Core 2 - Ch 7 - 1 - Sequences and Series - Solutions

8/13/2019 Core 2 - Ch 7 - 1 - Sequences and Series - Solutions

1/23

1. (a)

9

4

12

+ x

= 29+ 9 2

8

x4

1

+ 2

89

(27) 16

2x

+ 6

789

26 64

3x

M1 B1

(M1 for descending powers of 2 and ascending powers ofx; B1 for

coefficients 1 9 36 84 in an! for" as a#o$e)

= %12 + %76x + 288x2

+ 84x3

&1 &1 4

(#) x= 1'1

gi$es

(2'2%)9

= %12 + %76 + 288 + ''84 M1

= %72%64 &1 2[6]

2. (a) ar= 9

ar4= 112% M1

i$iding gi$es 8

1

9

12%13 ==rM1

*o r= 21

&1 3

(#) sing ar= 9 a= 21

9

= 18 M1 &1 2

(c) S

= 211

181 = r

a

= 36 M1 &1 2

[7]

3. (a)

32

,3

)2)(1(

,2

)1(k

nnnk

nn =

One coefficient (no

r

n

) M1A correct equation, no cancelling A1

eg 3k2= (n2)k3

Cancel at least n (n 1) M1

3 = (n- 2)k (*) &1 cso 4

(#) A= nk= 4 B1

3 = 4 - 2k M1

*o k = 21

and n= 8 &1 &1 4[8]

Holland Park School 1


2/23

4. (a)96'

1

12''

1=

=

rra

M1 &1

96' (1 r) = 12'' r= 41

(.) &1

(#) /9= 12'' (- '2%)

8(or /

1') M1

ifference = /9/

1'= ''1831'%0('''4%7760) M1

= ''23 (or ''23) &1

(c) * n =

( )(( )2%'1

2%'112''

n

M1 &1

(d) *ince nis odd ('2%)nis negati$e M1

so *n= 96' (1 + '2%n

) (.) &1[10]

5. (1 +px)n1 + npx + 2

)1(22

xpnn

+ 0 B1 B1

o"paring coefficients np= 18 2)1( nn

= 36 M1 &1

*o$ing n(n- 1) = 72 to gi$e n= 9;p= 2 M1 &1; &1 ft[7]

6. (2 -px)6= 2

6+

1

6

2%(-px) +

2

6

24(-px)

2M1

r

n

oa!

Coeff. of x or x2

= 64 + 6 2%(-px); + 1% 2

4(-px)

2&1; &1

o

rn

1% 16p2= 13% p2= 16

9

or p = 43

(on!) M1 &1

-632p=A A = -144 M1 &1 ft (t5eirp( ')) 7Con!one lost or extra "" signs for M #arks $ut A #arks #ust

$e correct.

%inal A1 ft is for 1&2x (t'eir p )[7]



3/23

7. (a) 1 + nax + 2

)1( nn

(ax)2+ 6

)2)(1( nnn

(ax)3+ 0 B1B1 2

accept 2*, +*

(#) na= 8 2

)1( nn

a2= 3' M1

$ot'

2

64

2

)1(

n

nn

= 3'

( )2

1 288 aaa

= 3' M1

eit'er

n= 16 a= 2

1

&1 &1 4

(c)

3

2

1

6

141%16

= 7' M1 &1 2[8]

8. (a) (S=) a+ ar+ 0 + arn-1

B1

S =- not require!. A!!ition require!.

(rS= ) ar+ ar2+ 0 + ar

nM1

rS =- not require! (M Multipl/ $/ r)

S(1 - r) = a(1 - rn) S= r

ra n

1

)1(

M1 &1 4

(M Su$tract an! factorise eac' si!e) (0)

(#) r= '9 B1

S2'

= 9'1

)9'1(1' 2'

= 878 M1 &1 3

(c) *" to infinit! = 9'1

1'

1 =

ra

= 1'' M1 &1ft 2

(ft onl/ for r 1)



4/23

(d) r

r

r

a

=

11 = 1' (t a= rin t5e for"a fro" (c) and eate to 1') M1

r= 1'(1 - r) r= 0 11

1'

(or e:act ei$aent) M1 &1 3[12]

9. (a) (x3)12

; 0 +

2

1)()(

2

12

2

1)()(

1

122

1'3113 +

+

x

xx

x

B1; M1

%or M1, nee!s $ino#ial coefficients,nC

rfor# O3 at least as

far as s'o4n5

orrect $aes fornC

rs 12 66 22' sed ("a! #e i"pied) B1

(x3)12

+ 12(x3)11 2

1)(22'2

1)(662

13

93

2

1'3 ++ xxxxx

=

24283236

2

%%

2

336 xxxx +

&2(1') %

(#) /er" in$o$ing

9

33

2

1)()(

xx

; M1

coeff =

9

2

1)(

123

1'1112

&1

=128

%%

(or -'429687%) &1 3[8]

10. (a) ar3= 12 ar

4= -8 r= 0 3

2 (or e:act ei$aent) M1 &1 2

(#) sing rwit5 ar3= 12 or ar

4= -8 to find a= 0 M1

a =-4' 1


5/23

11. (a)

2

1'2

1'2

%2

3

2

2

34%

%

+

+

++=

+

xk

xk

xkk

xk

M1 &1 2

eed not #e si"pified &ccept%

1

%

2

%

3

(#) 1'k3

2

2

x

= %4'x2wit5 or wit5otx

2M1

>eading to k= 6 (.) cso &1 2

?r s#stitting k= 6 into%

2k

3

2

2

x

and si"pif!ing to %4'x2

(c) oefficient ofx3is 1' 6

2

32

1

= 4% M1 &1 2[6]

12. (a) &pp!ing correct for"a @32% = 12' +%(n - 1)A M1

*o$ing to gi$e n= 42 (.) (or $erif!ing in correct eation) &1 2

(#) sing for"a for s" of & * = 2

42

24' + %(42 - 1)

or se 2 n a l+

M1&1

= 934% &1 3

(c) Cecognising D wit5 r= '98 M1

Eae ( in F ) = 72'' ( '98)24

M1

= 4434 ( on! t5is $ae) &1 3[8]

13. (3 + 2x)%= (3

%) +

1

%

34 (2x) +

2

%

33(2x)

2+ 0 M1

= 243+81'x+1'8'x2 B1 &1 &1 4

[4]

14. (a) ar72 ar3= %832 r2= 27

832%

(= '81) M1

r= '9 &1 2

(#) a=)(

27

a = 8 M1 &1 2



6/23

(c) s%'

= 9'1

))9'(1(8%'

M1

= 79%88 (3!p) &1 cao 2

(d) s

= 9'1

8

(= 8') M1

s

-s%'

= 8' - (c) = '412 &1 ft 2

[8]

15. (a) 1 + 12px +66p2x

2B1 B1 2

accept an/ correct equi6alent

(#) 12p= -q 66p2= 11q M1

%or#ing 2 equations $/ co#paring coefficients

*o$ing forpor q M1

p= -2 q= 24 &1&1 4[6]

16. (a) Grites down #ino"ia e:pansion p to and incding ter" inx3

aownC

rnotation 1 + nax+ n(n- 1) 6

)2)(1(

2

22 +

nnnxa

a3x

3

(con!one errors in po4ers of a)*tates na=1% B1

ts 6

)2)(1(

2

)1( 32 annnann =

dM1

(con!one errors in po4ers of a )

3 = (n2 )a

*o$es si"taneos eations in nand ato o#tain

a=6 and n= 2% M1 &1 &1 6

n.$. 7ust 4rites a = 8, an! n = 2.9 follo4ing no

4orking or follo4ing errors allo4 t'e last M1 A1 A15

(#) oefficient ofx3= 2% 1% '% 6

3H 6 = 67% B1 1

(or equals coefficient of x2= 2.9 : 1.9 : 8

2; 2 = 8


7/23

17. (a) (S=) a+ ar+ 0 + arn-1

B1

S =- not require!.

A!!ition require!

(rS=) ar+ ar2+ 0 + ar

nM1

rS =- not require!

(M Multipl/ $/ r)

S(1 - r) = a(1 - rn) S= r

ra n

1)-1(

M1 &1cso 4

(M Su$tract an! factorise) (0)

1 At least t'e + ter#s s'o4n a$o6e, an! no extra

ter#s.

A1 >equires a co#pletel/ correct solution.

Alternati6e for t'e 2 M #arks

M1 Multipl/ nu#erator an! !eno#inator $/ 1 r.

M1 Multipl/ out nu#erator con6incingl/, an!factorise.

(#) arn-1

= 3%''' 1'43= 39 4'' M1 &1 2

(M Correct a an! r, 4it' n = +, ? or 9).

M1 can also $e score! $/ a /ear $/ /ear- #et'o!.

Ans4er onl/ +& ? scores full #arks, +& +


8/23

18. (2 +x)6= 64 B1

(6 2%x) +

22

%6 24

x

+192x +24'x2

M1&1&1 4

B'e ter#s can $e "liste! rat'er t'an a!!e!.

M1 >equires correct structure "$ino#ial coefficients (per'aps

fro# ascals triangle), increasing po4ers of one ter#,

!ecreasing po4ers of t'e ot'er ter# (t'is #a/ $e 1 if factor 2

'as $een taken out). Allo4 "slips.

1

6

an!

2

6

or equi6alent are accepta$le,

or e6en

1

6

an!

2

6

Decreasing po4ers of x

Can score onl/ t'e M #ark.

8?(1 EFF), e6en if all ter#s in t'e $racket are correct, scores

#ax. 1M1AA.

[4

19. (a) ar = 4

2%1

= ra

(/5ese can #e seen esew5ere) B1 B1

a = 2%(1 r) 2%r(1 r) = 4 M Ii"inate a M1

2%r22%r + 4 = ' &1cso 4B'e M #ark is not !epen!ent,

$ut $ot' expressions #ust contain $ot' a an! r.

(#) (%r)(%r4) = ' r= %1

or %

4

M1&1 2

Special case

One correct r 6alue gi6en, 4it' no #et'o!

(or per'aps trial an! error) 1 .

(c) r= a= 2' or % M1&1 2M1 Su$stitute one r 6alue $ack to fin! a 6alue of a.

(d) r

raS

n

n

=1

)1(

#t

2%1

= ra

so Sn= 2%( rn) B1 1

Sufficient 'ere to 6erif/ 4it' Gust one pair of 6alues of a an! r.

(e) 2%(1 '8n) 24 and proceed to n= (or or J) wit5 no nsond age#ra M1



9/23

=> )42%14(

8'og

'4'ogn

n= 1% &1 2

Accept =- rat'er t'an inequalities t'roug'out, an! also allo4

t'e 4rong inequalit/ to $e use! at an/ stage.

M1 requires use of t'eir larger 6alue of r.

A correct ans4er 4it' no 4orking scores $ot' #arks.%or trial an! error- #et'o!s, to score M1, a 6alue of n

$et4een 12 an! 1H (inclusi6e) #ust $e trie!.

[11]

20. (a)

&ppropriate figre M1

1

1

sin i i

i i

r r

r r +

+

=

+(e:p for sin) M1 &1

ri+ r

i+1)sin= r

i- r

i+1

+i

i

r

r 1

M1

1 sin

ration of radii . (=r)1 sin

= + &1 cso %

(#) /ota area2 2 2

2 3 > r r = + + +

2 2 4(1 ) (correct KrK)> r r= + + + B12

2

2 2

1

1 1 sin1

1 sin

>>

r

= = + M1

2 2 2 2

2 2

(1 sin ) (1 sin )

4sin(1 sin ) (1 sin )

> >

+ += =

+ &1 3



10/23

(c) Ceired area = 2 &rea OA+ &rea "aLor sectorAO

&rea fond in (#) M1

&rea OA= 21

>(>cot ) B1

ange of "aLor sector to B 22

OA = = +M1

&rea sectorAO= 21

>2(+ 2) &1

Ceired area =>2(cot +

+++

sin

sinsin21

42

2

!

&1 cso %

=>2(+ cot-

sin4

eccos4

)

(d)

+=

cos

4cotcosec

4cosec1 22>

!

!s

M1 &1

2 2 cos( cot cos )

4 sin 4>

= +

2 2 2( cot cos (cosec 1))4

>

= + M1

2 2

cot ( cos 1)4>

= &1 (cso) 42(se of cot cosec 1) oe=



11/23


12/23

(a) 2nd

B1 for

2

9

(px)2or #etter ondone OP not O+P

(#) 1stM1 for a inear eation forp

2nd

M1 for eit5er printed e:pression foow t5rog5 t5eirp

B 1 + 9px2+ 36px

2eading top= 4 q= 144 scores B1B' M1 &1M1&' ie 4


13/23


14/23

4'x2(1

st&1) &1

-8'x3(2

nd&1) &1

&ow co""as #etween ter"s /er"s "a! #e isted rat5er t5an

added

&ow Treco$er!U fro" in$isi#e #racets so

32234% 213

%21

2

%21

1

%1 xxx

+

+

+

= 1 - 1'x + 4'x2- 8'x

3+

gains f "ars

1 + % (2x) +)2(

,3

34%)2(

,2

4% 32 +

+

xx

= 1 + 1'x + 4'x2+ 8'x

3+

gains B'M1&1&'

Misread first 4 ter"s descending ter"s if correct wod score

B' M1 1st&1 one of 4'x

2and -8'x

3correct; 2

nd&1 #ot5 4'x

2and

-8'x3correct

>ong "tipication

(1 - 2x)2= 1 - 4x + 4x

2 (1 - 2x)

3= 1 - 6x + 12x

2- 8x

3

(1 - 2x)4= 1 - 8x + 24x

2- 32x

3+ 16x

4

(1 - 2x)%= 1 - 1'x + 4'x

2+ 8'x

3+ 0

1 - 1'x

1 - 1'x "st #e seen in t5is si"pified for" in (a) B1

&tte"pt repeated "tipication p to and incding (1 - 2x)%

M1

4'x2(1

st&1) &1

- 8'x3(2

nd&1) &1

Misread first 4 ter"s descending ter"s if correct wod score

B' M1 1st&1 one of 4'x

2and -8'x

3correct;

2nd

&1 #ot5 4'x2and -8'x

3correct



15/23

(#) (1 +x)(1 - 2x)%= (1 +x)(1 - 1'x + 0)

= 1 +x- 1'x + 0 M1

1 - 9x (.) &1 2

se t5eir (a) and atte"pt to "tip! ot; ter"s (w5et5er correct

or incorrect) inx2or 5ig5er can #e ignored

f t5eir (a) is correct an atte"pt to "tip! ot can #e i"pied fro"t5e correct answer so (1 +x)(1 - 1'x) = 1 - 9x wi gain M1 &1

f t5eir (a) is correct t5e 2nd #racet "st contain at east (1 - 1'x)

and an atte"pt to "tip! ot for t5e M "ar &n atte"pt to "tip! ot is an atte"pt at

2 ot of t5e 3 ree$ant ter"s (B t5e 2 ter"s in x1"a! #e

co"#ined - #t t5is wi sti cont as 2 ter"s)

f t5eir (a) is incorrect t5eir 2nd

#racet "st contain a t5e ter"s

inx'andx

1fro" t5eir (a)

& an atte"pt to "tip! a ter"s t5at prodce ter"s in x'andx

1

B (1 +x)(1 - 2x)%= (1 +x)(1 - 2x) @w5ere 1 - 2x + 0 is ?/

t5e candidateUs answer to (a)A

= 1 -xie candidate 5as ignored t5e power of % M'

B /5e candidate "a! start again wit5 t5e #ino"ia e:pansion for

(1 - 2x)%in (#) f correct (on! needs 1 - 1'x) "a! gain M1 &1

e$en if candidate did not gain B1 in part (a) M1

B &nswer gi$en in estion &1

I:a"pe

&nswer in (a) is = 1+1'x + 4'x2- 8'x

3+

(1 +x)(1 + 1'x) = 1 + 1'x +x M1= 1 + 11x &'

[6]



16/23

25. (a) Sn= a + ar + 0 + ar

n- 1B1

rSn= ar + ar

2+ 0 + ar

nM1

(1 - r)*n= a(1 - r

n) dM1

Sn= r

ra n

1

)1(

(.) &1cso 4

Snnot reired /5e foowing "st #e seen at east one + sign

a arn- 1

and one ot5er inter"ediate ter" o e:tra ter"s (sa! arn) B1

Mtip! #! r; rSnnot reired

&t east 2 of t5eir ter"s on CV* correct! "tipied #! r M1

*#tract #ot5 sides >V* "st #e Q(1 - r)Sn CV* "st #e

in t5e for" Qa(1 - rpn + q

)

?n! award t5is "ar if t5e ine for Sn= 0 or t5e ine for rS

n= 0

contains a ter" of t5e for" arcn E !

Met5od "ar so "a! contain a sip #t not awarded if ast ter"

of t5eir Sn= ast ter" of t5eir rS

n dM1

o"petion cso

B &nswer gi$en in estion &1cso

Snnot reired /5e foowing "st #e seen at east one + sign

a arn- 1

and one ot5er inter"ediate ter" o e:tra ter"s (sa! arn) B1

?n CV* "tip! #! r

r

1

1

?r Mtip! >V* and CV* #! (1 - r) M1

Mtip! #! (1 - r) con$incing! (CV*) and tae ot factor of a

Met5od "ar so "a! contain a sip dM1

o"petion cso B &nswer gi$en in estion &1cso

(#) a= 2'' r= 2 n= 1' S1'

= 21

)21(2''1'

M1 &1

= 2'46'' &1 3

*#stitte r = 2 wit5 a = 1'' or 2'' and n = 9 or 1' into for"a for Sn. M1

21

)21(2''1'

or ei$aent &1

2'46'' &1



17/23

&ternati$e "et5od adding 1' ter"s

(i) &nswer on! f "ars (M1 &1 &1)

(ii) 2'' + 4'' + 8'' + 0 + 1'24'' = 2'46'' or

1''(2 + 4 + 8 + 0 + 1'24) = 2'46''

M1 for two correct ter"s (as a#o$e oe) and an indication t5at t5e

s" is needed (eg + sign or t5e word s") M1

1'24'' oe as fina ter" an #e i"pied #! a correct fina answer &1

2'46'' &1

(c) 3

1

6

%== ra

B1

S

= 3

1

1

6

%

1

=

Sr

a

M1

4

%=

oe &1 3

B SW

= r

a

1 is in t5e for"ae #oo

r =3

1

seen or i"pied an!w5ere B1

*#stitte a = 6

%

and t5eir r into r

a

1 sa res a#ot oting for"a M1

4

%

oe &1

(d) -1 J rJ 1 (or rJ 1) B1 1

B SI

= r

a

1 for X r J J1 is in t5e for"ae #oo

-1 J r J 1 or X r X J 1 n words or s!"#os

/ae s!"#os if words and s!"#os are contradictor! Mst #e J not Y B1[11]



18/23

26. (a) 1 + 6kx @&ow nsi"pified $ersions eg 16+ 6(1

%)kx

6C

'+

6C

1kxA B1

32 )(23

4%6)(

2

%6kxkx

+

+@*ee #eow for accepta#e $ersionsA M1&1 3

B /V* II ?/ BI *M>RI R?C /VI &1 (isw is appied)

/5e ter"s can #e TistedU rat5er t5an added

M1 Ceires correct strctre T#ino"ia coefficientsU

(per5aps fro" ascaUs triange) increasing powers ofx

&ow a TsipU or TsipsU sc5 as

3232

3232

23

4%6

2

%6

23

34%

2

4%

)(23

%6)(

2

%6

23

4%6

2

%6

xxkxkx

kxkxkxkx

+

+

+

+

+

+

+

=

Bt 1% + k2x

2+ 2' + k

3x

3or si"iar is M'

Bot5x2andx

3ter"s "st #e seen

3

6

and2

6

or ei$aent sc5 as6C

2and

6C

3are

accepta#e and e$en

3

6and

2

6

are accepta#e for t5e

"et5od "ar

&1 &n! correct (possi#! nsi"pified) $ersion of t5ese 2 ter"s

3

6and

2

6

or ei$aent sc5 as6C

2and

6C

3are accepta#e

escending powers ofx

an score t5e M "ar if t5e reired first 4 ter"s are not seen

Mtip!ing ot (1 + kx)(1 + kx)(1 + kx)(1 + kx)(1 + kx)(1 + kx)

M1 & f atte"pt to "tip! ot (power 6)

B1 and &1 as on t5e "ain sc5e"e

(#) 6k= 1%k2

k= %

2

(or ei$ fraction or '4) (gnore k= ' if seen) M1&1cso 2

M Iating t5e coefficients ofxandx2

(e$en if tri$iaeg 6k= 1%k)

&ow t5is "ar aso for t5e T"isreadU eating t5e

coefficients ofx2andx

3

&n eation is kaone is reired for t5is M "ar at5og5

condone 6kx= 1%k2x

2(6k= 1%k2) k= %

2



19/23


20/23

(d) Sn= '911

)'911('''%'

1

)1(1'

=

r

ra n

M1&1

F76' ''' (Mst #e t5is answer nearest F1'''') &1 3

M1 se of t5e correct for"a wit5 a= %'''' %''' or %''''' and

n= 9 1' 11 or 1%

M1 can aso #e scored #! a O!ear #! !earP "et5od wit5

ter"s added

(&ow t5e M "ar if t5ere is e$idence of adding 9 1' 11 or 1% ter"s)

1st&1 is scored if 1' correct ter"s 5a$e #een added (aow

Onearest F1''P)

(%'''' %4%'' %94'% 647%1 7'%79 76931 838%% 914'2

99628 1'8%9%)

o woring s5own *pecia case 76' ''' scores 1 "ar scored as 1 ' '

(?t5er answers wit5 no woring score no "ars)[9]

28. (a) o"pete "et5od sing ter"s of for" ark to find r M1

@eg Dividing ar6= 8' #! ar

3= 1' to find r; r

6- r

3= 8 is M'A

r = 2 &1 2

M1 ondone errors in powers eg ar4= 1' and


21/23

29. (a)

321'

2

1

3

1'

2

1

2

1'

2

1

1

1'1

2

11

+

+

+=

+ xxxx

M1&1

= 1 + %x; 4

4%+

(or 112%)x2+ 1%x

3

(coeffs need to #e t5ese ie si"pified) &1; &1 4

@&ow &1 &' if tota! correct wit5 nsi"pified singefraction coefficients)

Ror M1 first &1 onsider nderined e:pression on!

M1 Ceires correct strctre for at east two of t5e t5ree ter"s

(i) Mst #e atte"pt at #ino"ia coefficients

@Be generos aow a notations eg1'

C2 e$en

2

1'

; aow OsipsPA

(ii) Mst 5a$e increasing powers ofx

(iii) Ma! #e isted need not #e added; t'is applies for all #arks

Rirst &1 Ceires all three correct ter"s #t need not #e si"pified

aow 11'

etc1'

C2etc and condone o"ission of #racets arond

powers of \x

*econd &1 onsider as B1 1 5x

(#) (1 + 2

1

'.'1)1'

= 1 + %(''1) + ( 4

4%

or 112%) (''1)2+ 1%(''1)

3M1&1ft

= 1 + ''% + '''112% + '''''1%= 1'%114 cao &1 3

Ror M1 *#stitting their (''1) into t5eir (a) rest

@'1 '''1 '2% ''2%'''2% accepta#e #t not '''% or 1''%A

Rirst &1 (ft) *#stittion of (0.01) into their 4 ter!ed e:pression in (a)

"n#$er $ith no $or%ing scores no !ar%# (cacator gi$es t5is answer)[7]

30. (a) (1 + ax)

1'

= 1 + 1'ax (ot nsi"pified $ersions) B132 )(

6

891')(

2

91'axax

+

+

I$idence fro" one of t5ese ter"s is sfficient M1

+ 4%(ax)2 + 12'(ax)

3or + 4%a

2x

2 + 12'a

3x

3&1 &1 4



22/23

/5e ter"s can #e TistedU rat5er t5an added

M1 Ceires correct strctre T#ino"ia coefficientU (per5aps

fro" ascaUs triange) and t5e correct power ofx

(/5e M "ar can aso #e gi$en for an e:pansion in

descending powers ofx)

&ow TsipsU sc5 as

33232

23

789

2

91')(

23

91'

2

91'xaxaxax

Vowe$er 4% + a2x

2+ 12' + a

3x

3or si"iar is M'

3

1'and

2

1'

or ei$aent sc5 as1'

C2and

1'C

3

are accepta#e and e$en

3

1'and

2

1'

are accepta#e

for t5e "et5od "ar

1

st

&1 orrectx

2

ter"2

nd&1 orrectx

3ter" (/5ese "st #e si"pified)

f si"pification is not seen in (a) #t correct si"pified ter"s

are seen in (#) t5ese "ars can #e awarded Vowe$er if

wrong si"pification is seen in (a) t5is taes precedence

*pecia case

f (ax)2and (ax)

3are seen wit5in t5e woring #t t5en ost0

0 &1 &' can #e gi$en if 4%ax2and 12'ax

3are #ot5 ac5ie$ed

(#) 12'a3= 2 4%a

2a = 4

3

or ei$

7%'12'

9'eg

gnore a = ' if seen M1 &1 2

M Iating t5eir coefficent ofx3to twice t5eir coefficient

ofx20 or eating t5eir coefficent ofx

2to twice t5eir

coefficient ofx3 ( or coefficients can #e correct

coefficients rat5er t5an t5eir coefficients)

&ow t5is "ar e$en if t5e eation is tri$ia eg 12'a = 9'a

&n eation in a aone is reired for t5is M "ar

at5og5 condone eg 12'a3x

3= 9'a

2x

2(12'a3= 9'a2)

a= 4

3

Beware a = 4

3

foowing 12'a = 9'a w5ic5 is &'[6]



23/23

31. (a) B2'

=%

19

%

4

= ''72 (&ccept awrt) &ow % %

419

for M1 M1 &1 2

M Ceires se of t5e correct for"a arn-1

.

(#)2%

8'1

%=

=S

M1&1 2

M Ceires se of t5e correct for"a r

a

1

(a) and (#) orrect answer wit5ot woring scores #ot5 "ars

(c) 8'1

)8'1(%

k

249% (&ow wit5 = or J) M1

1 - '8k '998 (or ei$ see #eow) (&ow wit5 = or J) &1

k og '8 J og '''2 or k og'8

'''2 (&ow wit5 = or J) M1

k 8'og

''2'og

(.) &1cso 4

1stM /5e s" "a! 5a$e aread! #een T"anipatedU (per5aps

wrong!) #t t5is "ar can sti #e aowed

1st& & Tn"erica! correctU $ersion t5at 5as deat wit5

(1 - '8) deno"inator

eg 1 -

k

%

4

'998 %(1 - '8k) 499

2%(1 - '8k) 249% % - %('8

k) 499

n an! of t5ese %

4

instead of '8 is fineand condone

%

4k

if correct! treated ater

2nd

M ntrodcing ogs and sing aws of ogs correct!

(t5is "st incde deaing wit5 t5e power k so t5atp

k=k ogp).

2nd

& &n incorrect state"ent (incding eaities) at an!

stage in t5e woring oses t5is "ar (t5is is often

identifia#e at t5e stage k og '8 og '''2)

(*o a f! correct "et5od wit5 ineaities is reired)

(d) k =28 (Mst #e t5is integer $ae) ot k 27 or k J28 or k 28 B1 1[9]

Core 2 - Ch 7 - 1 - Sequences and Series - Solutions

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Transcript of Core 2 - Ch 7 - 1 - Sequences and Series - Solutions