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Transcript
'"
""
X100/901
~
.-
iii
NA TIONAL
QUALIijICATIONS
2003
MATHEMATICS
HIGHER
WEDNEI"SDAY, 21 MAY
9.00 AM - 10.10 AM
~
;
1jj
~
Units 1,2 and 3
Paper 1
(Non~calculator)
~
Read Carefully
1
Calculators may NOTbe u§ed in this paper.
2 Fullcreditwill'be givenonlywherethesolutioncontainsappropriateworkin£!.
3 Answersobtainedby readingsfromscaledrawingswill not receiveanycredit
IOi
'ii
i!I
'"
~
Eo
seams ')
;;
~
QUALIFICATION~/
AUTHORITY
LIB X100/301
6/3/26570
1I11I1~1111111111111111111~ IIII~IIII III11
@
~
ALL questions
should
be attempted.
Marks
1. Find the equation of the line which passes through the point (-1, 3) and is
perpendicular
to the line with equation 4x + y - 1 = O.
3
= x2 + 6x + 11 in the
2
2. (a) Writef(x)
form (x + a)2 + b.
(b) Hence or otherwise sketch the graph of y
3.
Vectors u and v are defined by u
Determine
=f(x).
2
= 3i + 2j and v = 2i -
whether or not u and v are perpendicular
3j + 4k.
2
to each other.
4. A recurrence relation is defined by Un+t=pUn + q, where -1 < P < 1 and Uo=12.
2
(a) If Ut = 15 and U2 = 16, find the values of p and q.
(b) Find the limit of this recurrence
'"
5.
Given that f(x)
relation as n ~
5
= -JX + x;, findf'(4).
6. A and B are the points (-1, -3, 2) and
A
(2, -1,1) respectively.
Band C are the points of trisection of
AD, that is AB = BC = CD.
Find the coordinates of D.
7.
2
00.
Show that the line with equation
with equation y = x2 + 3x + 4.
y
= 2x
3
+ 1 does not intersect
the parabola
5
t
8.
Find
4
Io(3xdx+ l)t .
9. Functions f(x) =-.L
x-4
and g(x)
= 2x + 3 are
defined on suitable domains.
=f(g(x)).
2
(b) Write down any restriction on the domain of h.
1
(a) Find an expression
for hex) where hex)
[Turn over for Questions
[Xl 00/301]
Page three
10 to 12 on Page four
'"
10. A is the point (8, 4). the line OA is inclined
Marks
y
at an al'lgle p radians to the x-axis.
(a) Find the exact values of:
A(8,4)
(i) sin(2p);
(ii) cos(2p).
x
0
5
~
The line OB is inclined at an angle 2p radians
to the x-axis.
(b) Write down the
gradient of OB.
exact
val'ue
y
B
of' the
1
"'
x
.~
11.
~
. 0, A and B are the centres of the three circles shown in the diagram below.
. The
.
two outer circles are congruent
and each touches the smallest circle.
Circle centre A has equatiopJx -12)2 -t",(y+",5)2=,25.
. The three centres lie on a parabola whose axis of symmetry is shown by tbe
broken line through A.
y
"
x
(a)
(i) Shte the coordinates
2
3
of A and find the length of the line OA.
(ii) Hence find the equation of the circle with centre B.
(b) The equation of the parabola can be written in the form y
Find the values of p and q.
=px(x
+ q).
2
\
12.
Simplify 3loge(2e) - 2loge(3e) expressing your answer in the form
A + loge B - loge C where",A, B andC are whole numbers.
[END OF QUESTION
iii
rX1'OO/3011
Pafle four
PAPER]
4
'"
I
X1'OO/3Q3
- ~-~->~= - C'~c..,
NATIONAL
QUALIFICA
2{)03
r
~~>
-_.~-~-WEDNESDAY,
10.30 AM
TIONS
-
21 MAY
12.00 NOON
~-~
,,-
_0
~~-
-~-~
MATHEMATICS
HIGHER
Units 1,2 and 3
Paper L
11'
Read,Car~fu'ily
0
1 Calculators may be usediii;l,tlilispaper.
2
Fullcreditwill be giVenonlywherethe solutioncontainsappropriateworking.
3 Answersobtainedby readingsfromscaledra)/Vings
will not receiveanycredit.
III
IiI
r,
!>
\v[
SCOTTISH./"'QUALIFICATIONS
AUTHORITY
LI B X 100/303
6/3/26570
l'IIIII.IIIIII:IIIIIIIIII.IIII.I!II;III!IIII~lllljllllIIII
@
ALL {'}.uestions should
be attempted'.
Marks
1. f(x)::::6x3 - 5x2 - 17x + 6.
(a) Show that (x - 2) is a factor off(x).
4
(b) Expressf(x) in its fully factorised form.
2.
The diagram s9-°ws a sketch of part
of the graph of a trigonometric
function whose equatioFl is of' the
fprm y = a sin(bx) + c.
Determine
- - ~- J - - - - - - - -~ - - - - - - - - - - - - -
5
- - - - - - - - - - - - - - - - -
3
the values of a, b aQd c.
x
- - - - ~- - - - - - - - - - -- - - - -- "--
-- --
-3
3.
The incomF>letegraphs off(x)
GC
x2'f- 2x
y
and g(x) = x3 - x2"",;6xare shown in the
diagram.
The
grawhs iEl}[email protected]>rseCF
at
A( 4, 24) and the origip.
FiEld the shaded area enclosed between
the curves.
5
x
y
4.
(a) Find
y
the
equation
= x3 + 2X2 -
(b) Show
that
of
the
tangent
to
the
curVe
= g(x)
with
equation
5
3x + 2 at the point where x::::1.
this
line
is also
x2 + y2 - 12x - 10y + 44
=
a tangent
0 and
state
to the
circle
the coordinates
contact.
with
equation
of the point
of
6
-
~'fu'rfl over
[X100/303]
Page three
Marks
5.
The diagram
a functionj.
y
shows the graph of
y = f( x)
f has a minimum turning point at
(0, -3) and a point of inflexion at
(-4, 2).
(a) Sketch the graph of y
=f(-x).
x
(b) On the same diagram, sketch
the graph of y = 2f(-x).
6.
7.
2
2
-3
4
Hf(x) = cos(2x)
- 3 sin(4x), find the exact value of 1'( ~).
'"
Part of the graph of
y = 2sin(x O) + Scos(x O)is shown in
the diagram.
y
y
= 2sin(x
O)+ Scos(x O)
(a) Express y = 2sin(xO) + Scos(XO)
in the form ksin(x o + a O)where
k > 0 and 0:::;a < 360.
4
0
(b) Find the coordinates
of the
minimum turning point P.
8.
3
An open water tank, in the shape of
a triangular prism, has a capacity of
108 litres. The tank is to be lined
on the inside in order to make it
watertight.
~
.......
.
..:-::<:»::::::::::-::-:-..,.
The triangular cross-section of the
tank is right-angled
and isosceles,
with equal sides of length x cm.
The tank has a length of I cm.
(a) Show that the surface area to be lined, A cm2, is given by A(x) = x2 + 432000.
x
(b) Find the value of x which minimises this surface area.
-----
[XIOO/303]
Page four
3
5
Marks
9.
The.diagram shows v~ctors a and b.
If Ia I =!5, Ib I == 4 anda.(a + b) ==36, find tBe
size of the acute angle e between a and b.
Ut
Solve the equation 3cos(2x) + l'Ocos(x)
~
4"
1 = 0 for 0 ::;;x ::;;1t, correct to 2 decimal
5
places.
[
11.
(a)
(i) Sketch the graph of y ;::.ax+ l, a> 2.
2
(ii) On the same diagram, sketch'Phe graph of y = a~+l, a> 2.
(b) Prove
that
the graphs
intersect
at a point
where
is
3
).
loga ( a ~ 1
[END OF QUESTION PAPER]
"
f
/
[XlOO/303]
the x-cooEdinate
Page five