Download - 13 June C3 UK Model
C3 Jurv- \3 (ncpa<-c-A po$€r) t.ttr*1. Given that
3xa-2x3-5xz-4 1 . dx+e
t__;rf
(4)
3r-* _-if- +}f i -8* t
TI-rrf +E:c -Lg i. {rLv
I
Given thatf(x):lnx, x>0
sketch on separate axes the graphs of
(i) y: f(x),
(ii) /: lf(n) 1,
(iii) y:-f(x-4).
Show, on each diagram, the point where the graph meets or crosses the x-axis,In each case, state the equation of the asymptote.
$ttx-)(7)
asU'\^P+ote-
a-= O
I
I
I
I
I
I
+'l
I
:
aYrnp\ote-
aSSrnOtofre-
*=O
U'te(*;l
g---(r+)
w*
3. Given that
2cos(x + 50)" : sin(x + 40)'
(a) Show, without using a calculatot, that
tan xo : l tan 4o'1J
(b) Hence solve, for 0 ( 0 <360,
2cos(20 + 50)" : sin(20 + 40)'
giving your answers to 1 decimal place'
O.CpS:--Cos-5O -e.Sqn,:r-S,1n 50 :- Srn>9(os QO- + CosI- Srn\O
e O,S-rv.reO --' e.\rrnX-,Co5 *O :- hr^nX (,s.4D +Srn
-- kr,*n)L + trrnYO
tt^-n 1=, tt"".SP
f(x):25x2e2'-76' re lR
(a) Using calculus, find the exact coordinates of the turning points on the
equation y:f(x)
(b) Show that the equation f(x) : 0 can be written as r : *1"*
The equation f(r) : 0 has a root a, where a: 0.5 to 1 decimal place'
(c) Starting with xo : 0.5, use the iteration formula
4Xn t: je-r'ri
of x, xrattd x, giving your answers to 3 dec
mate for a to 2 decimal places, and justify y
e))
curve with
(s)
(1)
ues
stin
to calculate the valt
(d) Give an accurate es
c)- L,o- i-O S- ,[X-r_;= O
' _!t55
d) +(-o'kgs): :O: *q < O_flCo.rr1s)'_ t o " trl__?o_
.7rz : O*frL- *tl-s -., o: tt89
5. Given that
x : sec2 3Y, 0 'Y 'L6
d"r(a) find ; in terms ofY.dy (2)
(b) Hence show that
dv- 1
d. - 6{.i,(4)
dzv(c) Find an expression for |f in terms of x. Give your answer in its simplest form.
(4)
cA >c*--r-( sec.st):
= l(Se-3",r) x 3Sec3,,tn^3\\-
x-:Scc1b-,;6-= bef3**,\a-r:Yf1_s-
;-t ctqA! -- Il.fioI-{I--\
(:-il =- [qp133
1
clx_ _-_ a(:c-r) a _utr;,_\_:f-. vr, 3o..)t
-'.4b--,'3
--
I
{oc-Da-&36 (x--\)
rl.:bCx.-rXr-t\L - 3r-,
--9--j6* Ct-t)T' 3b*(-1{;1'
e-3x=) tffit>zr> _b
6. Find algebraically the exact solutions to the equations
(a) 1n(4 - 2x) + ln(9- 3x) : 2ln(x + l), -l < x < 2
(b) 2'e"*' : 10
a+1nbGive your answer to (b) in the form
" + tn d where a. b, c and d are integers.
;;_W{L*.,Jai .,;I;k{iiirf I ",
e-5t':j2:r+3€ -O -> (s,c-+Xr:S) "s
rr) ,(*d'+r ) = ln \o
=) lnZ" + lr.re-H*\ = \rr \O
--=i---r( i;2) 3 -\ +htO
(s)
(s)
3+\n2-
7. The function f has domain -2 ( x ( 6 and is linear from (-2, 10) to (2, 0) and from (2' 0)
to(6,4).Asketchofthegraphofy:f(x)isshowninFigurel'
Figure 1
(a) Write down the range of f.
(b) Find ff(0).
The function g is defined bY
4+3xg:x-> 5---r'
(c) Find g-1(x)
(d) Solve the equation gf(x) : 16
(1)
(2)
(3)
(s)
xelR, x+5
a) esj_.tOb)
' G;$-"-
--- -Gto)is- i *--
4 -,--L- i+ '(sro;
c) Jc = t*3:r5=-S
5)e.-f,J = 4,+33 --
c) "\{t-1 = t6 =) t - 3+(=) =o - 5-*(r) t6
l;)
(+- ,+ x-= (> ,
8.
24mFigure 2
Kate crosses a road, of constant width 7 m, rn order to take a photograph of a marathon
runner, John, approaching at 3 m s-1.
Kate is 24 m aiead of John when she starts to cross the road from the fixed point ,4.
John passes her as she reaches the other side of the road at a variable point B, as shown in
Figure 2.
Kolte,s speed is Vms-l and she moves in a straight line, which makes an angle 0,
O < 0 < 150o, with the edge of the road, as shown in Figure 2'
You may assume that V is given by the formula
2l 0<0<150'V-24sin0 + 7 cos?'
(a) Express 24sin0 + 7cos7 in the form Rcos(0 - o), where R and a are constants and
where ft > 0 and 0 I a r-90o, giving the value of a to 2 decimalplaces.(3)
Given that 0 varies,
(b) frnd the minimum value of Z
Given that Kate's speed has the value found in part (b),
(c) find the drstance AB.
Given instead that Kate's speed is 1.68 ffi s-1,
(d) find the two possible values of the angle 0, given that 0 < P < 150o
(2)
(3)
(6)
ro) K tor($ -o() = IGu,dAi* * f sr6i,Ja1gff -*- zQsyd
trnrn c( = ?\, + &- *3.+\?-- L5
-=--t-68
-E-*--V*.,,tfhax (2-SCos{s:13:+q) 2.S,
--.+- -C-o-sIo -ei ?,$) :,L\ - ;-OSzsx(-68',