OCR MEI A LEVEL MATHS TEST WITH bb bb bb bb bb bb
COMPLETE SOLUTIONS
bb bb
what bare bthe b4 bstages bto bthe bproblem bsolving bcycle? b- b bcorrect banswer-
bb bb bb bb bb bb bb bb bb bb bb bb bb
bProblem bspecification band banalysis b
bb bb bb bb bb
information bcollection b bb bb
processing band brepresentation b bb bb bb
interpretation
bb bwhat bare bthe bdifferent bmethods bof bproof b- bbriefly bdescribe bthem? b- b bcorrect
bb bb bb bb bb bb bb bb bb bb bb bb bb
banswer- bproof bby bcounter bexample b- bfind bone bexample bwhere bthe bstates brule
bb bb bb bb bb bb bb bb bb bb bb bb bb
bdoes bnot bapply, btry bthe bmore bobvious, bnegative bnumbers, bzero band b1 bfirst b
bb bb bb bb bb bb bb bb bb bb bb bb bb bb
proof bby bexhaustion b- bjust bcontinuously busing ba bformula bor bother bmethod buntil
bb bb bb bb bb bb bb bb bb bb bb bb
bthe bstatement bcan bbe bdisproved b
bb bb bb bb bb bb
proof bby bdeduction b- busually binvolves bthe beasier bproofs bso busing balgebra bto
bb bb bb bb bb bb bb bb bb bb bb bb
bprove bthe bstatement bor bby bformulating ba blogical bargument botherwise
bb bb bb bb bb bb bb bb bb bb
proof bby bcontradiction b- bmain bmethod, bsuppose bthat bthe bstatement bis btrue, beg
bb bb bb bb bb bb bb bb bb bb bb bb
broot b2 bis brational bor bthere bis ba bfinite bnumber bof bprimes, bthen bfind ba
bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb
bcontradiction bto byour binitial bstatement
bb bb bb bb bb
, bb busing bproof bby bcontradiction, bprove bthat broot b2 bis birrational b- b bcorrect banswer-
bb bb bb bb bb bb bb bb bb bb bb bb bb
b1 b- bsuppose broot b2 bis brational band bcan bbe bwritten bas ba bfraction bin bits bsimplest
bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb
bform broot b2 b= ba/b
bb bb bb bb bb
2 b- blet b2 b= ba^2/b^2, btherefore b2b^2 b= ba^2
bb bb bb bb bb bb bb bb bb
3 b- bthis bmeans bthat ba^2 bmust bbe beven, btherefore ba bmust bbe beven, btherefore
bb bb bb bb bb bb bb bb bb bb bb bb bb bb
ba^2 b= b(2k)^2, bso ba^2 b= b4k^2, bso b2b^2 b= b4k^2, btherefore bb^2 b= b2k^2, bso bb^2
bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb
bmust bbe beven, bso bb bmust bbe beven b
bb bb bb bb bb bb bb bb bb
4 b- bTherefore ba band bb bboth bhave ba bfactor bof b2, bas bthis bis btrue, bm/n bis bnot bthe
bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb
bfraction bin bits bsimplest bform, btherefore bthis bis ba bcontradiction band broot b2 bis
bb bb bb bb bb bb bb bb bb bb bb bb bb bb
birrational
bb
bb busing bproof bby bcontradiction bprove bthat bthere bis ban binfinite bnumber bof bprimes b-
bb bb bb bb bb bb bb bb bb bb bb bb bb
b bcorrect banswer- bsuppose bthat bthere bis ba blist bof bthe bfinite bprimes,
bb bb bb bb bb bb bb bb bb bb bb bb bb
bp1xp2xp3xp4......pn b
bb bb
2 b- blet bP b= bp1xp2xp3xp4....pn b+1 bthis bway bP bis ba bprime band bnot ba bfactor bof
bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb
bany bof bthe bprevious bprimes.
bb bb bb bb bb
P bis beither ba bprime bnumber bor bnot ba bprime bnumber
bb bb bb bb bb bb bb bb bb bb
If bP bis bprime, bthen bP bis bnot bon bthe blist band bwe bhave bfound ba bnew bprime b
bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb
If bP bis bnot bprime bthen bP bmust bbe ba bfactor bof b1 b(as bit bis bnot ba bfactor bof bany
bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb
bother bprime) bthis bmeans bthat bP bmust bbe bless bthat bone, bthis bis bimpossible bas
bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb
bP bmust bbe bgreater bthan b1
bb bb bb bb bb bb
therefore bwe bhave bfound ba bcontradiction b bb bb bb bb bb bb
therefore bthere bare binfinitely bmany bprimes bb bb bb bb bb
bb bhow bwould byou bgo babout bfactorising ba bquadratic bwhere bthe bcoefficient bof bX^2
bb bb bb bb bb bb bb bb bb bb bb bb
bis bgreater bthan b1? b- b bcorrect banswer- beg b5x^2 b+ b11x b+ b2
bb bb bb bb bb bb bb bb bb bb bb bb bb bb
multiply ba band bb b bb bb bb bb
5 b* b2 b= b10 b
bb bb bb bb bb
find bnumbers bthat badd bto bmake b11 band bmultiply bto bmake b10 b
bb bb bb bb bb bb bb bb bb bb bb bb
10 band b1, bnow breplace b11x bwith b10x band b1x
bb bb bb bb bb bb bb bb bb
5X^2 b+ b10x b+1x b+ b2 bb bb bb bb bb
5x(x+2) b+ b1(x+2) b= b(5x+1)(x+2) bb bb bb bb
bb bwhat bare bthe bconditions bfor bthe bdiscriminant, bhow bdoes bthis bvary bwhen bthe
bb bb bb bb bb bb bb bb bb bb bb bb
bquadratic bis bin bcompleted bsquare bform? b- b bcorrect banswer- bb^2 b- b4ac b(let
bb bb bb bb bb bb bb bb bb bb bb bb bb bb
bdiscriminant b=D)
bb bb
when bD b> b0 bthe bquadratic bhas btwo bdistinct breal broots b
bb bb bb bb bb bb bb bb bb bb bb
when bD b< b0 bthe bquadratic bhas bno breal broots b
bb bb bb bb bb bb bb bb bb bb
when bD b= b0 bthe bquadratic bhas bone brepeated broot b(this bcan bbe bused bto bshow ba
bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb
btangent btouches ba bcircle) b
bb bb bb bb bb
when bin bcompleted bsquare bform bthe b+b bin b(x+a)^2 b+ bb bis balso blike bthe
bb bb bb bb bb bb bb bb bb bb bb bb bb bb
bdiscriminant bbut bin breverse:
bb bb bb bb
when bb<0 bthe bequation bhas btwo bdistinct broots b
bb bb bb bb bb bb bb bb
when bb b= b0 bthe bequation bhas ba brepeated broot b
bb bb bb bb bb bb bb bb bb bb
when bb b>0 bthe bequation bhas bno breal broots
bb bb bb bb bb bb bb bb
COMPLETE SOLUTIONS
bb bb
what bare bthe b4 bstages bto bthe bproblem bsolving bcycle? b- b bcorrect banswer-
bb bb bb bb bb bb bb bb bb bb bb bb bb
bProblem bspecification band banalysis b
bb bb bb bb bb
information bcollection b bb bb
processing band brepresentation b bb bb bb
interpretation
bb bwhat bare bthe bdifferent bmethods bof bproof b- bbriefly bdescribe bthem? b- b bcorrect
bb bb bb bb bb bb bb bb bb bb bb bb bb
banswer- bproof bby bcounter bexample b- bfind bone bexample bwhere bthe bstates brule
bb bb bb bb bb bb bb bb bb bb bb bb bb
bdoes bnot bapply, btry bthe bmore bobvious, bnegative bnumbers, bzero band b1 bfirst b
bb bb bb bb bb bb bb bb bb bb bb bb bb bb
proof bby bexhaustion b- bjust bcontinuously busing ba bformula bor bother bmethod buntil
bb bb bb bb bb bb bb bb bb bb bb bb
bthe bstatement bcan bbe bdisproved b
bb bb bb bb bb bb
proof bby bdeduction b- busually binvolves bthe beasier bproofs bso busing balgebra bto
bb bb bb bb bb bb bb bb bb bb bb bb
bprove bthe bstatement bor bby bformulating ba blogical bargument botherwise
bb bb bb bb bb bb bb bb bb bb
proof bby bcontradiction b- bmain bmethod, bsuppose bthat bthe bstatement bis btrue, beg
bb bb bb bb bb bb bb bb bb bb bb bb
broot b2 bis brational bor bthere bis ba bfinite bnumber bof bprimes, bthen bfind ba
bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb
bcontradiction bto byour binitial bstatement
bb bb bb bb bb
, bb busing bproof bby bcontradiction, bprove bthat broot b2 bis birrational b- b bcorrect banswer-
bb bb bb bb bb bb bb bb bb bb bb bb bb
b1 b- bsuppose broot b2 bis brational band bcan bbe bwritten bas ba bfraction bin bits bsimplest
bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb
bform broot b2 b= ba/b
bb bb bb bb bb
2 b- blet b2 b= ba^2/b^2, btherefore b2b^2 b= ba^2
bb bb bb bb bb bb bb bb bb
3 b- bthis bmeans bthat ba^2 bmust bbe beven, btherefore ba bmust bbe beven, btherefore
bb bb bb bb bb bb bb bb bb bb bb bb bb bb
ba^2 b= b(2k)^2, bso ba^2 b= b4k^2, bso b2b^2 b= b4k^2, btherefore bb^2 b= b2k^2, bso bb^2
bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb
bmust bbe beven, bso bb bmust bbe beven b
bb bb bb bb bb bb bb bb bb
4 b- bTherefore ba band bb bboth bhave ba bfactor bof b2, bas bthis bis btrue, bm/n bis bnot bthe
bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb
bfraction bin bits bsimplest bform, btherefore bthis bis ba bcontradiction band broot b2 bis
bb bb bb bb bb bb bb bb bb bb bb bb bb bb
birrational
bb
bb busing bproof bby bcontradiction bprove bthat bthere bis ban binfinite bnumber bof bprimes b-
bb bb bb bb bb bb bb bb bb bb bb bb bb
b bcorrect banswer- bsuppose bthat bthere bis ba blist bof bthe bfinite bprimes,
bb bb bb bb bb bb bb bb bb bb bb bb bb
bp1xp2xp3xp4......pn b
bb bb
2 b- blet bP b= bp1xp2xp3xp4....pn b+1 bthis bway bP bis ba bprime band bnot ba bfactor bof
bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb
bany bof bthe bprevious bprimes.
bb bb bb bb bb
P bis beither ba bprime bnumber bor bnot ba bprime bnumber
bb bb bb bb bb bb bb bb bb bb
If bP bis bprime, bthen bP bis bnot bon bthe blist band bwe bhave bfound ba bnew bprime b
bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb
If bP bis bnot bprime bthen bP bmust bbe ba bfactor bof b1 b(as bit bis bnot ba bfactor bof bany
bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb
bother bprime) bthis bmeans bthat bP bmust bbe bless bthat bone, bthis bis bimpossible bas
bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb
bP bmust bbe bgreater bthan b1
bb bb bb bb bb bb
therefore bwe bhave bfound ba bcontradiction b bb bb bb bb bb bb
therefore bthere bare binfinitely bmany bprimes bb bb bb bb bb
bb bhow bwould byou bgo babout bfactorising ba bquadratic bwhere bthe bcoefficient bof bX^2
bb bb bb bb bb bb bb bb bb bb bb bb
bis bgreater bthan b1? b- b bcorrect banswer- beg b5x^2 b+ b11x b+ b2
bb bb bb bb bb bb bb bb bb bb bb bb bb bb
multiply ba band bb b bb bb bb bb
5 b* b2 b= b10 b
bb bb bb bb bb
find bnumbers bthat badd bto bmake b11 band bmultiply bto bmake b10 b
bb bb bb bb bb bb bb bb bb bb bb bb
10 band b1, bnow breplace b11x bwith b10x band b1x
bb bb bb bb bb bb bb bb bb
5X^2 b+ b10x b+1x b+ b2 bb bb bb bb bb
5x(x+2) b+ b1(x+2) b= b(5x+1)(x+2) bb bb bb bb
bb bwhat bare bthe bconditions bfor bthe bdiscriminant, bhow bdoes bthis bvary bwhen bthe
bb bb bb bb bb bb bb bb bb bb bb bb
bquadratic bis bin bcompleted bsquare bform? b- b bcorrect banswer- bb^2 b- b4ac b(let
bb bb bb bb bb bb bb bb bb bb bb bb bb bb
bdiscriminant b=D)
bb bb
when bD b> b0 bthe bquadratic bhas btwo bdistinct breal broots b
bb bb bb bb bb bb bb bb bb bb bb
when bD b< b0 bthe bquadratic bhas bno breal broots b
bb bb bb bb bb bb bb bb bb bb
when bD b= b0 bthe bquadratic bhas bone brepeated broot b(this bcan bbe bused bto bshow ba
bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb
btangent btouches ba bcircle) b
bb bb bb bb bb
when bin bcompleted bsquare bform bthe b+b bin b(x+a)^2 b+ bb bis balso blike bthe
bb bb bb bb bb bb bb bb bb bb bb bb bb bb
bdiscriminant bbut bin breverse:
bb bb bb bb
when bb<0 bthe bequation bhas btwo bdistinct broots b
bb bb bb bb bb bb bb bb
when bb b= b0 bthe bequation bhas ba brepeated broot b
bb bb bb bb bb bb bb bb bb bb
when bb b>0 bthe bequation bhas bno breal broots
bb bb bb bb bb bb bb bb
