c9
1.
'lk
vw
The curve C has equationy: f(x) where
f(x) =
(a)
4x+7
x)2
x-2'
Show that
f,(x):
-9
@4f
(3)
Given that P
is. a
point on C such that f '(x)
(b) find the coordinates
:
-1,
of P.
A-?,D-i^t3a-+( - -V,:-:
yl;\
- (tx*rl
(3)
(r-+)z -= +i-8-Rr-t - -q
J-1t-)'-3dx=z)
ffi
*
2.
Find the exact solutions, in their simplest form, to the equations
(a) 2ln(2x + 1) -
(b) 3*e4':
10
:0
(2)
s7
(4)
nrytr =e3
b) 3r=e-+ +
:r . -l +r--s
-l*
jx=L :l- \xd-
=+-t -\x-
4z-t *!rr3 =fr;
:--Lt**tra3):-a
*=l--
-L
ry
4?[r
3.
The curve C has equation
x:8y
The point P has coordinates
tan2y
[".+)
\'8l
(a) Veriff that P lies on C.
(1)
(b) Find the equation of the tangent to C atP in the form ay: x *
a
and
b
are
b, where the constants
to be found in terms of z.
(7)
ff = 8{s}k^{€) :-rrr l-= fr --b-3'+
-")
2q
Uffi*v
ttt - oc^-C7.^
-.r o
v',r.Sex523
-*---,\A1.= I
-C* = -8to.,^e3-+ -t65sds3-
-.1+-
.(€)"ffi.Y
€:*-str-
4.
P(0, 11)
v
Figure
: f(x)
1
Figure 1 shows part of the graph with equationy: f(x), x e
1R'
The graph consists of two line segments that meet at the point
Q$, -1).
The graph crosses the y-axis at the point P(0, 11)'
Sketch, on separate diagrams, the graphs
(a)
v: lr@)l
(b)
y:2f(-x)+3
of
(2)
(3)
On each diagram, show the coordinates of the points corresponding to P and Q'
Given that f (x)
(c)
e,
:
alx
state the value of
- bl a
lc{ftr;}r
)tao-*hilal!-
1, where
a atd'b
are constants'
and the value of 6.
AlG51
-1;;r-
-4 L:bl--l
: ll
=yq1-O;[email protected]
q{ A
ct xt.b = lL
l>=b a,2L
A
5.
g(x) =
(a)
Show that g(x) =
x 3(2x + l\
x*3 x'+x-b
x+l x)3
x-2'
(4)
(b) Find the range of g.
(c)
(2)
Find the exact value of a for which g(a)
:
g-l(a)'
(4)
(x-7
2r+t
.)4-6)
-,tlr3
-1Cr-I:
;l-j;.l,sj--;-i5rt1-=sf!
2u\\.,
J
a-?
Llr
fr
ct
-\
i-r
\)
:) A2:3a =\ = t3
2(*L
^";3;ffi
Y\-
b
zx+\
6.
Figure
2
equation
Figure 2 shows a sketch of part of the curve with
\
(t!x'l+
, = -2cosl
---\2 )
-/
x3
-
3x
-2
a minimum turning point at R'
The curve crosses the x-axis at the point Q and has
(a)ShowthatthexcoordinateofQliesbetween2.land2,2
(2)
(b)ShowthatthexcoordinateofRisasolutionoftheequation
1+
2 . (t ,)
5xsmIr* )
(4)
Using the iterative formula
xn+t
(c)
l,
*l.,rt"(1r,')
,
x, = t.3
find the values of x, and xrlo 3 decimal places'
ou) ?C-
--:*x
c)
=
tro=
2'\ q= --O:tL
----fuZ^,X,
J
'- lt'5
(2)
s'*
^c*-a.rg*O,' S S * ---------l]+<-c<*-'-t---! --
t.5 X, g ['28S ae
c't^'
7. (a) Show that
cosec 2x
(b)
* cot2x:
Hence, or otherwise, solve, for 0
cosec
cotx,
x*90no,
n
eZ
(s)
( 6 < 180o,
(40 + 10')
+ cot(40+ 10"): ..6
You must show Your working.
(solutions based entirely on graphical or numerical methods are not acceptable')
c>-l .+fu
Srnax
5,nLx-
h)- Jae*-!xr-foip,
bLt=
_
'-
ft
Jx=4 g t tO -,_69* lrLoa *ffi
€.- l-Snliotso-
--lgSO
(s)
8.
Arare species of primrose is being studied. The population, P, of primroses
after the study started is modelled by the equation
"""-r,r, t>0,
P= 800e01'
I+3e""
(a)
time tyearc
/e]R
Calculate the number of primroses at the start of the study.
(b) Find the exact
a
(c)
at
and b are
value of r when
P:250, giving
(2)
your answer in the form a ln(b) where
integers.
Find the exact value of
dP
;
(4)
when
t:lO.
Give your answer in its simplest form.
(4)
(d) Explain why the population of primroses
can never be 270
(1)
-gI e:-a'- *' W-*?sOt+3
L
s)
ryY;tso:->
eo'v)sffi
t
tc(o
-+ eo'tv s I
A
-:
ffi;vl!-ffi
9. (a) Express 2sin0 - 4cos I in the formR sin(0ando
a), where R and a are constants, R > 0
7r
< a a,
Give the value of a to 3 decimal places.
(3)
H(6)
:
4 + 5(2sin 36
*
4cos30)2
Find
(b) (i)
the maximum value of H(d),
(ii)
the smallest value of 0, for 0
< 6 1r,
at which this maximum value occurs.
(3)
Find
(c) (D
the minimum value of H(d),
(ii)
the largest value of 0, for 0
< 0 l-r, atwhich this minimum
value occurs.
(3)
a\
2Si^9 - -- 4'Ldg
.1- ;*-or-;-hl0*
*2(9._ !:l*)
Z*tr-,S1^
4--tvu*te$-@
*5p:t;lffi;-E-g.ffi
hl^rrt .
Hr*x
= ++ S( tG)-= 4+rpt ;
-19:l'ffi'- 01f7$-** -, --{:---.-.
---ftffi,6{,*r
3