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Paper Reference(s)

6665/01

Edexcel GCECore Mathematics C3

Advanced Level

Monday 12 Jne 2006 A!ternoon

"ime# 1 hor 30 mintes

Materials re$ired !or examination %tems inclded &ith $estion 'a'ers

Mathematical Formulae (Green) Nil

Candidates may se any calclator E(CE)" those &ith the !acility !or sym*olic al+e*ra,

di!!erentiation and/or inte+ration- "hs candidates may ." se calclators sch as the

"exas %nstrments "% , "% 2, Casio C( 0G, 4e&lett )acard 4) G-

%nstrctions 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 0 /uestions in this /uestion paper% *he total mar' for this paper is 1$%

Advice 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%

.2351A *his publication may only be reproduced in accordance -ith .ondon 3ualifications copyright policy%

4566# .ondon 3ualifications .imited%

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1- (a) implify7

5"

5

5

−−−

x

x x

%

738

(b) 8ence or other-ise express7

5"

5

5

−

−− x

x x

9)7(

7

+ x x as a single fraction in its simplest

form%

738

2- :ifferentiate -ith respect to x

(a) e" x ; ln 5 x

738

(b) 5

"

)$( 5 x+ %

738

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3- i+re 1

Figure 7 sho-s part of the cur,e -ith e/uation y < f( x) x ∈ ℝ -here f is an increasing function

of x% *he cur,e passes through the points P (6 95) and Q(" 6) as sho-n%

=n separate diagrams s'etch the cur,e -ith e/uation

(a) y < f( x)

738

(b) y < f 97( x)

738

(c) y < 5

7

f(" x)%738

=ndicate clearly on each s'etch the coordinates of the points at -hich the cur,e crosses or meets

the axes%

N5"$07& "

Q

y < f( x)

x

y

O

(6 95)

P

(" 6)

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- & heated metal ball is dropped into a li/uid% &s the ball cools its temperature T °! t minutes

after it enters the li/uid is gi,en by

T < >66e 96%6$t ; 5$ t ≥ 6%

(a) Find the temperature of the ball as it enters the li/uid%

718(b) Find the ,alue of t for -hich T < "66 gi,ing your ans-er to " significant figures%

78

(c) Find the rate at -hich the temperature of the ball is decreasing at the instant -hen t < $6%

Gi,e your ans-er in °! per minute to " significant figures%

738

(d ) From the e/uation for temperature T in terms of t gi,en abo,e explain -hy the temperature

of the ball can ne,er fall to 56 °!%

718

N5"$07& >

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5 . i+re 2

Figure 5 sho-s part of the cur,e -ith e/uation

y < (5 x 9 7) tan 5 x 6 ≤ x ?>

π

%

*he cur,e has a minimum at the point P % *he x@coordinate of P is k %

(a) ho- that k satisfies the e/uation

>k ; sin >k 9 5 < 6%

768

*he iterati,e formula

xn ; 7 < >7

(5 9 sin > xn) x6 < 6%"

is used to find an approximate ,alue for k %

(b) !alculate the ,alues of x7 x5 x" and x> gi,ing your ans-ers to > decimals places%

738

(c) ho- that k < 6%511 correct to " significant figures%

728

N5"$07& $ "rn over

O

O x

P >

π

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6- (a) Asing sin5 θ ; cos5

θ ≡ 7 sho- that the cosec5 θ 9 cot5 θ ≡ 7%

728

(b) 8ence or other-ise pro,e that

cosec> θ 9 cot> θ ≡ cosec5 θ ; cot5 θ %

728

(c) ol,e for B6° ? θ ? 706°

cosec> θ 9 cot> θ < 5 9 cot θ %

768

- For the constant k -here k C 7 the functions f and g are defined by

fD x ln ( x ; k ) x C 9 k

gD x 5 x 9 k x ∈ ℝ%

(a) n separate axes s'etch the graph of f and the graph of g%

n each s'etch state in terms of k the coordinates of points -here the graph meets the

coordinate axes%

758

(b) Write do-n the range of f%

718

(c) Find fg

>

k

in terms of k gi,ing your ans-er in its simplest form%728

*he cur,e C has e/uation y < f( x)% *he tangent to C at the point -ith x@coordinate " is parallel to

the line -ith e/uation B y < 5 x ; 7%

(d ) Find the ,alue of k %

78

N5"$07& #

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- (a) Gi,en that cos A < >"

-here 516° ? A ? "#6° find the exact ,alue of sin 5 A%

758

(b) (i) ho- that cos

+

"5

π x ; cos

−

"5

π x ≡ cos 5 x%

738

Gi,en that

y < " sin5 x ; cos

+

"5

π x ; cos

−

"5

π x

(ii) sho- that x

y

d

d < sin 5 x%

78

""AL 9 )A)E9# 5 MA9:;

E.<

N5"$07& 1

05_c3_june_2006

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