NATIONAL SENIOR CERTIFICATE GRADE 12

Transcription

1 NATIONAL SENI CERTIFICATE GRADE MATHEMATICS P FEBRUARY/MARCH 0 MEMANDUM MARKS: 00 This memorandum consists of pages.

2 Mathematics/P DBE/Feb. Mar. 0 QUESTION. 6; 7 answer answer (). T Tk + Tk + k + T + + () + 5 T () + 0 k T Tk + Tk + k + ; T and k Tk Tk + k T k Tk Tk + k ; T and k [5] QUESTION. Total number of employees answer. Total amount needed ( ) + ( ) + ( ) + ( ) + ( ) + ( ) + ( ) + ( ) R Mean monthly salary R No. Only 0 employees in this company earn more than R The majority (40) of the employees earn below this amount. It is therefore not a good indicator of the average monthly amount earned by an employee. method answer () () answer () No 40 earn below the mean 0 of the 50 employees earn R7 500 or less. () [7]

3 Mathematics/P DBE/Feb. Mar. 0 QUESTION % 4% 4% 4% 4% % is standard deviation to the left of the mean percentage of teenagers who sent less than 8 messages 50% 4% 6%. 6 minutes is standard deviations from the mean 48% 5 minutes is standard deviation from the mean 4% Percentage of the teenagers who sent between 6 and 5 messages 48% + 4% 8% NOTE: Answer only: Full marks standard deviation 50% 4% 6% 48% 4% 8% [6]

4 Mathematics/P 4 DBE/Feb. Mar. 0 QUESTION 4 4. a 9,7 (9, ) b 4,9 ( 4, ) yˆ 9,7 4, 9x SCATTER PLOT SHOWING THE NUMBER OF SATURDAYS ABSENT AND THE FINAL MARK ACHIEVED a b equation y-intercept (9) Point (7 ; 57) Final Mark (as %) Number of Saturdays absent 4. r 0,87 ( 0, ) answer () 4.4 The greater the number of Saturdays absent, the lower the mark. number of Saturdays absent final mark () 4.5 yˆ 9,7 4,9 substitution 7,6% answer 7% () NOTE: Allow for the range 70% 74% for a student who reads off the graph. () []

5 Mathematics/P 5 DBE/Feb. Mar. 0 QUESTION 5 DO NOT PLAY PLAY SPT SPT TOTAL Male Female Total P(male) ,5(0, ) P (female and plays sport) ,8 (0, ) 5 No. From the table, P (male and do not play sport), which is greater 6 than zero. Since the probability of the intersection of these two events is greater than zero, these events are not mutually exclusive. 0 P(male) 6 00 P(NS) P(male) P( NS) , (0, ) 5 P(male and NS) 6 0, (0, ) So, P ( male) P( NS) P (male and NS) Therefore the events 'male' and 'do not play sport' are independent (correct to TWO decimal places). The events are not independent as there is a discrepancy from the third decimal place () () No probability of intersection greater than zero () are independent [0]

6 Mathematics/P 6 DBE/Feb. Mar. 0 QUESTION 6 6. first tier def (0,0) (A; D) second tier probabilities outcomes A (0,) non def (0,99) (A; ND) B (0,) def (0,0) (B; D) C (0,5) non def (0,98) def (0,06) (B; ND) (C; D) 6.. P(B; ND) 0, 0,98 0,9 non def (0,94) (C; ND) 0, 0,98 0,9 Accept: 0, P(defective) P(A; D) + P(B; D) + P(C; D) (0, 0,0) + (0, 0,0) + (0,5 0,06) 0,04 Accept: 0,08 multiplying probabilities adding probabilities answer [0] QUESTION ! different ways ! different ways 7. The items from each department can be arranged in! ways. The departments can be arranged in 4! ways. Advertisements can be arranged in!.4! different ways. answer () ! ()! 4! 44 [7]

7 Mathematics/P 7 DBE/Feb. Mar. 0 QUESTION 8 M S A 70 P N R B 8. Pˆ 90 ( in semicircle) Pˆ Pˆ 4 90 (vert opp s) BN is a diameter (chord subtends 90 ) () Pˆ 4 90 chord subtends 90 () 8.. Â 0 ( sum Δ) answer 8.. Pˆ6 70 (tan ch th) answer () () 8.. Pˆ 70 (vert opp s) Nˆ 70 (tan ch th) Pˆ 4 90 (proven) Bˆ 0 ( sum Δ) Pˆ5 0 ( s on str line) Bˆ 0 (tan ch th) Nˆ 70 Bˆ 0 Pˆ5 0 Bˆ 0 () () [7]

8 Mathematics/P 8 DBE/Feb. Mar. 0 QUESTION 9 R 0 O S x x T x P K 9. ST PT x (line from circ centre ch bis ch) S/R 9. In ΔRST and ΔPKT. Rˆ Pˆ ( in same seg). Ŝ Kˆ ( in same seg). R Tˆ K PTˆ K 90 (given) ΔRST ΔPKT ( ) 9. ST RT KT PT ( Δs) x 0 x x 4x 0 x 80 mm ST KT RT PT x x 0 x x 60 x 80 mm S/R S/R R ST RT KT PT substitution answer () [7]

9 Mathematics/P 9 DBE/Feb. Mar. 0 QUESTION 0 P S Q R 6 cm T W WS SP WS WT (ST PR; Prop th) SP RT 6 WT WT 9 cm WS WR (SR PQ; Prop th) SP RQ RQ RQ 0 cm WQ cm WS WT SP RT ST PR; Prop th answer WS WR SP RQ RQ RQ 0 WQ 5 [7]

10 Mathematics/P 0 DBE/Feb. Mar. 0 QUESTION. P S O T R Join RO and OP Let Ô x Ô 60 x ( s in a rev) Tˆ x ( circ centre circumference) Ŝ 80 x ( circ centre circumference) Ŝ + Tˆ x + 80 x 80 T P w x w 80 - w 80 - x 4 z 80 - z O x 80 - y z y y S R construction Ô x Ô 60 x Tˆ x Ŝ 80 x reason (6) Draw radii OP, OS, and OT Let Pˆ x, Ŝ w, Ŝ z and Rˆ y Tˆ x, Pˆ w, Rˆ z and Tˆ y ( s opp radii) Ô 80 w ( sum Δ) Similarly Ô 80 z, Ô 80 y, Ô 4 80 x By angles in a revolution + Ô + Ô + Ô 80 w + 80 z + 80 y + 80 x Ô ( w + z + y + x) ( w + z + y + x) 60 w + z + y + x 80 construction Let Pˆ x, Ŝ w, Ŝ z and Rˆ y Ô 80 w ( sum Δ) setting up equation simplification conclusion (6)

11 Mathematics/P DBE/Feb. Mar. 0 P T 90 - x x 90 - x x 80 - x y y O x 80 - y 90 - y y y R S Draw radii OP, OS, and OT Let Pˆ x and Rˆ y Tˆ x and Tˆ y ( s opp radii) Ô 80 y ( sum Δ) Similarly Ô 4 80 x Ŝ Pˆ and Ŝ Rˆ ( s opp radii) Tˆ + Tˆ + Ŝ + Ŝ x + y + 90 x + 90 y 80 construction Let Pˆ x and Rˆ y Ô 80 y ( sum Δ) Ŝ Pˆ and Ŝ Rˆ Tˆ + + Ŝ + Ŝ Tˆ conclusion (6)

12 Mathematics/P DBE/Feb. Mar. 0 A C x O F E B D.. Â x ( circ centre circumference) Fˆ 80 x (opp s of cyclic quad 80 ) Â x circ centre circumference Fˆ 80 x opp s of cyclic quad 80 reflex BÔC 60 x ( in revolution) Fˆ 80 x ( circ centre circumference).. Fˆ x (ext int opp ) Bˆ x Ê (corres s; EC BF) Fˆ Bˆ DF BD (sides opp s) reflex BÔC 60 x in revolution Fˆ 80 x circ centre circumference Fˆ x ext int opp Bˆ x sides opp s Ê x ( circ centre circumference) Bˆ Ê x (corres s; EC BF) Fˆ x (ext cyclic quad) Fˆ Bˆ DF BD (sides opp s) Bˆ Fˆ x x ext int opp sides opp s

13 Mathematics/P DBE/Feb. Mar. 0.. Ĉ x ( s opp equal sides in Δ) + Ĉ Ĉ + Ĉ x (corresp s CE BF) Ĉ Ĉ Ĉ + Ĉ x s opp equal sides in Δ Ĉ x + Ĉ corresp s CE BF Dˆ 80 x ( sum Δ) Ô + Dˆ 80 OCDB is a cyclic quad (opp s suppl) Bˆ (ext cyclic quad) Ĉ Bˆ Ĉ ( same seg) Ĉ Ĉ..4 Dˆ 80 x ( sum Δ) 80 x ( on straight line) Ô Dˆ Ô sin Dˆ sin Ô area ΔBFD area ΔAOC DF.BD.sin Dˆ AO.OC.sin Ô 5.5.sin Dˆ 6.6.sin Ô opp s suppl Bˆ Ĉ ext cyclic quad Bˆ Ĉ Dˆ Ô x 80 x DF.BD.sin Dˆ AO.OC.sin Ô [] TOTAL: 00