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Paper Reference(s)

6666/01

Edexcel GCECore Mathematics C4

Silver Level S1

Time: 1 hour 30 minutes

Materials reuired !or examination "tems included #ith uestion

$a$ers

Mathematical Formulae (Green) Nil

Candidates may use any calculator allowed by the regulations of the JointCouncil for Qualifications. Calculators must not have the facility for symbolicalgebra manipulation, differentiation and integration, or have retrievablemathematical formulas stored in them.

"nstructions to Candidates

Write the name of the examining body (Edexcel) your centre number candidate number

the unit title (!ore Mathematics !") the paper reference (####) your surname initials

and signature$

"n!ormation !or Candidates

% boo&let 'Mathematical Formulae and tatistical ables* is pro+ided$

Full mar&s may be obtained for ans,ers to %-- .uestions$

here are / .uestions in this .uestion paper$ he total mar& for this paper is 01$

%dvice to Candidates

2ou must ensure that your ans,ers to parts of .uestions are clearly labelled$

2ou must sho, sufficient ,or&ing to ma&e your methods clear to the Examiner$ %ns,ers

,ithout ,or&ing may gain no credit$

Su&&ested &rade 'oundaries !or this $a$er:

%( % ) C * E

6+ ,- ,3 4+ 40 34

Silver 1 his publication may only be reproduced in accordance ,ith Edexcel -imited copyright policy$3455064578 Edexcel -imited$

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1. f(x) 9)"(

7

x+ x: "$

Find the binomial expansion of f (x) in ascending po,ers ofx up to and including the term in

x8$ Gi+e each coefficient as a simplified fraction$

6

une 200-

2. (a) ;se the binomial theorem to expand

8

7

)8/( x x: 8/

in ascending po,ers of x up to and including the term in x8 gi+ing each term as a

simplified fraction$

(5)

(b) ;se your expansion ,ith a suitable +alue of x, to obtain an approximation to 8(0$0)$Gi+e your ans,er to 0 decimal places$

(2)

anuar 200

3. f(x) 9)"

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4. (a) Find the binomial expansion of

8 (/ < )x |x|:/

(b)78 4 8

7 7 1(0$0) 4 (5$7) (5$7) (5$7) $$$

" 84 0#/

Attempt to substitute5$7x= into a candidate*sbinomial expansion$

M7

4 5$541 5$5558741 5$55555#175"7##$$$=

7$

+ mar7s

il+er 7= 1>74 0

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QuestionNumber

Scheme Marks

3. (a) ( ) ( ) 7

4f $$$ $$$ $$$x x = M7

( )

7

4

# < $$$

= 7

4

#

or

4 d

d

ybx

x M7

4 d"/ $$$ 1" $$$

d

yy

x+ %7

4 4 d< < 7/

d

yx y x xy

x + or e.ui+alent C7

( )4 4d

"/ < 7/ 1" 5d

yy x xy

x+ + = M7

4 4 4 4d 1" 7/ 7/ #

d "/ < 7# 8

y xy xy

x y x y x

= = + + %7 ,

(b) 7/ # 5xy = M7

;sing8

xy

= or8

yx

=

4

8 8 87# < 1" 5y yy y

+ =

or

8

48 87# < 1" 5x xx x

+ =

M7

-eading to

"7# /7 7#4 5y + = or " "7# 4 5x x+ = M7

" /77#

y = or " 7#x =

8 8

4 4

y= or 4 4x= %7 %7

ubstituting either of their +alues into 8xy= to obtain a +alue of theother +ariable$

M7

8 8

4 44 4

both %7 +

=12>

il+er 7= 1>74 75

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dM7

7 7ln 755ln1

5$57 1t

= = 7#5$

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QuestionNumber

Scheme Marks

7 (a) %tA47 / 0 J ( 7) 7 (07)x y A= + = = = (07)A B1

(1)(b) 8 4/ x t t y t= =

4d 8 /d

xt

t=

d4

d

yt

t=

4

d 4

d 8 /

y t

x t =

heir

d

d

y

t di+ided by theird

d

xt M1

!orrectd

d

y

x A1

%tA 44( 7) 4 4 4

m( ) 8( 7) / 8 / 1 1

= = = = Tubstitutes for tto gi+e any of

the four underlined oe= A1

( ) ( )( )= their 7 their 0Ty m x = T

or ( )

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Question 1

his pro+ed a suitable starting .uestion and the maQority of candidates gained 1 or # of the

a+ailable # mar&s$ Nearly all could obtain the index as 74 but there ,ere a minority ofcandidates ,ho had difficulty in factorising out " from the brac&ets and obtaining the correct

multiplying constant of 74 $ !andidates* &no,ledge of the binomial expansion itself ,as good

and e+en if they had an incorrect index they could gain the method mar& here$ %nunexpected number of candidates seemed to lose the thread of the .uestion and ha+ing earlier

obtained the correct multiplying factor 74 and expanded

74

7"

x

+ correctly forgot to multiply

their expansion by4

7$

Question 2

n part (a) a maQority of candidates produced correct solutions but a minority of candidates

,ere unable to carry out the first step of ,riting ( )78/ 8x as

788

4 7/

x $ hose ,ho did so

,ere able to complete the remainder of this part but some brac&eting errors sign errors and

manipulation errors ,ere seen$

n part (b) many candidates realised that they ,ere re.uired to substitute 5$7x= into theirbinomial expansion$ %bout half of the candidates ,ere able to offer the correct ans,er to 0

decimal places but some candidates made calculation errors e+en after finding the correct

binomial expansion in part (a)$ % fe, candidates used their calculator to e+aluate the cube

root of 0$0 and recei+ed no credit$

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n part (a) most candidates manipulated 8 )74 71

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n part (a) those candidates ,ho ,ere able to separate the +ariables ,ere usually able to

integrate both sides correctly although a number integrated7

745 incorrectly to gi+e

( )ln 745 $ Many candidates substituted 5 745t = = immediately after integration tofind their constant of integration as ln755 and most used a +ariety of correct methods toeliminate logarithms in order to achie+e the printed result$ % significant number of

candidates ho,e+er correctly rearranged their integrated expression into the form

745 e tA = before using 5 745t = = to correctly findA$ !ommon errors in this partincluded omitting the constant of integration treating as a +ariable and incorrect

manipulation in order to fudge the printed result$ %lso a number of candidates struggled to

remo+e logarithms correctly and ga+e an e.uation of the form 745 e et c

= + ,hich ,as

then sometimes manipulated to 745 e $tA =

n part (b) most candidates ,ere able to substitute the gi+en +alues into the printed e.uation

and achie+e 7#7t= seconds$ ome candidates made careless errors ,hen manipulating theirexpressions ,hilst a number did not round their ans,er of 7#5$

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the direction +ector 4 4+ i ? 7and setting the result e.ual to 5 although some candidatesused OA 4 4+ +i ? 7or 78 /+ +i ? 7instead of 4 4 $+ i ? 7 ther errors included ta&ing thedot product bet,een OA and OB or deducing 7p= from a correct " " 5$p =

hose candidates ,ho attempted part (b) usually managed to find the magnitude of PAand

many dre, a diagram of triangle PAB correctly and deduced $PA AB= From this pointho,e+er many candidates did not &no, ho, to proceed further resulting in a lot of incorrect,or& ,hich yielded no further mar&s$ ome candidates ho,e+er ,ere able to form a correct

e.uation in order to find both +alues of $ t ,as unfortunate that a fe, ha+ing found the

correct +alues of 8= and 0= then substituted these into8 4

4 4

# 7

+

instead of the

e.uation for the line l$ he most popular method for finding correct +alues of ,as for

candidates to form and sol+e a Pythagorean e.uation in of #AB= or 4 8#$AB = thersuccessful methods for finding included sol+ing # 4PB= or sol+ing a dot product

e.uation bet,een either PA and PB or AB and PB $

Fe, candidates realised that the lengthAB,as t,ice the length of the direction +ector of the

line land applied t,ice the direction +ector 4 4+ i ? 7in either direction fromAin order tofind both positions forB$

Statistics !or C4 9ractice 9a$er Silver Level S1

Mean score for students achieving grade:

QuMaxscore

Modalscore

Mean%

ALL A* A B ! " #

$ 6 83 4.96 5.59 5.19 4.77 4.16 3.41 2.12

7 73 5.12 6.21 5.13 4.28 3.51 2.96 1.65

& 9 74 6.65 8.56 7.64 6.85 6.06 5.05 3.92 2.32

4 9 66 5.97 7.61 6.28 5.73 4.45 4.18 3.11 1.32

' 12 63 7.60 11.50 9.41 7.62 5.96 4.53 3.16 1.69

6 11 11 64 7.04 10.81 9.8 7.67 5.12 3.26 1.97 0.88

( 12 60 7.25 8.99 6.53 5.57 4.20 3.05 1.29

8 9 0 41 3.71 7.65 5.45 3.63 1.93 0.97 0.53 0.21

(' 64 48)& '+)&( 48)&' &8)$4 +)86 )$$ $$)48

il+er 7= 1>74 70