ZXTape! 0Created with Ramsoft MakeTZXMODULESE > > @ :1:1::"":20,31;:"MODULESE" E @' <Bf8B 8xB8x?B8 B<<8<<8D @>-!!! >>!!!!  <!!!!  B~<B| x<<!!"! ! @Bb888xB8<8hh8 D8xD8x88D8h88B8hxD88B8x8D88x88!!"! ! |@RD DDB DD TTD<27.1}L,-5:2:.4L,-20:984 P.1}L,35#:17,27;8;1;7;q$:986 Iq$<22.1}L,-5:2:.4L,-20:984 qu=qu+1 r=0::.1}L,35#:6;0,10 ;"MATHEMATICS";2;7;0,0;"Module ";qu+21:6000p+qu:a,qi1,qi2,qi3,qi4,qi5,qi6,qi7,qi8,qi9,qi10,qi11,qi12,qi13,qi14,qi15,qi16,qi17,qi18,qi19:20:.1}L,35#:17,0;3;7;1;" Do you wish to work through thenotes before you attempt the questions? Press Y/N. ":6100+20*qu:20 E.1}L,35#:65001,11 :65000:65040 ^n=1a:n=qi1n=qi2n=qi3n=qi4n=qi5n=qi6n=qi7n=qi8n=qi96100+n+20*qu .1}L,35#:65001,12 :65000:65040:12 ,0;"QUESTIONS";12 ,16;"HINTS" 7000X+30*qu+n:vval,g$,u$,x,y,nlsq:l=1nlsq:d$(l):12 +l,0;d$(l):l::l=1(g$+u$+*2):y,x+l-1;"*";:l:1100L an=qi10n=qi11n=qi12n=qi13n=qi14n=qi15n=qi16n=qi17n=qi18n=qi196500d+n+30*qu Jn:2000:8990#:65001,22:65000:984 L5ph=0:hh=0:bb=0:hint=0:i Q1hint120,1;"TRY AGAIN" Se0,9 ;"enter answer: ";h$:v$="":u$""0,0;"enter units of answer: ";v$ VCa$=h$+v$:a$>g$+u$Ʊa$>(16-x)a$=a$(116-x) X20,1;" ";y,x;5;b$(1̱g$+u$+1);1;7;y,x;a$:h$+v$=g$+u$1138r Zvval=01160 \4vct=0:vctt=0:h$=01180 ^1l=1̱h$:h$(l)=46.vct=vct+1 _$h$(l)=47/vctt=vctt+1 `-h$(l)<46.ůh$(l)>5791180 b-h$(l)>47/Ưh$(l)<58:1130j cl=h$1180 e#h$(l)=46.l=h$1180 f%h$(l+1)=47/1180 j)l:vct>1vctt>11180 k#h$(1)=47/1180 lh$=g$v$=u$1138r mh$ɰg$1180 n 1165 r20,1;2;6;1;1;"CORRECT";:" ":l=14:.1}L,34":.1}L,30:l:20:8990#:1150~ t 1160 ~hh=0r=r+1  1195 h$g$1180 5v$u$20,1;0;7;1;"UNITS!":.1}L,-5:2:.4L,-20:hint=hint+1:8990#::l=1nlsq:5;0;12 +l,0;d$(l):l:l=1(g$+u$+(*2)):y,x+l-1;"*":l:hint=2hh=hh+1:1200  hint=1:hh8ph=0 il=120:.01z# =,35#:.01z# =,25:l:m=1j:k$(m):hh=im=jbb=1 &bb;12 +ph+m,16;k$(m):m  ph=ph+j *20,7;5;" ": 65001,22:65000:5,0;" In this module you got:- answers correct without a hint. ";9 ,0;" If you need more help, work through the following sections in the PAN STUDY AID. " "a=0r=1:a=12 ҳ9 ;6,10 ;r;" out of ";a:r/a.8L3,11 ;"WELL DONE":j=15:.1}L,16:.1}L,24:.1}L,20:j 2000+10 *qu: 013 ,0;" Exercise 13a, page 111.": "13 ,0;" Page 111.": 013 ,0;" Exercise 13b, page 115.": E1,23;" ";13 ,0;" Chapter 17 page 149.": '13 ,0;" Pages 154-156.":  013 ,0;" Exercise 18c, page 165.": q10 ,6,.8L,.9ffff,.1}L,1.1 ,.2~L,.5,.5,.5,.1}L,.5,1.1 ,1.2,1.43333,1.9s333,2.4,2.5 ,.9ffff,.1}L r6,4,.8L,.9ffff,.1}L,1.1 ,.2~L,.5,.5,.5,.1}L,.5,1.1 ,1.2,1.43333,1.9s333,2.4,2.5 ,.9ffff,.1}L s19,10 ,.8L,.9ffff,.1}L,1.1 ,.2~L,.5,.5,.5,.1}L,.5,1.1 ,1.2,1.43333,1.9s333,2.4,2.5 ,.9ffff,.1}L t15,6,11 ,.9ffff,.1}L,1.1 ,.2~L,.5,.5,.5,.1}L,.5,1.1 ,1.2,1.43333,1.9s333,2.4,2.5 ,.9ffff,.1}L u4,3,.8L,.9ffff,.1}L,1.1 ,.2~L,.5,.5,.5,2,3,4,1.2,1.43333,1.9s333,2.4,2.5 ,.9ffff,.1}L v12 ,5,.8L,.9ffff,.1}L,1.1 ,.2~L,.5,.5,.5,.1}L,.5,1.1 ,1.2,1.43333,1.9s333,2.4,2.5 ,.9ffff,.1}L U1,4;6;"PROBABILITY-DEFINITION":.1}L,35#:8521I!: 65001,22:65000:1,35#:5,0;"2. A number is chosen at randomfrom:-"'" 1, 1, 1, 2, 2, 3, 4, 5": X1,3;6;"PROBABILITY-ADDITION RULE":.1}L,35#:8541]!: 65001,15:65000:1,35#:8,0;"2. A number is chosen at random from:-"'"1, 2, 2, 3, 3, 3, 5, 5, 5, 7, 7": 65001,15:65000:1,35#:8,0;"2. A number is chosen at random from:-"'"1, 2, 2, 3, 3, 3, 5, 5, 5, 7, 7": W1,3;6;"PROBABILITY-PRODUCT RULE":.1}L,35#:8561q!: 65001,15:65000:1,35#:7,0;"2. Books are chosen without replacement from a shelf containing 7 French books, 8 German and 5 Italian"'" (ENTER fractions; eg. 6/18).": %K1,9 ;6;"TRIGONOMETRY":.1}L,35#:8581!: *065001,22:65000:8700! +w1;11 ,21;"10cm";5,15;"7.5cm";9 ,14;"51.2":1,23;"B":1,35#:2,0;"2."'" In the triangle"'"ABC,BX is"'"perpendicular"'"to AC,AB=7.5cm,"'"AC=10cm and"'"5)= /10",1,3,"There are 4","numbers greater","than 5 in set E."," Pr(>5)=4/10" {0,"7","",10 ,14,2,"e."," Pr(>2)= /10",1,3,"There are 7","numbers greater","than 2 in set E."," Pr(>2)=7/10" ||0,"2","",10 ,14,2,"a."," Pr(2)= /8",1,3,"There are 2 '2's","in set E."," Pr(2)=2/8" }z0,"1","",10 ,14,2,"b."," Pr(5)= /8",1,3,"There is 1 '5'","in set E."," Pr(5)=1/8" ~0,"3","",10 ,14,2,"c.","Pr(even)= /8",1,3,"There are 3 even","numbers in set E."," Pr(even)=3/8" 0,"0","",10 ,14,2,"d."," Pr(>5)= /8",1,3,"There are no","numbers greater","than 5 in set E."," Pr(>5)=0/8" 0,"3","",10 ,14,2,"e."," Pr(>2)= /8",1,3,"There are 3","numbers greater","than 2 in set E."," Pr(>2)=3/8" 1,"3/7","",11 ,14,5,"a.","Pr(3 or 5)=","","ENTER fraction;","Eg. 5/9 .",2,2," Pr(3 or 5)","=Pr(3)+Pr(5)",2,"=1/7 + 2/7","=3/7" 1,"2/7","",11 ,14,5,"b.","Pr(2 or 3)=","","ENTER fraction;","Eg. 5/9 .",2,2," Pr(2 or 3)","=Pr(2)+Pr(3)",2,"=1/7 + 1/7","=2/7" 1,"3/7","",11 ,14,5,"c.","Pr(5 or 7)=","","ENTER fraction;","Eg. 5/9 .",2,2," Pr(5 or 7)","=Pr(5)+Pr(7)",2,"=2/7 + 1/7","=3/7" 1,"6/11","",11 ,14,5,"a.","Pr(3 or 5)=","","ENTER fraction;","Eg. 5/9 .",2,2," Pr(3 or 5)","=Pr(3)+Pr(5)",2,"=3/11 + 3/11","=6/11" 1,"5/11","",11 ,14,5,"b.","Pr(2 or 3)=","","ENTER fraction;","Eg. 5/9 .",2,2," Pr(2 or 3)","=Pr(2)+Pr(3)",2,"=2/11 + 3/11","=5/11" 1,"5/11","",11 ,14,5,"c.","Pr(5 or 7)=","","ENTER fraction;","Eg. 5/9 .",2,2," Pr(5 or 7)","=Pr(5)+Pr(7)",2,"=3/11 + 2/11","=5/11" u1,"5/12","",1,15,3,"a.","Pr(1st a toffee)","=",1,2,"Pr(1st a toffee)"," =5/12" 1,"5/12","",1,16,4,"b.","Pr(1st a toffee","& 2nd a mint)","= X .",1,2,"Pr(1st a toffee)"," =5/12" 1,"3/11","",6,16,4,"b.","Pr(1st a toffee","& 2nd a mint)","=5/12X",1,3,"[Only 11 left]","Pr(2nd a mint)"," =3/11" 1,"5/12","",1,16,4,"c.","Pr(1st a toffee","& 2nd a toffee)","= X .",1,2,"Pr(1st a toffee)"," =5/12" 1,"4/11","",6,16,4,"c.","Pr(1st a toffee","& 2nd a toffee)","=5/12X",1,5,"[Only 11 left,of","which 4 are","toffees]","Pr(2nd a toffee)"," =4/11" 1,"3/12","",1,16,4,"d.","Pr(1st a mint","& 2nd a mint)","= X .",1,2,"Pr(1st a mint)"," =3/12" 1,"2/11","",6,16,4,"d.","Pr(1st a mint","& 2nd a mint)","=3/12X",1,5,"[Only 11 left,of","which 2 are","mints]","Pr(2nd a mint)"," =2/11" 1,"3/12","",1,16,4,"e.","Pr(1st a mint","& 2nd a toffee)","= X .",1,2,"Pr(1st a mint)"," =3/12" 1,"5/11","",6,16,4,"e.","Pr(1st a mint","& 2nd a toffee)","=3/12X",1,5,"[Only 11 left,of","which 5 are","toffees]","Pr(2nd a toffee)"," =5/11" 1,"7/20","",1,16,4,"a.","Pr(1st is French","& 2nd is French)","= X .",1,2,"Pr(1st is French)"," =7/20" 1,"6/19","",6,16,4,"a.","Pr(1st is French","& 2nd is French)","=7/20X",1,2,"Pr(2nd is French)"," =6/19" 1,"7/20","",1,16,4,"b.","Pr(1st is French","& 2nd is Italian)","= X .",1,2,"Pr(1st is French)"," =7/20" 1,"5/19","",6,16,4,"b.","Pr(1st is French","& 2nd is Italian)","=7/20X",1,2,"Pr(2nd is Italian)"," =5/19" 1,"7/20","",1,16,4,"c.","Pr(1st is French","& 2nd is German)","= X .",1,2,"Pr(1st is French)"," =7/20" 1,"8/19","",6,16,4,"c.","Pr(1st is French","& 2nd is German)","=7/20X",1,2,"Pr(2nd is German)"," =8/19" ”1,"8/20","",1,16,4,"d.","Pr(1st is German","& 2nd is French)","= X .",1,2,"Pr(1st is German)"," =8/20" Î1,"7/19","",6,16,4,"d.","Pr(1st is German","& 2nd is French)","=8/20X",1,2,"Pr(2nd is French)"," =7/19" ĕ1,"8/20","",1,16,4,"e.","Pr(1st is German","& 2nd is Italian)","= X .",1,2,"Pr(1st is German)"," =8/20" Ő1,"5/19","",6,16,4,"e.","Pr(1st is German","& 2nd is Italian)","=8/20X",1,2,"Pr(2nd is Italian)"," =5/19" ѥ0,"2.57","cm",6,14,2,"a."," BX=",3,1,"40=BX/AB",1,"BX=ABx40",3," =4x0.6428"," =2.5712"," =2.57 to 3 sf." ҍ0,"3.06","cm",6,14,2,"b."," AX=",2,1,"AX=AB40",3,"AX=4x0.7660"," =3.064"," =3.06 to 3 sf." Ӎ0,"2.94","cm",6,14,2,"c."," XC=",2,1," XC=AC-AX",3," =6-3.064"," =2.936"," =2.94 to 3 sf." ԭ0,"41.2","",6,15,3,"d."," Angle BCX,"," 4 and H)=X=_, (2 of the 6 2 12 12)":8990# !|65001,22:65000:3,0;" The Product Rule is that if A and B are independent events,"'" Pr(A and B)=Pr(A).Pr(B)":9120# !}""8573}! !~+5:0:="Y"Ŧ="y"8564t! !"N"Ʀ"n"8573}! !65001,15:65000:1,35#:7,0;"1. Sweets are chosen from 5 toffees,4 chocolates and 3 mintswithout replacement."'" (ENTER fractions; eg. 6/18).": !.1}L,35#:65001,22:65000:5,0;" If we have a triangle containing a right angle,we may only need to use the definition of sine,cosine and tangent to find the required sides and angles.":8990#:65000 !b8700!:1;11 ,21;"6cm";5,15;"4cm";9 ,14;"40":1,35#:2,0;"1."'" In the triangle"'"ABC,BX is"'"perpendicular"'"to AC,AB=4cm,"'"AC=6cm and"'"pPp8A8y9BBB9PLB q #$RR| P pd<*]\"_\C*]\~}t[3*a\Þ*x\#"x MODULESF > > @ :1:1::"":20,31;:"MODULESF" F @ <Bf8B 8xB8x?B8 B<<8<<8D @>-!!! >>!!!!  <!!!!  B~<B| x<<!!"! ! @Bb888xB8<8hh8 D8xD8x88D8h88B8hxD88B8x8D88x88!!"! ! |@RD DDB DD TTD<30.1}L,-5:2:.4L,-20:984 P.1}L,35#:17,27;8;1;7;q$:986 Iq$<28.1}L,-5:2:.4L,-20:984 qu=qu+1 r=0::.1}L,35#:6;0,10 ;"MATHEMATICS";2;7;0,0;"Module ";qu+27:6000p+qu:a,qi1,qi2,qi3,qi4,qi5,qi6,qi7,qi8,qi9,qi10,qi11,qi12,qi13,qi14,qi15,qi16,qi17,qi18,qi19:20:.1}L,35#:17,0;3;7;1;" Do you wish to work through thenotes before you attempt the questions? Press Y/N. ":6100+20*qu:20 E.1}L,35#:65001,11 :65000:65040 ^n=1a:n=qi1n=qi2n=qi3n=qi4n=qi5n=qi6n=qi7n=qi8n=qi96100+n+20*qu .1}L,35#:65001,12 :65000:65040:12 ,0;"QUESTIONS";12 ,16;"HINTS" 7000X+30*qu+n:attmpts,vval,g$,u$,x,y,nlsq:l=1nlsq:d$(l):12 +l,0;d$(l):l::l=1(g$+u$):y,x+l-1;"*";:l:1100L an=qi10n=qi11n=qi12n=qi13n=qi14n=qi15n=qi16n=qi17n=qi18n=qi196500d+n+30*qu Jn:2000:8990#:65001,22:65000:984 L5ph=0:hh=0:bb=0:hint=0:i Q3hint120,1;"TRY AGAIN" Se0,9 ;"enter answer: ";h$:v$="":u$""1,0;"enter units of answer: ";v$ V<a$=h$:h$>g$Ʊh$>(16-x)a$=h$(116-x) X|20,1;" ";y,x;5;b$(1̱g$+u$);1;7;y,x;a$;v$:h$+v$=g$+u$1138r Zvval=01160 \4vct=0:vctt=0:h$=01180 ^1l=1̱h$:h$(l)=46.vct=vct+1 _$h$(l)=45-vctt=vctt+1 `Mh$(l)<480Ưh$(l)46.Ưh$(l)45-ůh$(l)>5791180 b#h$(l)=45-l=h$1180 d#h$(l)=46.l=h$1180 j)l:vct>1vctt>11180 kh$=g$v$=u$1138r lh$ɰg$1180 m 1165 r20,1;2;6;1;1;"CORRECT";:" ":l=14:.1}L,34":.1}L,30:l:20:8990#:1150~ t 1160 ~hh=0r=r+1  1195 h$g$1180 (v$u$20,1;0;7;1;"UNITS!":.1}L,-5:2:.4L,-20:hint=hint+1:8990#::l=1nlsq:5;0;12 +l,0;d$(l):l:l=1(g$+u$):y,x+l-1;"*":l:hint=attmptshh=hh+1:1200 (hint=attmpts-1:hh8ph=0 il=120:.01z# =,35#:.01z# =,25:l:m=1j:k$(m):hh=im=jbb=1 &bb;12 +ph+m,16;k$(m):m  ph=ph+j *20,7;5;" ": 65001,22:65000:5,0;" In this module you got:- answers correct without a hint. ";9 ,0;" If you need more help, work through the following sections in the PAN STUDY AID. " "a=0r=1:a=12 ҳ9 ;6,10 ;r;" out of ";a:r/a.8L3,11 ;"WELL DONE":j=15:.1}L,16:.1}L,24:.1}L,20:j 2000+10 *qu: "13 ,0;" Page 179.": /13 ,0;" Exercise 21, page 186.": $13 ,0;" Chapter 21.": q24,5,.8L,.9ffff,.1}L,1.1 ,.2~L,.5,.5,.5,.1}L,.5,1.1 ,1.2,1.43333,1.9s333,2.4,2.5 ,.9ffff,.1}L r10 ,3,5,6,8,10 ,.2~L,.5,.5,.5,5,10 ,1.1 ,1.2,1.43333,1.9s333,2.4,2.5 ,.9ffff,.1}L s12 ,7,.8L,.9ffff,.1}L,1.1 ,.2~L,.5,.5,.5,.1}L,6,12 ,1.2,1.43333,1.9s333,2.4,2.5 ,.9ffff,.1}L |1,0;6;"DETERMINANT AND INVERSE OF 2 X 2";2,12 ;"MATRICES":.1}L,35#:8521I!: 65001,21:65000:1,35#:3,0;"2. Work out the inverses of the following matrices:-";5,7;"1 2";5,23;"1 -2";6,2;" =";6,18;" =";7,7;"3 8";7,23;"3 4":557,135:0,-23,/3:88X,135:0,-23,-/3:183,135:0,-23,/3:216,135:0,-23,-/3 _9 ,8;"2 3";9 ,23;"1 -3";10 ,2;" =";10 ,18;" =";11 ,7;"-2 1";11 ,23;"2 -4":557,103g:0,-23,/3:88X,103g:0,-23,-/3:183,103g:0,-23,/3:216,103g:0,-23,-/3: $1,4;6;"TRANSFORMATION MATRICES":.1}L,35#:8541]!:4,4;"What is 5";'" the image under";6,6;"of 0";5,30;"?":102f,143:0,-23,/3:112p,143:0,-23,-/3: .1}L,35#:8,6;"5 4";10 ,6;"0 3":45-,111o:0,-23,/4:58:,111o:0,-23,-/4:93],111o:0,-23,/4:106j,111o:0,-23,-/4:68D,100d:16,0:-4,4:4,-4:-4,-4:10 :4,13 ;"0";6,13 ;"5": y.1}L,35#:8,19;"0 -3";10 ,19;"5 4":149,111o:0,-23,/4:162,111o:0,-23,-/4:197,111o:0,-23,/4:220,111o:0,-23,-/4:172,100d:16,0:-4,4:4,-4:-4,-4: 65001,22:65000:1,35#:4,0;"2.";4,5;"What is 5 0.8 0.6";5,3;"the image under";5,30;"?";6,7;"of 0 0.6 -0.8":167,144:0,-23,/3:232,144:0,-23,-/3:103g,144:0,-23,/3:112p,144:0,-23,-/3: .1}L,35#:8,6;"5 4";10 ,6;"0 3":45-,111o:0,-23,/4:58:,111o:0,-23,-/4:93],111o:0,-23,/4:106j,111o:0,-23,-/4:68D,100d:16,0:-4,4:4,-4:-4,-4:10 :4,13 ;"0";6,13 ;"5": y.1}L,35#:8,19;"0 3";10 ,19;"5 -4":149,111o:0,-23,/4:162,111o:0,-23,-/4:197,111o:0,-23,/4:220,111o:0,-23,-/4:172,100d:16,0:-4,4:4,-4:-4,-4: 1,7;6;"FINDING THE MATRIX";2,2;"TO DESCRIBE A TRANSFORMATION":.1}L,35#:8561q!: 65001,21:65000:1,35#:4,0;"2."'" Find the matrix that maps"''" 1 0 0 -1"'" into and into"'" 0 1 1 0":15,119w:8750.":24,119w:8751/":71G,119w:8750.":80P,119w:8751/":159,119w:8750.":168,119w:8751/":215,119w:8750.":232,119w:8751/": .1}L,35#:65001,22:65000:8730":80P:1,35#:12 ,31;"X":80P:1,35#:6,29;"X":502:.1}L,35#:173,77M:64@,480:208,76L:-10 ,19,/4:209,76L:-10 ,19,/4:5,27;"(4,3)" 100d:1,35#:2,21;"X":80P:1,35#:4,15;"X":502:.1}L,35#:172,77M:-480,64@:172,112p:-19,-10 ,/4:172,113q:-19,-10 ,/4:4,0;"The "'"transformation"'"is a ROTATION "'"through the "'"angle shown. ";3,12 ;"(-3,4)":8990#: 3.1}L,35#:65001,22:65000:8730":80P:1,35#:12 ,31;"X":80P:1,35#:6,29;"X":502:.1}L,35#:173,77M:81Q,27:-162,-546:5,27;"(4,3)" 3100d:1,35#:2,21;"X":80P:1,35#:20,27;"X":502:.1}L,35#:4,0;"The "'"transformation "'"is a REFLECTION"'"in the line "'"shown. ";21,25;"(3,-4)":8990#: <65001,21:65000:8740$":8990#:8,30;"X":1,35#:164,148:80P,-80P:13 ,25;"X":100d:.1}L,35#:3,25;"X":1,35#:8,20;"X":8990#: 465001,21:65000:8740$":8990#:8,30;"X":1,35#:243,108l:-39',39',/2:3,25;"X":100d:1,35#:-39',-40(,/2:8,20;"X":8990#: wa2,0,"-2","",6,14,2,"a."," ||=",1,2,"3x6-4x5"," =-2" xd2,0,"38","",6,14,2,"b."," ||=",1,2,"3x6-(-4)x5"," =38" yg2,0,"-2","",6,14,2,"c."," ||=",1,2,"3x6-(-4)x(-5)"," =-2" ze2,0,"2","",6,14,2,"d."," ||=",1,2,"3x(-6)-(-4)x5"," =2" {2,0,"2","",6,16,4,"a."," ( )"," =_( )"," ( )",1,2,"1x8-2x3"," =2" |2,0,"8","",8,14,4,"a."," ( )"," =_( )"," 2( )",2,1," Swap a and d.",1," 8" }|2,0,"1","",11 ,16,4,"a."," (8 )"," =_( )"," 2( )",1,1," 1" ~2,0,"-2","",10 ,14,4,"a."," (8 )"," =_( )"," 2( 1)",2,1," Change signs.",1," -2" }2,0,"-3","",8,16,4,"a."," (8 -2)"," =_( )"," 2( 1)",1,1," -3" 2,0,"10","",5,16,4,"b."," ( )"," =_( )"," ( )",1,2,"1x4-(-2)x3"," =10" 2,0,"4","",8,14,4,"b."," ( )"," =_( )"," 10( )",2,1,"Swap a and d.",1," 4" |2,0,"1","",11 ,16,4,"b."," (4 )"," =_( )"," 10( )",1,1," 1" 2,0,"2","",11 ,14,4,"b."," (4 )"," =_( )"," 10( 1)",2,1," Change signs.",1," 2" }2,0,"-3","",8,16,4,"b."," (4 2)"," =_( )"," 10( 1)",1,1," -3" 2,0,"8","",6,16,4,"c."," ( )"," =_( )"," ( )",1,2,"2x1-3x(-2)"," =8" 2,0,"1","",8,14,4,"c."," ( )"," =_( )"," 8( )",2,1," Swap a and d.",1," 1" |2,0,"2","",11 ,16,4,"c."," (1 )"," =_( )"," 8( )",1,1," 2" 2,0,"-3","",10 ,14,4,"c."," (1 )"," =_( )"," 8( 2)",2,1," Change signs.",1," -3" {2,0,"2","",8,16,4,"c."," (1 -3)"," =_( )"," 8( 2)",1,1," 2" 2,0,"2","",6,16,4,"d."," ( )"," =_( )"," ( )",1,2,"1x(-4)-(-3)x2"," =2" 2,0,"-4","",8,14,4,"d."," ( )"," =_( )"," 2( )",2,1," Swap a and d.",1," -4" |2,0,"1","",11 ,16,4,"d."," (-4 )"," =_( )"," 2( )",1,1," 1" 2,0,"3","",11 ,14,4,"d."," (-4 )"," =_( )"," 2( 1)",2,1,"Change signs.",1," 3" }2,0,"-2","",8,16,4,"d."," (-4 3)"," =_( )"," 2( 1)",1,1," -2" 2,1,"4","",9 ,14,4,"a. The image of"," (5) ( )"," ( )=( )"," (0) ( )",1,2,"0.8x5+(-0.6)x0"," =4" 2,1,"3","",9 ,16,4,"b. The image of"," (5) ( 4)"," ( )=( )"," (0) ( )",1,2,"0.6x5+0.8x0"," =3" 2,1,"-3","",8,14,4,"c. The image of"," (0) ( )"," ( )=( )"," (5) ( )",1,2,"0.8x0+(-0.6)x5"," =-3" 2,1,"4","",9 ,16,4,"d. The image of"," (0) (-3)"," ( )=( )"," (5) ( )",1,2,"0.6x0+0.8x5"," =4" 1,0,"1","",0,18,6,"e. Is the","transformation a","ROTATION"," (press 1)","or a REFLECTION?"," (press 2)",1,1," Rotation." 2,1,"4","",8,14,4,"a. The image of"," (5) ( )"," ( )=( )"," (0) ( )",1,2,"0.8x5+0.6x0"," =4" 2,1,"3","",9 ,16,4,"b. The image of"," (5) ( 4)"," ( )=( )"," (0) ( )",1,2,"0.6x5+(-0.8)x0"," =3" 2,1,"3","",9 ,14,4,"c. The image of"," (0) ( )"," ( )=( )"," (5) ( )",1,2,"0.8x0+0.6x5"," =3" 2,1,"-4","",8,16,4,"d. The image of"," (0) ( 3)"," ( )=( )"," (5) ( )",1,2,"0.6x0+(-0.8)x5"," =-4" 1,0,"2","",0,18,6,"e. Is the","transformation a","ROTATION"," (press 1)","or a REFLECTION?"," (press 2)",1,1," Reflection." s2,1,"0","",5,14,4,"a."," ( )"," ( )"," ( )",1,1," 0" t2,1,"-1","",7,14,4,"b."," ( 0 )"," ( )"," ( )",1,1," -1" t2,1,"-1","",4,16,4,"c."," ( 0 -1)"," ( )"," ( )",1,1," -1" r2,1,"0","",8,16,4,"d."," ( 0 -1)"," ( )"," (-1 )",1,1," 0" 1,0,"2","",0,18,6,"e. Is the","transformation a","ROTATION"," (press 1)","or a REFLECTION?"," (press 2)",1,1," Reflection." 2,0,"4","",1,19,6,"f. This reflects","in,","x-axis (press 1)","y-axis (press 2)","y-x=0 (press 3)","y+x=0 (press 4)",1,3,"Reflection in","the line;"," y+x=0" s2,1,"0","",5,14,4,"a."," ( )"," ( )"," ( )",1,1," 0" t2,1,"-1","",7,14,4,"b."," ( 0 )"," ( )"," ( )",1,1," -1" r2,1,"1","",5,16,4,"c."," ( 0 -1)"," ( )"," ( )",1,1," 1" r2,1,"0","",8,16,4,"d."," ( 0 -1)"," ( )"," ( 1 )",1,1," 0" 1,0,"1","",0,18,6,"e. Is the","transformation a","ROTATION"," (press 1)","or a REFLECTION?"," (press 2)",1,1," Rotation." 2,1,"90","",1,15,4,"f. This rotates","through,"," degrees"," anticlockwise.",1,3,"Anticlockwise","rotation through"," 90 degrees." !I""8521I! !J+5:0:="N"Ŧ="n"8539[! !K"Y"Ʀ"y"8521I! !L7.1}L,35#:65001,21:65000 !MV3,0;" Given a 2 x 2 matrix ,say";5,15;"a b";7,15;"c d";6,10 ;" =":119w,135:0,-23,/3:153,135:0,-23,-/3:9 ,0;"we call the expression ad-bc thedeterminant of ,written det  or||.":8990# !N'12 ,15;"2 3";14,15;"4 5";13 ,7;"If =":119w,79O:0,-23,/3:153,79O:0,-23,-/3:16,0;"||=2x5-3x4=":8990#:16,4;"2x5";12 ,15;"2";14,18;"5":8990#:16,4;"2x5-3x4";12 ,15;"2 3";14,15;"4 5":8990#:16,4;"2x5-3x4";12 ,15;"2 3";14,15;"4";16,12 ;"-2":8990# !Oq65001,8:65000:12 ,15;"2 -3";14,15;"4 5";13 ,7;"If =":119w,79O:0,-23,/3:153,79O:0,-23,-/3:16,0;"||=2x5-(-3)x4=":8990#:16,4;"2x5";12 ,15;"2";14,18;"5":8990#:16,4;"2x5-(-3)x4";12 ,15;"2 -3";14,15;"4 5":8990#:16,4;"2x5-(-3)x4=22 (taking care with - sign)";12 ,15;"2 3";14,15;"4":8990# !P|65001,8:65000:12 ,15;"2 3";14,15;"4 6";13 ,7;"If =":119w,79O:0,-23,/3:153,79O:0,-23,-/3:16,0;"||=2x6-3x4=0":8990#:17,0;" All matrices whose determinantsare zero are called singular matrices.":8990# !Qp65001,21:65000:3,0;" To find the inverse, of a 2 x 2 matrix ," !R5,15;"a b";7,15;"c d";6,8;" =":119w,135:0,-23,/3:153,135:0,-23,-/3:8990#:9 ,0;" First interchange a and d.";5,15;"d";7,15;"c a":8990#:10 ,0;"Then change the signs of b and c";5,15;" d-b";7,15;"-c a":8990# !S11 ,0;"Finally divide by ||.";6,8;"=__";5,16;"d-b";7,15;"-c a";7,11 ;"||":8990# !T^65001,21:65000:3,0;" To find the inverse, of ," !U5,16;"2 3";7,15;"-4-5";6,8;" =":119w,135:0,-23,/3:153,135:0,-23,-/3:8990#:9 ,0;" First interchange 2 and -5.";5,15;"-5";7,15;"-4 2":8990#:10 ,0;"Then change the signs of 3 and -4";5,15;"-5-3";7,15;" 4 2":8990# !V12 ,0;"Finally divide by ||.";6,8;"=__";5,15;"-5-3";7,15;" 4 2";7,11 ;" 2":8990#:13 ,0;" If ||=0, there is no inverse, so singular matrices have no inverse.":8990# !W65001,21:65000:3,0;" Always check that,";5,15;"1 0";7,15;"0 1";6,7;". =":119w,135:0,-23,/3:153,135:0,-23,-/3:80P:.1}L,35#:12 ,9 ;"-5 -3 2 3 1 0"'" Here, ";14,7;"2 4 2 -4 -5 0 1";13 ,22;"=":71G,79O:0,-23,/3:112p,79O:0,-23,-/3:127,79O:0,-23,/3:168,79O:0,-23,-/3:191,79O:0,-23,/3:224,79O:0,-23,-/3:8990#:8995## !X""8536X! !Y+5:0:="Y"Ŧ="y"8524L! !Z"N"Ʀ"n"8536X! ![1,35#:65001,21:65000:3,0;"1. Work out the determinant of the following matrices:-";5,7;"3 4";5,23;"3 -4";6,2;" =";6,18;" =";7,7;"5 6";7,23;"5 6":557,135:0,-23,/3:88X,135:0,-23,-/3:183,135:0,-23,/3:216,135:0,-23,-/3 !\_9 ,8;"3-4";9 ,23;"3 -4";10 ,2;" =";10 ,18;" =";11 ,7;"-5 6";11 ,23;"5 -6":557,103g:0,-23,/3:88X,103g:0,-23,-/3:183,103g:0,-23,/3:216,103g:0,-23,-/3: !]""8541]! !^+5:0:="N"Ŧ="n"8560p! !_"Y"Ʀ"y"8541]! !`7.1}L,35#:65001,22:65000 !ay8710":8715 ":.1}L,35#:13 ,0;"Multiplying by matrix 2 0";15,25;"0 2"'"maps P into P(6,8),"''"since,";17,11 ;"2 0 3 6";19,11 ;"0 2 4 8";18,18;"=":199,71G:0,-23,/3:224,71G:0,-23,-/3:87W,39':0,-23,/3:112p,39':0,-23,-/3:127,39':0,-23,/3:137,39':0,-23,-/3:159,39':0,-23,/3:168,39':0,-23,-/3 !b20,0;"(enlargement factor 2)":.1}L,35#::2,24;"(6,8) P":189,93]:45-,63?:190,93]:45-,63?:-7,-2:7,1:-2,-6:3,7 !cV8990#:65001,22:65000:8710":8715 ":.1}L,35#:13 ,0;"Multiplying by matrix -1 0";15,24;" 0 1":191,71G:0,-23,/3:224,71G:0,-23,-/3:16,0;"maps P into P(-3,4)"''"(reflection in the y-axis)" !d502:.1}L,35#:4,16;"P";5,15;"(-3,4)":188,91[:-23,32 :187,91[:-23,32 :2,-6:-2,7:6,-2:8990# !e65001,22:65000:8710":8715 ":.1}L,35#:13 ,0;"Multiplying by matrix 0 -1";15,24;"1 0":191,71G:0,-23,/3:224,71G:0,-23,-/3:16,0;"maps P into P(-4,3)"''"(rotation about the origin through +90 degrees in an anti- clockwise sense) " !f502:.1}L,35#:5,18;"P";6,15;"(-4,3)":188,91[:-31,24:187,91[:-31,24:2,-6:-2,7:6,-2:8990# !g65001,22:65000:8710":.1}L,35#:6,26;1;"X";0;13 ,0;"The addition of matrix 2";15,24;"-1":191,71G:0,-23,/3:208,71G:0,-23,-/3:16,0;"maps P into P(5,3)"'"(translation 2 units parallel tothe x-axis then 1 unit parallel to the y-axis in a negative direction) " !h44,26;"P";5,25;"(3,4)":8990#:1:6,27;"X":1,35#:6,28;"X";6,27;"X":1,35#:7,28;"X";6,28;"X":502:0:8,28;"P";9 ,26;"(5,3)":8990# !i65001,22:65000:8710":.1}L,35#:13 ,0;"We can combine many of these transformations.For example the image (x,y) of a point (x,y) is given by the above transformation:-;":8718" !j8990#:65001,11 :65000:13 ,0;"The"'"image"'"of,";13 ,6;" 2 0 1 2 3 4";15,6;" 1 1 0 1 1 3";14,9 ;"=";14,18;"+";14,22;"=":8720":9 ,25;"X":8722":8990#:1,35#:8,24;"X":60<:1,35#:7,27;"X":8990# !k65001,22:65000:8710":8718":13 ,0;"The"'"image"'"of,";13 ,6;" 3 0 1 3 3 3";15,6;" 0 1 0 0 1 4";14,9 ;"=";14,18;"+";14,22;"=":8720":10 ,26;"X":8722":8990#:1,35#:7,23;"X":60<:1,35#:6,26;"X":8990# !l65001,22:65000:8710":8718":13 ,0;"The"'"image"'"of,";13 ,6;" 0 0 1 0 3 5";15,6;" 2 1 0 2 1 1";14,9 ;"=";14,18;"+";14,22;"=":8720":8,23;"X":8722":8990#:1,35#:10 ,25;"X":60<:1,35#:9 ,28;"X":8990# !m 8995## !n+5:0:="Y"Ŧ="y"8544`! !o"N"Ʀ"n"8558n! !p65001,22:65000:1,35#:5,0;" Draw careful diagrams to solve the questions which will be given below, about the transformation matrices.":8990#:65001,19:65000:1,35#:4,0;"1.";4,21;"0.8 -0.6";6,21;"0.6 0.8":167,143:0,-23,/3:232,143:0,-23,-/3: !q""8561q! !r+5:0:="N"Ŧ="n"8577! !s"Y"Ʀ"y"8561q! !t7.1}L,35#:65001,21:65000 !u}4,0;" The matrix maps into";3,12 ;"a b";3,21;"1";3,28;"a";5,12 ;"c d";5,21;"0";5,28;"c":95_,151:8750.":121y,151:8751/":167,151:8750.":176,151:8751/":223,151:8750.":232,151:8751/" !v08,0;"since,";7,7;"a b 1 a";8,14;"=";9 ,7;"c d 0 c":557,119w:8750.":80P,119w:8751/":95_,119w:8750.":104h,119w:8751/":127,119w:8750.":136,119w:8751/" !w12 ,0;"and maps into .";11 ,10 ;"0 b";13 ,10 ;"1 d":79O,87W:8750.":88X,87W:8751/":135,87W:8750.":144,87W:8751/":8990# !x416,0;" If we know the images of";17,1;"1 0";19,1;"0 1";18,3;"and ,we can write down the"''"matrix of the transformation.":7,39':8750.":16,39':8751/":557,39':8750.":64@,39':8751/" !y8990#:65001,21:65000:8740$":3,30;"X";3,20;"X";4,0;"eg. Find the matrix"'"which maps";6,1;"1 1"'" into and"'" 0 1"''" 0 -1"'" into"'" 1 1":7,127:8750.":16,127:8751/":63?,127:8750.":72H,127:8751/":7,95_:8750.":16,95_:8751/":63?,95_:8750.":80P,95_:8751/":8990#:14,0;"The required 1 -1"''"matrix is, 1 1":103g,63?:8750.":144,63?:8751/":18,0;" This is an enlargement by a factor of 2 and a rotation through +45 degrees. ":8990# !z244,108l:-12 ,28,/4:.1}L,35#:12 ,12 :3,30;"X":8990# !{204,148:-28,-12 ,/4:.1}L,35#:-12 ,12 :3,20;"X":8990# !}W65000:3,0;" Matrices of the form a b";5,23;"-b a"'"describe rotations."'''"and those of the form a b";11 ,23;"b -a"'"describe reflections.":183,151:8750.":216,151:8751/":183,103g:8750.":216,103g:8751/":8995## !~""8574~! !+5:0:="Y"Ŧ="y"8564t! !"N"Ʀ"n"8574~! !.1}L,35#:65001,8:65000:16,0;" Draw careful diagrams to answerthe following questions.":8990# !65001,21:65000:1,35#:4,0;"1."'" Find the matrix that maps"''" 1 0 0 -1"'" into and into"'" 0 -1 1 0":15,119w:8750.":24,119w:8751/":71G,119w:8750.":88X,119w:8751/":159,119w:8750.":168,119w:8751/":215,119w:8750.":232,119w:8751/": !0192,160:-88X,-64@:144,0:-568,64@:0,-64@:8,0:0,8:-8,0:135,96`:-8,15,/4:1;1,23;"B";10 ,12 ;"A";10 ,31;"C";10 ,23;"X": "i=124|2528:i,89Y:0,67C:i:i=92\1568:124|,i:128,0:i:124|,91[:128,0:187,89Y:0,67C:i=082:1;11 ,23+i;i:i:i=-8-22:1;11 ,22+i;i:i:i=282:1;10 -i,22;i:i:11 ,30;"x";2,21;"y": " 7,26;"P(3,4)":189,93]:22,31:190,93]:22,31:-7,-2:7,1:-2,-6:3,7:3,0;" =";2,7;"3";4,7;"4"'"denotes the"'"position vector"'"of the point P"'"relative to the"'"origin.":557,159:0,-23,/3:64@,159:0,-23,-/3: "4,1;"x 0 1 x 3";6,1;"y 1 0 y 1";5,12 ;"+";5,3;"=":7,143:0,-23,/3:20,143:0,-23,-/3:39',143:0,-23,/3:65A,143:0,-23,-/3:79O,143:0,-23,/3:88X,143:0,-23,-/3:111o,143:0,-23,/3:120x,143:0,-23,-/3: "47/,71G:0,-23,/3:64@,71G:0,-23,-/3:87W,71G:0,-23,/3:112p,71G:0,-23,-/3:127,71G:0,-23,/3:137,71G:0,-23,-/3:159,71G:0,-23,/3:168,71G:0,-23,-/3:191,71G:0,-23,/3:200,71G:0,-23,-/3: "17,0;" The tansformation is a reflection in the line x-y=0, followed by a translation.":188,92\:64@,64@: "i=92\2528:i,12 :0,144:i:i=12 1568:92\,i:160,0:i:92\,77M:160,0:173,12 :0,144:1:i=05:12 -2*i,20;i;13 ,21+2*i;i:i:i=-5-12:13 ,20+2*i;i:i:i=-4-1:12 -2*i,19;i:i:0:13 ,30;"x";3,20;"y": "$i=1642448:i,68D:0,80P:i:205,68D:0,80P:i=68D1488:164,i:80P,0:i:164,109m:80P,0:9 ,31;"x";3,23;"y";1;9 ,19;"-1";9 ,24;"0";9 ,30;"1";13 ,23;"-1";3,24;"1";8,30;"X";3,25;"X": ".0,-23,/4: "/ 0,-23,-/4: #""8990# #d30:21,0;"press key when ready":2:0:="@"900 # H.1}L,35#:9 ;21,0;" ": ##17,0;" Do you wish to work through thenotes again before you attempt the questions? Press Y/N. ": #':.1}L,35#:7,0;" To use Module again, press R Press S to STOP ":9100# #("MODULESF"2 #)#"mathcode6"65000,600X # #""9100# #*5:0:="R"Ŧ="r"900 #"="S"Ŧ="s".5,20: # 9100# `mathcode6 X€FZD!X6(#!Y6(#!Y6(#!Z6(#!Y60#!Y60#!Y60#!Y60#!Z60#!0Z60#!PZ60#!pZ60#!Z60#!Z60#!Z60#!Z60# P P P 88 MMbXaXL8Wq !v>