Grade 11 Mathematics Paper 2 (PDF) - November 2023

Donwloaded from Grade 11 Exam Papers or Grade 11 Study Guides a.z co PROVINCIAL EXAMINATION NOVEMBER 2023 p. GRADE 11...

Donwloaded from Grade 11 Exam Papers or Grade 11 Study Guides a.z co PROVINCIAL EXAMINATION NOVEMBER 2023 p. GRADE 11 ee MARKING GUIDELINES D MATHEMATICS (PAPER 2) ra lt 14 pages U Donwloaded from Grade 11 Provincial Apps (KZN, Gauteng, EC, WC, NW, FS, etc)  Donwloaded from Grade 11 Exam Papers or Grade 11 Study Guides MATHEMATICS MARKING GUIDELINES (PAPER 2) GRADE 11 INSTRUCTIONS AND INFORMATION A – ACCURACY CA – CONSISTENT ACCURACY a S – STATEMENT R – REASON.z S & R – STATEMENT with REASON NOTES: co If a candidate answered a question TWICE, mark only the first attempt. If a candidate crossed OUT an answer and did not redo it, mark the crossed-out answer. p. Consistent accuracy applies to ALL aspects of the marking guidelines. Assuming values/answers in order to solve a question is UNACCEPTABLE. ee D ra lt U 2 Donwloaded from Grade 11 Provincial Apps (KZN, Gauteng, EC, WC, NW, FS, etc)  Donwloaded from Grade 11 Exam Papers or Grade 11 Study Guides MATHEMATICS MARKING GUIDELINES (PAPER 2) GRADE 11 QUESTION 1 1.1 Class interval Frequency Cumulative frequency ✓ Frequency values a 300 < x  400 4 4 ✓ Cumulative 400 < x  500 11 15 frequency values.z 500 < x  600 22 37 600 < x  700 13 50 co 700 < x  800 7 57 800 < x  900 3 60 (2) 1.2 70 p. ✓ (300 ; 0) ✓ End points ee 60 ✓ Shape 50 Cumulative Frequency D 40 ra 30 lt 20 10 U 0 0 100 200 300 400 500 600 700 800 900 1000 Daily Sales (3) 1.3 Q2 = R580  Answer Accept R560 to R600 (2) 1.4 On 10 days they made a loss.  Answer  50 days they made a profit. (2) 1.5 To be in the top 25% of the previous months, the tuck ✓ R650 shop would need to have sold more than R650. ✓ Conclusion Therefore, R725 would have been in the top 25%. (2) 3 Donwloaded from Grade 11 Provincial Apps (KZN, Gauteng, EC, WC, NW, FS, etc)  Donwloaded from Grade 11 Exam Papers or Grade 11 Study Guides MATHEMATICS MARKING GUIDELINES (PAPER 2) GRADE 11 QUESTION 2 2.1 2k + k + 1 + k + 2 + k − 3 + 2k − 2 ✓ 7k – 2 = 15 a 5 ✓ Divide by 5 7k − 2.z = 15 ✓ Manipulation 5 7k – 2 = 75 co 7k = 77 k = 11 (3) 2.2 22 ; 12 ; 13 ; 8 ; 20 p.  Answer  = 5,22 (2) ee 2.3 Mean would decrease by t units but the standard ✓ Mean decreases by t deviation would be unaffected. units ✓ Standard deviation unaffected (2) D ra QUESTION 3 3.1 0 − ( −2) ✓ Substitution into mMQ = − 2 − ( −4) correct formula lt 1 mMQ = − ✓ Answer U 3 (2) 3.2 1 ✓ Perpendicular gradients mMQ  − = −1 MP ⊥ MQ 3 ✓ Gradient of MP mMP = 3 ✓ Answer y = 3x + 6 (3) 3.3 xR + 4 y R + ( −2) ✓ Substitution into –2 = 0= midpoint formula 2 2 ✓ xR x R = −8 yR = 2 ✓ yR R (–8 ; 2) (3) 4 Donwloaded from Grade 11 Provincial Apps (KZN, Gauteng, EC, WC, NW, FS, etc)  Donwloaded from Grade 11 Exam Papers or Grade 11 Study Guides MATHEMATICS MARKING GUIDELINES (PAPER 2) GRADE 11 3.4 PQ = (0 − 4) 2 + (6 − ( −2)) 2 ✓ Substitute into distance formula PQ = 80 PQ = 4 5 ✓ Simplified surd (2) a E(–4 ; –6) ✓ x E = −4.z 3.5 ✓ y E = −6 (2) co 3.5 The diagonals of the parallelogram are perpendicular. ✓ Diagonals perpendicular p. ✓ Given parallelogram (2) QUESTION 4 ee 4.1 subst. x = 0 ✓ x=0 py – x – 5p = 0 ✓ Manipulation py – (0) = 5p ✓ x=5 D 5p y= p ra y=5 C(0 ; 5) (3) lt 4.2 py – x – 5p = 0 ✓ Standard form py = x + 5p ✓ Equate gradients U x y= +5 p 1 mAC = 2 1 1  = p 2 p=2 OR 1 A(–10 ; 0) mAC = ✓ A (–10 ; 0) 2 ✓ Substitute A p(0) – (–10) – 5p = 0 10 = 5p p=2 (2) 5 Donwloaded from Grade 11 Provincial Apps (KZN, Gauteng, EC, WC, NW, FS, etc)  Donwloaded from Grade 11 Exam Papers or Grade 11 Study Guides MATHEMATICS MARKING GUIDELINES (PAPER 2) GRADE 11 4.3 Equation of AC: 2y – x – 10 = 0 ✓ Substitute p = 2 1 y= x+5 2 ✓ Equation of BD a Equation of BD: y = – 2x – 10.z 1 ✓ Substitute y into other x + 5 = – 2x – 10 2 equation x + 10 = – 4x – 20 co 5x = – 30 x = –6 ✓ x = –6 y=2 ✓ y=2 B(–6 ; 2) p. (5) 4.4 1 ✓ Substitution into correct tan  = ee 2 formula  = 26,57 ✓ Answer (2) D 4.5 AĈD = 63,43 int 's of  ✓ AĈD  = 26,56 int 's of  ✓  = ra ✓ Rede  ABOD is a cyclic quadrilateral (converse 's in same segment) (3) lt 4.6 DB is a diameter (line subtends 90) ✓ x=0  0 + 0 5 + ( −10)  5 M CD =  ✓ y=– U ;  2  2 2   5 = M CD  0 ; −   2 (2) 6 Donwloaded from Grade 11 Provincial Apps (KZN, Gauteng, EC, WC, NW, FS, etc)  Donwloaded from Grade 11 Exam Papers or Grade 11 Study Guides MATHEMATICS MARKING GUIDELINES (PAPER 2) GRADE 11 QUESTION 5 5.1 5.1.1 2 ✓ Quadrant II sin  = a 3 y ✓ x=– 5 2 2 2 x + (2) = 3.z ✓ Substitute x=– 5 3cos2  – 1 x cos  co  5 2 ✓ Answer =3   –1  3  5 = –1 3 p. 2 ee = 3 (4) 5.1.2 tan (– –180) ✓ –tan  D = –tan  ✓ Substitution  2  ✓ Answer = –  ra − 5 2 = 5 (3) lt 5.2 5.2.1 t cos 15 = 4 t ✓ cos 15 = 4 4 U cos 15 = t t ✓ t 2 − 16 sin 15 ✓ t t − 16 2 15° = 4 t (3) 5.2.2 sin 75 ✓ Complement = cos 15 ✓ Answer 4 = t (2) 7 Donwloaded from Grade 11 Provincial Apps (KZN, Gauteng, EC, WC, NW, FS, etc)  Donwloaded from Grade 11 Exam Papers or Grade 11 Study Guides MATHEMATICS MARKING GUIDELINES (PAPER 2) GRADE 11 5.2.3 1 – tan2 15  t 2 − 16  2  t 2 − 16  2 ✓    4  =1–      4    a t 2 − 16 t 2 − 16 ✓ =1– 16.z 16 ✓ Correct 16 − t 2 + 16 = multiplication by co 16 32 − t 2 LCD = 16 (3) 5.3 p. cos(90 − α ) sin( − − 540 ) ✓ tan 45 ✓ sin .– sin  tan 225 + sin . sin(180 + ) ee ✓ sin .sin  sin α.sin α ✓ 1 – sin2  = tan 45 + (sin .)( − sin α ) 1 ✓ D sin2 α tan 2 α = 1 − sin2 α ra sin 2 α = cos 2 α = tan2  (5) lt 5.4 5.4.1 1 – cos = 2 sin2 ✓ 2 (1 – cos2) 1 – cos = 2 (1 – cos2) U 1 – cos = 2 – 2 cos2 ✓ standard form 2 cos2 – cos  – 1 = 0 (2 cos + 1)(cos – 1) = 0 (2) 5.4.2 2cos + 1 = 0 or cos – 1 = 0 1 ✓ cos = – 1 2 cos = – or cos = 1 2 ✓ cos = 1 RA = 60 or  = 0 + K.360 ; K  Z ✓ Both solutions for QII :  = 120 + K.360 ; K  Z 1 cos = – QIII :  = 240 + K.360 ; K  Z 2 ✓  = 0 + K.360 ✓ +KZ (5) 8 Donwloaded from Grade 11 Provincial Apps (KZN, Gauteng, EC, WC, NW, FS, etc)  Donwloaded from Grade 11 Exam Papers or Grade 11 Study Guides MATHEMATICS MARKING GUIDELINES (PAPER 2) GRADE 11 QUESTION 6 6.1 180 ✓ Answer (1) a 6.2 1 1 ✓ Endpoints – y.z 2 2 ✓ Notation (2) co 6.3.1 45  x  90 ✓ x  45 ✓ x < 90 (2) 6.3.2 –90< x < – 45 or 90< x < 135 p. ✓ –90< x < – 45 ✓ 90< x < 135 (2) ee 6.3.3 2 tan x + cos 2x = 2 ✓ 2 tan x – 2 = –cos 2x 2 tan x – 2 = –cos 2x 1 ✓ tan x – 1 = – cos 2x 1 2 tan x – 1 = – cos 2x D 2 ✓ Value of x  f (x) = g(x) where x = 45 ra (3) 6.4 sin x + cos x ✓ tan x + 1 h(x) = cos x ✓ Up by 2 units lt sin x h(x) = +1 cos x U h(x) = tan x + 1 h(x) = f (x) + 2  Vertical translation up by 2 units (2)  1  ✓ –2 cos 2x 6.5 p(x) = 4  − cos 2 x   2  ✓ Max of 2 p(x) = – 2 cos 2x  Maximum value of 2 (2) 9 Donwloaded from Grade 11 Provincial Apps (KZN, Gauteng, EC, WC, NW, FS, etc)  Donwloaded from Grade 11 Exam Papers or Grade 11 Study Guides MATHEMATICS MARKING GUIDELINES (PAPER 2) GRADE 11 QUESTION 7 1 1 7.1 tan p = ✓ tan p = a 3 3 p = 18,435 ✓ Value of p to 3.z decimal places (2) co 7.2 In BCD ✓ Correct substitution CD 121 into sine rule = ✓ Manipulation  18,4  sin135 sin  p. ✓ Length of CD  2  121 x sin 9,2 CD = sin135 ee CD = 27,36 m (3) D QUESTION 8 ✓ Substitute into volume ra 8.1 V = x2 h 490 = x2h formula 490 ✓ h in terms of x h= 2 x lt (2) 8.2 A = x2 + 4xh ✓ Base of x2 U  490   490  A = x2 + 4x  2  ✓ 4x  2   x   x  1960 3 A = x2 + cm x (2) 8.3 490 ✓ Substitute h = 10 10 = x2 ✓ x2 = 49 x2 = 49 ✓ x=7 x=7 (x > 0) ✓ Surface area 1 960 A = (7)2 + 7 2 A = 329 cm (4) 10 Donwloaded from Grade 11 Provincial Apps (KZN, Gauteng, EC, WC, NW, FS, etc)  Donwloaded from Grade 11 Exam Papers or Grade 11 Study Guides MATHEMATICS MARKING GUIDELINES (PAPER 2) GRADE 11 QUESTION 9 9.1 a Q.z co 1 2 O 1 R p. 2 P ee D Join PO and extend ✓ Construction In ΔPOQ ✓ SR ra Ô1 = Q̂ + P̂1 ext  of  ✓ SR but Q̂ = P̂1 ' s opp equal radii ✓ S  Ô1 = 2P̂1 lt Similarly in  ROQ Ô 2 = 2P̂2 ✓ S U  Ô1 + Ô 2 = 2P̂1 + 2P̂2 ✓ S ( Ô1 + Ô 2 = 2 P̂1 + P̂2 )  RÔQ = 2P̂ (6) 9.2 9.2.1 (a) D̂ = 55° ' s in the same segment ✓ S R (2) (b) Ô 2 = 110°  at centre = 2   at ✓ S R circumference (2) (c) Ê 2 = 110° – 90 ext  of  ✓ R ✓ S Ê 2 = 20° (2) 11 Donwloaded from Grade 11 Provincial Apps (KZN, Gauteng, EC, WC, NW, FS, etc)  Donwloaded from Grade 11 Exam Papers or Grade 11 Study Guides MATHEMATICS MARKING GUIDELINES (PAPER 2) GRADE 11 (d) Ê1 = 35° int ' s of  ✓ R ✓ S ✓R Ê 3 = 35° ' s in a semi-circle OR a Ê 3 = F̂1 ' s opp equal radii ✓ S R ✓ SR.z Ê 3 = 35° int ' s of  (3) co 9.2.2 OE = 5 equal radii ✓ SR HE = 4,7 line from centre perpendicular ✓ S ✓ R to chord OH2 = 52 – (4,7)2 p. Pythagoras' Theorem ✓ S OH = 1,71 units OR ✓ SR ee OE = 5 radii OH ✓✓ sin 20° sin 20 = 5 ✓ OH D OH = 1,71 units OR HE = 4,7 line from centre ✓ S R ra perpendicular to chord ✓ tan 20° OH ✓ OH sin 20 = 4,7 lt OH = 1,71 units (4) U QUESTION 10 10.1 B̂1 = x alt. 's, AC ‖‖ TV ✓ S ✓R V̂1 = x ' s opp equal sides ✓ S ✓R T̂2 = x tan chord theorem ✓ S ✓R (6) 10.2 Â = 2 x opp ' s parm ✓ S ✓R B̂3 = 2 x ext  cyclic quad ✓ S ✓R  Â = B3 both equal 2x ✓ R AT = BT sides opp equal ' s 12 Donwloaded from Grade 11 Provincial Apps (KZN, Gauteng, EC, WC, NW, FS, etc)  Donwloaded from Grade 11 Exam Papers or Grade 11 Study Guides MATHEMATICS MARKING GUIDELINES (PAPER 2) GRADE 11 OR B̂3 = 2 x ext  cyclic quad ✓ S ✓R ✓ SR T̂1 = 2 x alt 's, AC ‖‖ TV a  T̂1 = V̂1 + V̂2 both equal 2x.z ✓ S ✓R  BTVC is an isosceles trapezium (one pair parallel sides and one pair of base angles ✓ Logic co equal)  CV = BT sides of isos trap en CV = AT opp sides parm  AT = BT p. (5) 10.3 In  ATB, ✓ R ee 5x = 180° int ' s of  ✓ S x = 36° (2) D QUESTION 11 ra 11.1 D̂3 = Ĥ1 + Ê1 ext  of  ✓ S (1) 11.2 D̂3 = Ĥ1 + Ê1 ext  of  lt Ĥ1 = F̂2 tan chord theorem ✓ S ✓R ✓ SR U and Ê1 = F̂1 tan chord theorem  D̂ 3 = F̂1 + F̂2 (  DEFH is a cyclic quadrilateral (ext  = opp int  ) ✓ R (with D̂3 = F̂1 + F̂2 ) (4) 11.3 Ĥ1 = F̂2 tan chord theorem ✓ SR ✓ S ✓R Ĥ1 = F̂1 ' s in the same segment  F̂1 = F̂2 (3) 13 Donwloaded from Grade 11 Provincial Apps (KZN, Gauteng, EC, WC, NW, FS, etc)  Donwloaded from Grade 11 Exam Papers or Grade 11 Study Guides MATHEMATICS MARKING GUIDELINES (PAPER 2) GRADE 11 11.4 Ĥ 3 + Ĥ 4 = Ê1 + Ê 2 ext  cyclic quad ✓ S ✓R Ĥ 4 = K̂ 1 tan chord theorem ✓ SR a Ĥ 3 = K̂ 2 ' s in the same circle segment ✓ S ✓R.z  K̂1 + K̂ 2 = Ê1 + Ê 2 K̂1 = Ê1 given ✓ S co  K̂ 2 = Ê 2 KF is a tangent converse tan chord theorem ✓ R (if K̂ 2 = Ê 2 p. proven) (7) ee TOTAL: 150 D ra lt U 14 Donwloaded from Grade 11 Provincial Apps (KZN, Gauteng, EC, WC, NW, FS, etc)