2007-8 prelim
1.
Consider the isosceles triangle and the rectangle below.
The triangle has a base measuring 2x and a vertical height of x  k .
The rectangle has dimensions 2k  2 by x as shown.
All dimensions are in centimetres.
2k  2
xk
x
2x
(a)
Given that the area of the rectangle is 4cm2 more than the area of the
triangle, show clearly that the following equation can be formed.
x 2  (2  k ) x  4  0
2.
3
(b)
Hence find k, given that the equation x 2  (2  k ) x  4  0 has equal roots and
3
k  0.
(c)
Find x when k takes this value and calculate the area of each shape.
In the diagram A has coordinates (3,9) and the point
B has coordinates (3,-11) as shown.
A lies on the line with equation y  3 x .
(a)
A(3,9)
If line BC is perpendicular to the line
AD, establish the equation of BC.
(b)
Hence find the coordinates of D.
(c)
If D is the mid-point of BC, write
down the coordinates of C.
2
o
Find the equation of the circle passing
through the points A, D and C.
x
3
1
C
(d)
y
3
D
B(3,-11)
4
3.
A function is defined on a suitable domain as f ( x)  13 x 3  4 x 2  x .
(a)
Show that its derivative can be expressed in the form
f ( x)  ( x  p) 2  q , and state the values of p and q.
(b)
4.
4
Hence state the minimum rate of change of this function and the
corresponding value of x.
2
Triangle ABC has vertices (-2,-2), (1,7) and (19,1) as shown.
M is the mid-point of side BC.
y
B(1,7)
M
D
o
C(19,1)
x
A(-2,-2)
(a)
Establish the equation of the median AM.
3
(b)
The horizontal line through C intersects AM at D.
Find the coordinates of D.
3
Hence show clearly that BD is perpendicular to AM.
3
(c)
5.
2
Part of the graph of the curve with equation y  x3  15
is shown below.
2 x  12 x  18
The graph is not drawn to scale.
y
Q
o
x
P
6.
(a)
Find the coordinates of the stationary point P.
4
(b)
Find the coordinates of Q.
3
Two functions are defined on a suitable domains as f ( x)  x 2  a and g ( x)  x  1 ,
where a is a constant.
(a)
Find the value of a given that f g (2)  1
2
(b)
Hence solve the equation f  f ( x)  2
5
7.
Two circles, both with the same radius, touch extenally at T as shown below.
The circle with A as its centre has equation x 2  y 2  4 x  2 y  15  0 .
Line L1 is the common tangent to both circles through T and has as its equation
y  2x  5 .
y
B
T
o
x
A
L1
8.
(a)
Find the coordinates of T, the point of tangency.
3
(b)
Find the coordinates of B and hence write down the equation of the other
circle in the diagram.
3
Consider the diagram below.
D
2
AB
CD
DE
AB
5
2 units
5 units
2 units
1 unit
BCD
BAE
E
x
y
1
x
y
A
2
B
C
(a)
Calculate the lengths of AE and BC.
(b)
Hence show clearly that
cos ( x  y)  55
2
4
9.
Find the solution(s) of the equation 2 cos 2 a  cos a  1 for 0  a   .
10.
An amateur rockateer has built a rocket which he hopes will reach a height of at least
4000 feet when using his own home made liquid fuel.
5
He has modelled the height reached to the mass of fuel used
by the formula
H (m)  4m 
m2
,
1200
where H is the height reached in feet and m is the mass of fuel used in millilitres
(ml).
(a)
Find the mass of fuel he should use to propel his rocket to its maximum
height.
(b)
11.
Given
What is the predicted maximum height for this rocket when m takes this
value?
f ( x) 
9
1  4x
where x  14 , find the value of f (1) .
4
1
4
2007-8 mini prelim
12.
Triangle ABC has vertices A(4,-1,2), B(13,2,-10) and C(15,3,-5) as shown.
Point D lies on side AB.
C(15,3,-5)
B(13,2,-10)
D
A(4,-1,2)
(a)
(b)
Given that D divides the line AB in the ratio 2:1, find the coordinates
of D.
3
Hence calculate the size of angle CDA.
5
2008-9 prelim
1.
Part of the graph of the curve with equation y  3x 2  x 3 is shown below.
The diagram is not drawn to scale.
y
Q
P
x
O
2.
(a)
Establish the coordinates of the stationary point P.
4
(b)
The horizontal line through P meets the curve again at Q.
Find the coordinates of Q.
3
Two functions, defined on suitable domains, are given as f ( x)  x 2  1 and
g ( x)  2  x .
(a)
Show that f ( g (a)) can be expressed in the form pa 2  qa  r and write
down the values of p, q and r.
4
(b)
3.
Hence find a if f ( g (a))  8 and a  0 .
2
The diagram below shows part of the graph of y  sin 2 x  1 , for 0  x   , and the
line with equation y  12 .
y
A
O
Find the coordinates of the point A.
x
4
4.
Solve algebraically the equation
3 cos 2 x   4 sin x   1  0
5.
(a)
for
0  x  360 .
( x  1) 2
If k  2
, where k is a real number, show clearly that
x 4
(k  1) x 2  2 x  (4k  1)  0 .
(b)
6.
5
3
Hence find the value of k given that the equation (k  1) x 2  2 x  (4k  1)  0
has equal roots and k  0 .
5
Consider the diagram below.
The circle centre C1 has as its equation ( x  4) 2  y 2  52 .
The point P(0, k) lies on the circumference of this circle and the tangent to this circle
through P has been drawn.
A second circle with centre C2 is also shown.
y
P(0,k)
( x  4)  y  52
2
2
d
C1
O
C2
x
(a)
What is the value of k?
2
(b)
Hence find the equation of the tangent through P.
4
(c)
The tangent through P passes through C2 the centre of the second circle.
State the coordinates of C2.
1
(d)
Given that the second circle has a radius of 8 units, calculate the distance
marked d on the diagram, giving your answer correct to 1 decimal place. 3
7.
The diagram below, which is not drawn to scale, shows part of the graph of the
curve with equation y  x 3  x 2  5 x  3 .
Two straight lines are also shown, L1 and L2.
y
Q
P
x
O
L1
T
L2
(a)
Find the coordinates of P.
2
(b)
Line L1 has a gradient of  32 and passes through the point P.
Find the equation of L1.
1
Line L2 is a tangent to the curve at the point T where x  2.
Find the equation of L2.
4
(c)
(d)
Hence find the coordinates of Q, the point of intersection of the two lines. 3
8.
The floor plan of a rectangular greenhouse is shown below. All dimensions are in
metres.
The gardener places a rectangular wooden storage shed, of width x metres, in one
corner.
4m
SHED
GREENHOUSE
FLOOR
(a)
9.
3m
Given that the area of the shed is 3 square metres, show clearly that the
area of greenhouse floor remaining, A square metres, is given in terms
of x as
A( x)  12  4 x 
(b)
xm
9
.
x
3
Hence find the value of x which minimises the area of the greenhouse floor
remaining, justifying your answer.
5
Angle A is acute and such that tan A  36 .
(a)
(b)
Show clearly that the exact value of sin A can be written in
the form 15 k , and state the value of k.
3
Hence, or otherwise, show that the value of cos 2 A is exactly 15
3
2008-9 mini prelim
10.
Consider the diagram below.
S(3,13,4)
P(0,15,-3)
Q
R(6,3,0)
11.
12.
(a)
Given that Q divides PR in the ratio 1 : 2 , find the coordinates of Q.
3
(b)
Hence prove that angle SQR is a right angle.
4
Find the coordinates of the point on the curve y  x 3  x 2  4 x  2 where the
gradient of the tangent is 1 and x  0 .
4
 4
 6 
 


The diagram shows two vectors a and b where a   0  and b    3  .
 2
 0 
 


The angle between the vectors is  .
a
b
4
(a)
Show clearly that cos  
(b)
Hence, or otherwise, find the exact value of cos 2 .
5
.
3
2
2009-10 prelim
1.
The diagram below, which is not drawn to scale, shows part of the graph of the
curve y  4 x 2  x 3 and the straight line y  3 x .
The line intersects the curve at three points.
y
A
L1
O
(a)
Find the coordinates of the point A.
(b)
The line L1 is the tangent to the curve at A.
Establish the equation of the line L1.
2.
x
4
4
Angle x is acute and is such that cos x  102 .
(a)
Show clearly that the exact value of sin x is 7102 .
3
(b)
Hence show that sin( x  4 )  0  8 .
4
3.
An equation in x is given as
5x
4

, where k is a constant and non-zero.
2
x k
x  k2
2
(a)
Show that this equation can be written as
x 2  5k 2 x  4k 2  0
(b)
4.
2
Hence find the two values of k for which the equation x 2  5k 2 x  4k 2  0
has equal roots.
4
PQRS is a rectangle measuring 6 units by 4 units.
Points A, B, C and D are points on the sides of the rectangle such
that AQ  SC  2x and PD  BR  x as shown.
A
P
2x
Q
x
D
B
x
S
(a)
2x
C
R
Show that the area of ABCD is given by the function
A( x)  4 x 2  14 x  24 .
(b)
Hence find the value of x which minimises the area of ABCD and
calculate this minimum area.
4
5
5.
Kite ABCD has two of its vertices at A(3,8) and B(7,14) as shown.
B(7,14)
y
A(3,8)
E
C
O
x
D
6.
(a)
Given that the equation of the longer diagonal BD is y  4 x  14 , find the
equation of the short diagonal AC expressing your answer in the
form ax  by  c  0 and write down the values of a, b and c.
4
(b)
Find the coordinates of E, the point of intersection of the two diagonals.
3
(c)
Hence establish the coordinates of C.
2
Two functions are defined on suitable domains as f ( x)  4 x  1 and g ( x) 
(a)
If h( x)  f ( g ( x)) , show clearly that h(x) can be written as
h( x ) 
(b)
1
.
x 1
x3
.
x 1
3
Show that value of h( 5 ) can be expressed in the form p  5 and write
down the value of p.
4
7.
A function is defined on the set of real numbers as f ( x)  2 x 3  3x 2  12 x  7 .
Part of the graph of y  f (x) is shown below.
y
Q
O
x
P
Find the coordinates of the stationary points P and Q.
8.
3
Solve algebraically the equation
15 sin 2 x   10 cos x 
for
0  x  360 .
5
9.
An ice-cream manufacturer has decided on a new logo for her company.
It consists of a triangle and two circles representing
a wafer cone and two balls of ice cream.
Placed on a set of rectangular axes the logo is modelled in the diagram below.
The triangle has coordinates P(3, 6), Q(4,6) and R(8,2).
A is the midpoint of QR.
y
C
Q(4,6)
B
A
R(8,2)
O
x
P(3, 6)
(a)
Find the equation of PA
(b)
When PA is extended it intersects with the larger circle at B and C.
(c)
3
If the larger circle has as its equation x 2  y 2  10 x  20 y  105  0 ,
find the coordinates of C.
4
Given that C is the centre of the smaller circle and that its radius is
exactly half of the larger circle, find the equation of the smaller circle.
3
10
A curve has as its equation y  x 3  kx2  16 x  32 .
Part of the graph of this curve is shown below.
The diagram is not drawn to scale.
y
B( p,35)
A(2,0)
O
x
(a)
If the curve crosses the x-axis at A(2,0), find k.
3
(b)
The point B( p,35) also lies on this curve, find the value of p.
3
(c)
Calculate the size of the angle between the line AB and the x-axis in the
positive direction. Give your answer to the nearest degree.
2
2009-10 mini prelim
11
A function is defined on a suitable domain as f ( x) 
(a)
Show clearly that the derivative of this function can be written in the form
f ( x) 
(b)
12
 16
.
(2 x  1) 2
k
(2 x  1) n
and write down the values of k and n.
4
Hence find x when f ( x)  1 and x  0 .
3
In the diagram below A, B and C have coordinates (4 , 0 ,13) , (6,  3 , 4) and
(6 ,1,12) respectively.
P lies on BC and has coordinates (3 , 0 , k )
C
P
B
A
(a)
Find the value of k.
3
(b)
Hence calculate the size of angle APB.
5
2010-11 prelim
1.
The diagram below, which is not drawn to scale, shows part of the curve with
equation y  2 x 3  px 2  12 x , where p is a constant.
A is a stationary point and has –1 as its x-coordinate.
y
y  2 x 3  px 2  12 x
A
–1
1
o
x
C
B
(a)
(b)
(c)
By considering the derivative of y, and using the x-coordinate of
point A to help you, find the value of p.
4
Establish the coordinates of B the other stationary point.
(all relevant working must be shown)
4
The point C on the curve has 1 as its x-coordinate.
Find the equation of the tangent to the curve at C.
3
2.
Solve the equation 4 cos 2  6 cos   1 , for 0    2 .
3.
(a)
If x  1 is a factor of 3 x 3  kx 2  4 x  13 , find the value of k.
3
(b)
Hence find the x-coordinate of the single stationary point on the
curve with equation y  3x 3  kx2  4 x  13 when k takes this value.
4
6
4.
In the diagram below, which is not drawn to scale, triangle ABC is isosceles
with AB  AC.
D is the mid-point of BC. AB  12 units and AD  2 units as shown.
Angle BAD  x .
D
B
C
x
A
5.
2.
3
(a)
Show clearly that sin x 
(b)
Hence show that sin BAC 
2 2
.
3
3
3
Two functions, defined on suitable domains, are given as
f ( x)  3 px 
1
2
p and g ( x)  x(3 px  2) , where p is a constant.
(a)
Show clearly that the composite function f ( g ( x)) can be written in the form
f ( g ( x))  ax 2  bx  c , and write down the values of a, b and c
in terms of p.
4
(b)
Hence find the value of p, where p  0 , such that the equation
f ( g ( x))  0 has equal roots.
3
6.
In the diagram below triangle ABC has two of its vertices as B( 7,18 ) and C( 11, – 2).
M is the mid-point of BC. The line AM crosses the y-axis at ( 0, 5).
BN is an altitude of the triangle.
y
B( 7,18 )
M
5
A
O
N
C( 11,
7.
x
)
(a)
Find the equation of the median AM.
3
(b)
Given that the equation of side AB is y  x  11 , establish the coordinates
of vertex A.
3
(c)
Hence find the equation of the altitude BN.
3
The cost of laying one mile of service piping to a wind farm is
estimated by means of the formula
C 
16200
 450a ,
9a
where C is the cost in tens of pounds and a is the cross-sectional
area of the tube in square inches.
What cross-sectional area is the most economical to use?
5
8.
The diagram below, which is not drawn to scale, shows part of the curve with
equation y  x 3  11x 2  28 x and the line y  4 x .
The line and the curve intersect at the origin and the point P.
The curve also crosses the x-axis at Q.
y
P
Q
O
x
Find the coordinates of P and Q.
9.
A formula is given as
(a)
E  sin 2  
5
1
2
sin   1 for 0   
Express E in the form E  (sin   p) 2  q and write
down the values of p and q .

.
2
2
(b)
Hence, or otherwise, state the minimum value of E and the corresponding
replacement for  . Give your answer correct to 2 decimal places.
3
10.
The circle, centre S, has as its equation x 2  y 2  16 y  12  0 .
T( p,  12 ) is a point of tangency.
y
R
O
x
S
T( p,
(a)
)
Find the value of p, the x-coordinate of T.
2
(b)
Write down the coordinates of S, the centre of the circle.
1
11.
(c)
Find the equation of the tangent through T and hence state the coordinates
of R.
4
(d)
Establish the equation of the circle which passes through the points S, T
and R.
A function, defined on a suitable domain, has as its derivative f ( x)  3 x 2 
3
10
.
x2
(a)
Given that f (2)  3 , find f (x) .
5
(b)
Hence find f (1) .
1
2010-11 mini prelim
12.
Triangle ABC has vertices A( 1, 0 ,  3 ), B( 5,  4 ,  1 ) and C( 4,  16 , 4 ) respectively.
A, B and D are collinear such that
A( 1, 0 ,
AB 2
 .
BD 3
)
B( 5,
,
)
D
C( 4,
13.
14.
,4 )
(a)
Find the coordinates of D.
2
(b)
Hence show clearly that angle ADC is a right angle.
4
(c)
Prove that angle ABC is obtuse.
3
Given that ( x  1) and ( x  3) are both factors of 2 x 3  5 x 2  ax  b ,
find a and b.
(a)
Given that y  3 (sin 2 x  cos 2 x) , show clearly that
dy
 3 3 sin 2 x  .
dx
(b)
4
Hence find the gradient of the tangent to the curve
y  3 (sin 2 x  cos 2 x) at the point where x  6 .
3
2
2011-12 prelim
1.
The circle C1 has P( 1 , 3 ) as its centre and a radius of
5 units.
The circle C2 has as its equation x 2  y 2  18x  14 y  85  0 .
y
C2
Q
C1
P
O
(a)
(b)
2.
x
Find the coordinates of Q, the centre of C2, and the radius of
this larger circle.
3
Show clearly that C1 touches C2 at a single point.
3
A, B and C have coordinates (  4 ,  3 ), (  2 , 5 ) and ( 10 , 9 ) respectively as shown.
S is the mid-point of BC.
C( 10 , 9 )
S
B(
A(
,
, 5)
)
(a)
Find the equation of the line through S parallel to AB.
4
(b)
Find the coordinates of the point D where ABSD is a parallelogram.
2
3.
Three functions, defined on suitable domains, are given as
f ( x)  sin x , g ( x)  x 2 and h( x)  1  2 x .
(a)
(b)
4.
Show clearly that the function k, where k ( x)  hg ( f ( x))  , can
be written in its simplest form as k ( x)  cos 2 x .
3
5  .
Hence find the value of k 12
3
The line x  3 y  10 is a tangent to the circle with equation
x 2  y 2  4 x  8 y  20  0 at the point P.
A second line with equation y  kx  4 also passes through P.
Find the value of k, the gradient of this second line.
5.
(a)
7
A function f, defined on a suitable domain, is given as f ( x)  ( x  1) 2 .
A second function h is such that h( x)   f ( x  3) x 2 .
Show clearly that h can be written in the form h( x)  x 4  8 x 3  16 x 2 .
(b)
3
Part of the graph of y  h(x ) is shown below.
y
A
O
Find the coordinates of point A.
x
5
6.
Solve algebraically the equation
5 sin 2 x   4 sin x   0
7.
for 0  x  360 .
5
Part of the graph of y  x 3  6 x 2  12 x  8 is shown in the diagram.
y
P
O
x
Find the coordinates of P.
8.
3
Consider the diagram below. Angle ABC  angle DBA  p .
Triangle ACB is right-angled with BC equal to 3 and CA equal to 1 unit.
D
A
1
p
p
B
3
Show clearly that the exact value of cos DBC is 54 .
C
5
9.
The diagram shows two concentric circles with centre C( 2 , k ).
The larger of the two circles has the line with equation y  x  11 as a tangent.
The point P(  4 , 7 ) is the point of tangency between this line and the circle.
y
P(
, 7)
Q
C( 2 , k )
O
(a)
(b)
10.
x
By considering gradients, find the value of k, the y-coordinate of
the point C.
3
Hence find the equation of the smaller circle given that Q is the mid-point
of PC.
3
The tangent to the curve y  x 
p
, at the point where x  4 , is parallel
x
to the line with equation x  y  10 .
Find the value of p.
5
11.
From a square sheet of metal of side 30 centimetres, equal squares of side x
centimetres are removed from each corner.
The sides are then folded up and sealed to form an open cuboid.
30cm
30cm
x
x
(a)
Show that the volume of this resulting cuboid is given by
V ( x)  4 x 3  120 x 2  900 x .
(b)
(c)
12.
3
If the cuboid is to have maximum possible volume, what size of
square should be removed from each corner?
5
How many litres of water would this particular cuboid hold?
1
A designer is testing two model racing cars along a straight track.
Each car completes a single run and the following information is recorded.
(a)
Speed
Distance
Car A
kx
3
Car B
k
4x
Given that both cars completed the run in exactly the same time, show
clearly that the following equation can be constructed.
4 x 2  4kx  3k  0
(b)
(c)
3
Find the value of the constant k if the equation 4 x 2  4kx  3k  0 has
equal roots and k  0 .
3
Hence find x when k takes this value.
2
2011-12 mini prelim
13.
In the diagram P, Q, and R have coordinates P( 3 , 4 ,  1 ), Q( 0 , 6 ,  6 ) and
R( k , 8 ,  10 ) respectively.
Q( 0 , 6 ,  6 )
P( 3 , 4 ,  1 )
S
R( k , 8 ,  10 )
14.
(a)
Given that angle PQR is a right-angle, find the value of k.
4
(b)
Calculate the size of angle RPS where S is the mid-point of QR.
6
A function is defined on a suitable domain as h( x)  2 sin 2 x  3 cos 2 x .
Calculate the rate of change of this function at the point where x  3 .
5
2012-13 prelim
1.
Two functions are defined on suitable domains as
2( x 2  1)
.
f ( x)  2 x  1 and h( x) 
k
(a)
Find f ( f ( x)) in its simplest form.
(b)
Hence show clearly that the equation h( x)  f ( f ( x)) can be written as
2 x 2  4kx  (3k  2)  0
(c)
2.
3.
Find the two possible values of k so that h( x)  f ( f ( x)) has
equal roots.
2
3
5
A( 1 , 2 ), B( 7 , 6 ) and C(  7 , 14 ) are the vertices of a triangle.
(a)
Show clearly that triangle ABC is right-angled at A.
3
(b)
M is the mid-point of AC. Find the equation of the median BM.
3
(c)
If the median BM is extended it meets the line through C, with gradient 5,
at P. Find the coordinates of P.
4
A function is given as f ( x)  x 3  4 x  14 .
(a)
If f ( p)  14 , find the value of p if p  0 .
2
(b)
Prove that the function is increasing when p takes this value.
3
4.
The diagram shows the graph of y  a sin x  b for 0  x  2 .
y
P
Q
x
O
(a)
Write down the values of a and b (where a is a whole number).
(b)
Determine the exact x-coordinates for the points P and Q.
Your answers must be accompanied by the appropriate working.
5.
3
The graph of the curve with equation y  2 x 3  3x 2  3x  a crosses the x-axis at
the point (  1 , 0 ).
(a)
Find the value of a.
(b)
Hence write down the coordinates of the point at which this curve crosses
the y-axis.
1
(c)
This curve also crosses the x-axis at a further two points.
Find algebraically the coordinates of these other two points.
6.
2
2
4
Solve algebraically the following system of equations for 0  x  90  .
y  3 cos 2 x 
y  1 10 cos x 
5
7.
Part of the graph of y  x 3  3x 2  9 x  5 is shown in the diagram below.
The curve crosses the x-axis at the points (  1 , 0 ) and ( 5 , 0 ) as shown.
y
5
x
O
A
Find the coordinates of the stationary point A.
8.
4
Angle A is such that tan A  1 and both A and 2A are acute.
5
(a)
(b)
Show clearly that the exact value of sin 2 A is 35 .
3
Given now that cos 2 A  23 , show clearly that sin( 3  2 A) has an exact
value of 16 2 3  5 .
4


9.
The diagram below shows a circle, centre C, with equation
x 2  y 2  16 x  4 y  12  0.
The point Q( 4 , 6 ) lies on the circumference of the circle.
The line PQ is a tangent to the circle.
y
P
Q( 4 , 6 )
x
C
(a)
Find the equation of the tangent PQ.
4
(b)
Write down the coordinates of P.
2
(c)
Establish the equation of the circle which passes through the points P, Q
and C.
5
10.
An underground storm drain has a cross-section in the shape of a rectangle with a
semi-circular base.
The rectangle part of the drain measures 2x metres by h metres as shown.
2x
h
(a)
Show clearly that if the perimeter of the drain is approximately 3 8 metres
then h can be expressed as
h  1  9  x  12 x .
(b)
Hence show that the cross-sectional area, A, in terms of x, can be written as
A( x)  3  8x  2 x 2  12 x 2 .
(c)
2
2
Find the value of x which will produce the largest cross-sectional area.
Justify your answer.
5
2012-13 mini prelim
11.
In the diagram below PQRS is a parallelogram.
Points P, Q and R have coordinates (  8 , 0 , 5 ), ( 3 ,  2 , 12 ) and ( 7 , 6 , 6 )
repectively as shown.
T has coordinates ( 9 , 10 , k ).
T( 9 , 10 , k )
S
R( 7 , 6 , 6 )
P(  8 , 0 , 5 )
12.
Q( 3 ,  2 , 12 )
(a)
Establish the coordinates of S
1
(b)
Given that Q, R and T are collinear, find the value of k.
3
(c)
Calculate the size of angle TSQ.
5
Part of the graph of y  2  x 2  13 x 3 is shown below.
y
A
O
Find the coordinates of the point A.
x
5