ANSWERS Answers to questions taken from past examination papers are the sole responsibility of the authors and have not been approved by the Examining Boards. Exercise 1b-p. 3 Re-2 Ryowty 4 Spa Sayty 6 Pas txt7 1541-0 ML 3x24 18-20 Wt ab—2ae + cb 13, Uer—2er"~ ss? = +98 5. may + 28 ~ 16. SRS + SRF Exercise 1e-p. 4 7 22 .@t -5 ®t 4@t 0 3 Exercise 1d-p. 4 Le +6r+8 Rb eet 1s Rat be+a 4s 154 56 Se 41+ 66 62e + Iles 1. 5p 428+ 15 8 Gat a4 12 9. 38 + 861+ 48 10, 95:7 + #2 +6 2 28+ Se 20 4e 43 810 a Answers 2. 6g — 9-15 Boxy? 30.2 Far 6 Exercise 1e-p.5 Lend Pere 2 3 4. 92 + 30r + 25 5 ax + 28x +99 62 2e+t Txt 6r +9 Bae artt 9. 16x? — dar +9 10, 25x — 20+ 4 1, 9p — 421489 Berdyty 13, dp? + 36p + 8 14, 992 ~ 664 + 121 1S. dr? — 2017 + 25)" Exerse tg-p.7 Ix 2-12 bee 36-254 ae 4 lip? 3p~2 Opt ip +t ase tr? aap ent ise 3— 8 tay Hay 10. 16x?-9 1 oe! +204 49 WB 1s-R-2R° 13. a? ~ 6ab + 96° 14. de? — 20¢ + 25, 15, #0" — 46? 16, 90° + 300b + 256? 1. @) 6-22 () IS, Ow @ al Exercise 1h-p. 9 1 + S943) 2 w+ e+ 4) 3 @toet) 4 tae +3) 1 @— Dee=9) 6 @-3) 1+ OMe +2) 8-9-1) 9. @ + Ne -2) e+ 4u-3) Ste = She +) = +) (+72) «-P = e43) + 9-3) +o = De+ = 6 +3) te 2. (r= 4x +4) 2B. r= 390x43) B. tear Exercise 11 |. Gx = 40x +3) Ge = 396-2) ax — Dee Gx = 2) = er = 298 +2) I. (Sx 4) Sx + 4) G+a-2 . (x= Nex = 12) Gx + 5 6-00 +) 0+ 44-30) 1 (xy) Gx +2) ety) (= 2990 +29) 1. x= 9? 3G +20) rx 20. Gx 48x +3) ait812 21. (x + 30, - 5) 22. (6~ Sx)6+ 54) BG ie +y) 4. Ox=29)) 25. 06)? 26 (x - 350+ 39) 2 +5) arty) 2. Gx + 4)2x+9) 30. (pq 2)" Exercise Up. 11 1 2 3 4 5. not possibte 6 not possible 7, not possible 8% ~2 9. 3x —2yee+ 1) Wo. r= 344) 11, 36-4 +2) 2. B rs 1s. 6 1. 1 (~ oe +2) not possible 4er= 5x +5) see not possible not poss not possible Exercise 1k- p. 12 LO-# 4-2 2 Be ast? 340 8e + 1-5 4-28 +1 5. 2e 9 ~ 24-9 6 +60 + Ihr 6 Ie tae eet Pea ted 18 e+ 182-9 38 — Six +10 i? + 68x +210 Gc + 2B ~ “16,28 22, 6-1 Bae dey + dey ty Date dey + Gay ay yt Exercise 11-p. 14 Loo +904 2427 2 t= 84 2a He + 16 Rirtt4e doe tact 4 bet ne sort 5. x5 15x! + 90" — 270? + 405 ~ 248, Se" 1b ay *— Apa + 69a" — Apa + 4 36+ Ste +27 = oe + Tee ~ G40 + 12808 = 1024 9, hat ~ 108° + $42— IB #1 10, 1 + 200+ 130a! + Sa’ + 6250 1, Gta ~1920"b + 24a? ~ 16a"! + ab ~ 120d +B 12, B= te + IS ~ 125 Ma 2)ee- 1) 280 40 -396049) ae acer «5 7 CHAPTER 2 Exercise 2a-p. 17 Ay 2 +2) 3a} Answers rel OFD0-9 1 36H) x42 4, 16 1. ve @trsd 18 + not possible (e+ sKae 2p Ps el Besa ) ery Exercise 2b~p. 18 Lz ae 2 a 813 240+ 3) x3 3 x0 =m 15 sinB + sind Mad sin B sind + 008.4 M1 cos sind{ 815 Exercise 2e~ p. 23 aa ‘3 ae IE) eD 4 3ett e+) 26-0 7 ‘ 0 : ” _ Saq=3 Teed = e+ 1 a 1. 23-3 S248 . 25 + SVS 4 Z - See— 1) 3x #3) 24 Gn Mixed Exercise 2-p. 23 Po9 eth 1Exercise 39 - p. 38 1. bog p + log 2. togp + long + log i = 12 = O80 of ~1554 ae eee 13. x = 0804 oF 1554 Br =0orr =} 13 Sore = 4 _ 15.5 = tor = - w. tax 2ore = h - 2 : Leredye drat B REssy ease u. Raekye nee? 5 ereaysaeect %. eee n. @redye nbn Bray eres Exercise Af p. 50 ae Lox ft Exercise 3f- p. 36 Gh Re Peee ies a6 Ranaye3Answers Answers 819 ‘Multiple Cholce Exercise A-p. 58 : LA Lor LE = erent cn 2 IN = 2SemLN = 4Sem ae 3, BC = 2.25¢em ferent ae ve ae ds 7 oY ; ; ; Bex = Hy2l- Dy = 16-29 “4-4 CHAPTER 5 Exercise 5a-p. 66 12cm I 2 Sem 3 20cm 4 @) Hom () Trem 55:7 ome |ara (©) pespendicular perpendicular @) neither (©) parallel Exercise 64-p. 89 La=Ob=4 313 14, no; an angle and the side opposite to it are not known 74 on 821 or 5 (@) 18" (b) 126° 6 1. Idem, 68° Exercise 79-p. 115, ‘740m Bem 150m (2B = 8%sa = em @ = Ietem:e = 272m 28 3410 = S10em 1. ZC = 8b = deme = 135em Baeaieac 5 Coraeat Exercise 7e- 1. 529 29 239 4 408 % 120 1 640 aos cise 71 - p. 113 4 2BA0 = 14", ZCA0 = 52°; Rom?Answers 823 10, the unshaded region is the one required. 3 CHAPTER 10 Exercise 10a-p. 152 1. @) ZADE. ZACE—_(b) ZAFE (@) ZABC (0) ZCAD. ZCED. (2) 6.9),531° ay 5. P = $54°,Q = 312, = 24" wom we ou" Be-2y¥+7 = 0 xed 0.34419 = 0 CHAPTER 9 a Mixed Exercise 9~p. 149 on 3 st 2 * x. 10. Y, 3 uli S S 3 S RS Spt aaoswers 0, 8000 (©) = Ofer << 20000 (= 2000) forx > 20000 overeat x20 (yes fw () B2em" 5m, 139mm! ) 2S en? oom 18, 196m, 108 cn domain x > 0 butx < GNPH) range f(s)> 0 (&) 94t em (@) 688m CONSOLIDATION & Multiple Choice Exercise B- p. 173 Le mB * Ze 1.BC 4 xc 8. ' ae 2. ® 3 SA wAB ec nic a BAC * ap WE oA 25.7 Hs WA wT Oe c nF A wT BD BT 4D a0 7 1D BF t : : | 7 40.2455) 1S, (4Y2, 8/24 4).(2 + 4/2, 8/2, a 4 (o 0 5 (@) 4x> bande c-1 S.x>2+ylandx<2-y7 6 -E Sand mTand -Scx<-3 ~4ex<-} meprer 3Sandp 6018 <1 exe Tees tered x>-1 x>landx <-3 2 2h |. The curve y = 2° is translated 3 units Ges? +1) HBG +3) 9x? (©) Qn Dx. Qne pr Leas value is 2, greatest valve 0,60", 120%, 180° dant 409°, 2209° @) ty2.tv6 in the diection Ox and translated $ units in the direction yO followed by a stretch offacior 2 parallel to Oy. PQ) > 3x2 fla) = 34+ V+ u. 3. 26, ny. r-y4tB = 0 (a) 161° + 180"? (o) 272" + 34, 1528" + 360n° Onin dm 2m nn, 2am x ~4and “1 2 x= 72024 ‘odd with period» (&) odd even (@) even with period 2n (@) aad (€) CONSOLIDATION F ‘Multiple Choice Exercise Fp. 688 8 17. AB c AC A BAC gE DAC E WBC c Ra a BAB c wa 8 2k A 6.7 E mF B aT € BF Wc mF D LT Ee + ot? 2 1p nage gig Ow, paager Bon ix he lxi<3Mx = Vy = Pande = =v yech? Is x 2p = Sande Ry ot Worn ya hee ey a3 m72-2ia Inthe remaining questions, the degree of aceuracy depends upon the accuracy of the first approximation: these answers are 15, ~091735, 086366 16. 31038 17. ~14837, 053979 Mixed Exercise 38 - p. 716 OF (@) -4+ 1y-% ow -ies meen 2 Yo, v.2V6 ys 4A = 103.8 = anata B= 1+) 15, p> pata 39d-p. 738 ~ bai + 34+ 20) Grn tnAy 2-2 ae Exercise 39¢~ p. 742868. TL, (a) arecos (0) atccos j ) #6 -2) (46+) (@) contained in the plane, Exercise 39)-p. 771 1 © there is no special them, Exercise 40d - p. 790 ‘Arguments ate given in ten 6leos0+ i ® @ 0421 o Sleoss + isin) (i) 0-34 869 Mixed Exercse 40~ p. 793 Ge ow 142 CONSOLIDATION G Multiple Choice Exercise G--p. 799 B 7B A a c E 20.4 D mA a 2 BC D BBC ry WBC c 2A E 6B A 2.BC c 2. F .D BF D 2.7 Cc aL F E Re 3053" (@) S13", 287" bn, 067 rad, 248 rad870 1x = sy = Fors = By = 3" INDEX Abscissa 16 ofa triangle 728 ‘Acceleration S47 ‘of a volume of revolution 566, $67 BBs Wit j++ AM 3j—W;-2 “Abtimde of a triangle 72 Chain rule 324,377 i ‘Ambiguous case 10S ‘Change of variable 436, ‘Amplitude 358,787 Chord 222 ‘Angle 94,163, Circles 150,439 between a line and a plane 125,771 tangents to 157, 158, 482 between two lines 420, 754 touching 447 between two planes 125, 70 Circular (ie. trig) functions 260 between two Vectors 721, 752 fraps of 262,267, 269 enema 94 inverse 361,362, 363 small 366 Coefficient 3 ibn subtended by an are ISL Combination 619, 627, 628,634, 636, 637 #6-3) Common difference 594 i Common ratio 60 Comparative rate of change 403, Complementary angles 271 Complete primitive $40 Complex number _ 774,796, 777,791 argument of 787 bounded by a curve 462,467,468, conjugate 775 471,351 imaginary part of 776 ‘Argand diagram 782 modulus of | 787 ‘Asqument 787 real par of 776 | ‘Anthmedic mean 607 vector representation of 783, 784 Avithmetic progression 594, $96 zero 716 Acymptote 192 Complex roots of a quadratic Compound functions 206 difereniation of 324,373,379 Conic sections» 449 Conjugate 775 Constant of integration 460 Continuous 262,672 Convergence 606 theorem — 642, 80 price Canesian 76 Cartesian coordinates 76,417 parametric 389 ‘equation 437 Cosecant 273 equations of aline 748 Cosine function 266,267 i Centre of gravity 563 ratio 9 Centce of mass 563 rule 109, 112 Centroid 563 Cotangent 273 ‘of area 563, 564 Coverup method 505 8713872 Cubic curve 190 funetion 190 Cure sketching 186 187, 200,39, 456, 662 (Cyclic quadrilateral 158 tofind a volume of revolution $56 Delia pref (6) 26 Derivative 227,229 second 250 Derived function 227,229 Difference method 611 Dilference of two squares coefficient 538 equation 384, $38, $49 solution of 539, 540, 4 variables separable 539 Differentiation — 227,232, 233,234 from first principles 227 ‘compound functions 324,343,373, 379 ‘exponential functions — 332, 342,343, 188 function of a function $24,330 logarithmic functions 340,342,343, products 235,327 quotients 235,329 trigonometric functions 370 Direction ratios 736 vector 737 Displacement 546,718 equivalent 718 Distance ‘of plane from origin 761 of point from line 421 fine from plane 767 Element 463,471 Ellipse 449,451 gsation ‘approximate solution of 706,340,713 meaning of 136 ofacincle 439,440 ofan ellipse 451 ofa hyperbola 4s1 ‘afaparabola 450 ofa normal 241 quadratic 39 trigonometsic 277,284, 286,291, 358, 359, 697 Equations 136 te complex rots of 780 with repeated roots 52,702 Even function 670 Exponent 194 Exponential equation 700 function — 194, 332 Factor theorem Factorial 621, 628,631 First moment of area St fof volume of revolution 566 Frictions partial 21, S04, $05, $06, 307 proper 22 Frame of reference 75 Free vector 726 Function 181, 182 elie 262 even 610 194,332, 334 201, 202,208 logarithmic 339 Index modulus of | 674 odd 670 ofa function 207 Period of a 262, 671 Periodic "262, 71 Polynomial 191 quadratic 187 rational 192 sine 260,261 tangent 260,269 trigonometric 260,354 General angle 94 trig ratios of 94,97, 98 General sotution of tig equations Geometric mean 607 Geometric progression 601, 607 Gradient "83.223 of normal 237 of tangent 237 Hyperbola 449,451 rectangular 452 Identity 279 Inegrat 460 ‘approximate value of 472 Sefinite 465,490, 496, constant of 460 873 Timits of | 466 ‘aspect of 481,484, 519, S32 Intercept 140 Interection 147, $14,676 Traverse trig function 361,362,363, Inverse function 201,202 graph of 202 Irrational number 26 Terave methods 709 ‘equation of 139 gradient of 83 Tength of 79 midpoint of 80 of greatest slope 126 ‘angle between 420,754 gradients of perpendicular 86 non-parallel 86, 749) perpendicular 86 parallel 85,748 skew 748 Locus 437 Logarithm | 34 base of 34 ‘Mapping 180 Maximum value 2473874 Mean value of a function $70 2 ou ‘equation 616 of a complex number 787 ofa function 674 ofa vector 718,731 Moment first of area S64 Using Newton-Raphson 713, using = gtx) 710 044 fonction 670, Ordinate 76 Origin 76 Periodic function 262, 671 Permutation 619, 620, 22 634,636, 267.385 125, 70 number 26 Rectangular hyperbola 452 Regression lines 427 Remainder theorem S08 Repeated roots $2,515 Resolved parts 755 Resultant vector 720 Roots ube 25 general 25 square 2 polynomial equation $12,702 of 706 quadratic equation 39, $1,499 jon to coefficients 499 Sealar 717 Index Segment 150 Separating the variables 487 Sequence 591 nature of 248 Theorems 63, 67,68, 71 ‘Threedimensional space 729 ‘Transformation 196, 62 one-way stretch 265, 266, 663 reflecion 198, 62 197, 662 centroid of 728 Undefined 270 Unit vector 730, 796 Canesian 730 representation resolved parts of 755, resultant 720 tied 725 tnit 730,736 Vectors 717 angle berween 721.752 Volume of revolution 556Coré Maths for A-level has been written to cover the whaie of the Common Core syllabus for A level Hathematies, together withthe additional material required bY most Examining Boards © Because students enter A-level courses (rom a variety of backgrounds, the early] chapters provide transitional material of an introductory nature. Allchapters contain some new work however, to interest those students who come to the course With a higher level of skilis and knowledge. Exercises are included at frequent intervals, giving ample provision ot straightforward questions Consolidation Sections contain a summary.of the preceding work and multiple: choice questions as well as quéstiansat the level of sophistication found in ‘A-level examination papers. These questions are intended for practice when udent has, through experience, developed conficience and Skil