### 2003 Higher Paper 2

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X 1 00/303
NA TIONAL
QUALIFICA
2003
TIONS
WEDNESDAY,
21 MAY
10.30 AM - 12.00 NOON
MATHEMATICS
HIGHER
Units 1,2 and 3
Paper 2
L
1 Calculators may be used in this paper.
2 Fullcreditwillbe givenonlywherethe solutioncontainsappropriateworking.
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QUALIFICATIONS
AUTHORITY
LIB
X100/303
613/26570
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ALL questions
= 6x3 -
1. lex)
should
be attempted.
Marks
5x2 - 17x + 6.
(a) Show that (x - 2) is a factor off(x).
(b) Expressf(x)
4
in its fully factorised form.
2. The diagram shows a sketch of part
y
of the graph of a trigonometric
function whose equation is of the
form y = a sin(bx) + c.
Determine
~-- -
-----------
5
3
the values of a, band c.
~
x
-3
3.
The incomplete graphs off(x)
= x2 + 2x
y
and g(x) = x3 - x2 - 6x are shown in the
diagram.
The
graphs
intersect
at
A( 4, 24) and the origin.
Find the shaded area enclosed between
the curves.
5
x
y=g(x)
"-'
- -
4.
(a) Find
y
= x3
the
+ 2x2-
equation
of
the
3x + 2 at the point
tangent
where
x
to
the
curve
with
equation
5
= 1.
(b) Show
that this line is also a tangent to the circle with equation
x2 + y2 - 12x - 10y + 44 = 0 and state the coordinates
of the point of
contact.
6
[Turn over
[XI00/303]
Page three
-
Marks
s.
The diagram
a function j.
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).
2
2
-3
6.
4
Hf(x) = cos(2x) - 3 sin(4x), find the exact value of f'( ~).
...JJ
7.
Part of the graph of
y = 2sin(x O) + Scos(x O) is shown in
the diagram.
(a) Express y
= 2sin(x
y
y
= 2sin(x
O) + Scos(x O)
O) + Scos(x O)
in the form ksin(x °+ a O)where
k > 0 and 0 ~ a < 360.
4
0
(b) Find the coordinates
of the
minimum turning point P.
8.
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.
3
~
.......
.:-:<::<..:>:>:::-:.:..
.
~
.
)
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 l cm.
(a) Show that the surface area to be lined, A cm2, is given by A(x) = x2 + 43,2000.
x
(b) Find the value of x which minimises this surface area.
[XlOOj303]
Page four
~
3
5
Marks
9. The diagram shows vectors a and b.
If Ia I = 5, Ib I = 4 and a.(a + b) = 36, find the
size of the acute angle e between a and b.
10.
11.
Solve the equation
places.
3cos(2x) + 10cos(x) -1
4
= 0 for
0:::; x:::;It, correct to 2 decimal
5
(i) Sketch the graph of y = ax + 1, a> 2.
(a)
2
(ii) On the same diagram, sketch the graph of y = ax+l, a> 2.
(b) Prove
tL.
loga
that
the graphs
intersect
at a point
~
--
--
the x-coordinate
is
3
( a ~ 1).
[END OF QUESTION
[X100j303]
where
Page five
PAPER]
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