Mathematics/P DoE/November 008 NSC Memorandum NATIONAL SENI CERTIFICATE GRADE MATHEMATICS P NOVEMBER 008 MARKS: 0 This memorandum consists of ages. Coyright reserved
Mathematics/P DoE/November 008 Continued accuracy alies as a rule in the memorandum. If a candidate does a question, crosses it out and does not re-do it, mark the deleted attemt. QUESTION y D(0 ; A( ; 0 θ C( ; B( ;. M + ; ; 0 into midoint formula for both coordinates ( Answer only: mark er coordinate. Midoint BD + 0 + ; ; Midoint of AC and BD are the same oint therefore AC and BD bisect each other Wrong formula: 0 / into formula conclusion (midoints are the same (
Mathematics/P DoE/November 008 AM AM CM CM + 6, + 6, + 0 + + + / for answer on the left (because candidate did not show that M is on BD BM BM + 6, + + DM 0 + + + DM 6, AC and BD bisect each other. 0 m AD 0 + Note: If do: m AD m AD m CD mcd 0 mcd then / 4 if calculated the gradients correctly. m AD mcd If m AD m CD and conclude AD CD without any working, then / 4 AD CD A D ˆC 90 m AD m CD m m AD CD conclude A D ˆC 90 (4 tanθ m CD tanθ θ,69 tan DAC ˆ DAC ˆ,69 ADC ˆ,69,69 ADC ˆ 90 tan θ mcd θ, 69 D A ˆC, 69 A D ˆ C 90 (4
Mathematics/P 4 DoE/November 008 AD AD DC DC AC AC AD ( 0 + ( 0 ( ( ( + ( 0 ( 0 ( + ( 6 + DC + 6 AC AD DC A D ˆC 90 AD DC AC 6 conclusion (4.4 BD ( + 6 + (0 + for BD AC ( + (0 + for AC 6 diagonals are equal diagonals are equal diagonals bisect each other (Proved in. (i.e. ABCD is a rectangle m AC. m BD AC BD bisect each other m. AC m BD. AC BD (6 ( 0 + ( 0 ( AD AD DC ( ( + ( 0 DC The figure is a rectangle and one air of adjacent sides are equal in length it is a square. for AD for DC conclusion (6
Mathematics/P DoE/November 008 AD AD DC DC AB AB BC ( 0 + ( 0 ( ( ( + ( 0 ( ( + ( 0 ( ( ( + ( ( BC All four sides equal and one internal angle equal to 90 for AD for AB for DC for BC all four sides are equal one internal angle equal to 90 (6 The diagonals bisect one another A D ˆC 90 AD ( 0 + ( 0 ( AD DC ( ( + ( 0 DC adjacent sides equal in length ABCD is a square diagonals bisect each other A D ˆC 90 into distance formula for AD for DC conclusion (6. + tan θ 0 tan θ θ 6,09947 + 80 θ,7 gradient of CD tan θ ( tan DAO ˆ DAO ˆ,7 ADC ˆ 90 θ 90 +,7 θ,7 Penalty for incorrect rounding θ 90 + DA ˆ O tan D AO ˆ (
Mathematics/P 6 DoE/November 008.6 OC ( 0 + ( 0 OC OC OC,6067977 OC > C lies outside the circle ( OC ( 0 + ( 0 OC OC > 4 C lies outside the circle + y 4 ( + ( > 4 C lies outside the circle Answer only: 0 / [0]
Mathematics/P 7 DoE/November 008 QUESTION y Q( ; O R(t ; P ( ; y. r OQ ( 69 + ( substituting ( ; into + y 69 + y 69 + y 69 ( + y ( + ( 69. 0 mpq 0 mpq y + y r coordinates 69 ( Answer only: Full marks gradient c 0 (
Mathematics/P 8 DoE/November 008. P( ; (By symmetry y ( + y 69 + 69 44 + 69 44 46 69 46 44 ± y.4 tangent diameter m m PQ QR m PQ m QR PQ QR m QR m m PQ QR PQ QR ( (. y + c ( + c 69 c 69 y + y,4 +,8 y m + c substitution of gradient and ( ; calculation of c. (
Mathematics/P 9 DoE/November 008 y y ( m formula of gradient y ( and ( ; y (.6 y + 44 + y + 69 + y 69 0 69 y + t 74 t t 74 4, ( t 69 + equation in correct form ( substitution of (t ; answer ( m QO m QR 6 t t 4, PQ 76 + 00 + ( t 7 + 44 4t + t 48t 696 t 4, + QR PR + 6 ( t + t.7 ( + ( y OQ OQ ( 0 + ( 0 ( + ( y 69 + 6 + 4t + 44 + 6 69 6 t Pythagoras with substitution ( ( y 69 ( ( ( ( + ( y 69 By translating units right and units u + y ( ( 69 If answer only: ( + ( y 69 : / [7]
Mathematics/P 0 DoE/November 008 QUESTION ( ; /.. coordinate of P / P / y-coordinate of P (.. / P (, / coordinate of P / y-coordinate of P (.. / D ( ; ( / If rotated anti-clockwise: D ( ; No mark for / D ( ;.. / coordinates A / y coordinates B / coordinates C 4 / coordinates E C rotation correct.. // D (6 ; 9 - -4 - - - 4 - -4 // If rotated anti-clockwise: D ( 6 ; 9 B A E D A / ( ; B / ( ; 0 E / ( ; C / ( ; D / ( ; ( If all the oints on the sketch are correct and / labels are A etc: / If all the oints on the sketch are correct and labels at incorrect oint: 4 / Deduct marks for anti-clockwise direction If write down coordinates correctly and did not sketch: 4 / -coordinate y-coordinate (..4 ( ; y ( y ; ( y ; ( y ; ( y ; ( y ; ( ; y ( y ; (4 Answer only: 4 / 4 If answer ( ky ; k / 4 y ; If Answer: (
Mathematics/P DoE/November 008 4 / 4 If rotated anti-clockwise the answer would be: ( ; y ( y ; ( y ; ( y ; ( ; y ( y ;.. Area ABCDE : area A : : 9 ABCDE // // A B C 9 // D // E // // B // C // D // E // If // // // // A B C D E ABCDE 9 0 / // ( [8] QUESTION 4 cos( 4 y sin( 4 cos 4 + sin 4,4 and y P / + or cos( 4 + sin( 4 cos 4 sin 4 or ; or 0,7 formula 4 or of or for formula of for y (7 A enalty of marks for substituting 4 instead of 4. The answer will then be ; or ( 0,7 ;,4
Mathematics/P DoE/November 008 If a candidate rotates clockwise and substitutes 4 the formulae will be: cosθ + y sinθ cos 4 + sin 4 +,4 y cosθ sinθ cos 4 sin 4 formula for 4 for formula for 0,7 for (7 Let OP OP / r The -coordinate of P r cos( θ 4 r(cosθ.cos 4 + sinθ.sin 4 cosθ.cos 4 +. +.. sinθ.sin 4. The y-coordinate of P r sin( θ 4 r(sinθ.cos 4 cosθ.sin 4 P /.. + sinθ.cos 4 ;. cosθ.sin 4. formula r cos( θ 4 eansion for formula r sin( θ 4 eansion for y (7
Mathematics/P DoE/November 008 cos 4 y sin 4 y cos 4 + sin 4 y : y y + ( + ( : y + y + y - - - ( - - - ( ( ; ( r cos( θ 4 ; r sin( θ 4 r + tanθ θ 6,09947... r cos( θ 4 cos(6,... 4,4 r sin( θ 4 + y 0,7 r r sin(6,... 4 formula formula solving simultaneous answer y r tan θ θ 6,09947... r cos( θ 4, 4 y r sin( θ 4 y 0, 7 (7 (7 Answer only: 6 / 7 [7]
Mathematics/P 4 DoE/November 008 QUESTION Penalise mark for treating as an equation in this question... tan 480.sin 00.cos4.sin( sin04.cos tan0.( sin 60.cos4.( sin 4 sin 76.( cos4 ( tan 60.( sin 60.cos4.( sin 4 cos4.( cos4 (.. sin 60 sin 4 cos 4 tan 60 cos 4 or sin 76 Penalise mark for treating as an equation in this question. (6 tan 480.sin 00.cos4.sin( sin04.cos tan0.( sin 60.cos4.( sin 4 sin 76.( cos 4 ( tan 60.( sin 60.sin 76.tan 4 sin 76 (.. sin 60 sin 4 cos 4 tan 60 sin 76 (6.. cos7 cos(4 + 0 cos4.cos0 sin 4.sin 0.. 6 4 4. 4 ( 4 cos( 4 + 0 eansion simlification (4
Mathematics/P DoE/November 008 cos 7 cos(4 + 0 cos 4.cos 0 sin 4.sin 0... cos( 4 + 0 eansion simlification (4. cos(90.tan(80 + + sin sin.tan + sin sin sin.cos. + sin cos sin + sin sin (60 sin tan sin sin tan cos sin sin.cos sin (6 If uses cos instead of sin and then works correctly: ma /6 [6]
Mathematics/P 6 DoE/November 008 QUESTION 6 6.. (tan (sin cos sin cos sin cos (sin cos cos (sin cos (sin sin.cos + cos ( sin.cos (tan (sin cos sin cos sin (sin ( sin.cos (sin cos cos ( sin.cos cos ( sin.cos cos sin.cos sin.cos + cos ( cos (sin cos sin cos + cos ( sin cos + cos (sin cos ( sin cos (tan (tan.tan cos ( sin cos cos RHS sin sin cos cos ( sin cos (tan sin cos (tan (tan LHS (tan (sin cos ( sin.cos cos ( + cos sin cos (sin cos LHS sin tan cos sin sin. cos factorisation simlification sin + cos sin tan cos sin sin. cos simlification factorisation sin + cos sin + cos factorisation sin tan cos sin sin. cos simlification sin tan cos simlification sin sin. cos sin + cos factorisation (
Mathematics/P 7 DoE/November 008 6.. tan tan 6 simlification simlification tan 78, 7 78,7 + k.80 + k. 80 k Z k Z ( tan tan 6 tan 0, + k.80 k Z tan tan 6 tan 0, + k.60 or 8, + k.60 k Z If the candidate has used tan( 6 ma of / 6.. cos β r y tan β β third quadrant y answer If is negative:/4 (4
Mathematics/P 8 DoE/November 008 6.. cos cos β β ( sin cos β β sin cos cos β β β cos β or ( β sin or ( or ( β β sin cos or or ( [7]
Mathematics/P 9 DoE/November 008 QUESTION 7 7. tan 40 LB LB tan 40 LB,8 m (,7... trig ratio (, m ;,7 m ;,6 m ( LB sin 0 sin 40 sin 0 LB sin 40 LB,8 m (,7... sine rule ( 7. AB AB AB AL (. 4,40408 m AB 7,8 m + BL + (.8. AL. BL.cos Note: AB 7, m or 7,4 m: accet (.(.8 cos (7,79... use of cos rule AB 4,404 m (4 Do not enalise if units are omitted. 7. Area of ABL AL. BL.sin ALˆ B (.(.8 sin 8.680976 8,7 m Note: Area 8, or 8,6 : accet formula answer If cos AL ˆ B : 0/4 (4 [0]
Mathematics/P 0 DoE/November 008 QUESTION 8 8. cos sin sin(90 sin 90 + k.60 90 80 + k.60 k Z 4 90 + k.60 or 90 + k.60, k.90 k Z 4 k.80 k Z 67, ;, ;, 4 ; equating 90 + k. 60, k. 90 90 80 + k. 60 4 k. 80 values of (8 cos cos(90 90 + k.60 60 (90 + k.60 4 90 + k.60 or 70 + k.60, + k.90 k Z + k.80 k Z 67, ;, ;, 4 ; equating 90 + k. 60, + k. 90 60 (90 + k. 60 + k. 80 values of (8 cos cos(90 90 + k.60 90 + + k.60 4 90 + k.60 or 90 + k.60, + k.90 k Z 4 k.80 67, ;, ;, 4 ; k Z equating 90 + k. 60, + k. 90 90 + + k. 60 4 k. 80 values of Note: (8 If not all values for is given, the following alies 4 or values : marks values : mark value : 0 marks
Mathematics/P DoE/November 008 8. y -90-4 4 90 80-0 g f - (6 Penalise with - going beyond the domain. 8. 67, 4 [ 67, ; 4 ] From 67, u to and including 4 critical values notation ( Note: If 67, < < 4 : / Half of the inequality: / If 67, or 4 : 0/ If answer is,, then / If answer is 80 then / [7]
Mathematics/P DoE/November 008 QUESTION 9 9. 9. 0 Mean minutes 0 Time taken ( ( i 8-4 6-6 -6 6 4 4 8 6 6 0-4 0 0 9 7 49 9-9 Sum 6 sum of minutes number of runners answer ( Answer only: / setting u of table and correct values in column of ( i σ ( i n 6 0,9 (If the candidate used a calculator to answer QUESTION 9. and QUESTION 9., award full marks if answers are correct. If only one mistake in the calculation: / 4 Answer only: 4 / 4 If candidate uses n in the formula, the answer in formula answer (4 9. One standard deviation of the mean is in the interval ( -,9 ; +,9 which is (8,0 ;,9 6 runners comleted the race within one standard deviation of the mean. (List of times:, 4, 0,, 9, If candidate used σ 4, 6, then the interval is (7,84 ; 6,6 and the answer is 7 runners. ( Answer only : / [8]
Mathematics/P DoE/November 008 QUESTION 0 0. Daily Sales (in Rand Frequency Cumulative Frequency 60 rand < 70 70 rand < 80 6 80 rand < 90 8 90 rand < 00 00 rand < 0 7 8 0 rand < 0 6 Frequency Column cumulative frequencies ( If one wrong in the frequency column, deduct mark. 0. Sales for November and December 007 Cumulative frequency 70 60 0 40 0 0 0 0 0 0 40 60 80 00 0 40 Daily Sales (rands cumulative totals oints at uer limits of intervals shae ( If the ogive is NOT grounded, no enalty. If lotted as the midoint of the interval and the cumulative frequency: / 0. Median R 87 (Accet answers between 84 and 90 correctly read off ogive ( 0.4 R 96 sales R 0 correctly read off ogive ( [9]
Mathematics/P 4 DoE/November 008 QUESTION. 600 Scatter Plot of height above ground level vs time SCATTER PLOT OF HEIGHT ABOVE GROUND LEVEL VS Height above ground level (metres 00 400 00 00 00 0 0 4 6 7 8 9 Time (seconds all oints lotted correctly. ( No enalty if the oints are joined.. Eonential ( Straight line : 0 / Quadratic Hyerbola Decreasing steely then gradually. (Alicable descritions are accetable. Aroimately 90 m ( [4]
Mathematics/P DoE/November 008 QUESTION. The median, the maimum scores, IQR Note: Any two statements that are valid in the contet of the roblem aly. any two of the list (. IQR 90 7 8. formula (. No. In the calculation of the median only the value in the middle of an ordered data set is of imortance. The etreme values are not taken into account. In this case, % of the learners in Class A had a score of less than 66 marks. The minimum mark in Class B is 66 marks. Hence the erformance of the two classes differ significantly. No. The one is skewed to the left and the other is skewed to the right. The etreme values are not taken into account. Answer only: / No etreme values not taken into account minimum marks different ( [7] No. The lower quartile of Class A is below the minimum of Class B. The etreme values are not taken into account. No. The left whisker of Class A is much longer than the left whisker of Class B. The etreme values are not taken into account. TOTAL: 0 marks