Mathematics Grade 9 ANA Exemplars 2012 Memo (English)

Downloaded from hlayiso.com ANNUAL NATIONAL ASSESSMENT GRADE 9 MATHEMATICS MEMORANDUM SET 1: 2012 EXEMPLAR

Downloaded from hlayiso.com ANNUAL NATIONAL ASSESSMENT 2012 GRADE 9 MATHEMATICS EXEMPLAR MEMORANDUM REAL NUMBER SYSTEM 1. 1.1.1 4 - rational (1) 1.1.2 √2 - irrational (1) 1.1.3 0,2- rational (1) 1.2 NUMBERS REAL NON - REAL UNDEFINED 0 √ 7 7 √ 0 √7 √ √−7 √ (4) 1.3.1 7 (1) 0,7 = 10 1.3.2 13 (1) 0,13 = 100 1.3.3 201 1 (1) 2,01 = =2 100 100 1.4.1 2,5 × 10 × 7 = 1,75 × 10 (1) 1.4.2 0,007 (2) 2

Downloaded from hlayiso.com 1.4.3 1,12 × 10 × 3 × 10 = 3,36 × 10 (2) 1.5.1 1,6 (1) 1.5.2 −√5 (1) 1.6.1 0,152, 0,153, 0,154 anyone of the three (1) 1.6.2 0,71; 0,72; 0,73; 0,74; 0,75; 0,76; 0,77 anyone of the seven (1) 1.7.1 2 and 3 (2) 1.7.2 4 and 5 (2) 1.7.3 1 and 2 (2) 1.7.4 4 and 5 (2) 1.8 NATURAL WHOLE INTEGER RATIONAL IRRATIONAL REAL NUMBER NUMBER 7 √ √ 15 √ √ 1 2 8 0,081 √ √ 2 √ √ −√16 √ √ √ 0,528 √ √ 2,6 √ √ 6 √ √ √ √ √ 2 3

Downloaded from hlayiso.com 1.9.1 0,6 < 0,625 < 0,75 < 0,8 (1) 1.9.2 0,2 < 0, 2̇ 0̇ < 0, 2̇ < 0,24 (1) 1.9.3 √0,36 < √0,366 < 0, 6̇ < 0,69 (1) 1.9.4 −3 5 1 −2 (1) < − <− < 2 6 6 3 1.9.5 −0, 12 < −0,11 < ̇−0,1 < −0,01̇ (1) FINANCIAL MATHEMATICS 1. (3) 1.1 Selling price=22 × 1,50 = 33 Profit= Selling price – cost price = 33 − 20 = 13 1.2.1 a) 18% 75 000 = 13 500 (2) b) Tax = 12 000 + 20% ( 97 500 − 80 000) (2) = 12 000 + 2 100 = 14 100 c) Tax= 20 000 + 25% 30 000 (2) = 27 500 d) Tax= 42 000 + 35% 80 000 (2) = 70 000 1.2.2 Taxable income R140 000 R100 000 R230 000 Tax R25 000 R16 000 R45 4

Downloaded from hlayiso.com 500 1.3 Increase: 1 740 − 1500 = 240 % increase = × 100 = 16 (2) 1.4 = (1 + ) (2) = 1500(1 + 2(0,11)) = 1 830 or × × SI= = = 330 Total amount = 1500 + 330 = 1 830 1.5.1 Investment = (1 + ) (3) 11,5 = 2 750 1 + 100 = 5 891,92 1.5.2 Interest earned= 5 891,92 − 2 750 = 3 141,92 (1) 1.6.1 = (1 + ) (4) × 12 = 9500 1 + 100 × 4 = 24 463,29 Interest = 24 463,29 − 9 500 = 14 963,29 5

Downloaded from hlayiso.com 1.6.2 = (1 + ) (4) × 8 = 9500 1 + 100 × 2 = 17 793,32 Interest= 17 793,32 − 9 500 = 8293,32 1.6.3 = (1 + ) (4) × 6,5 = 9500 1 + 100 × 12 = 15 956,86 Interest= 15 956,86 − 9 500 = 6456,86 1.6.4 = (1 + ) (4) 7,25 = 9500 1 + 100 = 16630,37 Interest= 16630,37 − 9 500 = 7130,37 6

Downloaded from hlayiso.com RATIO AND PROPORTION 1 (3) 1.1 Speed= km/h = 300 km/h 1.2 (3) MINUTES 2 4 6 8 20 COST R1,60 R3,20 R4,80 R6,40 R16,00 1.3 (3) NO OF MEN TIME 5 45 9 45 × 5 = = 25ℎ 9 OR Time taken by 5 men = 45ℎ Time taken by 1 man = (45 × 5)ℎ × Time taken by 9 men = = 25ℎ 1.4 (3) TIME NO OF HOURS 8 3 2 2 =8×3 = 12 OR In 8 hours no of workers needed = 3 In 1 hour no of workers needed = 3 × 8 In 2 hours no of workers needed = 3 × 8 ÷ 2 = 12 7

Downloaded from hlayiso.com 1.5 (2) NO OF KG COST IN R 3 24 7 7 = 24 3 7 × 24 = 3 Cost = R56 OR Cost of 3 kg = 24 Cost of 1 kg = 24 ÷ 3 Cost of 7 kg = 8 × 7 = 56 1.6.1 Direct proportion (1) 1.6.2 =5 (1) 1.6.3 = 5(25) = 125 (1) 1.7 COST IN R NO OF (3) APRICOTS 5,60 15 10,08 15 10,08 = 5,60 10,08 × 15 = 5,60 = 27 8

Downloaded from hlayiso.com 1.8 , (3) Cost per ℓ = = 8,65 1.9 (3) NO OF KG COST IN R 0,35 25,10 1 1 = 25,10 0,35 2510 = 35 = 71,714 … ~ 71,71 1.10 (3) NO OF KG COST IN R 12,5 90 17,2 7,2 = 90 12,5 7,2 × 90 = 12,5 Cost= R51,84 1.11 (3) Distance in km No of ℓ 130 6,5 80 80 = 6,5 130 80 × 6,5 = 130 =4 No of ℓ = 4 1.12 (3) 9

Downloaded from hlayiso.com No of ℓ No of m 5 40 1 1 = 40 5 1 × 40 = 5 =8m SPEED/TIME /DISTANCE 1. × = 1.1 120 × = 600 = h = 5h (2) ∴time taken is 5 hours. 1.2 × = × 8 = 600 600 = /ℎ 8 = 75 /ℎ (2) 10

Downloaded from hlayiso.com 2. S (km/h) T (h) D (km) (3) car 100 x 1 0x air 300 6-x 300(6-x) 100 + 300(6 − ) = 1200 100 + 180 = 1200 −200 = −600 =3 Time by air = (6 − 3)ℎ = 3ℎ 3. = (120 × 1,5) km = 180 km (3) , km , = = 125 km/h h 660 z= h = 3h 220 4. d =s×t (3) = (5 × 0,5) km = 2,5 km s × 0,25 = 2,5 2,5 s= 0,25 s = 10 km/h 5.1 Average speed = km/h (3) = 7,5 km/h 5.2 Distance = × (1) = (7,5 × 6) km = 45 km 6. s×t=d (3) (p + 20)t = q t= h 11

Downloaded from hlayiso.com EXPONENTS 1. 1.1 99 × 1001 = 99 × 1 = 99 (1) 1.2 (1) × = = 1.3 ( ) (2) = = 1.4 (8 )(−4 ) −32 (3) = (2 ) 4 = −8 1.5 2 8 (3) 2 2 2 = 2 2 = 2 =2 1.6 15 . 3 . 25 (4) 9 . 125 . . . = . = =3 1 = 3 1.7 5 5 10 10 (3) (2 )× = (2 )× = = 10 = 1.8 1 (1) ( )( )= = = 1.9 1 (2) ( . . )÷ = ÷ = 1.10 (3 ) ( ) 27 ×1 (3) = = 27 = 27 ( ) 12

Downloaded from hlayiso.com PATTERNS, FUNCTIONS AND ALGEBRA 1 1.1 90 and 69 (2) 1.2 36; 72; 108; 144; 180 ; 216 (2) 1.3 39; 48 (2) 2 2.1 Add 6. (1) 2.2 =6 −3 (1) 2.3 6 − 3 = 45 6 = 48 =8 (1) The 8th term = 45 3.1 Number 1 2 3 4 5 6 7 (2) of panels Number 2 3 4 5 6 7 8 of poles Number 6 12 18 24 30 36 42 of slabs 3.2 = +1 (2) 3.3 =6 (2) 3.4 ( ) = 10 + 1 = 11 (2) ( ) = 6(10) = 60 4 (3) Input Rule Output 1 −2 + 3 1 2 −1 3 −3 5 Input Rule Output −2 2 −1 7 13

Downloaded from hlayiso.com −1 1 0 −1 1 1 2 7 (5) 6 6.1 = −2(0) + 6(0) − 8 = −8 (1) 6.2 = −2(−1) + 6(−1) − 8 = −2 − 6 − 8 = −16 (1) 6.3 = −2 +6 −8 6.4 = −2( + 2) + 6( + 2) − 8 (4) = −2( + 4 + 4) + 6 + 12 − 8 = −2 −8 −8+6 +4 = −2 −2 −4 7 7.1 =3 =1 (1) 7.2 = 3 = 27 (1) 7.3 1 (1) =3 = 9 PRODUCTS 1 1.1 6 −3−5=6 −8 (2) 1.2 5 −6 +3= − +3 (3) 1.3 5 −6 + 3 = −6 +8 (3) 1.4 −9 (2) 1.5 9 −4 (2) 1.6 3( + 2 − 3) = 3 +6 −9 (3) 1.7 −5(2 + − ) = −10 −5 +5 (3) 1.8 − 8 + 16 (3) 1.9 4 − 12 + 9 (3) 14

Downloaded from hlayiso.com 1.10 3(16 − 24 + 9) = 48 − 72 + 27 (3) 1.11 −5 +6−( + 4 + 4) = −9 + 2 (5) 1.12 3 − 6 − 12 + 12 − 5 + 10 = −14 + 6 + 10 (6) 1.13 2 1 2 2 (9 − 1) − ( − 1) = 3 − − + 3 3 3 3 7 1 = + (6) 3 3 1.14 1 1 1 1 1 1 4( − )−( + + )=4 − − − − 16 2 16 4 2 16 1 5 =3 − − 2 16 (6) 1.15. 1 2 1 − − ( − + ) 1 1 = − − +2 − 2 (6) =2 − FACTORS 1. 1.1 4(2 + 3 ) (2) 1.2 −5(2 + 3 ) (2) 1.3 + (1) 1.4 4(6 − 3 + 4) (2) 1.5 ( − 1) (2) 1.6 −2 (1 + ) (2) 1.7 2 (4 − 5) (2) 1.8 6 ( −2 + 3) (2) 1.9 4 (2 −3 − 5) = 4 (2 − 5)( + 1) (2) 1.10 ( − ) − ( − ) = ( − )( − ) (3) 1.11 ( − )( − ) = ( − )( − )( + ) (3) 1.12 − )( + ) (2) 1.13 ( − ) = ( − )( + ) (3) 15

Downloaded from hlayiso.com 1.14 ( + )( − )=( + )( + )( − ) (3) 1.15 (4 + )(4 − ) (2) 1.16 3( −9 ) = 3( − 3 )( + 3 ) (3) 1.17 1 1 (3) ( − 4) = ( − 2)( + 2) 2 2 1.18 1 1 (3) (4 − ) = (2 − )(2 + ) 8 8 OR 4 − (2 − )(2 + ) = 8 8 2 2.1 (97 − 3)(97 + 3) = (94)(100) = 9400 (2) 2.2 (101 − 100)(101 + 100) = 1(201) = 201 (2) 2.3 (55 − 25)(55 + 25) = (30)(80) = 2400 (2) 2.4 (20,5 − 15,5)(20,5 + 15,5) = (5)(36) = 180 (2) EQUATIONS 1. (2) 1.1 6 = 24 (3) =4 1.2 ( − 3) = 0 = 0 or =3 (2) 1.3 2 + 4 = 9 − 12 7 = 16 or 16 = 7 16 16 (3) = = 7 7 1.4 × 36: 9( − 5) − 4(3 − 4 ) = 468 9 − 45 − 12 + 16 = 468 25 = 525 = 21 (4) 16

Downloaded from hlayiso.com 1.5 × 15: 45 + 3 + 195 = 5(2 + 1) 48 + 195 = 10 + 5 38 = −190 = −5 (4) 1.6 5 =5 ∴ =4 (2) 1.7 2 =2 ∴− =5 (2) = −5 1.8 3 =3 (2) ∴ +1 =4 =3 2.1 + +1+ + 2 = 78 3 + 3 = 78 3 = 75 = 25 The numbers are 25; 26; 27 (4) 2.2 AGE NOW AGE IN 4 YEARS Ben +4 Patrick 5 5 +4 5 + 4 = 3( + 4) 5 + 4 = 3 + 12 2 =8 =4 Ben’s age is 4 years and Patrick’s is 20 years (4) 2.3 Area = × −12 +6 = −6 × −12 + 6 = −6 −6 (2 − 1) = −6 17

Downloaded from hlayiso.com = (2 − 1) (3) 2.4 Area = × 2 − 8 = × ( + 2) 2 −8 = +2 2(− 4) = +2 2( + 2)( − 2) = +2 = 2( − 2) (5) 2.5 =2 Area = × = 32 2 × = 32 2 = 32 = 16 = ±4 =4 (because > 0) ∴ =4 and = 2(4) = 8 (4) INEQUALITIES 1. 1.1 (2) 1.2 • (2) 18

Downloaded from hlayiso.com 1.3 • •• (2) 1.4 (2) 1.5 ••• (2) 2 2.1 <2 (2) 2.2 3≤ ≤ 7and ∈ (2) 2.3 √3 ≤ < 2,1and ∈ (2) 2.4 −3 ≤ ≤ 0 and ∈ or −4 < < 1 and ∈ (2) 2.5 < −2 or ≥ 0 and ∈ (2) ALGEBRAIC FRACTIONS 1. 1.1 2 (2) 1.2 − (2) 1.3 3 2 6 (5) × × = 19

Downloaded from hlayiso.com 1.4 ( − 4) 4 ( − 1) ( − 4)( − 1) (5) × × = 4 ( − ) ( − ) 1.5 2 (1) + = = 2 2 2 1.6 3(7 − 5) − 2(10 − 7) 21 − 15 − 20 + 14 −1 (3) = = 6 6 6 1.7 5(6 − 10) + 6(3 − 5) − 180 30 + 50 + 18 − 30 − 180 (4) = 30 30 48 − 60 = 30 16(3 − 10) = 30 8(3 − 10) = 15 1.8 3( − 1) − ( + 2)( + 1) 3 − 3 − − 3 − 2 (4) = ( − 1)( + 1) ( − 1)( + 1) − −5 = ( − 1)( + 1) 20

Downloaded from hlayiso.com GRAPHS 1 1.1 2 3 (4) 21

Downloaded from hlayiso.com 1.2 y   23 x  1 3 2 (4) 2. 2.1 = −2 − 2 (2) 2.2 = −2 + 3 (2) 2.3 1 (2) = +1 3 22

Downloaded from hlayiso.com 3. (3) Y 8 7 x  3 6 5 y  x 1 4 3 3y  x  3 2 B 1 A  -5 -4 -3 -2 -1 0 1 2 3 4 X  C 4. =2 +2 (3) 23

Downloaded from hlayiso.com SPACE AND SHAPE 1. 1.1 A.Trapezium B. Octagon C. Triangle D. Rectangle E. Parallelogram (5) 1.2 A. cylinder (4) B. rectangular-based pyramid C. cube D. triangular pyramid 2.1 (1) 2.2 70° East of North or 20° North of East or 70° North (2) 3. In ∆ and ∆ (5) = = 90°(given) is common = (given) ∴∆ ≡∆ (90°, ℎ , ) 4. + = (Pyth) (5) + (3) = (5) = 25 − 9 = 16 = 4unit = − = 4 − √7 unit = + (Pyth) = 7 + 9 unit = 16unit = 4 units 24

Downloaded from hlayiso.com 5. 2 + = 180° (co-int< ; || ) (4) + 120° = 180° 2 = 60° 2 + + = 180° ( sum of < of ∆) but = (< opp equal sides) 2 +2 = 180° 60° + 2 = 180° 2 = 120° ∴ = 60° 2 + + = 180°(sum of < of ∆) but = (< oppequal sides) 2 +2 = 180° 120° + 2 = 180° 2 = 60° ∴ = 30° 2 + + = 180° ( suppl< on a str line or is a str. <)60° + + 30° = 180° 2 = 180° − 90° = 90° 6. = (< opp. sides) 40° + + = 180° ( sum of < of ∆) (6) 2 = 180° − 40° 2 = 140° = 70° = = 70° ( alt < , || ) + = 180° (suppl< on a str line or K is a str. <) = 180° − 70° 2 = 110° 25

Downloaded from hlayiso.com 7. 7.1 In ∆ and ∆ (4) = ( common side ) = (alt. < , || ) = ( alt. < , || ) ∴∆ ≡∆ (<, <, ) ∴ = = ( . ∆) 7.2 In ∆ and ∆ (4) = (proved above) = (alt. < , || ) = ( alt. < , ||) ∴∆ ≡∆ ( <, <, ) ∴ = = ( . ∆) 7.3 The diagonals of a parallelogram bisect each other. (1) 8. = = = (prop. sides of similar quads) (6) 7 = 9 28 9×7 = = 2,25 28 6 7 = 28 6 × 28 = = 24 7 26

Downloaded from hlayiso.com 9. In ∆ and ∆ (6) = (alt < , || ) = (alt < , || ) = (vert. opp< ) ∆ ||| ∆ (<, <, <) ∴ = = (prop. sides of similar ∆s) 2 3 ∴ = 6 =9 , and = = 7,5 = + = 2,5 + 7,5 = 10 TRANSFORMATIONS 1. 1.1 (−1; −4) (1) 1.2 (3; −6) (1) 1.3 (−3; −4) (1) 1.4 (−4; 3) (1) 2. Reflection about the X-axis (4) 3. Vertical translation 3.1 Reflection about the Y-axis (2) 3.2 Reflection about the Y-axis and then reflection about the X-axis or reflection about the X-axis and then reflection about the Y-axis (4) 4. ′(−6; 9) ; ′(3; 15) ; ′(15; −6) (3) 27

Downloaded from hlayiso.com 5.1 (1) 5.2 (1) 28

Downloaded from hlayiso.com MEASUREMENT 1. 1.1 Area of ∆ = ℎ = × 3 × 4 cm = 6cm (2) 1.2 Area of ∆ = ℎ = × 6 × 4 cm = 12cm (3) 1.3 Area of ADEG= ( + ) = (9 + 6)4 cm = 30cm (3) 1.4 Area of BDEG= × ℎ = 6 × 4 cm = 24cm (3) 1.5 = + (Pyth) = 3 + 4 cm = 9 + 16cm = 25cm ∴ BE = 5cm 1.6.1 Perimeter of ∆ = (5 + 5 + 6)cm = 16cm (2) 1.6.2 Perimeter of ADEG = (9 + 5 + 6 + 4)cm = 24cm (4) 2. = + (Pyth) = 49 + 120 unit = 169 unit = 13 units Area = 1 13 =( × × )unit 2 2 = 66,36 unit (3) 3. 3.1 Total surface area= 2(8 × 12) + 2(12 × 18) + 2(8 × 18)cm = 912cm (6) 3.2 Volume= × ×ℎ = (8 × 12 × 18)cm = 1728cm (3) 29

Downloaded from hlayiso.com 4. Volume of cylinder= ℎ 22 550 = × (5) × ℎ 7 550 × 7 ℎ= 22 × (5) (4) ℎ=7 5. Volume = × ×ℎ 50 × 30 × ℎ = 24000 24000 ℎ=( ) 50 × 30 ℎ = 16 (3) DATA HANDLING 1. 1.1 20 30 30 40 50 50 60 70 70 70 70 80 80 90 100 (1) 1.2 1.2.1 Median = 70 (1) 1.2.2 Mode = 70 (1) 1.2.3 Range = 100 − 20 = 80 (1) 1.2.4 Mean = = 60,67 to decimal places (2) 2. 2.1 24 passengers (1) 2.2 Range = 70 − 1 = 69 (1) 2.3 Mean= = 26,7 = 27 (3) 2.4 AGES IN YEARS TALLY FREQUENCY (INTERVAL) MARKS 1 - 13 IIIIII 7 14 - 26 III 3 27 - 39 IIIIIII 8 40 - 70 IIIII 6 (4) 30

Downloaded from hlayiso.com 2.5 27-39 (1) 2.6 18 passengers (1) 3. 3.1 22 (1) 3.2 range= 15 − 2 = 13 (1) 3.3 SHOE SIZES TALLY FREQUENCY (INTERVAL) MARKS 1-5 IIII 4 6 - 10 IIIIIIIIII 12 11 - 15 IIIII 6 (4) 4. 4.1 18 babies (2) 4.2 15 babies (1) 4.3 11 – 15 months (1) 5.1 DIE COIN COMBINED OUTCOMES 1 H 1H T 1T 2 H 2H T 2T 3 H 3H T 3T 4 H 4H T 4T 5 H 5H T 5T 6 H 6H T 6T (6) 31

Downloaded from hlayiso.com 5.2 (2) 5.3 5.3.1 1 (1) 12 5.3.2 3 1 (1) = 12 4 6.1 1 (1) 10 6.2 1 (1) 10 6.3 2 1 (1) = 10 5 7. 7.1 18 3 (1) = 30 5 7.2 12 2 (1) = 30 5 8. 8.1 3 (1) 32

Downloaded from hlayiso.com 8.2.1 3 (1) 10 8.2.2 2 1 (1) = 10 5 8.2.3 1 (1) 10 H H T H H T T H H T T H T T 33