Mathematics
Higher Prelim Examination 2013/2014
Paper 2
Assessing Units 1 & 2
NATIONAL
QUALIFICATIONS
Time allowed - 1 hour 10 minutes
Read carefully
1. Calculators may be used in this paper.
2. Full credit will be given only where the solution contains appropriate working.
3. Answers obtained from readings from scale drawings will not receive any credit.
FORMULAE LIST
Circle:
The equation x 2  y 2  2 gx  2 fy  c  0 represents a circle centre ( g ,  f ) and radius
The equation (x  a )2  ( y  b)2  r 2 represents a circle centre ( a , b ) and radius r.
Trigonometric formulae:
Scalar Product:
sin  A  B 
cos  A  B 
sin 2 A
cos 2 A






sin Acos B  cos Asin B
cos Acos B  sin Asin B
2sin Acos A
cos 2 A  sin 2 A
2 cos 2 A  1
1  2 sin 2 A
.
a b  a b cosθ, where θ is the angle between a and b.
or
 a1 
 b1 
 
 
a b  a1b1  a2 b2  a3 b3 where a   a 2  and b   b 2 
a 
b 
 3
 3
.
Table of standard derivatives:
f (x )
sin ax
cos ax
Table of standard integrals:
f (x )
a cos ax
 a sin ax
 f (x)
f (x )
sin ax
cos ax

dx
1
cos ax  C
a
1
sin ax  C
a
g2  f 2  c .
ALL questions should be attempted
1.
In the diagram triangle ABC has vertices A( 8 , 4 ), B( 20 , p ) and C( 2 , 17 ) as shown.
AD is perpendicular to side BC.
y
C( 2 , 17 )
D
B( 20 , p )
A( 8 , 4 )
O
2.
x
(a)
Given that the gradient of side BC is  12 , find the value of p.
3
(b)
Find the equation of the altitude AD
2
(c)
By considering the gradients of side AB and the altitude AD, calculate the size
of the shaded angle DAB.
3
In the diagram below the triangle and the rectangle have equal areas.
2p
x
2 x
2x + p
(a)
(b)
Show clearly that the following quadratic equation, in x, can be constructed from
the information given.
x 2  (2 p  2) x  p 2  0
4
For what value of p does the above quadratic equation have equal roots?
4
3.
The diagram below shows parts of the graphs of y  x 2  3 x  4 and y  4  3 x 2  x 3 .
The curves intersect on the y-axis and at the point with x-coordinate a.
y
y  x 2  3x  4
y  4  3x 2  x 3
4.
x
a
O
(a)
Find, algebraically, the value of a.
4
(b)
Hence calculate the area enclosed between the two curves.
4
The diagram shows part of the quartic with equation y  g( x).
There are stationary points at x  2, x  0 and x  a.
y
y  g( x)
a
2
0
x
On separate diagrams sketch the graph of
(a)
y  g( x).
2
(b)
y  g( x  3).
2
5.
The diagram below shows part of the curve with equation y  2 x 3  5 x 2  4 x  3.
A
y
y  2 x 3  5 x 2  4 x  3.
O
B
6.
x
(a)
Find the coordinates of the stationary point marked A.
4
(b)
Find the coordinates of B, one of the points where the curve crosses the x-axis.
2
A circle has a radius of 10 units and the point (  3 , 2 ) as its centre.
(a)
Write down the equation of this circle.
1
(b)
Given that the point P( 5 , k ) lies on this circle, find k where k  0 .
4
(c)
Find the equation of the tangent to this circle at the point P.
4
(d)
Show clearly that this tangent passes through the centre of the circle with
equation x 2  y 2  4 x  16 y  43  0 .
2
7.
A closed wooden box, in the shape of
a cuboid, is constructed from a sheet of
h cm
wood of area 600 cm 2 .
The base of the box measures 2x cm by x cm.
x cm
The height of the box is h cm.
2x cm
(a)
Show that the volume (in cubic centimetres) of the box is given by
4
3
V ( x)  200 x  x 3
(b)
8.
3
Calculate the value of x for which this volume is a maximum.
Rectangle ABCD measures
5
6 units by 1 unit as shown. The diagram is not drawn to scale.
Angle BAC  p radians. Triangle BCE is isosceles with BC  BE .
6
D
C
E
1
p
A
B
(a)
Show clearly that  CBE  2 p .
3
(b)
Hence calculate the exact value of cos CBˆ E .
4
[ END OF QUESTION PAPER ]