1252299092_4A04_FS_01e

NSS Mathematics in Action 4A Full Solutions4 Exponential FunctionsReview Exercise 4 (p. 4.5)1.491419543012=⨯⨯=⨯⨯-2. 444441)3(232===÷-----3.64121)2()2()2(663232-=-=-=-=--⨯--4.6422)12()72(6)2()3(23203===⨯=⨯-⨯-----5.12527535353535335333221==⎪⎭⎫⎝⎛=⎪⎭⎫ ⎝⎛⨯⎪⎭⎫ ⎝⎛=⎪⎭⎫ ⎝⎛-⨯⎪⎭⎫ ⎝⎛-6.55)2(3231aa aaa ===⨯--+---7. 6)1(5150151a a a a a a a ==⨯=⨯----8. 16161)2(288)4(2442a a a a ==-=⎪⎭⎫ ⎝⎛---⨯--9.aa a a a a a a 993)3()3(672327237237==⨯⨯=⨯=÷-⨯----10.333331)(qp qp pq ==---11. 1284342432ba b a b a ==⎪⎪⎭⎫ ⎝⎛⨯⨯12.663323232027*******b b b b b a -=-=-=⎪⎪⎭⎫ ⎝⎛⨯-=⎪⎪⎭⎫ ⎝⎛--⨯---13.101010105252552232322)2(ba b a b a b a ===-⨯-⨯-14.23232)5(252252)(ba b a b a a b a a ab ====-----------15.3153153)3(5353)5(4)1(4351441)(][ba b a b a b a b a b a b a =====⎪⎪⎭⎫ ⎝⎛----⨯-----------4 Exponential Functions16.4101232)2()()4(----⨯b b a b a464414)2(3)2()2(21616214--⨯--⨯-⨯--⨯=⨯=b b a b b a104104)4(64ba b a b a ===--+-17. 12)1(12222++++==∙n n n n n18. 333333122)1()2(212===∙---++-+n n n nn n19. nnn n n n nn n 33333)3(39222====-20. 2322)22(12155555)5(525++++++==∙=∙=∙n n n n n nn n n21.8133333333)3(3)3(93274)22(632263)1(2)2(3122312===∙=∙=∙=∙+--+++++++++n n n n nn n nn n n n n nn22. nn n n n n n n n n nn n n 7)63(86263862)2(3)2(4)3(224)3(22322222222)2()2()2(8164==∙=∙=∙=∙--+-------ActivityActivity 4.1 (p. 4.22) 1.xx2. (a) (i) 1 (ii) 1 (iii) 1(b) Yes, the y -intercepts of the three graphs are equal to 1.3. (a) Yes(b) The function y = 1.8x > 0 for all real values of x .Therefore, the graph of y = 1.8x always lies above the x -axis. Similarly, the graphs of y = 2x and y = 3x always lieabove the x -axis. 4. (a) increase (b) zero 5. (a)(b) increase (c) YesClassworkClasswork (p. 4.7) 1.fig.) sig. 3 to (cor. 21.2244=2. fig.) sig. 3 to (cor. 31.4803-=-Classwork (p. 4.9) 1. (a) ✗ (b) ✗ (c) ✓ (d) ✗2. (a) 425232523a a a a ==⨯+(b) 2432141412141kkk kk ===÷⎪⎭⎫ ⎝⎛---NSS Mathematics in Action 4A Full SolutionsQuick Practice 4.4 (p. 4.12) (a) 201120115441544154415441111aa a a a aa a a ===∙=∙=∙---(b)21214543345232432365234323654327273)3(9)9(a a a aa aa a a ===∙=∙=∙---⨯--Quick Practice 4.5 (p. 4.13) (a)82)2(1616343443433434====⎪⎪⎭⎫⎝⎛=x x x (b)942323827827232332322323=⎪⎭⎫ ⎝⎛=⎥⎥⎦⎤⎢⎢⎣⎡⎪⎭⎫ ⎝⎛=⎪⎭⎫ ⎝⎛=⎪⎪⎭⎫ ⎝⎛=------x x xQuick Practice 4.6 (p. 4.16)∴223333)3(91272323-=-====--x x x xx4 Exponential FunctionsQuick Practice 4.7 (p. 4.16)25142222)2(2434)14(4344)4(4344442222222222==-====+=+=+-------x x x x x x x x x xQuick Practice 4.8 (p. 4.17) ⎩⎨⎧==-++)2(162)1(0331 yx y x From (1),∴0113303311=-+=+==-++y x yx yx y xFrom (2),∴ 4221624=+==++y x y x y xConsider the simultaneous equations: ⎩⎨⎧=+=-+)4(4)3(01 y x y x (3) + (4):2341240)()1(==++=++-+x x y x y x By substituting 23=x into (4), we have 25423==+y y ∴ The solution is 25,23==y x .Quick Practice 4.9 (p. 4.20)(a) fig.)sig. 3 to (cor.3.1838.05)8.1()8.1(8.18.1≈+=+p f (b) fig.)sig. 3 to (cor.0333.038.0)5(8)2()3(823=∙=-∙---p fQuick Practice 4.10 (p. 4.21)(a) Number of words remembered by the person after studyingfor 5 minutes = 200 (1 – 0.95)= 82 (cor. to the nearest integer)(b) Number of additional words remembered in the sixthminute= 200 (1 – 0.96) – 82= 12 (cor. to the nearest integer)Quick Practice 4.11 (p. 4.27)(a) From the graph, when y = 12, x = – 3.6.∴ The solution of 0.5x = 12 is x = –3.6 (cor. to thenearest 0.1).(b) (i) The rate of increase of y increases as x increases.(ii)27722128==From the graph, 2x = 11.4 when 27=x .∴ 4.11128= (cor. to the nearest 0.2)Further PracticeFurther Practice (p. 4.12) 1. (a) ∵ (–6)3 = –216∴62163-=-(b) ∵ 6251514=⎪⎭⎫⎝⎛∴5162514=NSS Mathematics in Action 4A Full Solutions 2.(a)1414196196221=== (b)31312431243155551=⎪⎭⎫ ⎝⎛==⎪⎭⎫⎝⎛ (c)64141442562563334434143===⎪⎭⎫ ⎝⎛=⎪⎪⎭⎫ ⎝⎛=----(d)49472723438343822332332=⎪⎭⎫ ⎝⎛=⎥⎥⎦⎤⎢⎢⎣⎡⎪⎭⎫ ⎝⎛=⎪⎪⎭⎫ ⎝⎛=⎪⎭⎫ ⎝⎛ 3.(a)4321232123212112121xxx x xx x x ==⎪⎪⎭⎫ ⎝⎛=⎪⎪⎭⎫ ⎝⎛=⎪⎪⎭⎫ ⎝⎛∙=∙+(b)12723245323324532459)3()3(27xxx x x x =⨯=÷=⎪⎭⎫⎝⎛÷-4. (a)25225232123213232121323yx y x yxy x y x yx xy ===∙=∙--+--(b)33531341323131334323131331322323332244141)4()()64()(64x y y x y x xyyx yx y xy x y x ===÷=÷=÷⎪⎪⎭⎫ ⎝⎛---------Further Practice (p. 4.18) 1.(a) ∴ 9131322)2()2(4831361236=====x x xxx(b)∴ 21012551)5(512550122-==+==∙=∙+x x x x x(c)∴2112444460)14(46044604)4(604161222222221221=====-=-=-=-+++x x x x x x x x x xx4 Exponential Functions2.⎩⎨⎧==-+)2(636)1(03912 y x x yFrom (1),∴223303303)3(039222=-===-=-=-x y xy x y x y x y x yFrom (2),∴ 122666)6(63612212212+====+++y x y xy x y xConsider the simultaneous equations: ⎩⎨⎧+==-)4(122)3(02 y x x y (3) + (4):1122)12(0)2()2(=+=+++=+-x y y x y x x yBy substituting x = 1 into (3), we have21012==-y y ∴ The solution is 21,1==y x .Further Practice (p. 4.21)1. (a) ∵40)3(=f∴540840)2(3===k k k(b) From (a), we have)2(5)(x x f =.0.625)(or 85815)2(5)3(515)2(5)0(30=⨯==-=⨯==-f f2. (a) (i)Value at the beginning of 2002000400$1000400$23000400$02.0=⨯=⎪⎭⎫⎝⎛⨯=⨯(ii) Value at the end of 2006000600$23000400$23000400$52.0=⨯=⎪⎭⎫⎝⎛⨯=⨯(b) Increase in value in 2007000 600$ 23000400$62.0-⎪⎭⎫⎝⎛⨯=⨯700 50$= (cor. to 3 sig. fig.)ExerciseExercise 4A (p. 4.14) Level 11. ∵ 25644=∴42564=2. ∵ 1000)10(3-=-∴ 1010003-=-3. ∵ 51227833=⎪⎭⎫⎝⎛∴ 83512273= 4. fig.) sig. 3 to (cor.15.2103=5. fig.) sig. 3 to (cor.77.15.175-=-6. fig.) sig. 3 to (cor.76.15484= 7. 9818121-=-=-8.414)4(64641131313131==⎥⎥⎦⎤⎢⎢⎣⎡=⎪⎪⎭⎫⎝⎛=----9.7371373)(xx x ==10. 3883183)(xx x =⎪⎪⎭⎫ ⎝⎛=NSS Mathematics in Action 4A Full Solutions11. 2525521511)(1-==⎪⎪⎭⎫⎝⎛=xx x x12.5252512521)(11-===xxx x13. 273)3(9323223===14.161212)2()8(4434334====---15.3433431717171491113323223=⎪⎭⎫ ⎝⎛=⎥⎥⎦⎤⎢⎢⎣⎡⎪⎭⎫ ⎝⎛=⎪⎭⎫ ⎝⎛=⎥⎥⎦⎤⎢⎢⎣⎡⎪⎭⎫ ⎝⎛=⎪⎭⎫ ⎝⎛-----16.94323224332252552=⎪⎭⎫ ⎝⎛=⎥⎥⎦⎤⎢⎢⎣⎡⎪⎭⎫ ⎝⎛=⎪⎭⎫⎝⎛17. 6234234)(a aa ==⨯18. 432)6(326326)(1a aa a ===⎪⎭⎫ ⎝⎛⎪⎭⎫ ⎝⎛-⨯----19. 472145214532626aa a a ==÷⎪⎭⎫⎝⎛---20.1211216543236543231aa aa a a ===÷⨯--⎪⎭⎫ ⎝⎛-+-21.61312131213aa aa aa===-22.8743814381438aa aa a a ==⨯=⨯+23.314134413143aa aa a a =⎪⎪⎭⎫ ⎝⎛=⎪⎪⎭⎫ ⎝⎛⨯=⨯24.223432333231314313123134312155)(51)(55)5(a a aa aa a a a ====⎪⎭⎫ ⎝⎛=÷---------4 Exponential Functions25.8133444141==⎪⎪⎭⎫⎝⎛=x x x 26.243131********5225255252=⎪⎭⎫ ⎝⎛=⎥⎥⎦⎤⎢⎢⎣⎡⎪⎭⎫ ⎝⎛=⎪⎭⎫ ⎝⎛=⎪⎪⎭⎫ ⎝⎛=x x x27.812)2(3232353553533535====⎪⎪⎭⎫⎝⎛=------x x x28.222122)(q p q p qpa a a a a a -++==∙∵12=∙qpa a a∴ 022a aq p =-+∴ )2(2022p q qp +==-+ When p = –1, q = 2[2 + (–1)] = 2. When p = 0, q = 2(2 + 0) = 4. When p = 1, q = 2(2 + 1) = 6.∴ Possible values of p and q are p = –1, q = 2 or p = 0, q = 4 or p = 1, q = 6. (or any other reasonable answers)29.(a)bb b b b b b =⎪⎪⎭⎫ ⎝⎛=⎪⎪⎭⎫ ⎝⎛=⎪⎪⎪⎭⎫ ⎝⎛=⎪⎪⎭⎫ ⎝⎛-661631216312163 (b) (i) We must have x > 0 for x to be well defined.∴ 0>x and 03>x∵ 3x x >∴13>xx∴ 11663>>⎪⎪⎭⎫ ⎝⎛x x xTherefore, possible values for x are x = 2, x = 3 or x = 4. (or any other reasonable answers)(ii) We must have y > 0 for y to be well defined.∴ 0>y and03>y∵ y y >3∴31yy>∴11636<⎪⎪⎭⎫⎝⎛>y y y Therefore, possible values for y are y =21,y =31or y =41. (or any other reasonable answers)Level 2 30.3333)3(327312123613236123==⨯=⨯=⨯---31.41222)2()2(482352323532335==÷=÷=÷-------NSS Mathematics in Action 4A Full Solutions32. 21666636636393193933==⎪⎪⎭⎫ ⎝⎛=⎪⎪⎭⎫⎝⎛=⎪⎪⎭⎫ ⎝⎛33. 515555)5()5(1252515225522215232341252324==⎪⎪⎭⎫ ⎝⎛=⎪⎪⎭⎫ ⎝⎛⨯=⎥⎥⎦⎤⎢⎢⎣⎡⨯=⎪⎪⎭⎫ ⎝⎛⨯-----34.601960193252433252433225431)(pp p pp p p p p===÷÷=÷---35. 12119342439342439341243)(qq qq q qq q ==÷∙=∙-+36. 24133187318731214732143aa a a a a a a a ==÷=÷⎪⎪⎭⎫ ⎝⎛=÷⎪⎪⎭⎫ ⎝⎛⨯-37.513513532125532125535525025021502150ss sss s s ss ==∙=∙÷∙∙=∙∙+-38. 1223122332432132432132432132********t t t tt ttt t==∙∙=∙÷=∙÷-⎪⎭⎫ ⎝⎛-+------39.1674147441214114121414433327927927aa a a a a a a a =⎪⎪⎭⎫ ⎝⎛⨯=⎪⎪⎭⎫ ⎝⎛⨯⨯=⎪⎪⎭⎫ ⎝⎛⨯⨯⨯=⨯⨯++40.23494923)2(41)2(432243412434ab ba bab a b b a b ===⎪⎪⎭⎫ ⎝⎛∙=⎪⎪⎪⎭⎫ ⎝⎛∙----⨯---41.ab a b abba ab ab ab 34134323132232323123232)(===÷=÷-----4 Exponential Functions42. ba baab b a b a ab b a b a 2212111212121212121212121==∙∙=∙⎪⎭⎫ ⎝⎛∙+-++-43. 616161611312112131312121311nm n m nm mn n m n m ===÷⎪⎪⎭⎫ ⎝⎛⎪⎪⎭⎫ ⎝⎛---+-+44.381217232613441313261312342324131232324131232324112)3(2272272272nm nm mn m n m m n m n m m n m n m =⨯⨯⨯=⨯⨯=⎪⎪⎭⎫⎝⎛⨯⎪⎪⎭⎫ ⎝⎛⨯=⎪⎪⎭⎫⎝⎛÷⎪⎪⎭⎫⎝⎛÷+-+--45. []602932602932523141313215231313241523131234152313241444164164166416yx y x yx y x y x xy y x y x xy yxy x xy ==∙∙÷=÷÷=÷÷=÷÷---⎪⎭⎫ ⎝⎛------46. (a)125454545)4(33333==⎪⎪⎭⎫⎝⎛==k k k k(b) 54)4(16223323=⎪⎪⎭⎫ ⎝⎛==k k k 47. (a) 40961677)7(49333222==⎪⎭⎫ ⎝⎛===m mmm(b)[]512727)2()7(16716)7(1673433423433443232333232======⨯⨯m m m m m m48. 169434364272764232332232323=⎪⎭⎫⎝⎛=⎥⎥⎦⎤⎢⎢⎣⎡⎪⎭⎫ ⎝⎛=⎪⎪⎭⎫ ⎝⎛==x x x x 49. 821211610580343443343434=⎪⎭⎫ ⎝⎛=⎥⎥⎦⎤⎢⎢⎣⎡⎪⎭⎫ ⎝⎛=⎪⎪⎭⎫ ⎝⎛==-------x x xx50.141411611613222322322332323232=⎪⎭⎫ ⎝⎛=⎥⎥⎦⎤⎢⎢⎣⎡⎪⎭⎫ ⎝⎛=⎪⎪⎭⎫⎝⎛===⨯x x x x xNSS Mathematics in Action 4A Full Solutions51.255)5()(1251255004232332232333======-x x x x xExercise 4B (p. 4.18) Level 1 1.∴ 42255)5(51255223232322=====x x x x x2.∴2332662166322-==-==--x x x x3.∴2112322333)3(2793223121===+===+++x x x x x x 4.∴2224)2(1682333====x x x x5.∴32322212)2(1)2(3214223252532523-=-=====-x xx x x6. ∴242222666636)6(22222212==+=+==⎪⎪⎭⎫ ⎝⎛=+++x x x x x x 7.∴1222102510)12(21022210221222====∙=+=+∙=++x x x x x x x x8.∴431327321638216)13(32163)3(32163)3(3311211121==-===∙=-=-=-------x x x x x x x x x9.∴1225152652626)15(526)5(552655265)5(26525022222222222222221221==-===∙=+=+=+=+=+--------x x x x x x x x x x x x x10.∴22412432343)12(2432)2(243)2()2(2434)2(42222222-====∙=-=-=-=--x x x x x x x xx x4 Exponential Functions11.∴76182115156322)2()2(3281515633352133321==-=-===------x x x x x x x x x x 12.∴34434422)2(2)2()2(164444222==+====+x x x x x x x x x x 13.∴[]33235)5(5)5(55)25(125532232333=+=====+x x x x x xxx x14.∴53356333)3()3(327)3(9621231221===+===++++x x x x x x xx x xLevel 2 15.∴1014141941919)144(4194)4(4)4(41944401121111211==-===∙=-+=-+=-+-------+x x x x x x x x x x x16.∴1661626122)12126(62)6(12)6)(6(2)6(62)6(12)6(2612212-====∙=--=--=---++x xx x x x x x x x17.∴12252551455529145151551455)5(51)5(5145)5()5(51)5(5145255)5(52222222222122=====∙=⎪⎭⎫ ⎝⎛+-=+-=+-=+--x x x xx x x x x x x x x x 18.[]9431294)231(394)3(2)3(3394)3(23394)3(2)3(394)9(29322222222222212212=∙=++=++=++=++=+++++x x x x x x x x x x x x x x∴ 23323271332-=-===-x x x 19.∴5191958109322)2)(2(2)2()2()2()16(488109381093245233253==+-=+==∙==+-+-+-+-+x x x x x x x x x x x x20.∴11313647933)3(3)3(9181276443932241333213==+-=-==⎪⎭⎫⎝⎛=+--------x x x x x x x x x xNSS Mathematics in Action 4A Full Solutions 21. ∴1141481422333333)3()3()3(381279381422899481524333242152332415-=-==+==∙∙=∙∙=∙∙++++++++++x x xx x x xx x x xx x x xx x x22. ⎩⎨⎧==++)2(125)1(125522 y x y xFrom (1),∴ 325125532=+==+y x y xFrom (2),∴ 022512502=+==+y x y xConsider the simultaneous equations: ⎩⎨⎧=+=+)4(02)3(32 y x y x 2 × (4) – (3):1333)0(2)2()2(2-=-=-=+-+x x y x y x By substituting x = -1 into (4), we have20)1(2==+-y y∴ The solution is x = -1, y = 2.23. ⎪⎩⎪⎨⎧=⎪⎭⎫ ⎝⎛=+-)2(27823)1(256443 y x y x From (1),∴1344256414443=-===-y x y xFrom (2),∴323322782333-=+⎪⎭⎫ ⎝⎛=⎪⎭⎫ ⎝⎛==⎪⎭⎫ ⎝⎛-+y x yx Consider the simultaneous equations:⎩⎨⎧-=+=-)4(3)3(13 y x y x(3) + (4):212431)()3(-=-=-=++-x x y x y xBy substituting 21-=x into (4), we have25321-=-=+-y y∴ The solution is 21-=x , 25-=y .24. ⎩⎨⎧==--+)2(55)1(422213 yx y x From (1),∴ 22322)2(24222312313+====+++y x y xy x y xFrom (2),∴ yx y x y x -=-=-=--4225522 Consider the simultaneous equations: ⎩⎨⎧-=+=)4(4)3(223 y x y x (3) + 2 ⨯ (4):2105)4(22223==-++=+x x y y x x By substituting x = 2 into (4), we have242=-=y y∴ The solution is x = 2, y = 2.25. ⎩⎨⎧=-=---)2(0)81(39)1(033512 yx y x From (1),∴ yx y x yx y x -=-=-==-----6251233033512512From (2),∴ []yx y x y x y x y x 41233)3(330)3(3)3(0)81(394124242+====-=-+4 Exponential FunctionsConsider the simultaneous equations: ⎩⎨⎧+=-=)4(412)3(62 y x y x (3) - (4):155)41(60==+--=y y y y By substituting y = 1 into (3), we have2552162==-=x x x ∴ The solution is 25=x , y = 1.26. ⎩⎨⎧==∙-)2(25)5()1(1281641y x y x From (1),∴ 742222)2()2(128164742742=+==∙=∙+y x y x y x y xFrom (2),∴442225)5(525)5(221221-=-====---y x y xy y xy xConsider the simultaneous equations: ⎩⎨⎧-==+)4(44)3(742 y x y x (3) + (4):133)44(7)42(==-+=++x x y x y x By substituting x = 1 into (4), we have4554441==-=y y y∴ The solution is x = 1,45=y .27. cb a cb ac b a cb a 32101010)10()10(101000100103232========∴2 :3 : 1: 2 :==c b b a∴ 2:3:6::=c b aExercise 4C (p. 4.27)Level 11. (a) fig.) sig. 3 to (cor.31605.7)4(4==f(b) fig.) sig. 3 to (cor.2825.7)8.2(8.2==f(c)fig.)sig. 3 to (cor.299.05.75353==⎪⎭⎫⎝⎛--f2. (a) fig.) sig. 3 to (cor.358.0)38.0(4)38.0(4)2()3(23-=-=-f f(b) fig.) sig. 3 to (cor.77.6)38.0(4)38.0(4)38.0()0(38.00=+=+f f(c)fig.)sig. 3 to (cor.0.48)38.0(4)38.0(4)2()6(26==----f f(d) fig.)sig. 3 to (cor.122)38.0(4)38.0(4)1.2()2.4(1.22.4=∙=∙--f f3. (a) fig.)sig. 3 to (cor.86.39.0532323232=+=⎪⎭⎫⎝⎛+⎪⎭⎫ ⎝⎛g f (b) fig.) sig. 3 to (cor.96.39.0352)7.2(3)7.2(27.27.2-=⨯-⨯=-----g f4.8.94.154.155)4.1(5)2(222=⨯====---A A f5. (a)28815.02.12.1)(15.02.1)3(333======b b b f(b) 0)2(15.02)2(15.0)4(2)5(45=⨯-=-f f6. Weight of the substance mg5625.1mg )5.0(400mg)5.0(400825200===7. Price of a lunchbox $0.1)nearest the to (cor.7.26$)995.0(120$300==NSS Mathematics in Action 4A Full Solutions8. (a) Number of students absent on the 4th dayinteger)nearest the to (cor.38)7.01(504=-=(b) Number of students absent on the 5th day42)7.01(505=-=Increase in number of students absentinteger)nearest the to (cor.43842=-=9. (a) Depth of the river on 1 January 2010fig.) sig. 3 to (cor.m 57.5m)93.0(85==(b) Depth of the river on 1 January 2015m 4 fig.) sig. 3 to (cor.m 87.3m )93.0(810<==Therefore the river will be in flood.10. (a) )9.0(000 400%)101(000 400t t V =-=(b) Value of the car after 5 years196236$)9.0(000400$5==11. The graph of f (x ) = ka x is symmetrical to that ofxa k x g ⎪⎭⎫⎝⎛=1)( for any k and a .(a) The required graph is )5.0(3213x xy =⎪⎭⎫⎝⎛⨯=.(b) The required graph is x xy 5.24.01-=⎪⎭⎫⎝⎛-=.12. (a) (i) From the graph, when x = -1.4, y = 7.0.∴ 0.5)nearest the to (cor.074141..=⎪⎭⎫ ⎝⎛-(ii) 2222414..-⎪⎭⎫⎝⎛=From the graph, when x = -2.2, y = 21.0. ∴ 0.5)nearest the to (cor.021422..=(b) From the graph, when y = 12, x = –1.8.∴ The solution of 1241=⎪⎭⎫⎝⎛xis x = –1.8 (cor. to thenearest 0.1).13. (a) (i) From the graph, when x = 2.5, y = 9.8. ∴ 8.95.25.2= (cor. to the nearest 0.2)(ii)2335.2258125=⎪⎭⎫⎝⎛=From the graph, when 23=x , y = 4.0.∴0.2)nearest the to (cor.0.48125=4 Exponential Functions(b) From the graph, when y = 9, x = 2.4.∴ The solution of 95.2=x is x = 2.4 (cor. to thenearest 0.1).Level 2 14. (a) (i)278322332239483)3()3(6633323333=∙=⎥⎥⎦⎤⎢⎢⎣⎡⎪⎭⎫ ⎝⎛∙=⎪⎭⎫⎝⎛∙=∙g f(ii) 1633232349839483)5.0()5.0(35.05.05.05.05.0=⨯=⨯=⎪⎭⎫ ⎝⎛=g f(b)6414922424983294832)1()1(11-=-=-==+⨯=⎪⎭⎫⎝⎛+⎪⎪⎭⎫ ⎝⎛=-+---k k k k g kf15. 3223222)2(4820)8(5)(2323=======a a a f aa a a16. 3512551000)15)(5(21000)5(2)5(2)()1(31====-=-=-++k k f k f k k k k17. (a) Number of bacteria initially = 1000 (1.242×0) = 1000 (b) Increase in bacteria in the 5th hourinteger)nearest the to (cor. 3005)24.1(1000)24.1(10004252=-=⨯⨯18. (a) When t = 0, L = 4.59)1(9)37.0(940=-=-=-=A A A A(b) Blood glucose level after 1.5 hoursfig.) sig. 3 to (cor.units 87.7units )37.0(595.1=-=19. (a) Number of cakes sold per day with $5500 spent onadvertising= 500 (1.035.5) = 588 (cor. to the nearest integer) (b) advertising= 500 (1.0310) 672≈ Advertising expense for each cake= $10 000 ÷ 672≈$14.9 < $50 The spending is wise because the earning generatedcovers the expense.20. (a) (i) (ii) The rate of increase of y increases as x increases. (b) (i) From the graph of xy ⎪⎭⎫⎝⎛=37, when y = 18,x = 3.4.∴ The solution of 1837=⎪⎭⎫⎝⎛xis x = 3.4 (cor.to the nearest 0.1).NSS Mathematics in Action 4A Full Solutions(ii) 277328738028738=⎪⎭⎫⎝⎛=⎪⎭⎫⎝⎛=-⎪⎭⎫⎝⎛xxxFrom the graph of xy ⎪⎭⎫⎝⎛=73, when y =27,x = –1.5.∴ The solution of 028738=-⎪⎭⎫⎝⎛xis x = –1.5(cor. to the nearest 0.1).21. (a) (i)(ii)(b) (i) From the graph, when 7.2,4-==x y . ∴ The solution of 453=⎪⎭⎫⎝⎛xis x = –2.7 (cor.to the nearest 0.1).(ii) Since 3x , 5x > 0 for all real numbers x , thisequation has no real solutions.22. (a) (i)(ii)(b) Number of bacteria initially =200(c) From the graph, when 1600=N , t = 3.5.∴ Time required =hours 5.3 (cor. to the nearest 0.1)Revision Exercise 4 (p. 4.33) Level 1 1. 228333==2.2)2(32555-=-=-3.3131811444=⎪⎭⎫ ⎝⎛=4.2166)6(36323223===5.45545412564131331=⎪⎭⎫ ⎝⎛=⎥⎥⎦⎤⎢⎢⎣⎡⎪⎭⎫ ⎝⎛=⎪⎭⎫ ⎝⎛---6.94949973737334327112232332=⎪⎭⎫ ⎝⎛=⎥⎥⎦⎤⎢⎢⎣⎡⎪⎭⎫ ⎝⎛=⎪⎭⎫ ⎝⎛=⎥⎥⎦⎤⎢⎢⎣⎡⎪⎭⎫ ⎝⎛=⎪⎭⎫ ⎝⎛-----7.2121312331233121131211aaa a a a a ===⎪⎪⎭⎫ ⎝⎛=⎪⎪⎭⎫ ⎝⎛=⎪⎪⎭⎫ ⎝⎛⨯-⎪⎪⎭⎫ ⎝⎛-⨯--+-4 Exponential Functions8.6762236223623aa aa a a===-9.154154573557355735753511)(1aa a aa a a a a===∙=∙=∙-+--10. 5151525352532225153251533133333333aaa a a a a a a ===⎪⎪⎭⎫ ⎝⎛∙=⎪⎪⎭⎫ ⎝⎛∙⎪⎪⎭⎫ ⎝⎛=⎪⎪⎪⎭⎫⎝⎛∙⎪⎪⎭⎫ ⎝⎛-+-----11. 81418141811438143436143)1(436111b a b a b aaba a baab a =====⎪⎪⎭⎫ ⎝⎛-----⎪⎪⎭⎫⎝⎛-⨯⎪⎪⎭⎫ ⎝⎛-⨯---12.[]ba b a b a ba b a ba 2121231331631363366464)8()8()8()8(==-=-=-=---⨯-⨯--13. 3636)6()6(16)5(561010562525325325253506b a a b a b b a b a ab ==⎪⎪⎭⎫ ⎝⎛=⎪⎪⎭⎫ ⎝⎛÷=⎪⎪⎭⎫⎝⎛÷-----14.2929132122213212221322112213212ba ba bb b a a a b a b b a a b a b ba a ==⨯⨯⨯⨯=⨯⨯⨯⨯=⨯⨯⨯⨯++++----15. 811632323227808274343344334343=⎪⎭⎫ ⎝⎛=⎥⎥⎦⎤⎢⎢⎣⎡⎪⎭⎫ ⎝⎛=⎪⎪⎭⎫⎝⎛⎪⎭⎫ ⎝⎛===-x x x x16.326146323262121161116⎪⎭⎫ ⎝⎛=⎥⎥⎦⎤⎢⎢⎣⎡⎪⎭⎫ ⎝⎛===∙--xxNSS Mathematics in Action 4A Full Solutions22121211233223322332=⎪⎭⎫ ⎝⎛=⎪⎭⎫ ⎝⎛=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎪⎭⎫ ⎝⎛=⎪⎪⎭⎫ ⎝⎛-⎪⎭⎫ ⎝⎛-⨯---x x17.∴4554222)2(32165454=====x x x x x 18.∴42232322332493222221-=-=⎪⎭⎫ ⎝⎛=⎪⎭⎫ ⎝⎛⎪⎭⎫ ⎝⎛=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎪⎭⎫ ⎝⎛=⎪⎪⎭⎫ ⎝⎛-x xx x x19.∴1)01.0(100)01.0(40)01.0(521-====-x x x20.∴1551553)25(553)5(2)5(553)5(2511-====-=-=--+x x x x x x x 21. ∴34431332234)21(234)2(223422342)2(3428334333343313331313131===-==+=+=+=+=+----+-+-+-x x x x x x x x x x x x x22.∴[]44233)3()3(927422231323-=+====+++x x x x x x x x x 23.∴34433142222)2(824314312212-=-=--====----x x x x x xx xx x24. (a) 6491636346)2()2(22=⎪⎭⎫ ⎝⎛=⎪⎭⎫⎝⎛∙=∙g f(b) 8)1(3)1(5343)6(5)0(3)0(50=+=⎪⎭⎫⎝⎛+=+g f(c)923461346)1()1(11=⨯=⎪⎭⎫⎝⎛÷=----g f4 Exponential Functions(d)[]37216827646834)6(834)3(8)3(333131=-=-⎪⎭⎫⎝⎛=-⎥⎥⎦⎤⎢⎢⎣⎡⎪⎭⎫ ⎝⎛=-------f g25. (a) )4(3)2(3]2[3)2(239)5.0(3)2(9)()()(2)(1x x x x x x x xx g x f x h ===⎥⎦⎤⎢⎣⎡===---(b) 23434321192)4(3)3(21213==⎪⎪⎭⎫ ⎝⎛=⎪⎭⎫ ⎝⎛-==-h h 26. (a)525525)5(2512525)125(31331===⎥⎥⎦⎤⎢⎢⎣⎡=⎪⎪⎭⎫⎝⎛=k k k k f(b) []100)10(5)10(5)10(5)10(5)1000(510005)1000()1000(3133133131=--=--⎥⎥⎦⎤⎢⎢⎣⎡=⎥⎥⎦⎤⎢⎢⎣⎡--⎪⎪⎭⎫ ⎝⎛=--f f27. Surface area of the person = 0.096 (660.7) m 2 =2m 80.1(cor. to. 3 sig. fig.)28. (a) Initial amount of drug = 25(0.850) mg = 25(1) mg =mg 25(b) Percentage of drug left fig.)sig. 3. to (cor.%2.52mg 25mg)85.0(254==29. (a) nnA 22)02.1(000 252%41000 25=⎪⎭⎫ ⎝⎛+=(b) Amount received by Mr Chan 3 years later = $25 000 (1.02)2×3=154 28$ (cor. to the nearest $1)30. (a) (i) From the graph, when x = -0.5, y = 1.2.∴ 0.1)nearest the to (cor.213250..=⎪⎭⎫ ⎝⎛-(ii)2333232278⎪⎭⎫ ⎝⎛=⎪⎭⎫ ⎝⎛=From the graph, when 23=x , y = 0.5.∴0.1)nearest the to (cor.50278.=(b) (i) From the graph, when 2=y , x = –1.7.∴ The solution of 232=⎪⎭⎫⎝⎛xis x = –1.7 (cor. tothe nearest 0.1).(ii) From the graph, when 21=y , x = 1.7.∴ The solution of 2132=⎪⎭⎫⎝⎛xis x = 1.7 (cor. tothe nearest 0.1).NSS Mathematics in Action 4A Full Solutions31. (a) (i)(ii)(b) (i) From the graph, when 8=y , x = 3.5.∴ The solution of 859=⎪⎭⎫⎝⎛xis x = 3.5 (cor. tothe nearest 0.1).(ii)5592.0952.09511=⎪⎭⎫⎝⎛=⎥⎥⎦⎤⎢⎢⎣⎡⎪⎭⎫ ⎝⎛=⎪⎭⎫⎝⎛--xx xFrom the graph, when 5=y , x = 2.7.∴ The solution of 2.095=⎪⎭⎫⎝⎛xis x = 2.7 (cor. tothe nearest 0.1).(iii) Since 9x , 5x > 0 for all real numbers x , thisequation has no real solutions.Alternative Solution598.15959)5(9)9(50)5(9)9(5-=-=-==+x xx x x x xThe graph of y = 1.8x , however, always lies above the x -axis. So, 1.8x can never be negative. We conclude that the equation has no real solutions.32. Yes, the two graphs intersect at P (0, 1) only.Consider a > 1.When x > 0, a x> 1 and 1122<⎪⎭⎫⎝⎛=-xxa aWhen x = 0, a x = 1 and 12=-xaWhen x < 0, a x< 1 and 1122>⎪⎭⎫⎝⎛=-xxa a∴ It is obvious that the graphs of y = a x and y = a –2x donot intersect except at the point P (0, 1). Similar argument applies for 0 < a < 1.Level 2 33. 155)5()5(125259433229433229432===⨯=⨯⨯+⎪⎪⎭⎫⎝⎛-⨯--34. 3133)3()3(9271412613412613461===÷=÷-⨯-⎪⎪⎭⎫⎝⎛-⨯--35. []2122)2()2(222116641611)3(61103143611031436146314363===⎥⎥⎦⎤⎢⎢⎣⎡⨯=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⨯⨯⎪⎭⎫ ⎝⎛=⎪⎪⎭⎫⎝⎛⨯⨯--⨯⎪⎭⎫ ⎝⎛⨯+⨯-----36. 350350334153343534333533927)(9)3()(9)3(aa a aa a a a a -=-=-=⎪⎪⎭⎫ ⎝⎛∙-=∙---+---。