Technical Mathematics P2 Nov 2025 MG Afr & Eng

NATIONAL SENIOR CERTIFICATE NASIONALE SENIOR SERTIFIKAAT GRADE 12/GRAAD 12 TECHNICAL MATHEMATICS P2/TEGNIESE WISKUNDE V2 NOVEMBER 2025 MARKING GUIDELINES/FINALE NASIENRIGLYNE MARKS/PUNTE: 150 CODE/ EXPLANATION/VERDUIDELIKING KODE A Accuracy/Akkuraatheid AO Answer only/Slegs antwoord CA Consistent accuracy/Volgehoue akkuraatheid I Identity/Identiteit M Method/Metode NPR No penalty for rounding/Geen penalisering vir afronding nie NPU No penalty for omitting units/Geen penalisering vir eenhede weggelaat nie R Rounding/Afronding RE Reason/Rede S Simplification/Vereenvoudiging F Formula/Formule SF Substitution in correct formula/Vervanging in korrekte formule ST Statement/Bewering ST/RE Statement with reason/Bewering met rede These marking guidelines consist of 32 pages. Hierdie nasienriglyne bestaan uit 32 bladsye. Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 2 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne NOTE: • If a candidate answers a question TWICE, only mark the FIRST attempt. • The method of consistent accuracy marking must be applied in all aspects of the marking guidelines where applicable as indicated by the marking code CA. • # Shows questions where a Tolerance Range will be applied: Q 2.1.3 ; Q 3.3 ; Q 6.5 & Q 10.2.3 LET WEL: • Indien 'n kandidaat 'n vraag TWEE keer beantwoord, sien slegs die EERSTE poging na. • Die metode van volgehoue akkuraatheid-nasien moet waar moontlik op alle aspekte van die nasienriglyne toegepas word, soos aangedui deur die nasien kode CA. • # Toon vrae waar Toleransiewydte (Verdraagsaamheidsomvang) toegepas word: V 2.1.3 ; V 3.3 ; V 6.5 & V 10.2.3 QUESTION/VRAAG 1 1.1 OA = 4 units/eenhede ✓ length of /lengte van OA A (1)  x + xB y A + y B  1.2 M= A ;   2 2   0 + ( −2 ) − 4 + ( −3 )  = ;   2 2   7 ✓ x - coordinate/koördinaat =  −1 ; −  OR / OF ( −1 ; − 3,5 ) A  2 ✓ y - coordinate/koördinaat A (2) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 3 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne y A − yC 1.3 mAC = xA − xC = ( − 4 ) − ( 2 ) OR / OF = ( 2 ) − ( −4 ) ✓ SF A ( 0 ) − ( 3) ( 3) − ( 0 ) = 2 ✓ value of/waarde van m CA OR / OF OR / OF y − y1 = m ( x − x1 ) ( −4 ) − ( 2 ) = m ( ( 0 ) − ( 3) ) ✓ SF A − 6 = m ( − 3) −6 ✓ value of/waarde van m m= =2 CA −3 OR / OF OR / OF y = mx + c y = mx −4 2 = m ( 3) − 4 ✓ SF A 3m = 6 m=2 ✓ value of/waarde van m CA AO: Full marks/Volpunte (2) 1.4 tan  = mAC  = tan −1 (2) ✓ SF CA   63, 43 ✓ value of/waarde van  CA AO: Full marks/Volpunte NPR (2) 1.5 equal / gelyk OR / OF the same / dieselfde ✓ equal/gelyk A (1) 1.6 y − ( −3 ) = 2 ( x − ( −2 ) ) −3 = 2 ( −2 ) + c ✓ gradient/helling CA ✓ Subst./Vervang B A y + 3= 2 x + 4 c= 4−3 OR / OF c= 1 y = 2 x +1 ✓ equation in form y = ... / vergelyking in vorm y =.. CA (3) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 4 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne 2 1.7 cos  = − 2 ref. / verw.  = 45  ✓ Ref. / Verwys.  A   = 180 − 45 = 135 ✓ value of/waarde van  CA  m = tan (135 ) = − 1 ✓ value of/waarde van m A OR / OF OR / OF 2 cos  = − 2 x + y = r2 2 2 ( − 2 ) + y = ( 2) 2 2 2 y 2= 4 − 2 y 2=  2 y= 2 ✓ Value of/Waarde van y A 2 tan  = − 2 ✓ tan ratio/verhouding CA tan  = −1  m = −1 ✓ value of/waarde van m CA (3) [14] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 5 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 2 2.1 2.1.1 x 2 + y 2 = (13) 2 ✓✓ equation/vergelyking A OR / OF OR / OF x + y = 169 2 2 ✓✓ equation/vergelyking A OR / OF OR / OF x= 169 − y 2 ✓✓ equation/vergelyking A OR / OF OR / OF ✓✓ equation/vergelyking A y = 169 − x 2 OR / OF OR / OF ( 13 ) + ( 0 ) = r 2 2 2 ✓ substitution/vervanging A 169 = r 2 x2 + y 2 = 169 ✓ equation/vergelyking A AO: Full marks/Volpunte (2) 2.1.2 ( − 5) + ( k ) = 169 2 2 ✓ substitution/vervanging CA  k 2 = 144  k = − 12 ✓ value of/waarde van k CA AO: Full marks/Volpunte (2) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 6 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne 2.1.3 Since / Omdat M (13 ; y ) ✓ value of/waarde van x A # 12 mON = ✓ gradient/gradįënt of/van 5 ON CA 5 mMN = − ✓gradient/gradįënt of/van 12 MN CA ( x − ( − 5 ) ) OR / OF −12 = − 5 5 y − ( −12 ) = − ( − 5) + c 12 12 5 25 25 169 y=− x− −12 c=− −12 = − 12 12 12 12 5 169 y= − x− ✓equation/vergelyking CA 12 12 5 169 y = − (13) − 12 12 39 y=− = −19,5 ✓value of/waarde van y CA 2  M (13; − 19,5) OR / OF OR / OF Since / Omdat M (13 ; y ) ✓ value of/waarde van x A x  x1 + y  y 1 = r 2 ✓F A ✓ subst/vervang CA − 5 x − 12 y = 169 − 12 y = 5 x + 169 5 169 ✓ equation/vergelyking y= − x− CA 12 12 5 169 y = − (13) − 12 12 39 y=− = −19,5 ✓value of/waarde van y CA 2  M (13; − 19,5) OR / OF OR / OF Since / Omdat M (13 ; y ) ✓ value of/waarde van x A  tangents from same pt. /  and/en d LM = d NM   ✓ statement/bewering A  Raaklyne vanuit dies pt  (13 − ( − 5) ) + ( y − ( − 12) ) = y 2 2 ✓ subst/vervang CA (13 − ( − 5) ) + ( y − ( − 12) ) = y 2 2 2 ✓ squaring both sides / kwadreer beide kante CA 324 + y + 24 y + 144 = y 2 2 24 y = − 468 39 y=− = − 19, 5 2 ✓value of/waarde van y CA M (13; − 19, 5) (5) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 7 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne 2.2 ✓x-intercept/afsnit A ✓ y-intercept/afsnit A ✓elliptical shape/ elliptiese vorm A (3) ACCEPT/AANVAAR: 25 and / en − 25 4 and / en − 4 [12] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 8 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 3 3.1 sin B + sec A = sin 30, 5  + sec 72   substitution/vervanging A 1 = sin 30, 5  + I A cos 72  1, 93 S CA NPR AO: Full marks/Volpunte (3) 3.2.1 5 sin  = − 13 x + ( − 5) 2 = (13) 2 2  Pyth. Theorem/Stelling A x = 12  value of/waarde van x CA 12 cos  = 13  ratio/verhouding CA OR / OF OR / OF cos  = 1 − sin 2  I A 2  −5  = 1−    ratio of/verhouding van sin  13  12 A = 13  ratio/verhouding CA (3) 3.2.2 cot  − cosec  cot ratio/verhouding CA  12   13  = −  − −   cosec ratio/verhouding A  5   5 1 = S CA 5 ACCEPT/AANVAAR: Correct decimal values./ Korrekte desimale waardes. (3) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 9 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne 3.3 cot x = − 0, 587 # 1 tan x = − OR / OF tan x = − 1, 704  tan ratio/verhouding A 0, 587 Ref.  59, 59  ref. angle/verw. hoek CA x  180 − 59, 59 or/of x  360  − 59, 59   value of/waarde van x CA x  120, 41 or/of x  300, 41   value of waarde van x CA (4) [13] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 10 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 4 4.1.1 tan x  reduction/ reduksie A (1) cos x 4.1.2 I A sin x (1) 4.1.3  0 ; 90 ; 180 ; 270 ; 360   any two values/ enige twee waardes A (2) sin(180 + x )  sin(360 − x ) + cos(2 − x)  cos x 1 4.1.4 + sin x tan(180 + x ) − sin x  − sin x + cos x  cos x 1 = +  − sin x A sin x tan x  − sin x A sin x + cos x cos x 2 2  cos x = + A sin x sin x 1 cos x = + sin x sin x  I (1) A 1 + cos x = OR / OF = cosec x + cot x S CA sin x (5) 4.2.1 sin  (1 − cos  )  factors/ faktore A (1) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 11 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne sin  − cos   sin  4.2.2 LHS = cos  − (1 − sin 2  ) sin  (1 − cos  ) =  I ( cos 2  ) A cos  − cos 2  sin  (1 − cos  ) = cos  (1 − cos  )  common fact/ gemeensk. fakt A sin  = S A cos  = tan  = RHS OR / OF OR / OF sin  − cos   sin  LHS = cos  − (1 − sin 2  ) sin  − cos   sin  = cos  − 1 + sin 2  sin  (1 − cos  ) = cos  − 1 + 1 − cos 2   I ( 1 − cos2  ) A sin  (1 − cos  ) = cos  (1 − cos  )  common fact/ sin  gemeensk. fakt A = S A cos  = tan  = RHS (3) [13] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 12 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 5 5.1 m = 2 and / en p = 45  value of/waarde van p A  value of/ waarde van m A (2) 5.2 180  period/periode A (1) 5.3 1  value/waarde A (1) 5.4.1 165  x  180  endpoints/ eindpunte A  notation/notasie A OR / OF OR / OF x  ( 165 ; 180 ]  endpoints/ eindpunte A  notation/notasie A OR / OF OR / OF x  165 and / en x  180  endpoints/ eindpunte A  notation/notasie A OR / OF OR / OF Greater than 165 and smaller or equal to 180 . /  endpoints/ Groter as 165 en kleiner of gelyk aan 180 . eindpunte A  Correct interval/ Korrekte interval A (2) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 13 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne 5.4.2 0  x  90  critical values/ kritiese waardes 135  x  180 A  notation/ notasie A  critical values/ kritiese waardes A  notation/ notasie A OR / OF OR / OF x  [ 0 ; 90 ]  critical values/ kritiese waardes A or / of  notation/ notasie A  critical values/ x[135;180] kritiese waardes A  notation/ notasie A OR / OF OR / OF x  0 and / en x  90  critical values/ kritiese waardes A or / of  notation/ notasie A x  135 and / en x  180  critical values/ kritiese waardes A  notation/ notasie A OR / OF OR / OF  endpoints/ From 0 to 90 / Vanaf 0 tot by 90 eindpunte A  Correct interval/ Regte interval A or / of  endpoints/ eindpunte A From 135 to 180 / Vanaf 135 tot by 180  Correct interval/ Regte interval A (4) 5.5 h ( x ) = sin 2 x − 1  sin 2 x − 1 CA OR / OF OR / OF h ( x ) = sin mx − 1  sin mx − 1 A (1) [11] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 14 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 6 NOTE: If an angle of 90° is assumed – BREAKDOWN, hence no marks may be awarded. NOTA: Indien ‘n hoek van 90° aangeneem word = AFBREEK, vervolgens mag geen punte toegeken word nie.  6.1 M R P = 180 − 120 − 30  size of/grootte van = 30  MRP A (1) 6.2 Isosceles triangle/Gelykbenige driehoek  isosceles/ gelykbenig A OR / OF OR / OF Obtuse angled triangle/Stomphoekige driehoek  obtuse angled/ Stomphoekig A (1) MR MP 6.3 =   sine rule/sinusreël A sin P sin M R P MR 20 = sin120 sin 30  Subst. / Vervang. CA 20 sin120 MR = sin 30 MR = 20 3 units  34, 64 units  length of/lengte van MR CA OR / OF OR / OF MP = PR = 20 m  length / lengte PR A MR = (20) 2 + (20) 2 − 2(20) (20) cos120  Subst. into / Vervang. in cosine rule/cosinusreël = 20 3 units  34,64 units CA  length /lengte MR CA (3) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 15 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne  6.4 M R S = 30 ( alt. s/verw. e; SR||MP )  ST A  RE A (2)  6.5 SM 2 = SR 2 + MR 2 − 2SR  MRcosM R S # ( ) − 2 ( 30 ) ( 20 3 ) cos 30 2 SM 2 = ( 30 ) + 20 3 2  SF cosine rule/ = 300 cosinusreël CA SM = 10 3  17,32  length of/lengte van SM CA ( ) + ( 30 ) 2 2 SM 2 + SR 2 = 10 3 = 1200  value of/waarde van ( ) SM 2 + SR 2 2 MR 2 = 20 3 CA = 1 200  value of/ waarde van SM + SR = MR 2 2 2 MR 2 CA  SM + SR = MR 2 2 2 MRS is a right-angled triangle CA MRS is ' n reghoekige driehoek OR / OF OR / OF  SM = SR + MR − 2SR  MRcosM R S 2 2 2 ( ) − 2 ( 30 ) ( 20 3 ) cos 30 2 SM 2 = ( 30 ) + 20 3 2  SF cosine rule/ cosinusreël CA = 300  length of/lengte van SM = 10 3  17,32 SM CA (10 3 ) + ( 30 ) − ( 20 3 ) 2 2 2   SF cosine rule/ cos S = 2 (10 3 ) ( 30 ) cosinusreël CA   0   0  S = cos  −1  cos −1   CA   600 3   600 3    Size of/ S = 90  grootte van S CA MRS is a right-angled triangle MRS is ' n reghoekige driehoek OR / OF OR / OF Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 16 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne  SM 2 = SR 2 + MR 2 − 2SR  MR cosM R S  SF cosine rule/ cosinusreël ( ) − 2 ( 30 ) ( 20 3 ) cos 30 2 CA SM 2 = ( 30 ) + 20 3 2 = 300  length of/lengte van SM CA SM = 10 3  17,32   sin S sin R = s r   SF sine rule/sinusreël sin S sin 30 CA = 20 3 10 3   sin −1 (1) CA sin S = 1  Size of/grootte   S = sin −1 (1) van S CA  S = 90 MRS is a right-angled triangle MRS is ' n reghoekige driehoek OR / OF  sine ratio/verh. CA OR / OF h sin 30 = 20 3 h CA h = 20 3.sin 30  SF cosine rule/ h = 10 3  17,32 cosinusreël CA  length of/lengte van ( 30 ) + ( 20 3 ) − 2 ( 30 ) ( 20 3 ) . cos 30 2 2 SM = SM CA SM =10 3  17,32  Conclusion/  SM = Perp. height/loodregte hoogte Gevolgtrekking CA (5) Therefor SMR is a right-angled triangle. / Daarom is SMR ‘n reghoekige driehoek. [12] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 17 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 7 7.1 7.1.1 The angle subtended by the diameter at the circumference of the  RE A circle is 90 . (  in semi-circle ) / Die omtrekshoek wat deur die middellyn onderspan word, is 90 . (  in halwe sirkel ) OR / OF OR / OF Diameter subtends right angle. / Diameter onderspan regtehoek.  RE A (1)  7.1.2 A 2 = 70 ( co-int s / ko - binne e; BC|| AE )  ST A   opp s of cyclic quad /   ST CA  D = 110    teenoorst e van kdvk   RE A OR / OF OR / OF  C1 = 70 ( Int. s  / Binne e ; BC|| AE )  A 2 = 70 ( alt. s / verw. e; BC|| AE )  ST A   opp s of cyclic quad /   ST  D = 110   CA  teenoorst e van kdvk   RE A (3) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 18 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne 7.2 7.2.1 equal./gelyk. OR / OF the same./dieselfde.  ST A (1) 7.2.2 (a) s in the same segment/ e in dieselfde segment  RE A OR / OF OR / OF s subt. by same arc/chord./  RE A e onderspan deur dieselfde boog/koord. (1) 7.2.2 (b) Vertical opposite s / regoorstaande e  ST A (1) 7.2.2 (c)  ,  ,  OR / OF A, A, A / H, H, H  RE A OR / OF OR / OF Equiangular triangles/Gelykhoekige driehoeke  RE A (1) [8] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 19 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 8 8.1   8.1.1 G1 = G 2  ST A (1)    at centre = 2   at circ. /   ST A 8.1.2 G 2 = 20    Mdpt .  = 2  omtreks   RE A (2) 8.1.3 s opp = sides / e teenoor gelyke sye  RE A (1)    8.1.4 A1 + B1 + A C B = 180  ( Int. s  / Binne e  )  RE A  180° − 40° A1 = OR / OF 2  A1 = 70° ( s opp. = sides / e teenoor = sye )  A (2)   opp. s cyclic quad/   ST CA 8.1.5 D A C + 70 + 40 = 180    teenoorst. e kdvh   RE A   D A C = 70  ST CA (3) 8.2.1 exterior / buite  ST A ACCEPT/AANVAAR: outside/buitekant (1) 8.2.2 interior / binne  ST A ACCEPT/AANVAAR: inside/binnekant (1) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 20 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne 8.3   ST A 8.3.1(a) C3 = 79 ( Ext.  cyclic. quad / Buite  kdvh.)  RE A (2)   ST CA 8.3.1(b) D 2 = 49 ( Ext.   / Buite   )  RE A OR / OF OR / OF  B C D = 101 ( s on str. line / e op reguitlyn )  ST CA  D 2 = 49 ( Int. s  / Binne e  )  ST CA OR / OF OR / OF    s in same segm./  B2 = A 2 = 30    ST CA  e in dies. segm.    A1 = 49    s in same segm./  A1 = D 2 = 49    ST CA  e in dies. segm.  (2)  8.3.1(c) D1 = 180 − 79 − 49 ( Co - int. s suppl. ; AB||CD )  ST CA   RE A D1 = 52 ( Ko - Binne e sup pl.; AB || CD )  ST CA OR / OF OR / OF  B1 = 49 ( alt.  's =; verw.  ' e =; AB||CD )  ST CA   RE A  D1 = 52 ( Int.  's  / Binne  ' e  )  ST CA (3) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 21 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne    s in same segment /   ST 8.3.2 C1 = D1 = 52   CA  e in dies. segment   RE A    s on str. line /  C 2 = 49     e op reguitlyn     sides opp. = s /   ST CA D 2 = C 2 = 49    RE  sye teenoor = e  A CE = DE OR / OF OR / OF     s in same segment /  C 2 = B1 = 49    ST CA  e in dies. segment   RE A    sides opp. = s /  D 2 = C 2 = 49    ST CA  sye teenoor = e   RE A CE = DE (4) [22] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 22 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 9 9.1 AD AE  ST A 9.1.1 = ( Prop. Theorem / Eweredigheid ; DE|| BC )  RE A CD BE (2) AD AE 9.1.2 = CD BE AD 4 =  SF CA 12 8 4 AD = × 12 8 = 6 units / eenhede 6 CA AO: Full marks/ Volpunte (2) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 23 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne 9.2 DE 3 2 GH 7,5 5 = = = 0, 4 OR / OF = = = 2, 5  Ratio/verhouding GH 7,5 5 DE 3 2 A EF 4 2 HI 10 5 = = = 0, 4 OR / OF = = = 2, 5  Ratio/verhouding HI 10 5 EF 4 2 A DF 5 2 GI 12, 5 5 = = = 0, 4 OR / OF = = = 2, 5  Ratio/verhouding GI 12, 5 5 DF 5 2 A  DEF ||| GHI ( Sides in prop. / Sye in verhouding )  RE A OR / OF OR / OF  E = 90 Conv. Pyth. / Omg . Pyth.  Conv. Pyth./  Omg. Pyth. A H = 90 Conv. Pyth. / Omg . Pyth.  3  3 tan F = tan I =  Any trig ratio/ 4 4   Enige trig F = 36,87 I = 36,87 verhouding A   E=H ( Proven / Bewys )    ST A F=I ( Proven / Bewys ) DEF ||| GHI ( ;  ; )  RE A (4) [8] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 24 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 10 10.1 10.1.1  = 2 n F A 377 = 2 n  SF A 377 n= or / of 60 rev / s  377 or / of 60 CA 2 2 NPU NPR (3) 10.1.2 v = r F A = ( 377 )( 300 )  SF A  value of/waarde = 113100 mm / s van v CA OR / OF OR / OF v =  Dn F A  377  =  ( 600 )   OR / OF   ( 600 )( 60 )  SF  2  CA = 113100 mm / s  113097, 34 mm / s  value of/waarde van v CA NPU NPR (3) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 25 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne 10.1.3 4 h 2 − 4 dh + x 2 = 0 F A  SF A 4 h 2 − 4 ( 620 ) h + (130 ) = 0 2 4 h 2 − 2480 h + 16 900 = 0 S CA 2480  5880 000  both vertical h= 8 heights/beide h  613,11mm or / of h  6,89 mm vertikale  minor / klein h  6,89 mm hoogtes CA  minor height / klein hoogte CA OR / OF OR / OF  Line from centre of circle, perp. to chord  AC = BC = 65    BC = 65 A  Lyn mdpt . sirkel , loodreg op koord  CD2 = 3102 − 652 M A CD = 303,11 mm  Length/lengte CD CA h = 310 − 303,11 M A h = 6,89 mm  minor height/ klein hoogte CA NPU NPR (5) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 26 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne 10.2   10.2.1 B O C = 115  180 23 23 =   2, 01rad   or/of 2,01 A 36 36 (1) NPU NPR Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 27 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne r 2 10.2.2 Area = F A 2 ( 4, 5 )  23   2 ( 4, 5 ) ( 2, 01) 2  SF CA =  36  OR / OF 2 2 207  area/oppervlak =   2 0, 32 m 2  2 0, 35 m 2 32 CA OR/OF OR/OF 115 Area =  r2 360 F A 115 =   (4, 5) 2  SF A 360 207 =   20, 32 m 2  area/oppervlak 32 CA OR/OF OR/OF s = r  23  = 4, 5    OR / OF = 4, 5 ( 2, 01 )  36  23  arc =   9, 03 m  9, 05 m 8 length/booglengte rs CA Area = 2 F A  23  4, 5     8  4, 5 ( 9, 05 ) = OR / OF 2 2 207 =   20, 32 m 2  20, 36 m 2  area/oppervlak 32 CA NPR NPU (3) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 28 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne 10.2.3 # 3  ratio/verhouding DO =  4, 5 = 2, 7 m 5 A  length of/lengte van DO CA  23  (2, 7) 2     36  OR / OF A 115 A DOE = DOE =   (2, 7) 2 2 360 1863 =   7, 32 m 2  Area of/Oppv. van 800 DOE CA 207 1863 A shaded/ =  −  geskakeer 32 800 207 =   13, 01m 2  Area of the 50 shaded part/ Oppv. van geskakeerde gedeelte CA  Seedlings needed / Saailinge benodig = 13  4 = 52  Number of seedlings/Aantal saailinge CA OR/OF OR/OF Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 29 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne 3 DO =  4,5 = 2, 7 m 5  ratio/verhouding A  length of/lengte sDE = r  van DO CA  23  = 2, 7     36  69 =   5, 42 m 40 rs Area DOE = 2  69  2, 7    =  40  2 1863 =   7, 32 m 2 800  Area of/Oppv. van DOE CA 207 1863 A shaded/ =  −  geskakeer 32 800 207 =   13, 01m 2 50  Area of the shaded part/ Oppv. van geskakeerde gedeelte CA  Seedlings needed / Saailinge benodig = 13  4 = 52  Number of seedlings/Aantal saailinge CA NPU NPR (5) [20] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 30 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 11 11.1 11.1.1 30 cm ✓ 30 cm A (1) 11.1.2 y = 50 + 20 = 70 cm ✓ value of/waarde van y A (1) o +o  11.1.3 A T = a  1 n + o2 + o3 + ... + on−1  ✓F A  2   x + 2x  ✓ SF CA 7200 = 30  + 70 + 50 + 60   2  240 = 1,5 x + 180 ✓S CA ✓ value of /waarde van x CA  x = 40 cm OR / OF OR / OF ✓F A AT = a ( m1 + m2 + m3 + ... + mn )  x + 70 70 + 50 50 + 60 60 + 2 x  ✓ SF CA 7200 = 30  + + +   2 2 2 2  ✓S CA 240 = 1,5 x + 180 ✓ value of /waarde van x CA  x = 40 cm (4) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 31 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne 11.2 11.2.1 0, 5 m  100 = 50 cm ✓ 50 cm A (1) 11.2.2 r = 40  2 = 20 mm ✓ M 40  2 A  r = 20  10 = 2 cm ✓ r value in/waarde in cm CA OR / OF OR / OF d = 40 10 = 4 cm ✓M A  r = 4  2 = 2 cm ✓ r value/waarde in cm CA AO: Full marks/Volpunte (2) 11.2.3 TSA of Cylindrical rod / Buite-oppervlakte van silindriese staaf: = 2 r 2 + 2 r h = 2 (10 ) + 2 (10 )( 50 ) 2 ✓ SF CA = 1200   3769,91 cm2 ✓ total surface area/totale buite- oppervlakte CA (2) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P2/Tegniese Wiskunde/V2 32 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne 11.2.4 Volume of one steel ball bearing/van een staal bal 4  ( 2) 3 = ✓ SF CA 3 32 =   33,51 cm3 ✓ Vol. of ball bearing/Vol. van 3 staal koellaër CA Volume of Cylindrical rod/van silindriese staaf =  (10 ) ( 50 ) 2 = 5000   15707,96 cm3 ✓ Vol. of cyl. rod/Vol. van sil.staaf CA Volume Recovered / Herwin = 5000 − 18%  5000 OR / OF ✓M A = 82%  5000 = 4100  ✓ Vol. recovered/herwin CA  12880,53 cm 3 4100   32 32    400  3 3 12800 = 384, 375 OR / OF =  3  384 = 13404,13 cm 3  More than 400 balls can NOT be made / Meer as ✓ conclusion/gevolgtrekking CA (6) 400 balle kan NIE gemaak word NIE. [17] TOTAL/TOTAAL: 150 Copyright reserved/Kopiereg voorbehou