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ANSWERS234Number of bees, b Length of bee’s journey, x km Mass of honey, m g10 150 1020 375 50100 750 500500 300 100010 6 1000 6.67 3 10 7b x bx 4 150b 150m 4 b m150m 4 x x mNumber of grass-cutters, n Number of rugby pitches, r Time, t hours2 1 0·51 1_ 3 2 1.54 4 1r 4 t r tn tn tn r r 4 nNumber of houses, h Mass of waste, w kg Time, t weeks1 20 1100 20 000 109615 10 6 52h w w 4 20hw 4 20t w th 20ht t5 a 107333 b __ 3445 c __ 2566 d ___ 2036 254713 198 Exercise 21 a y 5 9x b y 5 90 c x 5 52 a y 5 0.4x 2 b y 5 90 c x < 6.123 a p 5 20√ __ q b p 5 160 c q 5 6.254 a p 5 ___ 50q b p 5 2.5 c q 5 2.55 a 5 20bb 10 5 30a 200 300 6006 a d 5 5t 2 b d 5 20 m c t < 4.24 s7 a C 5 1.5d b $97.50 c 53.3 cm8 a p 5 1.5n 2 b p 5 £216 c n 5 202
Exercise 2*1 a y 5 4x 3 b y 5 256 c x 5 62 a y 5 ___ 80x 2 b y 5 5 c x < 12.63 a p 5_____ 2500√ q b p 5 250 c q 5 25004 a p 2 5____ 8003 b p < 3.54 c q < 3.17q5b 125 8 1a 2 5 10ANSWERS6 a e 5 0.5v 2 b e 5 1250 kJ c v 5 1414 m/s7 a n 5 ____ 2502 b n 5 62.5 c t 5 15.8 yrst8 a v 5 3.1√ __d b i v 5 21.9 m/s ii v 5 98.0 m/sc 0.104 md d 5 5011 mExercise 31x 22 21 0 1 2 3 4y 217 21 3 1 21 3 192x 22 21 0 1 2 3 4y 219 0 1 24 23 16 653x 24 23 22 21 0 1 2 3 4y 23 24 26 212 — 12 6 4 34x 1 4 8 12 16 20y 100 25 12.5 8.3 6.25 55 a yb yc ydy x 3y 1 xy x 3yy 1 xOxOxOxOx6 at 1 2 4 6 8 10 12p 20 10 5 3.3 2.5 2 1.7b i 1 1_ 3 months ii $2222c x 5 803
v504030201001 2 3 4 5 6 7 8 9 10c i v 5 20 m/s, t 5 2. Time 5 12:02ii t > 4. Time for speed to be at least 25 m/s is: 12:04 < time < 12:10tANSWERSExercise 41 62° 2 55° 3 124° 4 132° 5 54° 6 66° 7 30° 8 222°9 12 10 3 11 8 12 8 13 4 14 3 15 5 16 4Exercise 4*1 57° 2 112° 3 68° 4 42° 5 44° 6 42° 7 228° 8 449 6 10 3 11 8 12 5 13 4 14 4 15 √ __2 16 4.85Exercise 51 16 30 2 2 11DGGB2 4 61 2 71613 3 50 4 8 39BFPV7 3 128 4 1428135 aABbAB5
cABdABANSWERSeAB6 aABbABcABdABeAB7 aABbABcABdABeAB6
8 acAABBbdAABBANSWERSeAB9 a A 5 {21, 0, 1} b B 5 {3, 4, 5}c C 5 {21, 0, 1} d D 5 {1, 2, 3, 4}10 a A 5 {x: x , 5} b B 5 {x: x > 28}c C 5 {x: 22 , x , 4} d D 5 5 {x: 3 < x < 8}Exercise 5*1H 188Q5n() 5 31 2 50B 1923x4 x256RG7a 4 2 xb x 5 33 40P 33CT 1210 x 12 xOa 12 2 x 4 60a 6 2 xb x 5 5SMb x 5 310 6 x 7x4 9 x1215PN5 aABbAB7
Examination practice 11 a 4 girls b 40 cars2Number of men, m Number of humous tubs, h Time, t mins2 120 54 480 10ANSWERS6 1080 15m h h 4 (12m)h 4 (12t) h tm 12mt t3 a 5_ 9 b ___ 37990 c ___ 290849954 ___ 263329 5 a y 5 8x b y 5 80 c x 5 56 a x 5 4 3 √ __y b y 5 51235 117d ____33 300 7 a y 5 48 ___x b y 5 6 c x 5 48 a N 5 9000 _____d 2 b N 5 2250 c d 5 39 a A 5 15h 2 b A 5 135 m 2 c h 5 6 m10 at 0 1 2 3 4 5 6w 25 24 17 10 9 20 49bw50403020100 1 2 3 4 5 6 tii 12.6 mphii 01:40, 05:009
11 at 1 2 3 4 5 6ANSWERSbw 2000 1000 667 500 400 333w20001500ii 571 approxii March 15 th approx.100050001 2 3 4 5 6t12 a x 5 60°, y 5 60°, z 5 55° b x 5 40°, y 5 70°, z 5 40°13 a x 5 16 b x 5 10 14 a 9 b 5 c 715 a b ABABcAB(A ∩ B)dAB(A ∪ B)(A ∩ B)(A ∪ B)Exercise 61 a 5 3 10 6 mm b 8 3 10 5 cm c 20 km d 400 km2 a 500 m 2 b 3 3 10 6 m 2 c 4 3 10 6 mm 2 d 2 km 23 a 2 3 10 6 cm 3 b 4 3 10 9 m 3 c 80 mm 3 d 70 m 34 a m 1 n 5 a m 2 n 6 a mn 7 18 1100 9 ___ 11000 10 1_ 8 11 1_ 912 125 13 ___ 1125 14 __ 127 15 __ 16416 181 17 ___ 1256 18 ___ 1125 19 __ 13620 1125 21 ____ 110 000 22 __ 132 23 __ 124 a 6 25 1 26 c 5 27 d 249 10
28 e 4 29 f 2 30 g 3 31 h 1532 1_ 233 21 34 22 35 2336 1_ 3 37 2 1_ 2 38 2 1_ 3 39 2 1_ 440 1 41 2 42 3 43 344 25 45 24 46 24 47 2348 1 49 1_ 2 50 1 51 1_ 452 2 53 2 54 21 55 10 000ANSWERSExercise 6*1 a 8.3 3 10 6 mm b 5700 km c 19 km d 3.5 3 10 4 km2 a 5 3 10 6 cm 2 b 9.6310 3 km 2 c 3.8 3 10 12 mm 2 d 0.84 m 23 5.3 3 10 19 mm 3 b 3.3 km 3 c 0.1 m 3 d 0.001 m 34 1_ 3 5 1_ 4 6 1_ 5 7 1_ 6 8 1_ 2 9 1_ 3 10 1_ 4 11 1_ 5 12 1_ 4 13 1_ 9 14 __ 116 15 __ 116 1_ 8 17 1_ 8 18 ___ 11000 19 120 125821 2197642522 412523 24 3_ 4 25 4_ 5 26 __ 310 27 4_ 9 28 a 22 5__ 1 a 2 29 b 21 5__ 1 b30 c 2 31 d 232 e 21 5__ 1 33 f 22 5__ 1 e f 2 34 g2 1_ 3 5___ 13 √ g 35 h236 x 5 __ 310 , y 5 2 __ 910 37 x 5 3_ 4 , y 5 0 38 x 5 2, y 5 2 3_ 5 39 x 5 3_ 5 , y 5 7_ 3 25 ___125 ___27 Exercise 71 1, 3 2 24, 23 3 1 4 22, 55 23, 2 6 22, 4 7 23, 27 8 22, 99 23, 8 10 21 11 24, 0 12 0, 713 22, 21 14 23, 2 15 25, 22 16 23, 517 3 18 26, 2 19 21, 0 20 0, 421 22, 2 22 23, 3 23 23.73, 20.268 24 0.697, 4.3025 21.45, 3.45 26 23.79, 0.791 27 25.24, 20.764 28 0.298, 6.7029 20.898, 3.90 30 21.86, 20.537 31 22.77, 0.271 32 20.231, 7.2333 23.31, 20.393 34 20.345, 0.637 35 x(x 1 1) 5 7, 2.19 m by 3.19 m36 1_ 2 x(x 1 3) 5 25, 5.73 cm 37 x2 1 (x 1 1) 2 5 (x 1 2) 2 , x 5 338 x 2 1 (x 1 1) 2 5 145, 8 and 9 39 x(30 2 x) 5 210, 11.1 cm by 18.9 cm11
ANSWERS40 x(x 1 2) 5 2 3 3, x 5 1.65 41 22 < x < 2 42 x , 27 or x . 743 24 , x , 4 44 x < 23 or x > 3 45 22 , x , 1 46 x < 24 or x > 2247 21 < x < 6 48 x , 3 or x . 4 49 21 , x , 5 50 x < 1 or x > 2Exercise 7*1 25, 22 2 25, 3 3 2 4 2, 65 1, 9 6 27, 2 7 26, 8 8 212, 239 4 10 0, 19 11 21, 4 12 26, 313 23, 22 14 22, 1 15 24, 23 16 22, 417 2 18 28, 3 19 22, 0 20 0, 321 26, 6 22 27, 7 23 1.59, 4.41 24 24.35, 0.34525 20.436, 3.44 26 20.135, 2.47 27 22.84, 1.17 28 20.257, 2.5929 212.7, 20.315 30 3.11, 21.61 31 21.82, 0.823 32 257.9, 24.2933 23.30, 0.379 34 22.45, 20.147 35 x 2 1 (x 1 2) 2 5 202, 9 and 1136 ___ x21 5 ___ 21 , x 5 14.85 cm2x 37 x2 1 (5 2 x) 2 5 16, 1.18 cm and 3.82 cm38 x(x 1 3) 5 9 3 5, x 5 5.37 39 p(x 1 2) 2 2 p2 2 5 p2 2 , x 5 0.828 m40_____( 1200x 2 20 )(x 1 3) 5 1200, x 5 12 days 41 x < 24 or x > 442 22 , x , 2 43 2 < x < 6 44 x , 25 or x . 3 45 28 , x , 2146 22 < x < 7 47 x < 29 or x > 7 48 25 , x , 8 49 x , 23.45 or x . 1.4550 21.65 < x < 3.65Exercise 81 a 22.6, 2.6, y 5 7 b 0, 1, y 5 x c 21.3, 2.3, y 5 x 1 3d 22.6, 1.6, y 5 4 2 x e 1, y 5 2x 2 1 f 22.4, 0.4, y 5 1 2 2x2 a 21.6, 0.6, y 5 0 b 21, 0, y 5 21 c 22, 1, y 5 1d 22.6, 1.6, y 5 3 e 23.3, 0.3, y 5 22x f 21, 1, y 5 x3 a y 5 2 b y 5 x c y 5 1 2 x d y 5 3 1 2x4 a y 5 0 b y 5 23 c y 5 3 d y 5 x5 a x 2 1 1 5 0 b 2x 2 1 x 2 3 5 0 c x 2 2 4x 1 3 5 0 d 2x 2 1 2x 2 4 5 06 a 21.4, 0, 1.4, y 5 2x b 0, y 5 23xc 1.4, y 5 4 2 x d 21.9, 0.3, 1.5, y 5 3x 2 17 a 21.5, y 5 22 b 21.7, 1.7, y 5 x c 1, 3, y 5 4 2 x d 22.3, 1.3, y 5 x 1 18 20.6, 1.69 21.3, 2.310 (1.5, 2.5)12
ANSWERSExercise 101 a 1_ 4 b 1_ 2 c 1_ 4 2 b i ___ 13 b 3_ 8 c __ 15 b i ___ 1225 __ii ___ 196225 __10 4 b 949 c 2449 200 ii __ 825 iii __ 425 6 c i __ 121 ii __ 314Exercise 10*1 b i __ 815 ii 2_ 53 b P(H > 2) 5 1_ 2 so she is not especially lucky.4 b i 310 ii 3_ 5 iii __ 710 5 b i 1_ 4 ii __ 724 iii 3_ 8 6 b i 72543ii 75iii 1825 28iii ___225 13iii __42 iii __ 1415 2 b i 1_ 4 ii 1_ 2 iii 3_ 4 Examination practice 21 a x b x 3 c 1 d x2 a x 22 b x 22 c x d x3 a 4 b 8 c 1_ 9 d __ 164 4 a 1.8 3 10 3n 1 1 b 5 3 10 n 2 15 a 64 b 0, 16 c 216, 06 a 24, 23 b 1, 3 c 22, 4 d 25, 17 a 3 b 62 c 0, 28 a 21, 2 b 22, 2 1_ 2 c 1_ 3 , 39 a 25.83, 20.172 b 0.438, 4.56 c 20.732, 2.73 d 0.279, 2.3910 x 5 311 c x 5 0.69712 a 210 < x < 10 b x , 24 or x . 4 c 21 , x , 2 d x < 25 or x > 313 a y 5 6 b y 5 2x 1 1 c y 5 2 2 x14 3x 2 2 x 2 1 5 015 a 3 3 10 7 mm b 10 km 2 c 1.2 3 10 7 cm 316 P 5 20.1 cm, A 5 30.2 cm 217 a 5 cm b x 5 53.1° c 23.2 cm 2 d 224 cm 318 a 135 cm 3 b 10.8 cm c 12 cm 2 and 55.7 cm 219 a i 0.6 ii 0.2 b i 0.04 ii 0.05514
ANSWERS5 Takes off at (0, 4), lands at (1.5, 2.5)6 a 1 b 11 c 77 a 2 b 0 c 3_ 5 8 a 2x 1 2 b 2x 1 19 2510 2, 411 a x 5 22 b x 5 1_ 2 c x , 1 d x , 2112 a All x b x > 21 c x > 013 a i 2√ __x 1 1 ii √ ______2x 1 1 b i x , 0 ii x , 2 1_ 2 c 4x 1 314 a _____ x 1 45 b 3 2 x c __ 1 2 3 xd √ _____x 1 3 15 a i x ii xb Inverse of each other.c pExercise 12*1 a (3, 5) or (24, 12) b (3, 2) or (23, 22) c (2, 3) or (3, 2)2 a (1, 1) b (1, 1) or (23, 5)3 a (0.586, 2.83) or (3.41, 22.83) b (24.16, 26.16) or (2.16, 0.162)4 a (21.92, 6.92) or (3.92, 1.08) b (21.24, 23.24) or (3.24, 1.24)5 (71.7, 14.3)6 (2, 3)7 a i 1.5 ii 21.5b 1_ 0 is undefined.8 a 2, 3 b 21, 6 9 a 8 1 6x b 4 1 6x10 1 or 2 2_ 3 11 a x 5 2 3_ 4 b x 5 1 c x . 2.512 a x > 2 1 b x > 0 c x > 013 a i √ ______3x 1 3 ii 3√ _____x 1 1 1 2b i x < 21 ii x < 21 c 9x 1 814 a i 2 2__ x ii x 7 2 2 5 iii √ __4x 2 1 iv _____x 1 2 b i f 21 (x) 5_____ xx 2 1 iii Self inverse15 a i x ii xb Inverse of each other.c √ ___19 Exercise 13Note these are calculated answers. Students’ answers will differ.1 a 1.17 b 23.5 c 0.672 a 21.79, 1.12 b 22.81, 2.14 c 22.36, 1.693 (1.1, 1.8), (21.8, 23.2)16
ANSWERSExercise 14*1 a i6( 23 ) ii ( 3 9) b m 5 1, n 5 3 c r 5 2, s 5 9 d u 5 21, v 5 22 m 5 2, n 5 213 abt654321t 0 1 2 3 4 5r ( 1 5) ( 3 4) ( 5 3) ( 7 2) ( 9 1) ( 11 0)c 8050 km/hr, 117°0 2 4 6 8 10 12 r4 a ___ ›RS 5 2r 1 s b ___ ›OP 5 3_ 2 r c ___ ›PQ 5 2 3_ 2 r 1 2s d ___ ›OM 5 s 2 3_ 4 r5 a i ___ ›PQ 5 2p 1 q ii ___ ›PR 5 1_ 3 (q 2 p) iii ___ ›OR 5 1_ 3(2p 1 q)b i k 5 3_ 5 ii ___ ›OS 5 1_ 5(2p 1 q)6 a i ___ ›MP 5 2_ 5 p ii ___ ›PQ 5 2p 1 q iii ___ ›PN 5 2_ 5 (q 2 p) iv ___ ›MN 5 2_ 5 qb ___ ›MN 5 2_ 5OQ, OQ is parallel to MNExercise 151 a32.5Length, l mmF.D.0 , l < 5 120 30 40 50 60 70 802F.D.5 , l < 10 21.510 , l < 20 2.2120 , l < 30 2.50.530 , l < 50 0.90 10 20 30 40 50 60Length mmb 7c 22.0 mm2 a3Time, t mins F.D.0102.515 , t < 20 2.42F.D.20 , t < 30 1.21.530 , t < 40 1140 , t < 70 0.20.5Time (mins)b 2 c 17 (16.8)18
3 a bMass, m g Frequency F.D.60 , m < 70 12 1.270 , m < 80 22 2.280 , m < 100 40 2100 , m < 120 20 1120 , m < 160 16 0.4ANSWERS32.52F.D.1.510.5040 60 80 100 120 140 160 18050 70 90 110 130 150 170Mass gramsc 414F.D.1412108642Time, t mins Frequency F.D.0 , t < 6 12 26 , t < 10 26 6.510 , t < 12 20 1012 , t < 14 18 914 , t < 22 24 30 2 4 6 8 10 12 14 16 18 20 22 24Time minsExercise 15*1 a504030F.D.20c 60%d 11.2 minsLength, l mm F.D.5 , l < 6 186 , l < 6.5 306.5 , l < 7 427 , l < 8 268 , l < 10 1010b 76045 6 7 8 9 10 11Length mmc 7.10 mm19
ANSWERS2 aTime, t mins F.D.20 , t < 30 0.530 , t < 35 0.835 , t < 40 1.840 , t < 50 2.250 , t < 70 0.3570 , t < 100 0.132.521.5F.D.10.5b 71.2%c 43.2 mins0 25 50 75 100Time mins3 a, c b x 5 166Life, t hrs Frequency F.D.60 , t < 80 24 1.280 , t < 90 26 2.690 , t < 95 20 495 , t < 100 25 5100 , t < 115 54 3.6115 , t < x 51 120
6F.D.543ANSWERS214 a, b050 70 90 110 130Life hoursMass, m kg Frequency F.D.2 , m < 3 52 523 , m < 3.25 34 1363.25 , m < 3.5 30 1203.5 , m < 4 60 1204 , m < 4.75 24 32150 170F.D.c 0.831401201008060402001 1.5 2 2.5 3 3.5 4 4.5 5Mass kgd 3.37 kgExamination practice 31 $446.50 2 £166.25 3 €5464 a x 5 4, y 5 1 or x 5 24, y 5 23 b x 5 2, y 5 0 or x 5 26, y 5 85 a i 27 ii 13 b _____ 3 2 x5 c 1_ 2 6 b i y > 0 ii y > 1 iii y > 217 a ff(x) 5 x b Self inverse8 a ( 1 5) b ( 1310 ) c 5.10 d 052.4°21
ANSWERS9Time, t 10 , t < 20 20 , t < 25 25 , t < 30 30 , t < 40 40 , t < 50 50 , t < 70 70 , t < 95Frequency 7 15 25 18 12 8 5Frequency density 0.7 3 5 1.8 1.2 0.4 0.210 a $3 000 b $35 000 c $72 500 d 7.94%11 a x , 0 b i fg(x) 5 1 2 √ ______2 1 3x ii gf(x) 5 5 2 3√ __x c i x 5 20.539 ii x 5 0.38212 1, 413 a 1800 mb i 222 ms 21 ii 250 ms 21 iii 228 ms 21c Accelerates to terminal velocity of 250 ms 21 in 11 secs, then falls at constant velocity, opensparachute at 45 secs, then slows to constant velocity of 25 ms 21 from 55 secs on.14 A is (20.411, 1.82), B is (0.911, 20.823)15 a i a ii 1_ 2 (a 1 b) iii a 2 b iv 1_ 2(a 1 b)b Parallel and equal lengthsc PQRS is a parallelogram16Consumption, m, mlFrequency0 , m < 50 950 , m < 100 42100 , m < 125 30125 , m < 150 25150 , m < 200 25200 , m < 300 22300 , m < 500 271.210.8F.D.0.60.40.20 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500Consumption mlc 134 ml d 50.1%22
Exercise 161 a, c, and e2 a e.g. 2.5 b e.g. 2.83 a e.g. √ ___17 b e.g. √ ___23 4 a 6 b a c 18 d 6 e 2_ 3 5 a √ ___27 b √ ___28 c √ ___80 d √ ___18 6 a 4√ __5 b 12√ __5 c 160 d 27 a 3√ __2 b 2√ __5 c 3√ __3 d 2√ __7 8 a 5√ __2 b 2√ __2 c 5√ __6 9 a 5 b 2√ __2 c 5 d 3_ 5 10 a 8 1 2√ __7 b 11 2 4√ __7 c 2611 a 2√ __2 b 2√ __6 c √ __6 d 1 2 √ __5 ANSWERSExercise 16*1 a, b and d2 a e.g. 4.2 b e.g. 5.43 a e.g. √ ____137 b e.g. √ ___9.9 4 a ab b __ a c ___ 1b 4a d ___ 8ab 5 a √ __3 b 2 c 2 d 4√ __2 e 9√ __3 6 17√ __3 b √ __7 c 5√ ___10 7 2√ __2 2 1 b 11 2 6√ __2 c 248 a a√ __b 14 a √ _____ 3 27 b 2 √ __2 15 n 5 4Exercise 17b a 1 b 1 2√ __a √ __b c 1 2 ad 4a1 9 2 x 2 1 1 3x 3 4 4 x5 2x 6 x 1 2 7 x 1 1 8 _____ x 1 2x 2 5 9_____ 2 1 x 410___ 3x 43x 2 111 ______612______ 3x 2 710 13 2x 14 1 15___ 43x 4x 1 3y16 _______xy 217 _______x(x 2 2) 18 ____________ 5x 1 1(x 1 1)(x 1 2) 19 _______ x 1 2 x(x 1 1)x(12 2 x)20 ________5(x 2 2) 21 6 22 5 23 22 24 2625 7.2 26 23 27 2 14928 24429 20 30 29 31 25 32 1833 2 2_ 334 2, 3 35 24, 5 36 22, 2123
ANSWERSExercise 17*1 x 2 2 2 2 1 3x 3 7 4 3x5 25x 6 x 1 y 7 x 2 2 8 _____ x 2 5x 2 4 9_____ x 1 2 10_______ 5x 2 13 11________ 93 2 11x 3 123512_______ x(x 2 1)x 1 1 13 (x 2 7)(x 2 3) 14 ___ 16x 15 ____________ 5x 1 2(x 1 3)(x 2 2) 16 ____________ 3x(1 2 x)(2 1 x) 17______________ x 1 7(2x 2 1)(3x 1 1) 18 _________ x2 1 x 1 1 19 ____________ 221 1 x(x 1 4)(x 1 6) 20 ______ 4xx 2 2 9 21 12 22 2 71123 27 24 21525 11 1_ 226 31927 2728 2 3613 29 530 2 31 28 32 2 or 101333 24 or 2 1_ 234 26 or 4 35 62 36 2.4Exercise 181 a dy ___dx 5 3e dy ___dx 5 5x42 a dy ___dx 5 6x2 1 10xe dy ___dx 5 3x2 1 10x3 a dy ___dx 5 4b dy ___dx 5 0f dy ___dx 5 12x5b dy ___dx 5 14x 2 3f dy ___dx 5 2x 1 2b dy ___dx 5 25c dy ___dx 5 3x2g dy ___dx 5 15x4c dy ___dx 5 15x2g dy ___dx 5 4x 2 9c dy ___dx 5 26d dy ___dx 5 4x3h dy ___dx 5 160x7d dy ___dx 5 12x3 2 10xh dy ___dx 5 2x 1 4d dy ___dx 5 194 a dQ ___dt 5 3t2 2 16t 1 14 b i 14 m 3 /s 2 ii 26 m 3 /s 2 iii 9 m 3 /s 25 a dy ___dx 5 3x2 2 3 b (1, 0), (21, 4) c (21, 4) is max, (1, 0) is min.6 a dy ___dx 5 3x2 1 6x 2 9 b (23, 28), (1, 24) c (23, 28) is max, (1, 24) is min.7 a v 5 24t m/s, 48 m/s b a 5 24 m/s 2 , 24 m/s 28 a v 5 3t 2 1 8t 2 3 m/s, 377 m/s b a 5 6t 1 8 m/s 2 , 68 m/s 2Exercise 18*1 a dy ___dx 5 2x22 5 2 1 __x 2 d dy ___dx 5 23x24 5 2 3 __x 4 b dy ___dx 5 22x23 5 2 2 __x 3 e ___ dydx 5 216x25 5 2___ 16x 5 c ___ dydx 5 __ 1 x 2 1_ 2 5 ____ 12 2√ x f dy ___dx 5 2x23 5 2 1 __x 3 g___ dy 5 4x 1 3 2 4x22 h___ dydx dx 5 2 1 3x222 a 22 b 2 c 0 d 22 1_ 8 24
3 4y 5 3x 1 44 y 5 23x, y 5 3x 2 95 a y 5 4x 2 7 b x 5 0 is a max, x 5 4_ 3 is a min.6 p 5 17 a ___ dCdt 5 4 2 16t22 5 4 2___ 16t 2 b t 5 2, C 5 16c 212 °C/monthANSWERS8 a ___ dPdt 5 10t 2 10 000t22 5 10t 2______ 10 0002 b t 5 10, P 5 1500t9 t 5 5, s 5 125 m10 a v 5 __ ds 5 3t 2 2 300 km/s, a 5___ dv 5 6t km/s2dt dtb at t 5 5, v 5 2225 km/s, a 5 30 km/s 2c t 5 10 sExercise 191 9.22 cm 2 18.0 cm 3 15.6 cm 4 13.5 cm 5 11.9 cm6 19.4 cm 7 46.5° 8 38.2° 9 55.1° 10 36.2°11 20.7° 12 59.0° 13 a 68.9 m 2 b 120 m 214 a 22.4 cm b 26.4 cm c 32.1° d 35.0°Exercise 19*1 a C 5 42.2°, a 5 6.96 m b C 5 44.7°, a 5 5.84 m2 a 25.5 km b 022.7° c 202.7°3 a 42.2 km b 022.7° c 14.1 km/h4 92.1 m5 a 70.7 m b 60.4 m c 41.8° d 67.5° e 9038 m 2Exercise 201 a i 125 ii __ 825 b ___ 13125 2 a i 511 ii __ 611 b i __ 1625 ii __ 925 c ___ 28693597 5 0.798 (3 sig.figs.)3 a 0.2 b 0.1625 c 0.097254 a k 5 0.1 b i 0.8 ii 0.5 c 0.9Exercise 20*1 a 217 b ___ 15525 c ___ 725525 d ___ 64425 2 a 0.03 b 0.44 c i 0.315 ii 0.392 d 0.58425
3 a0.8SANSWERSWS0.900.60.2FS0.40.4F0.10W0.6Fb i 0.72 ii 0.0288 iii 0.85684 a 15 diamonds, 5 rubies b 1538 5 a r 5 1_ 5b Five options per question.Examination practice 41 a 7√ __3 b 30 c 6 d 11 1 6√ __2 e 7 f 32 a 3√ __√2 b _____ 2 3 c 7 1 2 √ __________ 3 373 √ ___65 c _______ x 1 y2(x 2 y) d _____ x 2 yx 1 y 5 a 2x b____________ x 1 3(x 1 5)(x 2 3) c _____ x 2 2x 2 5 d _______ 4(x 2 y)x 6 a_______ 7x 1 5y b ____________ 8(x 2 2)xy(x 1 3)(x 2 5) c ____________ 7(x 2 5)(x 1 2) d ____________ 1(x 1 6)(x 2 1) 4 a 3_ 2 b 1 _____x 1 5 7 a x 5 4 b x 5 3 c x 5 2 or 258 a (2, 1) min b (22, 19) max, (2, 213) min c (23, 28) max, (1, 24) min9 a v 5 24t m/s, 72 m/s b a 5 24 m/s 2 (constant), 24 m/s 210 a v 5 3t 2 2 500 km/s, a 5 6t km/s 2 b at t 5 10 s, v 5 2200 km/s, a 5 60 km/s 2c t 5 √ _________ 500 s, 12.9 s311 PQ 5 10.4 cm, angle P 5 42°, angle Q 5 83°12 b AB 5 41.4 km c 103.0°13 a 16.7° b DF 5 94.3 m c 9.03° d 2.19 m/s14 107.6 m15 a i 1330 ii 2_ 7 b i __ 115 34ii Paper 11 a 35% b 1 : 2.535 2 a 12y 2 8 b x 2 1 3x 2 10 c 4(2y 1 5z) d x(x 2 5) e x 5 33 a (5, 3) b 7.21 cm4 a x . 2 b02x5 a 2 b 2.626
6 a 0.57 b 307 a 42 cm 2 b 630 cm 38 a i {1, 3} ii {1, 2, 3, 4, 5} b 3 is a member of the set A9 a 29.25 b c.f. are 8, 46, 74, 78, 80 c median ,2910 a____ 360° 5 11.25°32b 84.375° c 424 m d 27 minutes11 a 3_ 4b (3, 21.5)12 a i 21 ii 1_ 7 b i 3 __ x 2 2 ii 0 c 6113 a ii F.D. for 2000 2 2500 is 0.28 if F.D. for 0 2 1000 is 0.09ii Frequencies are 90, 130, 140, 120b UQ is 360 th person which is at 2500 calories.14 a i R 5 ___ 3.6r 2 15 b 4.73b 0.4 ohms16 a (1, 25) b 21.2, 3.2 c Line is y 5 x 1 2, solutions 4.4 or 21.4d i ___ dy 5 2x 2 2dxii Gradient is 1017 a 3.90 cm b 10.2 cm 218 a 1_ 6 b __ 719 15 cm420 _____x 2 3 18 ANSWERSPaper 21 712182 units at £0.0821 per unit £14.94429 units at £0.0704 per unit £30.20Total amount £45.14Tax at 5% of the total amount £2.26Amount to pay £47.403 a 4x b x 1 6 5 4x 2 6 c 44 a 632 cm b 148 cm5 a Tin 40 grams, Lead 80 grams b 75 grams6 a 2 3 3 3 b 2 3 3 2 3 5 c 6 d 3607 a g 5 10, h 5 0.8 b h 5___ v22g 8 a Kitchen chiarsb i {1, 2, 3, 4, 5, 6, 7, 8, 9} ii Yes, an integer cannot be both odd and even9 a 38.2° b i 7.65 cm ii 7.55 cm10 a 72 s b 4.17 cents c 16.7%11 a x 5 3 b x 5 1, y 5 22 c 3x 2 1 7x 2 612 a 7.5 b 10 c 135 cm 227
ANSWERS13 4.7 cm14 a ( 5 2) b i k( 5 2) ii k( 5 2) 2 ( 1 2) iii ( 4 0) 2 k( 5 2)c k 5 1_ 2 d X is halfway along OB as k 5 1_ 2 . ___ ›AX 5 ___ ›XC means AX 5 XC and the vectors are parallel, so AXCis a straight line with X as its midpoint.15 a x 5 4.9t 2 b x 5 44.1 c 1.43 s16 b i 28 2 4x ii x 5 7 iii Graph is a negative parabola c 98 m 217 a 138 cm 218 aFirst attemptSecond attemptb __ 1720 25Pass3519 (1, 0) or (2 1_ 3 , 2 4_ 3 )Fail3414Pass20 a v 5 3 2 2t 2__ 8 2 b a 5 22 1 ___ 16t t 3 c t 5 2Paper 31 240°Fail2 a width 5 6x cm, height 5 4x 1 28 cm b i 6x 5 4x 1 28 ii x 5 14 cm3 a 3 1_ 73b i 12 ii 6 __12 4 4.225 5 £76.80, a decrease of 4%. 6 Cement: 10 kg, Sand: 45 kg7 x > 6 8 a 0.4 b 0.35 9 a 12 cm b 340 cm 210 a Triangle vertices are: (0, 6), (2, 6), (2, 10)b Triangle vertices are: (6, 3), (6, 4), (8, 3)11 a CF: 41, 67, 87, 97, 1001009080706050403020100 10 20 30 c IQR 5 9 years28
12 a gradient 5 2 b y 5 2x 2 1 c y 5 2x 6 c, where c is a positive constant.13 a 3 4 b m 5 23, n 5 214 a ___ dy 5 5000 2 1250x b x 5 4, y 5 10 000dxc i (4, 10 000) max ii curve is an inverted parabola.d i x 5 £4 ii Produces the max profit of £10 00015 a r 5 10.3 cm b V 5 2289 cm 316 a i w 5 5 ii x 5 7 iii y 5 9 b i 3 ii 15 iii 017 a i f(3) 5 3_ 2 ii f(23) 5 3_ 4 b x 5 1c i ff(x) 5 x ii f(x) is its own inverse.18 (6, 5), (20.4, 27.8) 19 a 7 2 x b x 5 320 a 3n b ___ n3n 3 ______ n 2 13n 2 1 5 ___ 1 ⇒ n 5 7, 3n 5 21 people10ANSWERSPaper 41 2.4 3 10 12 a i 15x 2 20 ii 6x 3 2 8x 2 iii 9x 2 2 16 iv 9x 2 2 24x 1 16b i 2xy(1 2 6xy 2 ) ii 3x 2 y(1 2 3xy 1 5x 2 y 2 )c i x 4 y 6 ii xy 5 z 23 a 150 cm 3 b 320 cm 34 a p 5 2(n 1 1) b n 5_____ p 2 2 5 a 7.06 km/h b 1.96 m/s26 a i 1, 2, 3, 4, 5, 6 ii 2, 3, 4, 5 iii 7 a 0.162 m 3 b $43 200 8 509 a 18 red b 12 red beads 10 360 5 2 3 3 3 2 3 511 a Rotation of 90° clockwise about (2, 0) b i F ii T iii T12 x 5 3, y 5 2113 a 3.7 3 10 10 b 0.000 075 c 2 3 10 21014 a y max 5 1.05 b y min 5 0.5315 n 5 5 16 992 cm 2 17 a 12.6 cm b 2.6 cm 218 abx 0.5 1 1.5 2 3 4 5y 25.75 22 0.25 2.5 8 15.25 24.420y1001 2 3 4 5 x10c x 5 1.4 d y 5 2x, x 5 2.529
ANSWERS19 aw kg0 , w < 2 1282 , w < 3.5 1503.5 , w < 4.5 1364.5 , w < 62 72b 3.5 , w < 4.5 block is 136 high (6.8 blocks high)20 a (4 1 x) 2 5 (6 2 x) 2 1 (5 2 x) 2 ⇒ x 2 2 30x 1 45 5 0b x < 1.58 cmf30
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