C3 Jurv-
1.
\3 (ncpa<-c-A po$€r)
t.ttr*
Given that
3xa-2x3-5xz-4
1 .
dx+e
(4)
t
3r-*
__;rf
-rrf
_-if+E:c
+}f i -8*
TI
t
-Lg i. {rLv I
Given that
f(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.
(7)
$ttx-)
aSSrnOtofre-
*=O
U'te(*;l
asU'\^P+otea-=
I
aYrnp\ote-
w*
I
I
I
I
I
+'l
I
:
O
g---(r+)
3.
Given that
2cos(x + 50)"
(a)
Show, without using a calculatot, that
tan xo
(b)
: sin(x + 40)'
Hence solve, for 0
(
0
: l tan 4o'
<360,
2cos(20 + 50)"
giving your answers to
:
sin(20 + 40)'
1 decimal place'
O.CpS:--Cos-5O -e.Sqn,:r-S,1n
e
1
J
O,S-rv.reO --'
50
:- Srn>9(os
QO-
+ CosI- Srn\O
e.\rrnX-,Co5 *O :- hr^nX (,s.4D +Srn
--
kr,*n)L
tt^-n
+
trrnYO
1=, tt"".SP
f(x):25x2e2'-76' re
lR
(a) Using calculus, find the exact coordinates of the turning points on the curve with
equation
y:f(x)
(s)
(b)
Show that the equation f(x)
:
0 can be written as r
: *1"*
(1)
The equation f(r)
(c)
:
0 has a root a, where
a:
0.5 to 1 decimal place'
Starting with xo : 0.5, use the iteration formula
4
Xn
to calculate the valtues of
(d) Give an accurate
e))
x,
mate
esstin
xrattd
t: je-r'ri
x,
giving your answers to 3 dec
for a to 2 decimal places, and justify y
c)-
L,o-
i-O S- ,
[X-r_;=
.7rz :
- *tl-s
O
_!t55
O*frL
-.,
'
o:
tt89
d) +(-o'kgs): :O: *q < O
_flCo.rr1s)'_ t o trl__?o_
"
5.
Given that
x
d"r
(a) find ;
dy
(b)
:
sec2
3Y,
0
'Y 'L6
in terms ofY.
(2)
Hence show that
dvd. - 6{.i,
1
(4)
(c)
Find an expression for
dzv
|f
in terms of
x.
Give your answer in its simplest form.
(4)
cA >c*--r-(
sec.st):
x 3Sec3,,tn^3\
= l(Se-3",r)
\-
x-:Scc1b-
,;6-= bef3**,\
a-r:Yf1_s-
;-t
ctq
A!
oI-{I--\
--
Il.fi
(:-il
=- [qp133
1
clx_ _-_
a(:c-r)
a _utr;,_\_:f-. vr, 3o..)t
'3
-'.4b--,
-&
I
{oc-Da
36 (x--\)
--
rl.
:bCx.-rXr-t\L
--9--
=)
j6*
e-3x
Ct-t)T'
tffit>zr>
_b
- 3r-,
3b*(-1{;1'
6.
Find algebraically the exact solutions to the equations
(a)
1n(4
-
2x) +
(b) 2'e"*' :
ln(9-
3x)
:
2ln(x +
l),
-l
<x<2
(s)
10
Give your answer to (b) in the form
a+1nb
"
+ tn d
where a. b, c and d are integers.
(s)
;;_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*\
--=i---r(
i;2) 3 -\
3+\n2-
= \rr \O
+htO
7.
-2 ( x ( 6 and is linear from (-2, 10) to (2, 0) and from
to(6,4).Asketchofthegraphofy:f(x)isshowninFigurel'
The function f has domain
Figure
(a) Write down
(2' 0)
1
the range of f.
(1)
(b) Find ff(0).
(2)
The function g is defined bY
g:x-> 4+3x
5---r'
(c)
Find g-1(x)
(d)
Solve the equation gf(x)
xelR,
x+5
(3)
:
16
(s)
a)
esj_.tO
' G;$-"-
b)
---
c)
Jc = t*3:r
5=-S
c) o"\{t-1 = t6
is- i *--
-Gto)
i
4 -,--L'(sro;
+
5)e.-f,J = 4,+33 --
- 3+(=) t6
=) t- 5-*(r)
=
(+l;)
,+
x-= (> ,
8.
24m
Figure
2
Kate crosses a road, of constant width 7 m, rn order to take a photograph of a marathon
runner, John, approaching at 3 m s-1.
Kate is 24 m aiead of John when she starts to cross the road from the fixed point ,4.
shown in
John passes her as she reaches the other side of the road at a variable point B, as
Figure 2.
Kolte,s speed is Vms-l and she moves in a straight line, which makes an angle 0,
O < 0 < 150o, with the edge of the road, as shown in Figure 2'
You may assume that V is given by the formula
V-
2l
24sin0 + 7 cos?'
0<0<150'
7cos7 in the form Rcos(0 - o), where R and a are constants and
where ft > 0 and 0 I a r-90o, giving the value of a to 2 decimalplaces.
(3)
(a) Express 24sin0 +
Given that 0 varies,
(b)
frnd the minimum value of Z
(2)
Given that Kate's speed has the value found in part (b),
(c)
find the drstance AB.
Given instead that Kate's speed is 1.68 ffi
(3)
s-1,
(d) find the two possible values of the angle 0, given
that 0 < P < 150o
(6)
ro) K
tor($ -o()
=
IGu,dAi* * f sr6i,J
a1gff -*- zQsyd
trnrn c( = ?\,
+ &- *3.+\
?-- L5
-E-*--V*.,,t
fhax
(2-SCos{s:13:+q)
-=--t-68
--.+- -C-o-sIo -ei ?,$) :,
L\ - ;-OS
zsx(-68',
2.S,