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higher education & training Department: Higher Education and Training REPUBLIC OF SOUTH AFRICA T950(E)(A2)T APRIL EXAMINATION NATIONAL CERTIFICATE | MATHEMATICS N3 (16030143) - - -13200-16:00 n paper consists of 7 pages and a formula sheet of 2 pages. Downloaded from hlayiso.com Copyright reserved Please turn over

(16030143) 2 T9SQ(E)A2)T DEPARTMENT OF HIGHER EDUCATION AND TRAINING REPUBLIC OF SOUTH AFRICA NATIONAL CERTIFICATE MATHEMATICS N3 TIME: 3 HOURS MARKS: 100 NOTE: Diagrams are NOT drawn to scale. 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. i 4. Show ALL the calculations and intermediary Stops... 5. Questions may be answered in any order but sibbsections of questions must NOT be | separated. soy AN , 6. ALL final answers nitust be accurately approximated to THREE decimal places. 7. ALL graph siork «must be done in the ANSWER BOOK. Graph paper is NOT supplied. ce 8. A forinula sl ect is attached to this question paper. The list is NOT necessarily ete, Any other applicable formula may be used, Downloaded from hlayiso.com Copyright reserved Please turn over

3 T9S0(E)(A2)T (16030143) ; 3 QUESTION 1 Ll Simplify the following expression without using a calculator. i od 1 xr+x? {# -x | (4) 12 Determine the value of p if x—2 isa factor of the following function: f(s) =2x° + px? —4x4+5 } (3) 1.3 Factorise as far as possible in prime factors: 1.3.1 ay -2x-2y @) 1.3.2 5 2 dx? -3-— ft lig «) 1.4 Simplify: p-l2l, p-2_, 2p-22 pa—4 <Qpt22 pr+2p+i(pt2). - ©) [20] QUESTION 2 2A Solve for x: 21d. AP x9? 51 (3) 12 oF RFF =12 (4) So Jog ¥ + log 3—log(3x—1) = log 4 (4) ‘; . \ *s - ‘The aréa of a rectangular swimming pool, 10 m long and 3 m wide, doubles if its length and width are increased by the same amount. Calculate the dimensions. (4) 2.3 Make m the subject of the following formula: W- 2mgh \ mr? +1 (4) 2.4 Make T the subject of the following formula: ‘=0.le" Downloaded from hlayiso.com ° Copyright reserved Please turn over

(16030143) 4 T9SO(EMA2)T 2.5 Determine x by completing the square: 3x =10~x° (4) [27] QUESTION 3 3.1 In FIGURE 1 below L(-5;-2), M(-1;-6) and K(5;4) are vertices of trianglé KLM in a Cartesian plane. “ Y¥ K(5;4) a x “ 7 L(-5:-2) M(-1:-6) y FIGURE 1 Calculate the coordinates of the midpoint of MK. (2) Determine the gradient of LM. (2) 3.1.3 Determine the length of LM. Leave the answer in surd form, Q) 3.1.4 Determine the equation of the line parallel to LM passing through the point K. Leave your answer in general form. GB) 3.2 Determine the equation of the straight line that passes through the points A(2;4) and B(-1;-3). Leave your answer in gradient-intercept form. GB) Downloaded from hlayiso,com 3.3 Calculate the angle of inclination of the line in QUE: (3) Copyright reserved Please tum over

(16030143) -S- T9SO(EWA2)T 3.4 Determine the equation of the line AB that is perpendicular to the line f(z) = ne in FIGURE 2 below. AB cuts the x-axis atx = 6. Leave the answer in surd form and in gradient-intercept form. Y i ’ (3) | FIGURE 2 fda} indices and in surd form. 4.1.1 yeni a Wx (3) | +x? 4.1.2 = 5 @) Downloaded from hlayiso.com Copyright reserved Please tum over

(16030143) -6- T9SOKE)(A2)T 42 Consider FIGURE 3 below. Determine the x-coordinates of the turning points P and Q of foyae 9x Q FIGURE 3, 9] QUESTION 5 : 5.1 The design of a bri ein FIGURE 4’shows an arch represented by a parabola, which is defined by the equation y =2x - 0,252". ; ww.dragoart.com FIGURE 4 5.1.1 Draw the graph of the parabola. QB) 5.1.2 Use the graph to determine the height of the bridge (in meters) if the line y = 0 represents the water level of the river. qa 5.13 Dewnleaded.from.hlayise.com. (1) Copyright reserved Please turn over

(16030143) -7- TISOCE AQT 5.2 Draw the graphs of the following trigonometric functions on the same system of axes for 0° <x <180°: y=3sin2x and py =2cos3x ALL values at the point of intersection with the system of axes and the co-ordinates of the turning points must be shown. net QUESTION 6 ; 6.1 Simplify the following without using a calculator: tan(180° ~ 4)Vi—sin? A cos’ (180° + A) +sin? (360° ~ A) ; ; (4) 6.2 Make use of basic trigonometric identities to prove thai: bcos f sin B =2e0secB o sinfi i+cosf . : (4) 6.3 Calculate the values of a that will Satisfy the following equation for 0° < @ <360°: sina =—2cos a GQ) 64 Consider FIGURE 5. From the top of a cliff 60 m above sea level the angles of depression of two ships ori'the same vertical plane as the observation point are 20° and 25° respectively: Calculate the distance between the two ships. a FIGURE 5 4) Downloaded from hlayiso.com total: —_—_199 Copyright reserved

(16030143) MATHEMATICS N3 FORMULA SHEET Any applicable formula may also be used. 1. Factors/Faktore “le T9SOCEKADT 2. Logarithms/Logaritmes ab =(a—-bYe’ +ab +5?) ae +b? =(a+ 6a’ ab +b?) 3. Quadratic formula/ Kwadratiese formule pa lbt vb? = 4ae 2a 4. Parabola/Parabool year tbx+c _ 4ac~b? , da -b xo 2a 5. Circle/Sirkel logab=logatlogb a log—=loga—logb og oga-log loga” =mloga log, a= log, 6 log, a@=1..Ine=l qlee! =te.e"™ =m 6. Straight Line/Reguitlyn ey ar? x D=—+h 4h x= 4Dh- 4h? yoy smlx- x) Perpendicular: Loodreg: m,®m, =-1 Parallel Lines: ° Ewewydige lyne: m, =m, Distance: Afstand: D= (x, -x)/ +(y,-»,) Midpoint: Middelpunt: P= Gemeaee Angle of inclination: Hellingshoek:, @= tan” m Downloaded from Hlayiso.com——____ Copyright reserved Please turn over

ta (16030143) 7. Differentiation/Differensiasie dy _™ f{e+h)- se) dx 30 h Sle" )= nx” Max/ Min Maks/ Min For turning points: Vir draaipunte: f'(x)=0 8, Trigonometry/Trigonometrie sin@=2 = u r cosec? cos@=—= ! r secO tno=2-—1 x cote. sin? 6 +cos’ @=1 1+ tan’? O=sec’ 0 1+cot? @= cosec’@ sin 0 cos@ cote = cos@ sind sind sinB_ sinC a 5 ce a? <b? +c? 2d wnloaded from Copyright reserved hlayiso.com TOSOCEXKA2)T

higher education & training Department: Higher Education and Training REPUBLIC OF SOUTH AFRICA MARKIN G GUIDELINE — Ces: NATIONAL CERTIFICATE APRIL EXAMINATION MATHEMATICS N3 2 APRIL 2015 This marking guideline consists of 9 pages. Downloaded from hlayiso.com

MARKING GUIDELINE -2. T930(E)(A2)T MATHEMATICS N3 QUESTION 1 it tt toa foot es waxy 2b x2-x? x24x2-) x2? -—x 2? 1 1 a aVx4—~| vx-7= Hxttuet~x? 4px? Te oy? -22 (=n) v v = 2x vx (vx 2 We _x+l-x+1 v x x dx 2 wr y vx x 1.2 wx=2Q f(x) = 2x3 + pr? 4045 f (2) = 22) + p(2y —4(2) +5 ¥ =16+4p-8+5 =4p4+13 4p+13=0 13 4 v p= 13 13.1 vo -y-2x—2y = (x- yet y)-2(et y) = (xt y)(x-y-2) v 13.2 ; 2 v 2x* 3x? -2 v “Vr _Qe+DG'=2) v x2 14 peri2l po? 2p-22 p-4 2p+22 p> +2p+1l(p+2) _(P-1)(p+1) | p-2 Pet 2+ Mp +2) (p-2)(p+2) Up+1) 4 p-11) y _(p-U)(p41D_ (p-2)_ + ZT v ~i) Rewaloaded- trem hlayiso.com _ptll v 4

MARKING GUIDELINE eae MATHEMATICS N3 QUESTION 2 21 14 33 32-4 = 3° P, 3x+2x-4=0 5x=4 ¥ 4 v x== 5 Dl fe en ZI a se 27427 =12 2742?2=12 Vv 2? (1+2)=12 ¥ v 2? =2? x=4 v 2.1.3 log x + log 3—log(3x—1) = log 4 log x + log 3—log(3x—1) = log 4 v log Ba =log4 (3x-1 Y 3x=12x-4 y wat v 9 2.2 (10+ x) +x) =30x2 30 +13x-+ x? =60 v 2 = x +13x-30=0 E x=-15 orx=2 New length = 12 m y New width = 5 m v Downloaded from hlayiso.com T930(E)(A2)T

MARKING GUIDELINE -4. MATHEMATICS N3 23 we 2mgh V mr +I wmr? +I = 2mgh V wir’ —2mgh =-w I mwr? —2gh)=-wl v as wil aw 9 =" ” wr? —2gh mm 2gh-wr? 24 En i=0,le? t 900, t 309 n= Ine? 0, v int = 900 01 T p 200. ___ 900 v v nf. Int—In01 0,1 2.5 2 2 vs3c(3) -0-(3) 2 2 2 (=+3) =1042 v 2 x42at 10+2 2 4 xo~2+ 40 v 2 4 -347 2 2 x=2 or x=-5 v v QUESTION 3 3.1 3.1.1 midpoint of MK. Downlaaded from hlayiso.com =p aade viv = (2;-1) T930(E)(A2)T [

MARKING GUIDELINE -5- 3.2 3.3 3.4 MATHEMATICS N3 3.1.2 ~64+2 3.13 IM =f145y 4(-642)° =Vi6+16 v =V2x16 =4/2 3.1.4 yoy, =m(x—x,) y-4=-la@-5) prd=-x+5 Vv prx-9=0 7 y-4=2 (2-2) Vv or y= 2,333x - 0,667 Vv @ = tan"! (7) v Vr-Yy=m(x-»%) y-02-V3(x-6) v y=—v3x+6/3 v Downloaded from hlayiso.com T930(E)(A2)T

MARKING GUIDELINE 6 a T930(EY(A2T | MATHEMATICS N3 | QUESTION 4 4.1 4.1.1 pense x =x! oy D a ax ax v =2 224 Ye wv if 4.1.2 etx) Sx” v =tvy4, » 5 B® 3a,t v v ax 5 ! 4.2 pax ~9x ® a3 ~9 de 13? -J=0 Y x? =3 xaty3 ”.X-coordinate of P =—/3 v v X coordinate of Q=/3 Downloaded from hlayiso.com

T930(E)(A2)T 7. MATHEMATICS N3 MARKING GUIDELINE QUESTION 5 4 meters 5.1.2 8 meters 5.1.3 Downloaded from hlayiso.com

MARKING GUIDELINE 5.2 -3- MATHEMATICS N3 7 7 T930(E)(A2)T 1 fi i ' fl oe iaieiatiaiaianel att 120° y=3sin2x x-intercepts: 0°,90°,180° y-intercept:0° maxpoint:(45°, 3) minpoint : (135°, ~3), QUESTION 6 6.1 tan(180° — 4)VJ1—sin? A cos* (180° + A)-+sin?(180° — A) ~tan Avcos* A ~ (-cos Ay +(sin Ay __atan Acos A cos’ A+sin? A =—sinA v v y=2c083x x-intercept : 30°,90°, 150° y-intercept : 2 maxpoint:(0° 2), (120°, 2) minpoint : (60° 2), (180°, -2) Downloaded from hlayiso.com

MARKING GUIDELINE -9- MATHEMATICS N3 62 (1+ cos B)(1 + cos 8) + (sin £)(sin f) sin B(1 + cos f) _1+2cos B+ cos” # +sin* B v sin B(1+ cos £) _ 20+ cos B) v sin B(1 + cos 8) v 2 sin J v =2cosecf . LAS = RAS 6.3 sina _—2cosa@ cosa cosa tana =-2 v a = tan” (-2) reference o = 63,4° a =116,6° or a = 296,6° 64 In A ABD tan 20° = 22 BD v BD = 164,849 m IA ACD tan25°= 20, cD CD =128,67 m a v BC=BD - CD=36,179 m Downloaded from hlayiso.com T930(E(ADT TOTAL: { U 100

Copyright reserved higher education & training Depaniment: Higher Education and Training REPUBLIC OF SOUTH AFRICA T940(E)(NI8)T NOVEMBER EXAMIN. on paper consists of 6 pages and 1 formula sheet of 2 pages. < is a Downloaded from hlayiso.com Please turn over

(16030143) TOOCE)NIS)T DEPARTMENT OF HIGHER EDUCATION AND TRAINING REPUBLIC OF SOUTH AFRICA NATIONAL CERTIFICATE MATHEMATICS N3 TIME: 3 HOURS MARKS: 100 INSTRUCTIONS AND INFORMATION 1. Answer ALL the questions. 2. Show ALL the calculations and intermediary ste 7. A formula, sheet i complete. Any othe ached thisgquestion paper. The list is NOT necessarily x applicable: ormula may be used. ue Downloaded from hlayiso.cgm Copyright reserved Please turn over

(16030143) 3. QUESTION 1 1.1 Simplify the following WITHOUT using a calculator: Ld (log, 2) (og, b) 1.1.2 ve x Vx 113 aig -3/35 J2+¥8 1.2 Make use of the factor theorem to prove that xe afactor of the function f(x)=x? -39x +70 1.3 Factorise the following two expressions pi 4m? +4in—1 14 QUESTION 2... | 2.1 “ Solve for x in the following equations : 2A See 5 =x <3 2.4.2 log, x +log, x’ tlog, x° =18 2.13 1 tae (t) on 3 2.2 Make + the subject of the following formula: P=0,4e"" 23 Make x the subject of the formula: _ [xrl x+2 Downloaded from hlayiso,com a ease turn over Copyright reserved T940(E)(N18)T G3) (3) (3) @) @) [20] GB) GB) ™) (4) 4

(16030143) 4. T940(E)(N18)T 2.4 — Solve for x by completing the square: x? =12x+10 2.5 Calculate the coordinates of the points of intersection of the graphs defined by the following two equations: x =+2y=17 rane Jara 4 2.6 The length of the rectangular field is 6m longer than snes QUESTION 3 exzaxis at E and P and the 3.1 Consider FIGURE 1. The parabola 7 ips it Re skelebgouts ¢ y-axis at M. The straight line ¢ pr x ya a E and M respectively. The functions fand g are defined “a yes “> rand g(x)smxte. Shee N | x FIGURE 1 Determine the following: 3.1.1 The co-ordinates of M, E and P which are the intercepts of the graphs with the system of axes . 3.1.2 ‘The co-ordinates of the turning point N of the graph of f 3.1.3 The values of mand cin g(x)=mx+e 314 Renmilonsagiiaempdalayiso.com Copyright reserved . Please turn over @)

(16030143) -5- T940(E)(N18)T 3.2 Determine the equation of a straight line that passes through the point (-1,2) and which is perpendicular to the line y= 4x+ 2..Write your answer in general form. 3.3 Determine the value of p if: 3.3.1 p isthe x-coordinate of the nsidpoint of the line segment joining B,-D and (1,-3) 3.3.2 p is the angle of inclination of the line y= 3x4 3.3.3 3.3.4 3.3.5 (5x2) QUESTION 4 41 ar Determine lim hod . Write your answer with QUESTION 5 5.1 Solve the following equation if 0° <9 <360°:: 3cotO+2=9,5 9.2 Use trigonometric identities to prove that tan? - sin? @ = tan? O.sin’ @ Downloaded from hlayiso.com . f ' _ Copyright reserved ' . Please turn over (3) (10) [22] @) @) @) [ty @G) 4)

(16030143) 6 | T940(E)(N18)T 5:3 5.4 5.5 Consider FIGURE 2. A window washer on a ladder looks at a nearby building 100 m away, noting that the angle of elevation of the top of the building is 18,7° and the angle of depression of the foot of the building is 6, 5°. How tall is the nearby building? 18,7° @ Fig 4) Consider FIGURE 3. Calculate the length Fusing the sine rule. ae A 58° 72° B 17m : c FIGURE 3 (2) Draw the graphs which are represented by the following trigonometric equations on s the same system of axes for 0° < x <180°: s f(x) =cosx g(x) =sin(x—30°) (4) [17] Downloaded from hlayiso.com yoraL: 100 Copyright reserved

(16030143) 7. Differentiation/Differensiasie -2e dy tim flx+h)— f(x) ax hs h ("=n Max/Min Maks/Min For turning points: Vir draaipunte: f! (x) =0 8. Trigonometry/Trigonometrie sin? -2. | r cosecP cos@= aa | r secO tan 0 = ca u x coté sin? + cos? @=1 1+tan’ @=sec? 0 1+cot? O=cosec?@ sind tanO= cos0 cotd= £08 é sin 8 sind sinB sinc a b ec a =b? +c? —2becos A Copyright reserved T930(E)N18)T Downloaded from hlayiso.com

16030143 FORMULA SHEET Any other applicable formula may also be used. 1. Factors/Faktore -1- T930(E)(N18)T 2. Logarithms/Logaritmes a-b =(a~ba? +ab +b?) a +6? =(a+ Va --ab+b?) 3. Quadratic formula/ Kwadratiese formule —btvb? -4ac 2a x= 4, Parabola/Parabool ae year tbxt+e _ dac~b? y 4a 2 logab = loga + logb a log—=loga ~logh oF g 08 log, a log, b log, @= ¢ loga” =mloga log, a= = b log, log, a=1..Ine=1 e Inve all ape, =m 6. Straight line/Reguitlyn Downloaded fr Copyright reserved YrW= mlz =x) Perpendicular: Loodreg: m, em, =-L Parallel lines: Ewewydige lyne: m =m, Distance: Afstand: D= VG, — x y + (y, “yy y Midpoint: Middelpunt: P= St %e a he 2 2 Angle of inclination: Heil G2 Om & Please turn over

higher education &training Department: Higher Education and Training REPUBLIC OF SOUTH AFRICA MARKING GUIDELINE NATIONAL CERTIFICATE NOVEMBER EXAMINATION MATHEMATICS N3 18 NOVEMBER 2014 This marking guideline consists of 9 pages. Downloaded from hlayiso.com

MARKING GUIDELINE -2- T940(E)(N18)T MATHEMATICS N3 QUESTION 1 Ll LAE (log, a) (log, &) log, b ¥ apply law 1 loeb orlog, ax log,a ¥ ¥ simplify and = 084, 1080 v * write it as ONE logx loga = log. log _ logb v logx v =log, 5 1.1.2 ve + * * “common _ axle tay ¥ denominator nfs ¥ factorisation x x+y) ¥ divide and at ed v rationalized xx wx denominator _fl+y) vx _ved+y) ” x 113 2x3V2 —3x 42 , Vorime factors nn aay Time tractors 10 v2 + av2 roots _ V2(6-12) v Y factorisation J2(1+2) ¥ value =-2 , ¥ 1.2 Let x=-7 = “value of x S(-ND=-7 -3%-7) +70 v v . : =—343 + 273+70 ¥ substitution =0 eduction “x+7 isa factor of f(x) v v 13 13.1 p-4m +4m-1 ¥ factorise =p -Qm-)Qm-1) v , ¥ factorise =p? ~(2m-ly ¥ simplify ¥Y s[p-2m+l][pt+2m—-i] v Downlaaded from hlayiso.com 13.2 3|(4a? — 9p? 1] v ¥ factorise =3[(20 + 3b\(2a — 36)| “factorise

MARKING GUIDELINE 23. MATHEMATICS N3 v 14 x? 42x x3 x3 + x +4 x +dy-4 x42 _ x42), ee +4@-D) x-3 x +4 x~3 x12 v =X +2) +4 G@-D x3 v +4 x3 x+2 =x(x~1) v QUESTION 2 x+5=(x43) v X+5 55° +6x4+9 x +5x+4=0 v (x+(x44)=0 x=-lorx#-4 4 2.1.2 log, xt+log, x’ +log; x° =18 log, x” =18 Glog, x =18 log, x =2 v x= v x 25 v 2.1.3 ye gag? fo] an] () gel 4+ 3%? 3° =11 ¥#(34+9-1I)s1l vv F=l 3 23° v x= v T940(E)(N18)T ¥ division sign V factorise first term ¥ grouping Y answer ¥ square both sides ¥ factorise solve (penalise with 4% mark if -4 is not shown as invalid) v write it as one log by applying law Y applying definition of log “value Y applying exponent law ¥ factorising ¥use exporient rule to find value ¥ value Downloaded from hlayiso.com

MARKING GUIDELINE 4. ' MATHEMATICS N3 2.2 P=0,4e" Ppt y 0,4 0,4 v 23 xl x+2 H? (x4+2)=x~1 HPxt2H? =x-1 HW?’ x—x=-2H? ~1 x(H? —1l) =-2H? -1 v _ ~(2H? +1) Pl 24 x? -12x-10=0 v 2 2 x? -120-(-2) = iox[-2) 2 2 (x-6) =46 x-6= 4/46 x= 64/46 x=12,782 or x=—0,782 v v Downloaded from hlayiso.com T940(E)(N18)T ¥ manipulate “using logs “simplify “make r the subject v square both sides ¥ cross multiply V take out t as common factor ¥ simplify VY write LHS as perfect square ¥ x subject of the formula ¥ both values ¥ correct rounded off-penalize only with one here

MARKING GUIDELINE -5- . MATHEMATICS N3 2.5 S+2y=17 ._(l) V taye7 (2) 4 v x+l0y=85 2.x =85—-l0y v Replace x by 85-10y in equation 2: y44x=28 y+4(85-10y) = 28 v y+340—40y = 28 —39y=-312 y=8 Replace y by 8 in equation 2: Y 8 ee7 4 x=§ 2.6 Let breath be equal to x Then length (3x)+6 Perimeter = 2(1+b) 188 = Ax+3x+6] y 8x +12 =188 x=22m v 3x+6= 72m v QUESTION 3 3.1 3.1.1 let y=0 letx =0 > Osx —4x4+5 y=0-0+5 gn 5 yess ee 4 v orx=1 “4M =(0;5) Vv «FE =(-5;0) and P = (1;0) 3.1.2 -(- ee) 2(-1) TO40CE)(N18)T ¥ simplify ¥ substitutions “simplifying ¥ value of x ¥ value of y ¥v Any steps v answer y¥M YE ¥x-coordinate ¥y-coordinate x=-2 Dopnletdpd from hlayiso.com “y= oN =(-2:9)

MARKING GUILIELINE 6 ; MATHEMATICS N3 3.1.3 _ 5-0 yrmrte ee ors S=1@)+o v smal v .c=5 32 3.3.1 3.3.2 3.3.3 3.3.4 3.1.4 EM =J(%,-4)Y +0. -"y = (-5-0)? +(0-5)” =V254+25 = 50. 5V2 v 1 =4 n, =—-— m, m, r yry, =m@—x,) y-2=-Vi(x+l) vy Vv . po-5 Downloaded from hlayiso.com . T940(E)(N18)T ¥ Vanswer in surd form ¥ gradient ¥ substitution ¥ simplify vv value of p ¥v value of p vv value of p vv value of p

MARKING GUIDELINE QUESTION 5 5.1 3cotd+2=9,5 cotO =2,5 tan@ = 0,4 -8- MATHEMATICS N3 0=21,801° or @=201,801° v v 5.2 tan” @—sin? @ = tan? @.sin’ @ LHS = tan’ @ —sin’ 8 _ sin’ @ z= ~sin’ 6 cos’ @ _ sin? @—sin? Ocos’ 0 cos’ @ _ sin? 6(1— cos’ 0) 7 cos’ 0 = tan? @.sin? 6 =RHS wv 5.3 tan18,7° = BC 100 BC = 33,848 tan6,5° = ep 100 CD =11,394 Height = 8C+CD = 45,242m 5.4 in 72° ap alsin? sin 50° =21,106m v 4 v v T940CE)(N18)T ¥ manipulate for 0 ¥ ¥ two answers ¥ applying quotient identity ¥ finding die LCD ¥ factorise ¥ proof ¥ v determining BC “determining CD VYadding BC and CD to determine the height “replacing Y answer Downloaded from hlayiso.com

7 foe] og i ne 2g eB RIN 2 Bo R a > Bo = gg ¢ 28 £ & Ff 22 z 5 g Ca 3 8 B os & 1 8 8 Paes) if 2 DS 5 - 8 ga 41 8 2S = ~ S#8 EES BS £A255 88 z S SSE 62868 wv Soe 6&8 7 : : £ — t : : - : : : : _ — ieee n define : 2 H S fem eee ok tee y = » ' as : ma 1 EE piateteteted Tr 3 a On aQ i tees wow ee ew He YUL 4 a th a 1 an — 23 : za i a 1m Dp - 5 my ie) ' 6 : : i 5.5 graph TOTAL: Downloaded from hlayiso.com

MARKING GUIDELINE 7 MATHEMATICS N3 3.3.5 Loy ag or y-intercept 4 Oy? =81 anes oy? =81 x vo 3 ac =! y =9 p=B vv y=s3 p= QUESTION 4 4.1 _ jim ft Fe) A h ~4x? —8xh—4h? —(4x7) = lim 0 h = vw tim AHOX +A) v 30 h =-4(2x+0) =—8x v 4.2.1 1 yaz-4¥x x 1 pox —4xt v 1 a4 Ay ayt! * v =—-4x5-x . v 41 ~ 33 fe v 4.2.2 _ x(x? -D x-1 v x{x+1(x-1 yo2le DG =D x-l per +x . ® oy st T940(E)(N E8)T vw value of p ¥ multiplying of 2¢c+h)? and substituting ¥ factorising ¥ simplifying “answer ¥ rewriting by applying exponential rules y ¥ applying differentiation rule ¥ writing it as positive exponent and in surd form ¥ factorise vv differentiate Bownloaded from hiayiso.com 1 i 1

higher education & training Department: Higher Education and Training REPUBLIC OF SOUTH AFRICA T930(E)(M31)T Downloaded from hlayiso.com Copyright reserved Please turn over

(16030143) je QUESTION 1 Ll Simplify the following WITHOUT using a calculator. Lil Ly 7 92 _ 3H ae 1.1.2 log V27 +log V8 —log Vi25 log6-log5 1.2 Factorise as far as possible in prime factors: 1.2.1 22-2) £9(x—2)-5 122 (m-=ny36x" + 49ny? — 49m” 13 Given that x+1 isa factor of f(x), Determine the other factors if /( 14 Simplify: V3x410-x=2 a2 axes 23 x x 2.2 Make 'x' the subject of the formula by completing the square: x? +8x—S5a=0 : Downloaded from hlayiso.cem T930(H)(M31)T GQ) G) @) @) GB) (3) 6) B35] @) (3) Q) wa

(16030143) 4 T930(E)OM31)T 23 Make ‘t’ the subject of the formula: ” ¥, = Vie : (4) 24 Thomas is 4 times as old as John. In 12 years’ time Thomas will be twice as old as John. What are their present ages? QB) 2.5 Determine the points of intersection of the graphs represented by @ following equations algebraically: xy? =25 . i yext5 (5) | (21) QUESTION 3 L, 3. Complete the following sentences by fii Sissine values, expressions of | equations. Write only the missing ans ot fe question’ number (3.1.1-3.1.3) in the ANSWER BOOK. “ty ! 311 P(-};-1) and QC3, foordinates of the midpoint of o PQare... (2y 412 The acute anglBg ; equal t (2) | 3.13 The & () 3.2 In the dj pram be e ‘ is atangtint to thyrtite al'point P. Y-axis em P(-3,2) oN X-axis : Downloaded from hlayiso.com

(16030143) 5 ( T930CE)(M31)T 3.21 Calculate the length of OP. - ; (2) 3.2.2 Determine the equation of the circle. (2) 3.2.3 Determine the-equation’of the taagent AB.. 4 3.3 Determine the equation of the straight line that passes through point O (0;0) and which is parallel to the line 18x —3y +9=0 . GB) [16] QUESTION 4 4.1 Consider the function y =1~-4x" . 4. yp 4. Determine the derivative, =, che @) 112 Detormins the gra Gy 4.2 Determine a if ak A) mown system of axes. Calculations need NOT be shown. ALL oints @¥ intersection with axes and the coordinates of the turning points Cauiyemiust be shown, i 2 ane ae 256 dd Q) yo3x? 2x3 (4) 433 x yy 4°43 (2) (17) Downloaded from hlayiso.com .

(16030143) -6- T930(E)(M3 1I)T QUESTION 5 | 5.1 5.3 5.4 5.5 Calculate the exact value WITHOUT using a calculator: sin? 120°x sec? 150° + sin 150° * cosec 30°x cos 60° G) Calculate the value(s) of 8 which will satisfy the equation if 0° <@<360°: 2sin@cos@—sin@ =0 é 6) From the top of a tower, 100 m high, the angles of depression 0 5 ghia gon the ground, due west of the tower, are 41,45° and 21,68° x 5 Calculate the distance between the TWO object (5) Given : AABC with, A= 60°, AB= V2 ag Calculate: sy A 60° Va V8 B A c The value of ‘x’ 63) 5.4.2 The area of the triangle. Q) Sketch the graph of 3y =sin3@ for 0< 8 <120° (3) [21] TOTAL: 100 Downloaded from hlayiso.com Copyright reserved

MARKING GUIDELINE “2- T930(E)(M31)T MATHEMATICS N3 QUESTION 1 Ll Ltd Lo 9X Q 9g? 3"! oa _ ee xe? — 4 sa . ee TG > LL (ey 3 j ° 5 tO on = ye 4 a uae 2 ~. > - > 38 343°! - L$ ay 2 tr 8 2 1 : fs) Y 3 —) hee 4 Ss 4 ut x Quit, 39 ence 2 gov € oa a Moe obY Saye, ye 9 3 9 27 ( 3) 1.1.2 log V27 + tog V8 —log S125 log (Gy + log 4/2) —log Jy’ log 6—log5 ioe6 “ee? ; . log ee, joo VO y (le ao} ___V125 °8 ia ns eee 6 ~ 6 loc) br log 5 v a : log 105 loo ay ay ae = 5 (5) | oe a Ic log 6 = 6 J PLS log} ~ fp “LS 4 lose 5 bee 6y ov ; log 6 _ 5 log z 5 oe( | v 3 6 5 gl 3 6 2S aS 285) 3 6) *2 ef 24 af . log 5 lo (§ 2 *\s (5) Downloaded from hlayiso.com °

MARKING GUIDELINE -3- YO30(EKM3 IIT | MATHEMATICS N3 or 2(e—2)? +9(x-2)-5 12 Lat Mx 2F + 9(x-2)-5 lei x-2=k = Uxt dv +4) 9x 18-5 % ode +9k-5 = 2x? 8x48 49x-18-5 =(2k-IMk+5) ¥ & =I ¢x-td Factors of 2(x—2) +9(x-2)-5 = (2x -5\(x +3) =[2(x—2)~H[(r 2) +5] 7 | =(2x-S)(x4+3) v , B) | 1.2.2 Gn—n)36x" +49nyr —49my" = (mn —ny36x7 — 49)" (m1 -n) ‘x =(m—n)(36x" 499") a : =(m—1)(6x+ Ty)(6x-7¥) v @) 13 : xo-x-12 v yeljx? -13x-12 i - x?-13x -xi- x -12x-12 -12x - 12 2 f(s) = (et DOP =x -12)= (x4 De 4043) 14d ( 1 1 [: ‘| soy Flat xy yx yx vy x+y xy OF NO), xy yr-x xy r = xy (x+y) v Downloaded from hlayiso.com

MARKING GUIDELINE ade MATHEMATICS N3 12 4x 2x SX 3x¢1 1-3x 9x’ -1 4x 2x 5x = ! + v ¥ “3x4l 3x-1. Gx+)Gx-1) — 4xGe-1)-2xGx+D45x7 0 y Gx+DGx-D _ Lax? ~4x— 6x" - 2x +5x° Y Gx+NGx-1 Lx? - 6x * Gx4DGx—-1) QUESTION 2 2.1 2.1.1 Px+410-x=2 v3x+10=(«42)) Y o3x41l0sx 44x44 ow +x-6=0 v T930(H)(M3 7 (x+3}(x-2)=0 exb3=0 of 6-2=0 | ; ao 0” x=-3 or x=2 g wo? ae) X #-3 (1.a) extraneous solution “ x=2 is the only solution any clear rejection of the root is acceptable. 2.1.2 3 x43 . 2 ataet5= : mc ae xt} _ = A Btx 4 Sxex43 =e - wx? +4x=0 y a ate wex(xt4)=0 v a x=0 or xt4=0 e mm ~ on an x#0 7 x=4 VY oe . . Downloaded from hlayiso.com Ns nin (3)

MARKING GUIDELINE “5 T930(EMM31)T MATHEMATICS N3 Ae 2.2 vy +8x-Sa=0 ~ my wx 48x = Sa CX “OS Wa iS Lx £8x416=Sa+16 % Ne ean J (xt4y =5a416 \ wv va - a, ~ wxtdat/Satl6 ’ fu aye ) C oxe-diJ5a+16 4% < oe" Ss (3) Vv cf 2.3 Babe ay Jub se v ort o Inf, -In¥, a r a a (4 2.4 AGE NOW AGE IN 12 YEARS John x (x+12) Thomas 4x (4x+12) age of Thomas=2 x age of John 4x+12 =2(x +12) v 4x+12=2x+24 2x =12 v x=6 present ages = 6 & 24 yeas VW . : GB Downloaded from hlayiso.com

T930(E)(M31)T MARKING GUIDELINE -6- MATHEMATICS N3 2.5 Vey sie (1) YH OES coerce (2) replace y by x+5in (1) x4 (x45) =25 vex 410xt25=25 Vv 2x? +10x=0 2x(x+5) =0 r=0% or x=-5 i. ¥ If x=0 then y=5 orifx=-5 theny=0 ” QUESTION 3 cm 3.1.1 ae ve (-2;1) 3.1.2 ; aA 45° 3b m= 2 v 32 Bed oP =V(-3" +P wee . aa We =/9+44 =J13 units due er set \ pw ag et ee 3.2.2 e4yer’ twee o attr s(/By 7 vi arey sl 2 3.2.3 Mop *Myy =—) (OP 1 AB) 2 ay mae =-] a ©. May = ¥ - uel AB 2 oe . _—_ yr y, =m(x-%) x td 3 wyr-2=—(x+3) Vv y 5 ) a 3? . Downlodded from hlayiso.com 13 . 3 13 . “Rosa 5 v K G) [21]

MARKING GUIDELINE “1 T930(B)(M31)T MATHEMATICS N3 3.3 - [8x -3y4+9=0 v . - y ue ft (ec 0) p= 6x43 “4 - “Required line: p=6x 9) / . cee . : f G [16 QUESTION 4 41 4.1.1 pelea? oath hh 40 A — d(x? +2xh +h ?)-1 44x? = lig —— had A _ 14x? ~8xh—4h? 14402 = lim hod hh _ -8xh— 4h? = lin, ——- v ho0 fh hat h =-8x a Vv (4) 41.2 . if x= then ® gq) = 8 Y he a 4.2 2 oy fi -Vx-=— > x 1 v \y - of) sx? -14+2x! v ce ay wf) aan? 2x7 - oa . . 2 a : 1 2 vy oe X)= Re - T v Vv . Downloaded from hlayiso.com

TO3O(E)(MB INT fe! MARKING GUIDELINE 8. MATHEMATICS N3 43 ms “ — flare i i . : ' H if / | a aN yy 16“ x a -12 _. MARK ALLOCATION ___ | MARKS x iniercents at (16;0) and (- 16.0 50) 1 ee y intercepts at (0:12) and (0:-12) i rn 43.2 or j-x#90 or vel N a A 0,0) or Gt} i Iw I = 0 -x?(3-2x) =0 - y=Qorv=Oor3-2x=0 x =0 ~ ye y-intercept y=0 Downloaded from hlayiso.com

MARKING GUIDELINE ~9- MATHEMATICS N3 Y a 3 v v 2 1) 1 a NN “ ‘ 2 \ 0] — oy 1 — : a , —e -2 15 1 -0.5 0 05 { uy (0:5) \ : v : 2 & points) “PO3OCEM3 LYE 1 mark for tursing point(0;0)/ I | y-imtercept al O/x-interceptatQ oe . 3 \ x-intercepts at 5 and 0 Downloaded from hlayiso.com (4)

MARKING GUIDELINE -10- MATHEMATICS N3 T930(E)(M31)T 4.33 ¥ 7 3 v “ed 6 a | ALLOCATION OF MARKS MARKS ee L mark for x-intercept (439) i Limark for y-iniercept ai (0,3) en se ee QUESTION 3 3d sin® 120°xsec? 150° + sin 150° 20» eos BAP | v 2 v 5.2 2s8in @ cos@ —sin @ =0 v.sin@Qcos@-l=0 “sind =OQor v - 2cos@-1=0 “4 8 =0° or 180" or 360° 0=60° of300 YY v vv Downloaded from hlayiso.com (2) le [7] 3)

MARKING GUIDELINE “lie T930(E\(M31)T MATHEMATICS N3 5.3 C Tan6832=2- Tan48,55=—~~ 100 100 . x= 251 544m y=113,228m v .. Distance between objects = 251,544 m— 113,228 m ; = 138,316m v i 1 100 a | ortan 41,45° = 100 y= x ian 21,65 i . xa 190 _ = 251,929 0 } fan 41, 45° we “Distance= yx , = 113,228 = 133,701 (5) 5.4 aA veer + O2y —2V8N2Cos0 . 7 , «ff +2 wf ' v = V6 Ol 2,449 we Q) 542 a : Area =~ beSind 2 bog Se . see V2SOUS ¥ =1,732 mi? v () Downloaded from hlayiso.com

MARKING GUIDELINE -12- T930(E)(M3 1)T MATHEMATICS N3 $5 3y=sin30 _ sin3@ 3 | nineteen aed ne ca. 9ae | { i ra | mark for y-intercept at OP i oo T TOTAL: 100 Downloaded from hlayiso.com |

aa% yp higher education & training | _. Department ——-——. - --------—- Higher Education and Training: REPUBLIC OF SOUTH AFRICA T930(E)(I28)T AUGUST EXAMINATIO Downloaded from hlayiso.com Copyright reserved Please tum over

(16030143) 2 T930(E)I28)T DEPARTMENT OF HIGHER EDUCATION AND TRAINING REPUBLIC OF SOUTH AFRICA NATIONAL CERTIFICATE MATHEMATICS N3 - 2 cee ve. TIMED. 3-HOURS -—--- - -- MARKS: 100 INSTRUCTIONS AND INFORMATION 1. Answer ALL the questions. NO Show ALL the calculations and intermediary stey separated, 4. ALL final answers must be accuratel, 5. All graph work must be done i ‘question paper. The list is NOT necessarily a may’ be used. 6. A formula sheet is attaclied tot complete. Any other applicabie for Write neatly and-Jegi Downloaded from hlayiso.com Please turn over Copyright reserved

(16036143) “3. ‘7930(2)028)T QUESTION 1 ll Simplify the following WITHOUT using a calculator: (5) Qt x 3 gt 1.2 Prove that (log, a) (Jog, c)(log, b) =! 6) 1.3 Determine the factors of the following fimetion if x ~1 is one of the factor: fsx) 43x -x-3 (4) 14 Factorise the following expressions as far as possible in prj 14.1 x (x-1)+(-x) GB) 14.2 @ +2a—3+ab—b G3) 1.5 Simplify the following: ab @ ~2ab+ Qa-b 4a? -B2 4 [22] QHESTION 2 G) 3) “x4 “Y -1 (4) 2.3 Make ‘1‘ the subject of the formula : P=100e°" i) 24 Make ‘r? the subject of the formula: a=nr'+,420Wnloaded from hlayiso.com 0) Copyright reserved NN Please turn over * ape WS

(16030143) “4. T930(CE\U28)T 2.5 Calculate the coordinates of the points of intersection of the graphs defined by the following two equations: y= 2x? -8x-10 (4) p= 9x 2.6 The product of two consecutive odd numbers is 143. Calculate the numbers. (3) [26] QUESTION 3 3.1 AdBC hias vertices A(2;3), B(-2;-1) and C(4;)). 3.1.1 Draw AABC ona set of axes. qd) 3.1.2 Determine the gradient of AB. () () 2) @) @) G) 3.2 iH COLUMN B that matches an item in COLUMN A. Write xt to the question tumber(3.2.1-3.2.4) in the ANSWER COLUMN B y A x+y =10 B =+%=100 1 10 Q) lo 7 * 2 2 ~ 47 2100 1 10 Downloaded from hlayiso.com Copyright reserved Please turn over

(16030143) ‘F930(E)28)T Lo Ne bo D x=y y Copyright reserved Downloaded from hlayiso.com Please tum over

(16030143) 6- T930(E)(J28)T QUESTION 4 Determine o by making use of the rules of differentiation. Write your answer with positive exponents and in surd form when applicable. 4LL 2 : . = ae Lt * x @) 4.1.2 x8 —1 arn Q) 42 Determine the gradient of the tangent to the following 4 Ren neu : ya (x 341) @) » ) AS : $.1 Make use of basic trigonometric identifiés to prave thas, I+sin@ + cos? é : QUESTION 5 =2sec0 cos@ 1 +sin8 ae (5) 5.2 Q) A 259 80 46 B oo Cc . 5.3 Solve the following equation if 0° < B <360°: 26+4=0Dewnloaded from hlayiso. com _ Copyright reserved Please tum over

MARKING GUIDELINE -2- T1020(E)(528)T MATHEMATICS N3 QUESTION 1 ; MARKING : INSTRUCTION &NOTES i Ll gett y agai ¥ writing 8 as power with | — v base 2 . a wo ¥ writing 4 as power wit 28x22? 2 base 2 ! ape v ¥ simplifying root 2x2 Y applying multiplication | v rule of powers by adding ' aq DoHGH-2at2 74 16 exponents ¥ ¥ simplifying () i | 12 LHS =(log, a)(log,c)(log,b) RUS =1 ¥ ¥ applying log rule where \ 1 logb bases changed = OBE OBEY 108”, ¥ ¥v ¥ simplifying loge logb loga =1=RHS v i G3) i ; 1.3 AK 43 v v¥ for correct quotient x= xP 43x? - 2-3 ¥~ two factors- 2 v (x4+3)(x+1) 3 1 2 wie no credit to (x-1) it ; t4x?-x was given | 4x? - 4x +3x-3 3x-3 0-0 v v Factors : (x-1)(x+3)(x+1) i) L4 LAA 2 (e-D+(1-x) ¥ taking out -1 as common 2 v factor ax(x-1)-(x-D v Y factorise =(x-DOr-1) Y applying factorisation of = (x-D(e-D(e+1) v difference between two squares (3) 1.4.2 a +-2a—3+ab—b . Y factorising of quadratic ° . trinomial _ 2 =(@ +2a—3)+ab—b “taking cut of common =(a+3)(a—1)+b(a-)) v factor =(a-Dla+3+d] Yoev ¥ taking out of common factor Q) Downloaded from hlayiso.com Pléase turn over . : . Copyright reserved

MARKING GUIDELINE -3- MATHEMATICS N3 15 a’? a? —2ab+b' a-b Qa-~b da -B a tb v v ¥ _ (a+bXa~b) | Qatb\Qa-b) a~b 2a~b {a-bya-—b) atb =2at+b FT v QUESTION 2 21 da’ ~12a-7=0 4a -(2a=7% a—3a2e 4 3V. 7,9 a ~3a+|-=|)=—4+2 ( [-3)) a4 Sf («-2) ~16_4 / 2) 4 3 aq~—~=42 5 f a=242 2 7 1 Qa=—- Of a=~ > 2 2 2.2.1 yt araars( 2 =42 5 v 22 +2727 +27 =42 B[P+reil=42 v 2* [5,25] =42 v 3 v 2% =2 x=3 Copyright reserved TLO20(E)323)T ¥ change of division sign to multiplication sign Y (Vx2) factorisation using difference between two squares Y factorisation of trinomial simplifying 4) [22] “taking out 4 as common factor ¥adding the . 2 oe: of :) both sides v writing [hs as a square and simplification of RHS ¥ get rid of squares by drawing square roots both sides- Mark will not be awarded if tis left out. ¥ (x2) two answers (5) Y simplifying of negative power ¥ taking out common factor ¥ simplification of power ¥ solving by equating like powers 4) Downloaded from hlayiso.com Please turn over

(16030143) MATHEMATICS N3 FORMULA SHEET Any applicable formula may also be used. 1, Facters/Faktore 2, Logarithis/ Logaritmes T9300E)(J28)T a -8 =(a-D(@+ab+b’) a+b = (at bya -ab+ 8?) log ab=log a+ log b log slog a ~ log b, 3. Quadratic formula/ Kwadratiese formule log, a log, a= log r ~bt yb? ~ 4ac | 2a 4. Parabola/ Parabool | 2 ! voar* +hxr+e _ dae — b° da 5. Circle/ Sirkel ght line/ Reguitlyn 2 2 2 [ x + po =P gs, Perpendicular: Loodreg: m,-imy= ~] Parallel lines: Ewewydige lyne: my =m Distance: Afstand: = D= (ez ~ xy)? +@ - y,)? Midpoint: Middelpunt: P = (72 : arm 2 2 Angle of inclination: Hellingshoek: 6=tan/m - ° Copyright reserved Downloaded from hlayiso.com Please tum over

(16030143) 2 T930(E)(I28)T 7, Differentiation/ Differensiasie dh _ lim fern) - se) dx hoo h Max/Min Maks/Min For turning points: Vir draaipunte: f ‘@)= 0 8. Trigonometry/ Trigonometrie el sing=%=—+ 7 rk cosecO i COSO = = a : vr sec 1 7 tan@ == = —~— i x coid i sin?@ + cos’6 =1 D+tan?6 = see? ind _sinB _ sinC a .b° ¢ Pap pe? 2be cosA opyright reserved Downloaded from hlayiso.com

' MARKING GUIDELINE “4. MATHEMATICS N3 2.2.2 2x _ 4 4-x x-1 xt) x1 | 2xx+)—-4(x-1) 4x i (z-DY@t) — («-Darh ; 2x? 42x—-4e+4a4~x 2x*-x=0 x(2x-D=0 y X=U or ral ‘ 2 ; yf v ; 2.3 P=[00e°°" i Pam Vv 100 In a =In(e*™) | 100 In P—1In 100 = —0, 32Ine v i In P—1n.100 i oo = OF , ~0,3 v In f00—1n P on SI OF 0,3 | (2) ' AP lL, v 0,3 24 az ar +7rs 2 v ar’ +ars-a=0 pe estas’ +40 v 20 25 y= 2x? -~8x-10 y=-9x > 2x" -8x—10=—9x 2x? +x-10=0 (2x4+5)(x-2) $0 xa or xa] 2 v ¥ Copyright reserved Downloaded from hlayiso.com T1020(E)28)T ¥ correct LCD(denominator) Y rewrite fractions to LCM(numerator) ¥ simplification ¥ (¥x2)-solution Check the roots @) ¥ manipulation Yusing logarithms to manipulate ¥ / simplify co) ¥ using correct values in the place of a,b,c in the quadratic formula ¥ manipulation and simplification 2) ¥ v finding the x-coordinates vv finding the y-coordinates of points of intersection 4) Please turn over

MARKING GUIDELINE -5- 2.6 MATHEMATICS N3 5 nol -2;22—Jand (2; -18) vo ov (3322 )and (2; -18) XX(x+2)=143 - x’ +2x~-143=0 Vv («+ 19G-1)=0 x= -l3 x=I]l vw vw Two numbers: 1] and 13 Or -Ll and -13 QUESTION 3 34 3.14 T1020(E)(328)T Yany steps to show how the student determined the answers vv answers Students could determine answers also using inspection. 3.1.2 ~ May = +s ts X,-X, _3-CD 2-(2) 4 = t=] 4 v 3.13 My, = a=tan'(1 an” (1) y . =45° (3) [26] ¥ sketch of triangle (hy ¥ substituting values and simplifying q) “substituting value and simplifying (1) Downloaded from hlayiso.com Copyright reserved ° Please turn over e >

MARKING GUIDELINE -6- MATHEMATICS N3 3.1.4 M.= ¥,-¥, eX, - Xe MyXM,. =Mx(-D=~ Therefore the two lines are perpendicular. 3.1.5 Parallel lines have equal gradients. The gradient of the line through C(451) and parallel to AB with M,, =1 y Y-Yo =M(X- Xe) y-l=MGa-4) “ ysx-4t+l yox-3 2 2 Downloaded from hlayiso.com Copyright reserved TLO20(E)028)T finding the gradient of line AC ¥ multiplication of gradients to show that lines are perpendicular (2) Yusing identity on parallel lines to find gradient of line AB ¥ finding y-intercept by substituting equation (3) ¥ substituting “simplify Y substituting ¥ simplify 4) Please turn over

MARKING GUIDELINE -7- MATHEMATICS N3 SAT Dag =p) +04 9) i v¥2-(-2)" +3-C by = V¥16+16 v = 32 Y Dog = (%4~ Hn) + (Y4-e) = ¥B-)?+@2-07 v = V4+4 v = B AB=V32=J4x8=2V8 v AB =2DE 3.2 3.2.1 G v v 3.2.2 B wv vw 3.2.3 D v v 3.2.4 I v v QUESTION 4 41 4.1.1 2 ya2vx-2 x t =2x2-at ¥ v j voev YL x 242x7 ° dx . -L,2 vx x v v T1020(E)(528)T Yusing the distance formula and substitute ¥ simplify “using the distance formula and substitute v simplify “rewriting surds to prove that AB=2DE ¥ ¥ Choosing the correct equation to describe the gtaph ¥ ¥ Choosing the correct equation to describe the graph ¥ “Choosing the correct equation to describe the graph “¥ Choosing the correct equation to describe the graph Vrewriting square root rewriting a fraction ¥ differentiate function V differentiate function rewriting function rewriting function G) (2) @) [25] (4) Downloaded from hlayiso.com Copyright reserved Please turn over

MARKING GUIDELINE -8- T1020(E)(J28)T MATHEMATICS N3 4.12 xt] Y (xv )simplify fraction ' eae ¥ differertiation (xt -1y(x +1 i x +1 i ys x'-1 V : dy 3 | —i=4y dx v Q) 42 y= (x? —3x+D ¥ multiplication =p4p—3e-3 V differentiation POR EH Od v “substitution i f ()=3x +2x-3 Y FQ) =3(2Y +2(2)-3 =3(2) 44-3 | 213 ¥ (3) [9] | i | | QUESTION 5 i 3.1 l+sin@ cos? ¥LCD | cos + ising 2secd v 2 rting fractions to LHS = (ysind)(l a A+ cos “cos RHS =2sec@ “applying square identity cos @(1+sin A) y cos? x=1—sin? x “simplifying _L+2sin@+sin?@+(I-sin’?@) ~ Vindicating that ths = chs cos(1-++sin 8) _ _2+2sin@ cos@(l+sind) * ° _ 2(i+sind) . . cos A(1+sin A) _ 2 cos? =2secO LHS = RHS v . ° (5) ”“Bownldaded from hlayiso.com Copyright reserved . : Please turn over

MARKING GUIDELINE -9- 5.2 5.3 5.4 5.5 Copyright reserved MATHEMATICS N3 “ad? = 46° +80? -2x46%80%Cos25 a’ =2116+ 6400-2 46x80x0,906 a? =1845,575 . v a= 42,96 units Y x 2SecH+4=0 sec B=~-2 reference angle = 60° .B=120° and B=240° v v ¥ cD 40 CD#=14,558 yy tan 24° =f? 40 AD = 17,809 AD~-CD = AC(lenght of antenna) 17,809 -14,558= AC tan 20° = AC =3,250m v A=(45°:2) v B=(225°:2) v P=(180°;-D v T1020(E)(J28)T “substituting using cosine tule ¥ simplifying ¥ simplifying QB) ¥ manipulating Vv finding solutions (3) “ trigonometric equation Vlength CD Yilength AD “length AC (4) ¥(Vx2) ¥ (v¥x2) ¥ (4x2) (3) [18] TOTAL: 100 Downloaded from hlayiso.com

higher education & training Department: Higher Education and Training REPUBLIC OF SOUTH AFRICA TLOZO(E\AS)T APRIL EXAMINATION NATIONAL CERTIFICATE MATHEMATICS N3 (16030143) 3 April 2013 (X-Paper) 09:00-12:00 Non-progranmuable and non-graphical calculators may be used This question paper cousists of 6 pages and a 2-page formula sheet. cats Downloaded from hlayiso.com Copyright reserved Please turn over

“26 T1O7OCEMAS)T QUESTION 1 Ll Simplify the following WITHOUT the use of a calculator: Lil (7 +32) : (2) Lt2 log, x x log, 8 (3) 113 24-28 +f54 3 oe (3) 2 Factorise as far as possible in prime factors: 124 5x*8 ~ 807 @) 1.2.2 2’ (p~q)4 3xlq~ p)~18(p~ y) By SQx +1) -i7(e +46 @) i Use the remainder theorem to determine the remainder when «+ is divided jnto the followin La 3) phe 4) (25] Downloaded from hlayiso.com 4 4 oO

QUESTION 2 ne we Solve for x in each of the following: 2 Poe 5 211 a a -3 “3 =a a 2.4.2 2.1.3 Make “d" the subject of the formula: 8 = Rasen D2] Make "£" ithe subject of the formula: The sum of thres consecutive uatoral numbers TIO7OCE)\CAS)T is 33. Detennine the smallest of the Downloaded from hlayiso.com al @) @) {#8]

~5- T1070(E)(A5)T QUESTION 3 31 In the accompanying sketch, the graph of y=: -~x? + 3x +10 is represented, DE is parallel to the x-axis 3.1.1 Calculate the length of AB Q) 3.1.2 Calculate the value of the x -coordinate of the turning point at C (2) 3.1.3 Calculate the length of DE (2) iv i oe oe ao 4B f : / H / | i \ ! 32 Sketch the graphs of the following equations in the ANSWER BOOK. h graph must be drawn on its own sysiem of axes. ALL vahies at the poinis of . : te 2 | intersection with the system of axes must be shown. Name the type of graph below each sketch, SA 6) 3.2.9 ae @) 3.2.3 ol —_ @) Downloaded from hlayiso.com PTO

~6- TLOTOQR) (AST / 3.3 In the diagram, the circle with the origin 0 as centce cuts the straight Hne I at the, points A(0;5) aud C. The point M(2;4) Is the midpoint of AC and @ is the size of the j acuie angle that J makes with the x-axis, BL (2B i ‘ 332 JJsterimins the coordinates of CO (2) i Prove, by using analytical methods, that OM LAC (23 | Saloulate the sie af O correct to ave decimal place (2) ME SGG Glefgtata, ALNY i, A tg Ue Poult (324) end 1313 tae potat L2), atermine the following: 3.41 The co-crd The length of AD in surd form 1) 4.3 Downloaded. from hlayiso.com PTO

-T- T1070(E)(A3)T QUESTION 4 44 Determine /'(x) in each of the following by using the rules of differentiation. Leave the answer(s) with positive exponents and in sud form. Z a 401 f(s)=(- 2x2) Q) 4.1.2 i f(xje-tve $2 Deienmins the gradient of the tangent to the following curve at the point whers x= 3: 43 The diagram below represents the gfaph of: yaa Gx? 49% Determine ihe co-ordinates of the local turning points A and B. B oN 4 / \ / \ i / \ t / / awa a errr eran aS A WTR REE nope Nn soak saad Wh ew 0 A | (4) [10] Downloaded from hlayiso.com Pra

TLOTO(EY(ASYT QUESTION § 54 Calculate the exact value of the following WITHOUT the use of a calculator: cot? 30° + cosec? ag? ~tan 4s? (3) 5.2 Simplify the following: i cos(180° ~ x)tan(360°~ x) sin(goe.- x}tanflgg°+ 80° + (3) | 5.3 Make use of basic trigonometric identities ta prove the following: : I . i / Irian? x lec @) i satisfy the following irigcnometrin equation for; 2M ys QB) 33 Two towers, CD and X ae separated by a distance DY = . The angle of i elevation from Cte X is 60° and the angle of depression fom C: “ va ig 35° : ~ { Pont i a Determine the following: 5.5.4 Ths height of the tower CD (2) 55.2 The heigl ‘oad tow : (3) Downloaded from hlayiso.com PrO

5.6 9. TLO7O(EYAS)T The following graph represents the function f and g for xe[0°s190°] where J (x) =acosx and g(x) =sinw Find the values of a and ¢ TOTAL: Downloaded from hlayiso.com (2) (19 400 7 i

QUESTION i Il Ld (432) 12) 1 2 . © 34 ne)) . 112 flog, aJ® + in ve = ay? +5hn g vf f . me / aes i 3) 13 Ri) j P ¢ (3) \ ; v Y 3) Copyright reserved Please turn over Downloaded from hlayiso.com

Fisuiyesil: MATHEMATICS Fy 1.2 E21 5x8 ~ 80)? =5(x% ~16y") y =5(x" dye ~4y) v 12.2 x3 (p~q)+3x(g— p)-18(p ~ 9) J =x"(p~ 9)- 3x(p 4) 18(p ~q) =(p ~q){x? -3x~ 18) v =(p-a)=- Oe +3) y 123 s(x 1) 17+ +6 let(x+i)=k Sk -17E+6 V (5k —2Xk ~3) v a . 13 H4+i443 =? y La idl I V Copyright reserved Please turn over Downloaded from hlayiso.com G) (2) G)

PIVOGS 8 Sab 1.4.2 45 ox a 13 ’ | x-3 x42 x? -~7-6 aif 5 _ 9x +13 i x~3 x42 (x-3hx+2) ; 14x +2) ~ 5(x-3)~ (02 413) v = eS) (Pa 4 13) (x-3)e+2) | _ 14x + 28~Sx415—-9x 13 _ 30 130} QUESTION 2 21 QA i But =6 ' i whey ! @ j 21.2 i ( 243 (3) Copyright reserved Please turn over Downloaded from hlayiso.com

Vxe5=x-1 xe5e(e-i)? x+55x?-2x4] x? -3x-4=0 (x -4Xx+41)=0 wxed xe-l 2.2 P=PR+VT UR+Vi-P=0 . pa eye’ ~4ac — 2a 2.3 Asta; Ped Smallest no.=10 Copyright reserved PPogR as @) n=? os tO (3) (20) Please turn over Downloaded from hlayiso.com

Ro ee Ce VUES A SIH QUESTION 3 31 3d @) 3.4.2 ~b +=— 2a -3 yeot -2 veSN esis ! 33 ! Q} | 32 3.24 j Q Copyright reserved Please turn over Downloaded from hlayiso.com

ve = iv 3.43 OULESTION 4 44 411 Copyright reserved B=G3) AD =(-3-1)' +42) AD=4f(-4) +5? AD = VAI Units yy, =m(z~x,) 3-(-2)=mn{-3-1) Elicit vie ie) @) [28] (2) Please turn over Downloaded from hlayiso.com

QUESTION 3 * eee Sneceme =3+4-1 =6 5.2 00s x*— tan x cos x4 tanx =1 3.3 t I =] rt 2 see” x cosec cos? x+sin?x=1 “LHS = RAS 54 tanx=-l xetant~} we 45° Copyright reserved Q) (3) 13] TOTAL: 106 Downloaded from hlayiso.com

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