GRAAD 12 NATIONAL SENIOR CERTIFICATE GRADE 12

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1 GAAD NATIONAL ENIO CETIFICATE GADE MATHEMATIC P3 FEBUAY/MACH 0 MAK: 00 TIME: hours This questio paper cosists of pages, 3 diagram sheets ad iformatio sheet. Please tur over

2 Mathematics/P3 DBE/Feb. Mar. 0 INTUCTION AND INFOMATION ead the followig istructios carefully before aswerig the questios This questio paper cosists of questios. Aswer ALL the questios. Clearly show ALL calculatios, diagrams, graphs et cetera, that you have used i determiig your aswers. Aswers oly will ot ecessarily be awarded full marks. You may use a approved scietific calculator (o-programmable ad ographical), uless stated otherwise. If ecessary, roud your aswers off to TWO decimal places, uless stated otherwise. Diagrams are NOT ecessarily draw to scale. THEE diagram sheets for aswerig QUETION 4., QUETION 8, QUETION 9, QUETION 0, QUETION. ad QUETION. are attached at the ed of this questio paper. Write your cetre umber ad examiatio umber o these sheets i the spaces provided ad isert the sheets iside the back cover of your ANWE BOOK. A iformatio sheet, with formulae, is icluded at the ed of the questio paper. Number the aswers correctly accordig to the umberig system used i this questio paper. Write eatly ad legibly. Please tur over

3 Mathematics/P3 3 DBE/Feb. Mar. 0 QUETION The first FOU terms of the sequece of umbers are ; 5; 0 ad 7.. Write dow the ext TWO terms i the sequece. (). Write dow a recursive formula for the sequece. (3) [5] QUETION A large compay employs several people. The table below shows the umber of people employed i each positio ad the mothly salary paid to each perso i that positio. POITION NUMBE EMPLOYED IN POITION MONTHLY ALAY PE PEON (IN AND) Maagig director Director Maager Forema killed workers emi-skilled workers Uskilled workers Admiistratio Calculate the total umber of people employed at this compay. (). Calculate the total amout eeded to pay salaries for ONE moth. ().3 Determie the mea mothly salary for a employee i this compay. ().4 Is the mea mothly salary calculated i QUETION.3 a good idicator of a employee's mothly salary? Motivate your aswer. () [7] Please tur over

4 Mathematics/P3 4 DBE/Feb. Mar. 0 QUETION 3 The umber of M messages set by a group of teeagers was recorded over a period of a week. The data was foud to be ormally distributed with a mea of 40 messages ad a stadard deviatio of messages. ±68% ±96% ±00% x 3σ x σ x σ x x + σ x + σ x + 3σ Aswer the followig questios with referece to the iformatio provided i the graph: 3. What percetage of teeagers set less tha 8 messages? (3) 3. What percetage of teeagers set betwee 6 ad 5 messages? (3) [6] Please tur over

5 Mathematics/P3 5 DBE/Feb. Mar. 0 QUETION 4 A group of studets atteded a course i tatistics o aturdays over a period of 0 moths. The umber of aturdays o which a studet was abset was recorded agaist the fial mark the studet obtaied. The iformatio is show i the table below ad the scatter plot is draw for the data. Number of aturdays abset Fial mark (as %) CATTE PLOT HOWING THE NUMBE OF ATUDAY ABENT AND THE FINAL MAK ACHIEVED 90 Fial Mark (as %) Number of aturdays abset 4. Calculate the equatio of the least squares regressio lie. (4) 4. Draw the least squares regressio lie o the grid provided o DIAGAM HEET. () 4.3 Calculate the correlatio coefficiet. () 4.4 Commet o the tred of the data. () 4.5 Predict the fial mark of a studet who was abset for four aturdays. () [] Please tur over

6 Mathematics/P3 6 DBE/Feb. Mar. 0 QUETION 5 The sports director at a school aalysed data to determie how may learers play sport ad what the geder of each learer is. The data is preseted i the table below. DO NOT PLAY PLAY POT TOTAL POT Male Female Total Determie the probability that a learer, selected at radom, is: 5.. Male () 5.. Female ad plays sport () 5. Are the evets 'male' ad 'do ot play sport' mutually exclusive? Use the values i the table to justify your aswer. () 5.3 Are the evets 'male' ad 'do ot play sport' idepedet? how ALL calculatios to support your aswer. (4) [0] QUETION 6 I a factory, three machies, A, B ad C, are used to maufacture plastic bottles. They produce 0%, 30% ad 50% respectively of the total productio. %, % ad 6% respectively of the plastic bottles produced by machies A, B ad C are defective. 6. epreset the iformatio by meas of a tree diagram. Clearly idicate the probability associated with each brach of the tree diagram ad write dow all the outcomes. (4) 6. A plastic bottle is selected at radom from the total productio. 6.. What is the probability that it was produced by machie B ad it is ot defective? (3) 6.. What is the probability that the bottle is defective? (3) [0] Please tur over

7 Mathematics/P3 7 DBE/Feb. Mar. 0 QUETION 7 Three items from four differet departmets of a major chai store will be featured i a oepage ewspaper advertisemet. The page layout for the advertisemet is show i the diagram below where oe item will be placed i each block. A B C D E F G H I J K L 7. I how may differet ways ca all these items be arraged i the advertisemet? () 7. I how may differet ways ca these items be arraged if specific items are to be placed i blocks A, F ad J? () 7.3 I how may differet ways ca these items be arraged i the advertisemet if items from the same departmet are grouped together i the same row? (3) [7] Please tur over

8 Mathematics/P3 8 DBE/Feb. Mar. 0 I the ext FOU questios, esure you give reasos for EACH statemet you make. QUETION 8 I the diagram below, AM is the diameter of the bigger circle AMP. P is a commo taget to both circles at P. APB ad MPN are straight lies. M A 70 P N B 8. tate the size of Pˆ. () 8. Hece, show that BN is the diameter of the smaller circle. () 8.3 If Mˆ = 70, calculate the size of each of the followig agles: 8.3. Â () 8.3. Pˆ 6 () Bˆ () [7] Please tur over

9 Mathematics/P3 9 DBE/Feb. Mar. 0 QUETION 9 I the diagram below, O is the cetre of the circle with diameter K. P K. K itersects P at T. O T P K 9. If P = 4x, write dow the legth of T i terms of x. () 9. Prove that ΔT ΔPKT. (3) 9.3 If it is further give that TK = x ad T = 30 mm, calculate the value of x. (3) [7] Please tur over

10 Mathematics/P3 0 DBE/Feb. Mar. 0 QUETION 0 I ΔPQW, is a poit o PW ad is a poit o QW such that PQ. T is a poit o QW such that T P. T = 6 cm W : P = 3 : P Q 6 cm T W Calculate: 0. WT (3) 0. WQ (4) [7] QUETION. I the diagram below, O is the cetre of the circle. PT is a cyclic quadrilateral. Prove the theorem that states P Tˆ + PŜ = 80. P O T (6) Please tur over

11 Mathematics/P3 DBE/Feb. Mar. 0. I the diagram below, O is the cetre of the circle. AB is a diameter of the circle. Chord CF produced meets chord EB produced at D. Chord EC is parallel to chord BF. CO ad AC are joied. Let Ô = x C A 3 O x F E 3 B D.. Determie, i terms of x, the size of Fˆ. (4).. Prove that DF = BD. (4)..3 how that Ĉ = Ĉ3. (4)..4 If DF = 5 cm ad OA = 6 cm, calculate area ΔBFD : area ΔAOC. (4) [] TOTAL: 00

12 Mathematics/P3 DBE/Feb. Mar. 0 CENTE NUMBE: EXAMINATION NUMBE: DIAGAM HEET QUETION CATTE PLOT HOWING THE NUMBE OF ATUDAY ABENT AND THE FINAL MAK ACHIEVED 90 Fial Mark (as %) Number of aturdays abset QUETION 8 M A 70 P N B

13 Mathematics/P3 DBE/Feb. Mar. 0 CENTE NUMBE: EXAMINATION NUMBE: DIAGAM HEET QUETION 9 O T P K QUETION 0 P Q 6 cm T W

14 Mathematics/P3 DBE/Feb. Mar. 0 CENTE NUMBE: EXAMINATION NUMBE: DIAGAM HEET 3 QUETION. P O T QUETION. C A 3 O x F E 3 B D

15 Mathematics/P3 DBE/Feb. Mar. 0 b ± x = b 4 ac a A = P( + i) A = P( i) INFOMATION HEET: MATHEMATIC INLIGTINGBLAD: WIKUNDE A = P( i) A = P( + i) i= = i= ( + ) i = T = ar a( r ) = F = f x [( + i) ] i f ( x + h) f ( x) '( x) = lim h 0 h r T a + ( ) d = = ( a + ( d ) ; r x[ ( + i) ] P = i ( ) ( ) x + x y + y d = x x + y y M ; y = mx + c y y = m x ) ( x a) + ( y b) = r I ΔABC: si a A area Δ ABC ( x b c = = a = b + c bc. cos A si B si C = ab. si C ) a = ; < r < r y y m = m = taθ x x ( α + β ) = siα.cosβ cosα. si β si( α β ) = siα.cosβ cosα. si β si + cos ( α + β ) = cosα.cos β siα. si β cos ( α β ) = cosα.cos β + siα. si β cos α si α cos α = si α si α = siα. cosα cos α ( x ; y) ( x cosθ + y siθ ; y cosθ x siθ ) ( x ; y) ( x cosθ y siθ ; y cosθ + x siθ ) ( xi x) = σ = i= fx x ( A) P( A) = P(A or B) = P(A) + P(B) P(A ad B) y ˆ = a + bx ( ) b ( x x) ( x x) ( y y) =

NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P FEBRUARY/MARCH 0 MARKS: 50 TIME: 3 hours This questio paper cosists of 9 pages, 5 diagram sheets ad iformatio sheet. Please tur over Mathematics/P DBE/Feb.