iny =x*” Ine Inn y) = atinx 1 Iny.1 ay _ 1, aden dale + ox vv = = yiny[2zing + a] V — oper [oxinx +x] ¥ (4) [25] QUESTION 3 31 fG) = ~tx? + 10x? - 7x42 3.11 f'@) = 12x? + 20x—7 =0 (6x -7\(2x—1) = 0 an axa 1,167 x = 0,5 V and aye 1,093 y=O,5 resp. J @) 3.1.2 x 3-2 |-1 [0 1 2 3 f(x) | 221 | 88 | 23 (2 1 4 | -37 ¥ (2) 3.1.3 T 120 V (1,167;1,093) <2 _i wea , t f + (0,5:-0,3f[ 1 Xe -30 Hr) = -4x" 4 1027 ~ 7242 +60 ro @) 3.1.4 stem #0) Let/laat *1 — 16 Fy). ¢ 0,04 f0,6= 0,016 V¥ =1,6-\- 57207y F°Q,0= - 5,720 ¥ %: =1,6037 (4) Copyright reserved Please turn over Downloaded from hlayiso.com

MARKING GUIDELINE a 6. TLO90(E)(NI OT MATHEMATICS NS 4.1.2 oe _ lL pse* dx Saxzs"* slams y q jinte™* +5)+C y (3) 4.13 fo. elt-4) ay = “<4 v dus-4x4dx Vy age te y (3) 4.1.4 dx -{ az J 7+ 16x? (WP + (4 y 1 _y fhe Fee v Ey faa? =10583°™ *(S)+e y QB) 4.2 Jeosza. cosx. ax 421 _ 2 Cosa. CosB = 2Icos(4 + B}+ cask(A — BN Bey Lysin3x =3\ 4 $sing]4+C _ sin3x sine c =e Fate vy (3) 4.2.2 f COS2X. COSK. GX = fia — 2sin*x)cosx.dx J u=sine = {ua ~ Ju?) du 2 =uw—-—yF =u-gu +¢ Vv du cosx. dx ; 2s =siax-ssin?xt€ 3 Fy N , 3) Copan reserve Quynioaded from hlayiso.com 2:08 tun over

MARKING GUIDELINE +5. T1090(E)(N19)T MATHEMATICS NS 3. ; (6) /\ nh 34 40 -r To 34 . Verrth = rr?(34 ~ 3p) A : ve 34rr4 - 37r8 v —! yo _ 1 Gy = O80r - onr7 = 0 v n(68—9r}= 0 réfond y= 758m 7 “h=34-37 Vonrth =11,32cm § =%7,567(11,32) V=2032,544 cm* J 33 Bott RU RTE, apa rhe GO\(100) | “RoR eas 204 ion = 28570 v voy ~1,4R_ 2 GR, 1 ak, 1 L _ 7 50F = 9,02) ~ 100*(-0,03) 1 aa “285702 —o, “1 Ten 0,0132. 5 v (5) [21] QUESTION 4 4.LL 1 if dx 2vaxt” 23 Bxt 1 =a =—[x2. aay 87 -8e =3 1 oxztt c “we 2yqt z a2 _t rz ae ar a wey ly 1 -2 Sree ete. 2 “2 et Er tC Ya to y (3) Copyright reserved Please turn over Downloaded from hlayiso.com

MARKING GUIDELINE -7- MATHEMATICS NS 43 a =f sx+4 ~ [_Adx Bde +3e-10 J G-NG48 Jaa ley v then x=-5 and x=2 Ifx=5 then ¥ Ffx=2 then 1 es | Lo v QUESTION 5 5.1 4a “1 [rare f BE VE+ 4) dx usViFie a 1 Copyright reserved TIO90(E)QN 19)T Please turn over Downloaded from hlayiso.com (5) (23)

MARKING GUIDELINE i -8- T1090(E)(N19)T MATHEMATICS N3 4 du. i = [ a du = ———— dx no OY, 20+169= — orlof 1; : = [ - V x+16—ui oy 1 _ s[ (a? - LF du 2 fE=Vu? —16 _ 1 at - 10% =4-—j— du 2.2 v 2 ay = [-2eu - 187], v ifx = 1 then u =717 and eS hall ~ [-2a - 10937] x= 4thenu = ¥20 =2 (4) 5.2.1 (2) x=2e%4H1 (1,018;4,727) Enclosed ¥ Strip ¥ (1,486; 1,415) 512.2 8A = ey — 1, )8y a4 = EG y? + ey—5)- (2e-¥ + 1d 727 sae [(-y9] + 6 -6 = 2e ay J L315, a72F ay -| Z +3y? ~ 6+ 26] , ats A= 6.424 units’V (3) 5.2.3. ov= «(ad -aD ay Copyright reserved Please turn over Downloaded from hlayiso.com

MARKING GUIDELINE -9- MATHEMATICS NS THOPOENIOYT 6 = r[C a 1 6y 5)? (2e-¥ 1 4)2]ay y sven ats ‘T- 12y5 + 46y? — —60y 4 24~ ge-y ~ 4e-¥)ay V ys 46 4727 Vey r- 3y4 to — 0p? + 24y 4 2e-e 4e~¥ V = 92,191 units? wee vy (5) Mass per unit area a =m y 10 adm= te Vv _5 QUESTION 6 6.1 dy : dx 7? ~fRay= fom ax y v¥ i ae vhiy= Te + (3) . id? 1 3 62 fast = 7gU ts gitEn ye nat ay 47 dx? * 7 v ay 3, a ~ at pt +20x $C V a Zz 2nx* Rae ten Ft xe € Vv 1 x" atx? S-Mt Set gree, 12° "2 v (4) [7] Copyright reserved Please turn over Downloaded from hlayiso.com

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we ThosogR Kary ‘DEPAHTMENT OF HIGHER EDUCATION AND TRAINING REPUBLIC OF SOUTH AFRICA NATIONAL CERTIFICATE ‘MATHEMATICS NS TIME: 3 HOURS MARKS: 100 3TRUCTIONS AND INFORMATION Answer ALL the questions. Read ALL the questions carefully,” Number the answers correctly according to the numbering system used ir this question paper. Questions may be answered in any order, but subsections of questions must be kept togeti:ex. Show ALL the intermediate steps. Simplify where possible. ALL ‘fraph work nmust be done in the ANSWER BOOK. Ques:ions must be answered in blue or black ink. Write seatly and legibly. | Copytighteservea Please turn over 3 T1O90GEKAIYT QUESTION 1 Detemmine the values of the following limits: Lj - ( ef al 7 2x? ~2x @ 12 1 ‘a |_ x. a Foremes ) > 17 QUESTION 2. A sees 5x+4 aos. 21 Determine the derivative of y = from first principles. 2-3x ) 22 Calculate & of the following, by meking use of the derivatives of sin x and cos.x as well as the rules of differentiation: y= tan x * G6) 23° _ Determine 2 im each of the following cases. (Simplification is not required.) L_,ft 231 yea+n(4) : 2.3.2 =sec7| = oes (=) 2.33 y= v¥veotx—x? 24 Calculate sic y= (ce? ~4)" with the aid of logarithinic differentiation. 25 Find % of the implicit function: yf +x? = fay DHET Copyright reserved Q) @) 8) 4) 4 (24) Please turn over Downloaded from hlayiso.com

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TIOIOCEMANT \.THEMATICS N5 RMULA SHEET y applicable formula may also be used. IGONOMEFRY . cos’x = 1 ~sin’x x cos? x << . . sin’ = 1 - cos’ tan?x=sectx ——__». tans = sec*x- 1 cof x= cor::*x——» cot’x = cosee’r - | 2A = 2 sin A cos A 24 =cos74 - sin? A=M4-Ycos 24 = VA(1 - cos2A) A=%+% 20824 = %4 (1+ costa) A+ B)=sin A cos Bt sin Bcos A A+B)=c03A cos BF sind sini B dapye fen A tanB ) 1¥ tanAtanB cos B= ¥, [sin (4 +B) + sin (4 - BY] sin B= % vin (A + B)-sin (4 - B)} cos B* % (cos (A + B) + cos (4 - BY] sin B= % [20s (4 » B)- cos (4 + B)] sinx 1 i = ;siax= 3 S0sx = —— cosx -cosec x seex cosx 1 1 =—— | cos cox = ——— ; seex = sinz sin x cosx Copyright reserved T1O9QEKANT a fe) ZI [Fteyae ; =) x" ne"? +c (n#~1) n+l an" ost al ¥ de act ee em Se +2) a £ +C a (ax +b) eve ant ota A fe + 6) +e os na. (a+) Infax) tt xin aw—x+C ax dx d Jed fey e Nf _— af a”? tna ey rs — Loa In fs) —_— te) 705) a _ sin ax a@cos ax Be ee a COs ax -asin ax sm ec @ tan ax a sec? ax diatseo(ax)] + c a cot ax - acosec? ax “n{sintaxy]4+¢ a see ax @ sec ax-tan ax Ain {sec ax + tan ae] +C a cosec ax a cosec axcot ax 4 wo(=)] +C a DHET Copyright reserved Downloaded from hlayiso.com Az

higher education & training Department: iG) Higher Education and Training Sd REPUBLIC OF SOUTH AFRICA i MARKING GUIDELINE NATIONAL CERTIFICATE AUGUST EXAMINATION MATHEMATICS N5 1 AUGUST 2012 This marking guideline consists of 11 pages. Downloaded from hlayiso.com 44.45. uno Copyright reserved

MARKING GUIDELINE QUESTION 1 La gv #31] 5.2_9, 2x*° -2x gt | 5 2x" —2x 2+ MATHEMATICS N5 [5] = («) =) 1.2 1 x sin x ie xe zd cotx— + xCOS x— xsin x a o|o 1 =. tim x °°! cotx—x cos x x(—sin x) + (cos x)(1) — xcos x + (sin x)(1) QUESTION 2 5x+4 2-3x S(xt+h)+4 +h)=———-- farm 2—3(x+h) _Sx+5ht4 ~ 2-3x—3h 21 f= Sxt+Sh+4 Sxt+4 fOr W~ $0)= aan 23x _ 22h ~ (2-3x—3h)(2 -3x) _fathy-f@)_ 22 “ h ~ (2—3x— 3h M2 — 3x) lim) f@+*D-f@)__ 22 :-Downloadeéd f¥om hlayiso.com Copyright reserved T1OSOE (ADT (2) (5) @) Please turn over

MARKING GUIDELINE 3. MATHEMATICS N5 sin x 2.2 ystanx= cos x dy _ cos x(cos x) — sin x(—sin x) ax cos? x _ cos? xtsin? x cos” x do dx cos? x = sec” x 4 (tan x) = sec? x dx 1 1 2.3 2.3.1 =—-+in = one “3) ® 2 npg? Ah? dx 1 x -—t * xén?x x” et ane xtn?x x af x 2.3.2 y=sec (2) 2.3.3 y= veotx—x° a , 2,Jcot x . Copyright reserved —cosec?x— ») Downloaded from hlayiso.com T1O9O(EMANT (3) (3). Q) 3) Please turn over

MARKING GUIDELINE 4 MATHEMATICS N35 24 y=? aye eny = (4— 2)én{x? — 4) Viator —.— =(-1é -4)+ » 2x (4—x) y de (-l)én(x* — 4) yaa dy 1 nm = n(x? - 4) + ya 2x (4—x) di x 2x(4—x) Bat -4) {- tnx? -9+(22)| 2.5 yenxt x? = fny tnx, 42 43x? = lay dx x ydx 2 tne) =-a0 -2 3x22 ys dx QUESTION 3 3.1 f(x) =3x° + 6x? 452-4 3.11 f(x) =9x? +1245 fil @x) =182 412 for point of inflection: O=18x+12 “= — 0,667 y=-5,556 (- 0,667;— 5,556) 3.1.2 x -2 |-1| 0 1 | 2 3 fi) | -14 |-6 | -4 | 10 | 54 | 146 Downloaded from hlayiso.com Copyright reserved T1090CE)(ALT @) (4) Q) Q) Please turn over

MARKING GUIDELINE -5- MATHEMATICS NS 3.1.3 a‘ Bed 3 40 hy = Se + bx Sx -& te mac ao we" ae ia ar or war Sh. 3.14 letflaat x, =0,5 e= f@ F(0,5)=0,375 f'(@) f'(0,5) =13,75 f(%) =a fi) =05- 0,375 13,25 x, = 0,472 4 r=130+3 3.2 Vol fhemisphere V=—ar° olume of hemisphere 3 r =133dm WV 4 gt eager, & dt 3 t dt = 4m (133)? (-025) “ = = —55571,632dm'Vhr it Downloaded from hlayiso.com Copyright reserved T1090(EXALT (2) (4) (5) Please turn over

MARKING GUIDELINE -6- MATHEMATICS N5 3.3 ala xyal6 ays S=ax’+y? 3 4096 x S=x* +4096x7 ax + SS Loy —12288.x- =0 dx 2x5 =12288 “.4= 5,724 16. Y="y (given) _ 16 5,724 “y= 2,795 QUESTION 4 41 4.4.1 free. dx f(x)=x fiGx)=1 ~~, 1,2 1 -2s = he mae fl - ax a_ly pt lp, sare tale «dx ~ 1, ,2 il _1,» Rage ta get te —_1ly 4-2 _ 1-2 Bate qe.lfe TIN9O(E (ALT 6) gi(x)=e™ g'(z)=-fe* (4) Downloaded from hlayiso.com Copyright reserved Please turn over

MARKING GUIDELINE -7- 4.1.4 Copyright reserved MATHEMATICS NS jase a x =ltnfz Lax Let/Laat u = fnvx wldu=t . dx J2*- cot( *) de = Joot(2*) . 2* , dx Let/Laat u =2* =, u=2*. dx 1 te foot. u. du = = én(sin x) +¢ i er =Fa" én(sin 2 ) +e [¥25—927 «de = 3f Baw as 25 cd = 3 2-sint 4 +5 Bx +¢ Downloaded from hlayiso.com T1N90(E)(A 1)T Please turn over ©)) @) (4)

-8- MARKING GUIDELINE MATHEMATICS N5 4.1.5 Joos? 4x. dx - f4ah = [4+ fooser) a 1 sin8x +5. te 8 hath i dat bsin8x +e 42 x+1 xt 3 be = Serica) -x) xtl - A + B (24+ x\l-x) 2+x (l-x) xtl =A(l—x)+ B+ x) LetvLaat x=1 B= 4 x=? wA=-4 a1lpi 2p! 4 sean es lara =z én(2+ x)= tn(t- x)te Downloaded from hlayiso.com Copyright reserved TIONNE\ADT @) (5) Please turn over

MARKING GUIDELINE -9- MATHEMATICS N5 QUESTION 5 31 f e .ftjat if f@)=4 = of ee". 4dt k e* ai _ im |= = poe 5 a[-4.4 tule 5| =o+4 5 -4 5 52 5.21 yeeFe yale Intersection/Snyding: of =ve* et ae" 4a x? ox -4x=0 x(x-4)=0 “x=0 orof x=4 ficf: fO)=1 ff @=Ve* f(a)=e? =7389 =f, (0)=V1=1 f(t)=e' =2,7183 (0) = 1,648 f,(2) = 2,718 Copyright reserved Downloaded from hlayiso.com TLOSOEMADT Please turn over (4)

MARKING GUIDELINE 3.3 5.2.2 -10- MATHEMATICS N5 3.2.3 dA=(y, 7 Ya dx a= [e® ~ve*lax -[re —264F — | x 4 10 =[29,556—14,778-14,778|- [0-2-2] =4 UNITS” 2 j Taser 2 eq tam a 4fiu 2du u 4 h rope =4 2+ | =8./ve+4] =8\V6 - V5| =8(2,449 — 2,236) = 1,704 Downloaded from hlayiso.com Copyright reserved Let/Laat TLO90E ADT GB) Please turn over @

MARKING GUIDELINE -1l- T1O90(E)(ADT MATHEMATICS NS 5.4 dV amy; dx—my} dx yy ae sy ax? = mx ‘dx-mx'dx. Vaz), (xt —x*)dx =n7|—-— 5 7h -a|i-4 = 0,057 UNITS® 6) QUESTION 6 61 “ = —2y(4x?) wdy=4x . dx <n fny=x'+e = SS =(1)* +c planta wie (4) 62 a at dx 9 10 10 6 yp=t_-2 48x? +exte 10 60 (3) TOTAL: 100 Downloaded from hlayiso.com Copyright reserved

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QUESTION 1 = FTIUEYAT Dy termine the values of the following limits: 3a lit @) _ 12 3.2 Q) . x-l . x 12 Determine whether the function I= 7 48 continuous at x = — Qx- x) 2 Q) 7] 33 QUESTION 2 21 Determine * of the following, by making use of the derivatives of sinx and cosx, ‘we as well as ihe rales of differentiation: ys tans @) 22 2.21 Make a neat sketch of = arc cosx for the range [o, x). 2 223 . a 4.2, Derive a formula to determine 7 if » = are cosx, @) 2.3 Deiermine # in each of the following cases: (Simplification is NOT required.) BB ne inet) @ Q) 23.2 vex tay 23.3 GB) 24 Deteanine Fa if » = (in z)" with the cid of logarithmic differentiation, y 25 Determine of the implicit function cos y- x = y? 4 a G) 125) PTO TIIIQ{EMA1T QUESTION 3 A closed rectangular tank is constructed io Sccupy 70 dm? and its length is thrice its breadih. Caiculate the height that will minimise the quantity of material io be used. HINT: First draw the rectangular tank and then develop the formulas for volume and (6) area. A conical shaped ice artwork is exposed to ihe sun, The volume of this cone decrea- ses at a rate of 100 cm?.s~! and the radins decreases at a rate of lems”. Caleulati the rate ar which the height changes when the hight is 10 cm and the radius is3 cm . unt: y=taeh iso.com (4) 3. GIVEN: e faj=-1y 1, +2x-] 6 24 — 331 Determine the coordinates of the turning points of f(x). rs (2) 33.2 Draw up a table of values of x and J(%), with x ra nging from x = -3 to (o) x=3, = @ 3.3.3 Draw a neat graph of f(x) between these values and show the suring points on it. tg Q) 334 Use the table and the graph to eslimate a value for the root between x = rS and x=2 of the equation -t 2 ae +2x-1=0 and then nse (OG Taylor’s/Newton's method to determine 4 belter approximation of thi Toot {rool correct to THREE decimal figures). G) 119} Downlo PTO NV!

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TAVIO(EMAI)T QUESTION 6 6] Solve the differential equation: dy --3e” dr—-9 dy = 0) G) Detennine the particular sol +X, given thar ay = =_ we 0. sh and x I 6) (8 TOTAL: = “400 MATHEMATICS Ns FORMULA SREET Any applicable formula may also be used. TRIGONOMETRY sin? x + cos? x= 1 + tan? x= sec? x 1 + cot x= cosec? x sin24™2sind cos 4 COs 24 = cost - sin2y 2 tan 4 1~1an tan 24 = sin? = 4% cox 2a Cos! d=" 4% cos 24 sin (A+ B)=sin 4 cos Bt sin Bcos 4 cos (4 + B)= cos 4 cos BF sin A sin B tan 4 + tanB * = BO adm sin 4 cos B= % [sin (4 + BY + sin (4 - By} cos4 sin B= % [sin (4 + 8)~ sin (4 - By 625 A cos B= % [eos (4 + B) +cos (A - 3)] sind sin B= {cos (d - B)- eos (44+ B)) sinx 1 J +819 X = —— cosy = cos x cosec x secx tanx = PTO TOU ANT Downloaded from hlayiso.com

Old 7 = .28809 = I- ope T X 098 awd ~ i x 309 T- = were 7 x 0) = vt = X 803 T = Ty 7 x Uys > + (4109 - x99309) U] X109.x 99809- x oaso9. 2 4 Prue) 4 xoa8} a] X UOLx das xXoag 24 (rus) uy x ,oasa0- xy 2 +(x 998) uy x pos x Tay a4 XUS xus- x S05 24x 809- x S09 xs oux = l 3°80] t _ x = ¥°'Bo| Lf oor P a a Fs < ny 2 pret eo id i ors oy JE VVOTELE -e- LOWMDOTL Old ie & =a — aan jn@oud 5 2@~4 f of =e @y~ NOLLVILLNG Waa +H, _ +0 ewe iy ex) WaOUILL TYINONI ET

INTEGRATION IAs) etsy de = M0) gs) - J ray a(x) de SFP a= im fea) +e" +] fear fey a= FOr" nti fa) ~ 4, 8 (art bYex+ a) arth td 4 8 + Oy. (eta Gta)” ta? @eap APPLICA VIONS OF 1 NTEGRATION LREAS is if ye; 4, = i" 04 = ate = f Xe! 4, =f (4 - x)dy a T1130(E)(4 1)T S fede “Gta PTO VOLUMES ty =nf ae: ry =afl (2 ~ 39 \e Kerf dv it, =nf (? -3)y SECOND MOMENT OF AREA nef dds Ly = ft rad MOMENTS OF INERTIA Mass = density x volume Af= py DEFINITION: = m 72 GENERAL: J = t Pam = pf gy (ANT Downloaded from hlayiso.com

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Mt Jor nok tains = ah dy = Ei es 8, fab) aw = bean Xe ~3.= zh)- eo - 2)" + 6, = Aw =e aw © 2b, = 188717 - : () F Al = . rn ats = ho (a0) 0 — Ko = Te = es 27 = iw. ou x — " I Se ie Fa] T= a | x, dw — e 15) aCe, L= Eby fe + ole) + PY =e [3 I” ~ an are 10, = Lal a Lo= wa” Ao. mavtde = 6 v Ys Ga ~ eh ow + 29890 + 117 = Li] : cE Gawatiow / Thang 6 7 z +7 9 y L t an he = OE de = yt dy “3 aly : <i Te = peri x T. ow = ltyta).d (J si ; fa (3) = dt = w- 8 + oe 7 Jy. dw = | fey 4p). dy < alu a Y i y x j * y s = Fy. pe. yt +t = Wt te =~ eto, mte 24" t day G) E Seale ee] Teas 108 Vy {tT } are Crp Mt -02 05

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Mathematics Wiskunde NS Menorandum August 2011 Mathematies(Wiskun: Memorandum . z-i + | " we jt) = G@@-7F = a . i L22 b= arctan x Z ig (2) _ tt] (oy = % L7d 2 ee ec dy a Casali ‘rang 2 a any whew tiny =f To wy O ~~? ds i “s ron a {= cea*y phir y= wu Lel { = dan «= “eon — i : : : ners > dg fer) to% — (sind oie 8 di = (cctx)* hs a ae nid , a 2.3.1 db = tL] artidin (e-™ a (ema LE hic 4 el ia (3) i 4 im va = en Lae = (ux? } fan - Ea) a : 3 te i i Tai T igo Ge = Aten dus fi a 7 eayae (Phat) le e- 23.3 de =O”. ak o 9 EI j Lk ye —— v J =o AN 2) hogs Qo io~! V7 TE ——z > yeRrours — Aeey —_— _ i z 14a. —~—-- ; a : ‘ yk Ft Aw lan) Tt Bao wai. 0” = ! \ aa -t t w | $a

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i ql ea 7 + ye Liye ee a of” ae highei: education Z & training : Department: Higher Education and Training REPUBLIC OF SOUTH AFRICA T1100(E)(A2) T AUGUST 2010 NATIONAL CERTIFICATE MATHEMATICS N5 (16030175) 2 August (X-Paper) 09:00 — 12:00 This question paper consists of 7 pages and a 5-page formula sheet. Downloaded from hlayiso.com

eM, ie or , 4) T1400(E)(A2)T nad, : cn rn ae Ry : a 7 DEPARTMENT OF HIGH. R EDUCATION AND TRAINING REPUBLIC OF SOUTH AFRICA NATIONAL CERTIFICATE MATHEMATICS N5 TIME: 3 HOURS MARKS: 100 INSTRUCTIONS AND INFORMATION 1. 2. Answer ALL the questions. Read ALL the questions carefully. Number the answers correctly according to the numbering system used in this question paper. Questions may be answered in any order, but subsections of questions must be kept together. Show ALL the intermediate steps. Simplify where possible. ALL graph work must be done in the ANSWER BOOK. Questions must be answered in BLUE or BLACK INK. “Write neatly and legibly. PTO Downloaded from hlayiso.com

QUESTION 1 ‘ 1 Determine the values of the following limits: ne ; im 1-tan? x xoS 4 | zi x 8 2 1.1.2 jim | x-2 x32 i =) 2 12 Determine the value(s) of x where f (x) = —_—— is discontinuous. x? 45x+6 QUESTION 2 2.1 Given: y= arccos x 2.1.1 Make a neat sketch of the graph y = arccos x for the range [o; n}. . db 1 12 P that if y= arccosx, then —- = -—=——= =. 2.1 rove y x az oo 2.2 Calculate 2 of the following, by making use of the derivative of sin x and cos x, as well as, the rules of differentiation: ystanx 23 Determine 2 in each of the following cases: (Simplification is NOT required.) e 2.3.) = Tas) In* x 2.3.2 y=x? arc cotx 2.33 y=afx* +sec 7x PTO Downloaded from hlayiso.com (2) (2) {8} (2) (3) Q)

ie . . FAVOGCEMAQT a ” ‘ 24 Determine the slope of the tangent line at the point (1:2) of the graph: devin ; 2.5 Calculate 2 if y=4" ‘with the logarithmic differentiation. QUESTION 3 3.1 The sum of the slant height (4,) and the perpendicular height (A , ) of a right circular cone is 20cm. Calculate the radius and the perpendicular height of the cone, if the volume is to be a maximum. HINT: Volume ofa cone smh, and h? = hk? +r? 3.2 The motion of a point on a spring is modeled according to the equation, 5" cos mt s(t) =3e where 5 is measured in centimetres and ¢ in seconds. Determine the following: 3.2.1 The displacement after TWO seconds 3.3.2 The velocity after ¢ seconds 3.3 Given: f(x) =~ -2x? 43x41 3.3.1 Determine the coordinates of the turning points of 7(x). 3.3.2 Draw up a table of values of x and f(x), with x ranging from x=-4 to x=2. : 3.3.3 Draw a neat graph of f(x) between the values and show the turning points on it. 3.3.4 Use the TABLE and the GRAPH to estimate a value for the root berween x=-] and x=0 of the equation —x7-2x? +3x+1=0 and use Taylor/Newton method to determine a better approximation of this root (the root must be correct up to THREE decimal figures). PTO Downloaded from hlayiso.com (4) (4) 25} (6) (2) (2) @) 2) (4) (20}

Bee ; Trro(Enaaysee! | ny : i ‘QUESTION 4 4] Determine foae ift 4.1.1 yse™ sinx (4) 4 2 a ae =—— a @) 4.1.3 -—1_ “ 7754 bax (3) 4.1.4 y# cos 7x.sin 5x (3) cos 2x 4.1.5 y= ee sin.x + cosx (4) 42 Determine fy.dr by resolving the integrals into partial fractions: x+] rr ers x —10x+425 (5) [22] PTO Downloaded from hlayiso.com

QUESTION 5 ine [e~* fat if fy=—t 5. Determine [e Sat if N= 2. «) 5.2 The sketch below shows the area bounded by the curves: Loge yere, y= +1, x=3 and the y-axis. 2 \ fF 7 \ ‘ / \ \ 4 SSS i Sr 2 a eT -a} 5.2.1 Draw the sketch in the ANSWER BOOK and show on the sketch the enclosed area, the representative strip, as well as, the lower and upper limits. (3) 3.22 Calculate the magnitude of the area in QUESTION 5.2.] (3) 5.2.3 Calculate the volume generated when the area in QUESTION 5.2.2 rotates about the x-axis. (4) 5.3 Determine the second moment of mass of a rectangular lamina of mass m about an axis parallel to ONE side of the lamina. (4) (18) PTO Downloaded from hlayiso.com

TH1OOKENAZIT:. QUESTION 6 6.1 If the gradient of a curve is 2 = = , find the equation for the curve if it is known y that y=1 and x=. (4) 6.2 Determine the general solution of the following: 2 ay = fe" + iy dx? (3) 7] TOTAL: 100 Downloaded from hlayiso.com

ad fs) = f(x) J f(xpae ae n+l a nax”! te n+l a axte s x x ete x : a a dina te Ina i logex = _ x | ] og ax Bax xina = sinx cos x -cosxte cos x -sin x sinx+¢ tanx sec" x In (sec x) +c cot x -cosec? x In (sin x) +e sec x sec x-tan x In [secx + tanz] +s coset x “cose x*cot x In (Cosecx = cotr) + ¢ : | sin” x = _ Ti -x° 4 -l cos” x = 1-x* “ | tan” x z 1+x* ~ . -f cot" x 5 —_ J+x sec’ x ! xx? -1 ~~ ty . -) cosec” x PTO Downloaded from hlayiso.com