Mathematics
New Higher Prelim Examination 2014/2015
Paper 1
Assessing Units 1, 2 and 3
Time allowed -
NATIONAL
QUALIFICATIONS
1 hour 10 minutes
Read carefully
Calculators may NOT be used in this paper.
1. Full credit will be given only where the solution contains appropriate working.
2. Answers obtained by readings from scale drawings will not receive any credit.
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New Higher All Units
FORMULAE LIST
Circle:
The equation
represents a circle centre
The equation
and radius
represents a circle centre ( a , b ) and radius r.
Trigonometric formulae:
Scalar Product:
or
Table of standard derivatives:
Table of standard integrals:
dx
All questions should be attempted
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New Higher All Units
.
1.
2.
(a)
Show that (x – 2) is a factor of
(b)
Write
4.
2
.
.
2
Find the equation of the line which is perpendicular to 3y + 7x +5 = 0 and passes through
the point (2, – 1).
A function is defined on a suitable domain as
Calculate the rate of change of
5.
3
in its fully factorised form.
A function f is defined as
Find the value of
3.
.
.
when x = – 2.
The vectors a and b have components a =
3
3
and
b=
If a and b are perpendicular, calculate the value of z.
2
y
6.
Part of the graph of the function
shown opposite.
is
Make a sketch of the diagram and on the same diagram,
sketch the graph of the related function
3A
o
x
-4
B
showing clearly the images of points A and B.
3
3
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-1
New Higher All Units
7.
Evaluate
4x(x² – 1) dx .
3
8.
For what value(s) of p does the equation
9.
The function f is defined as
have real roots?
has a stationary value when
4
.
Calculate the value of a.
2
10.
Given that
find the value of x where x > 0.
11.
Triangle ABC has B(8,11) as one of its vertices.
3
The line through A which meets side BC at D has as its equation
.
y
B(8,11)
A
D
x
O
C
(a)
If side BC has a gradient of
(b)
Establish the coordinates of D.
3
(c)
If D is in fact the mid-point of side BC, write down the coordinates of C.
1
(d)
Show clearly that this triangle is right-angled at B.
3
(e)
Hence find the equation of the circle passing through the points A, B and C.
3
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, find the equation of BC.
2
New Higher All Units
12.
(a)
(b)
Express
and k > 0.
in the form
, where k and a are constants
4
Now given that a function h is defined as
, where
,
calculate the minimum value of h. and state the corresponding value
of x at which this minimum occurs.
13.
14.
A sequence is defined by the recurrence relation
p and q are constants.
(a)
Given that
(b)
Given that
,
and
3
, where
, find the value of p.
, calculate the value of
2
.
3
The small box below is in the shape of a cuboid.
2
The box has dimensions
,
and 2 as shown.
All the lengths are in centimetres.
The volume of the box is 60 cubic centimetres.
(a)
By forming an equation for the volume of the box, and simplifying it, show
that the following equation can be formed
.
(b)
2
Hence find the value of k, given that the above equation has equal roots.
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[ END OF QUESTION PAPER 1 ]
New Higher All Units
4
Mathematics
New Higher Prelim Examination 2014/2015
Paper 2
Assessing Units 1, 2 and 3
Time allowed -
NATIONAL
QUALIFICATIONS
1 hour 30 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
© Pegasys 2014
Circle:
New Higher All Units
1.
The diagram below, which is not drawn to scale, shows part of the graph of the
curve
.
y
A
B
x
O
(a)
(b)
Find the coordinates of the point A, the maximum turning point
of this curve.
4
The line through A, with gradient 5, intersects the curve at a further two
points, one of which is B.
Find algebraically the coordinates of B.
Your answer must be accompanied with the appropriate working.
2.
Solve algebraically the equation
, for
3.
4.
5
.
6
A scientist studying a large colony of bats in a cave has noticed that the fall in the population
over a number of years has followed the recurrence relation
, where
n is the time in years and 200 is the average number of bats born each year during a concentrated
breeding week.
(a)
He estimates the colony size at present to be 2100 bats with the breeding week just over.
Calculate the estimated bat population in 2 years time immediately before that
years breeding week.
2
(b)
The scientist knows that if in the long term the colony drops, at any time, below 700
individuals it is in serious trouble and will probably be unable to sustain itself.
Is this colony in danger of extinction?
Explain your answer with words and appropriate working.
A curve has as its derivative
(a)
.
Given that the point ( ,16 ) lies on this curve, express y in terms of x.
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4
New Higher All Units
4
5.
(b)
Hence find p if the point T(
, p) also lies on this curve.
(c)
Find the equation of the tangent to this curve at T.
1
3
On a recent survey mission to Mars an orbiting probe deployed four beacons
onto the surface of the planet. All four beacons were set down within the
same crater. The orbiting probe then activated locating signals for each
beacon.
Their positions relative to each other, within a three dimensional
framework, are shown below.
C( k ,
B( 8,
A( 2,
,
,
)
)
)
D(
,
, )
(a)
Given that A, B and C are collinear, find the value of k.
4
(b)
If the actual distance between A and B is 42 kilometres, how far
away from A is C?
2
Show clearly that angle ABD is obtuse.
3
(c)
6.
,
The diagram below shows a sketch of part of the graphs of
and
.
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New Higher All Units
The curves intersect at the point (0,3) and at A.
The dotted line shown is parallel to the x-axis and passes through (0,3) and A.
y
3
A
o
x
(a)
Establish the coordinates of the point A.
2
(b)
Hence calculate the area enclosed between the two curves.
5
7.
Solve
, for t, giving your answer correct to 2 significant figures.
8.
Calculate the gradient of the tangent to the curve
when
4
.
4
k
9.
(a)
Given that
show clearly that the exact value of k is
.
0
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New Higher All Units
4
(b)
10.
Hence show that
value of p.
can be written in the form
and state the
2
For the cylinder below the sum of its height h and its diameter d is 24 centimetres.
d
h
(a)
(b)
Write down expressions for r, the radius of the cylinder, and h, the height of
the cylinder, in terms of d.
1
Hence show clearly that the volume V of the cylnder can be expressed as a
function in terms of d as
2
(c)
(d)
Find the values of d and h for which the volume of this cylinder will be
a maximum and calculate this maximum volume. Justify your answer.
5
If h + d = N, express V in terms of N and d . Hence show that the maximum
volume always occurs when
.
3
[ END OF QUESTION PAPER 2 ]
New Higher Prelim Paper 1 2014/2015
Give 1 mark for each
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Marking Scheme
Illustration(s) for awarding each mark
New Higher All Units