Slk Ca(fa)
1.
Express
3t6
2x+3 2x-3'4x2-9
as a single fraction
__
(
in its simplest form.
\
'(zx,a)Oz+3)
w)
rr+?)C2.xr3)_
-;--l- /r
G
-'-'
TxIZ +-
s)
A curve C has equationy
(a)
:
eo*
+ xo + 8x + 5
Show that the x coordinate of any turning point of C satisfies the equation
x3
: -2-
e4'
(3)
(b)
On the axes given on page 5, sketch, on a single diagram, the curves with equations
(i) y:
x3.
(ii) v:-2-e4'
On your diagram give the coordinates of the points where each curve crosses the
y-axis and state the equation of any asymptotes.
(4)
(c)
Explain how your diagram illustrates that the equation x3 :
-2-
eo'has only one root.
(1)
The iteration formula
x,*t = (-z
-.0")1,
Jo
=
-l
can be used to find an approximate value for this root.
(d)
Calculate the values of x, and x2, givingyour answers to 5 decimal places.
(2)
(e)
Hence deduce the coordinates, to 2 decimal places, of the turning point of the
curve C.
Q)
-t
-*e?*
q)*ro
)
-*x\,-S
#*-nsv**\t.3.-r6
.'. X\
er**
"---L'
.41
b)
L-
c) 0wtt1 errr.l- ptn* U
lntcrscotton
d)
.
A,o ='1
?!t
''!: U-5+$
*r:--!:16t16
e L -hLbt- 2' ss
-
3. (i) (a) Show that2tanx - cot.tr:
5 cosec
x
acos'x+6cos
stating the values of the constants a.
(b)
Hence solve, for 0
( xI
may be written in the form
xlg:g
b
and, c.
2tr,the equation
2tanx-cotx=5cosecx
giving your airswers to 3 significant figures.
(ii)
Show that
tan0+cot0=)cosec20,
^
a*
n1f
2',
neZ
stating the value of the constant 2.
(or
LrO
--a'3-h=S
Cs__:2-
-lErrz3'c;-s"o'i52'2
eqe
sn0
__
__.ffi
d--L.
__-__
_S*gtDL
3r"o
--^-.a
2
4.
(i) Given that
x:
7t
0<y<;+
sec' 2y,
show that
dv_ |
dx - +x"{@i
(4)
(ii)
Given that
y:
find the exact value
.f
*atx
=
(x2
|,
+ x3)ln2x
eivireyour answer in its simplest form.
(s)
(iii) Given that
f(x)=-,3cosx
x*-l
f'(x)='(xg(i)-'
x*-I
(x + 1):
show that
+ 1):
where g(x)"is an expression to be found.
+ Seal Zgla+3gl-
C'|l &*S-
-brn2rtt*=
3" rSer33
a
7
:-o3y,,=---.*A-d.t- ef-J L_1t _
-l
V
*lvt}*
:-e
L1r301nrr-r- :"L:f,
l&
_-"----rl-
d't- t
tg
a.
4 (,xtD-Z f -S (:c+l) Srnr - Cos r- ]
C*-t) a
a
3(:c+r) Srn)r- 1-C}
[v+t)
L.
o
5. (a) Sketch the graph with equation
y: l+x - zl
stating the coordinates of any points where the graph cuts or meets
the axes.
Q)
Find the complete set of values of x for which
(b)
l+x-tl
>
2-2x
(4)
(c)
a
l+x-tl
>
.,
-2 -2x
(2)
)Lqt
"?
*x-4
=
L-L*
I --
4*'3'7'*4
Lt-s
6x" =S
flr3 A
I
ilFL
x,(t
or
*rL
LeiL
x4L
6*"\
?
F
ec!tt' ,9
-
3t
6.
The function f is defined by
f : x --- e'* + #,
(a)
xe
1R,
k is a positive constant.
State the range of f.
(1)
(b) Find f-iand
state its domain.
(3)
The function g is defined by
gix--+1n(2x), x>0
(c)
Solve the equation
g(x)+g(x')+
g(x\:6
giving your answer in its simplest form.
(4)
(d) Find fg(x), giving your answer in its simplest form.
(2')
(e) Find, in terms of the constant k, the solution of the equation
fg(x):2#
Q)
gL* +lt*
1: [rL'
dorrrrrrrn
FI; (zr) t ln (zr,')
-g ffi&5ffi
+ *r[
+
ln(er3
e* *exl*i*P-I-J^G
:-
r^" e l*.*uf T
7-=
x 'g-;'*i'ft'
7.
Figure
Figure 1 shows the curve c, with equationy
(a)
1
: 6 cosx + 2.5 sinxfor 0 ( x {
Express 6 cosx + 2.5 sinx in the form R cos(x
with R > 0 and 0 <
a. {. cir"
2
-
2r
a), where R and a are constants
your value ofa to 3 decimal places.
(3)
(b) Find the coordinates of the points on
coordinate axes.
the graph where the curve
C crosses
the
(3)
A student records the number of hours of daylight each Sunday throughout
the year. She
starts on the last Sunday in May with a recording of 18 hours,
and continues until her final
recording 52 weeks later.
She models her results
with the continuous function given by
H = t2 + r*"(T)*
z
s,n(4), o < ( 52
where -EI is the number of hours of daylight and r is the number
of weeks since her first
recording.
Use this function to find
(c)
the maximum and minimum values of rrpredicted by the
model,
(3)
(d)
the values for / when 11= 16, giving your answers to the
nearest whole number.
lYou must show your working. Answers based entirely on graphicar or numericar
methods
are not acceptable.l
(6)
: ( cr'/. fut o(
(( br (r- *)
-7o)
/( S rnc'\ *
6br
t
L'S ,)
,
ban 4o ll-
[t (tnd.= o
(L] t'st *(oL
(L
"'
t
+ L S,/* S r^ o(
+
2,.s s
l*
6.
6'5
SCor (7*
B(
6'S ct, (:
R**^r
,. lanS ,
6,5
Hr,n.n
-'-
?L- 0.315
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-- Znb
s2-
trr to.34s " -
E+o'3ts
, t$ +o'r15'- -
t 5'5
3) (t
)"e'
2ru- /
@ @
ry
(r-o'31s)
"&
O'Qoq t5'31> .ta.' a/tu -
"' DL,l'i1]fr/S'??
e
o)
hlaer,\ Lg O'b"St
) 6rr. = -6'5 olar* ,-,
\6* \L + 6'J(or (x+'3qs)
d)
trlaar"'
'o'xts)."
* t6'5
-+
hL 0,3r1S...
l'qil)
C (s.tt,
H - \?-r. 6'S (,or u--0'3qs"')
-o'31s" .)
A (oro)
JL=O 5, b
c)
M
s(= o'3gs
V-- lo.+E
i 4?"?J
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