Past Paper Questions - 2001 - 2008 – by topic
1.
Index
Decimals, Fractions & Percentages
Decimal, Fractions, Various calculations ............................................................................................... 3
Using Percentages ................................................................................................................................. 4
Reversing the Change (Price without VAT, Original price of cars before depreciation) .................................... 4
Standard Form calculations .................................................................................................................... 5
2.
Algebra 1 – Basic algebraic operations
Evaluation
Simplification, removing brackets, FOIL, squares
Factorisation, common factor, difference of two squares, quadratic (trinomial)
Solving linear equations
Simultaneous Equations
Functions, evaluating, finding values
Quadratic equations – using factorisation, using the formula
Inequalities – solving
Changing the subject of the formula
Algebraic Fractions – simplifying, common denominator
Algebraic fraction equations – solving .................................................................................................. 7
Indices and Surds ........................................................................................................................... 8 - 9
3.
Data Handling
Simple Probability .............................................................................................................................. 10
Probability from relative frequency ...................................................................................................... 10
Statistical Diagrams ...................................................................................................................... 11 - 12
Standard Deviation ....................................................................................................................... 12 – 13
4.
Area & Volume
Volumes of Cuboids, Cylinders and Prisms ................................................................................. 14 - 16
5.
Similar Shapes & Similar Triangles
Similar Shapes – Area and volume scale factors
Similar Triangles .......................................................................................................................... 17 – 18
6.
Pythagoras
Using Pythagoras in circles (oil tanker)
Converse of Pythagoras ................................................................................................................ 19 – 21
7.
Circle
Area of sector, arc length, angle of sector .................................................................................. 22 - 23
8.
Trigonometry – SOH-CAH-TOA
Calculating sides and angles in right angled triangles ........................................................................ 24
9.
Trigonometry – Non-right angled triangles
Using sine rule, cosine rule, area of triangle ................................................................................. 25 – 29
10.
Gradients and the Straight Line
Finding gradients, equations of a line
Applications, graphs, line of best fit .............................................................................................. 30 – 33
-1-
11.
Simultaneous Equations
Making and solving simultaneous equations ............................................................................... 34 - 35
12.
Functions and the Parabola (Quadratic)
Properties of the parabola
Applications, using quadratic equations for modelling ................................................................. 36 - 38
13.
Making and Using Formulae
Modelling using formulae
Substituting into formulae, making formula from information in tables
Making and using formulae derived from geometric shapes .......................................................... 39 – 44
14.
Trigonometry – Graphs and Equations
Graphs, triangles, maxima and minima
Solving Trigonometric equations .................................................................................................. 45 – 46
15.
Ratio & Proportion
Working with simple ratios ................................................................................................................. 47
16.
Variation & Proportion
Making proportionality statements, inverse, direct and joint variation
Making equations, finding constants of proportionality
Using equations to find different values
Halving and doubling .......................................................................................................................... 48
17.
Distance, Speed & Time
Calculations ....................................................................................................................................... 49
18.
Sequences
Working with sequences ..................................................................................................................... 50
-2-
Decimals, BODMAS, Fractions & Percentages
2008 – Paper 1
1.
Evaluate
24.7 - 0.63 × 30
2 KU
2007 – Paper 1
1.
Evaluate
6.04 + 3.72 × 20
2 KU
2.
Evaluate
3 16 ¸ 123
2 KU
56.4 - 1.25 × 40
2 KU
2006 – Paper 1
1.
Evaluate
2.
Evaluate
1 35 + 2 74
2 KU
3.8 - (7.36 ¸ 8)
2 KU
2005 – Paper 1
1.
Evaluate
2.
Evaluate
3.
Evaluate 12.5% of £140
2 13
5
6
+
of
1 25
3 KU
2 KU
2004 – Paper 1
1.
Evaluate
2.
Evaluate
6.2 - (4.53 - 1.1)
2
5
of
3 12
+
2 KU
4
5
3 KU
2003 – Paper 1
1.
Evaluate
2.
Evaluate
5.04 + 8.4 ÷ 7
2 KU
(1
2 KU
2
7
3
4
+
3
8
)
2002 – Paper 1
1.
Evaluate
2.
Evaluate
2 KU
7.18 - 2.1 × 3
118 ¸ 34
2 KU
3.1 + 2.6 × 4
2 KU
2001 – Paper 1
1.
Evaluate
2.
Evaluate
3 58 + 4 23
2 KU
-3-
Using Percentages
2008 – Paper 2
1.
A local council recycles 42 000 tonnes of waste a year.
The council aims to increase the amount of waste recycled by 8% each year.
How much waste does it expect to recycle in 3 years time?
Give your answer to three significant figures.
4 KU
2007 – Paper 2
1.
Alistair buys an antique chair for £600.
It is expected to increase in value at the rate of 4.5% each year.
How much is it expected to be worth in 3 years?
3 KU
2004 – Paper 2
4.
250 milligrams of a drug are given to a patient at 12 noon.
The amount of drug in the bloodstream decreases by 20% every hour.
How many milligrams of the drug are in the bloodstream at 3 pm?
3 KU
2003 – Paper 2
1.
Bacteria in a test tube increase at the rate of 0.6% per hour.
At 12 noon there are 5000 bacteria.
At 3 pm, how many bacteria will be present?
Give your answer to 3 significant figures.
4 KU
2001 – Paper 2
3.
In 1999, a house was valued at £90,000 and the contents were valued at £60,000.
The value of the house appreciates by 5% each year.
The value of the contents depreciates by 8% each year.
What will be the total value of the house and contents in 2002 ?
3 KU
Reversing the change
2008 – Paper 2
3.
In a sale, all cameras are reduced by 20%.
A camera now costs £45.
Calculate the original cost of the camera.
3 KU
2007 – Paper 2
5.
Mark takes some friends out for a meal.
The restaurant adds a 10% service charge to the price of the meal.
The total bill is £148.50
What was the price of the meal.
3KU
2006 – Paper 2
3.
Harry bids successfully for a painting at an auction.
An “auction tax” of 8% is added to his bid price.
He pays £324 in total.
Calculate his bid price.
-4-
3 KU
2004 – Paper 1
6.
Marmalade is on special offer.
Each jar on special offer contains 12.5% more
than the standard jar.
A jar on special offer contains 450 g of marmalade.
How much does the standard jar contain ?
3 KU
2002 – Paper 2
2.
A microwave oven is sold for £150.
This price includes VAT at 17.5%
Calculate the price of the microwave oven without VAT.
3 KU
Standard Form – Scientific Notation
2006 – Paper 2
1.
The orbit of a planet around a star is circular.
The radius of the orbit is 4.96 ´ 107 kilometres.
Calculate the circumference of the orbit.
Give your answer in scientific notation.
3 KU
2005 – Paper 2
1.
E = mc2
Find the value of E when m = 3.6 ´ 10-2 and c = 3 ´108 .
Give your answer in scientific notation.
3 KU
2004 – Paper 2
1.
Radio signals travel at a speed of 3 ´ 108 metres per second.
A radio signal from Earth to a space probe takes 8 hours.
What is the distance from Earth to the probe?
Give your answer in scientific notation.
4 KU
2002 – Paper 2
1.
A spider weighs approximately 19.06 × 10-5 kilograms.
A humming bird is 18 times heavier.
Calculate the weight of the humming bird.
Give your answer in scientific notation.
2 KU
2001 – Paper 2
1.
How many chocpops will be eaten in the year 2001.
Give your answer in scientific notation
-5-
2 KU
Algebra – Basic algebraic operations
2008 – Paper 1
5 x 2 - 45
2 KU
2.
Factorise fully
3.
Change the subject of the formula to H
5.
Express as a single fraction in its simplest form
13.
A new fraction is obtained by adding x to the numerator and denominator
of the fraction
W = BH 2
where
1
2
+
p p +5
17
2
. This new fraction is equivalent to .
24
3
2 KU
2 KU
Calculate the value of x.
3 RE
2007– Paper 1
P=
2 ( m - 4)
3
Change the subject of the formula to m.
4.
Given
5.
Remove the brackets and simplify
( 2 x + 3)
2
- 3 ( x2 - 6 )
3 KU
3 KU
2007 – Paper 2
2.
4.
Solve the equation
3 x 2 - 2 x - 10 = 0
Give your answer correct to 2 significant figures.
x 1
- <5
4 2
Solve the inequality
4 KU
2 KU
2006– Paper 2
4.
( x + 4 )( 3 x - 1)
(a) Expand and simplify
1 KU
2006 – Paper 1
5.
(a) Factorise
4x 2 - y 2
1 KU
4x - y
6x + 3 y
2
(b) Hence simplify
2
2 KU
2006 – Paper 1
6.
Solve the equation
x - 2 ( x + 1) = 8
3 KU
2005 – Paper 2
4.
Solve the equation
x2 + 2 x = 9
Give your answers correct to 1 decimal place.
3 KU
2005 – Paper 1
6.
Solve the equation
2
+1= 6
x
3 KU
-6-
2005 – Paper 1
11.
f ( x) = 4 x + 2
(a) Find the value of f (72) as a surd in its simplest form.
3 KU
(b) Find the value of t, given that f (t ) = 3 2 .
3 RE
2004 – Paper 1
3.
A = 2 x2 - y 2
4.
Simplify
Calculate the value of A when x = 3 and y = -4
3
4
+
m ( m + 1)
2 KU
3 KU
2003 – Paper 1
3.
4.
5.
3(2 x - 4) - 4(3 x + 1)
Simplify
3 KU
f ( x) = 7 - 4 x
(a) Evaluate f (-2)
1 KU
(b) Given that f (t) = 9, find t
2 KU
Factorise 2 x 2 - 7 x - 15
2 KU
2002 – Paper 1
3.
4.
5.
Solve the inequality
f ( x) = x 2 + 5 x ,
a) Factorise
3 KU
evaluate f (-3)
2 KU
p 2 - 4q 2
b) Hence simplify
1
2
5 – x > 2(x + 1)
(h - t)
1 KU
p 2 - 4q 2
3 p + 6q
6.
L=
9.
Two functions are given below:
2 KU
Change the subject of the formula to h
f ( x) = x 2 + 2 x - 1
2 KU
g ( x) = 5 x + 3
Find the values of x for which f(x) = g(x)
3 RE
2002 – Paper 2
3.
Solve the equation
2 x2 + 3x - 7 = 0
Give your answers correct to 1 decimal place.
4 KU
2001 – Paper 1
3.
Given that f (m) = m 2 - 3m ,
4.
Solve algebraically the equation
evaluate f (-5)
2x -
( 3x - 1) = 4
4
-7-
2 KU
3 KU
Indices and Surds
2008 – Paper 1
9.
Simplify m3 ´ m
2 KU
2008 – Paper 1
11.
A right angled triangle has dimensions as shown.
Calculate the length of AC, leaving your answer as a surd in its simplest form.
3 KU
2007 – Paper 1
7.
Remove brackets and simplify
æ 12
ö
a ç a - 2÷
è
ø
1
2
2 KU
2007 – Paper 1
9.
A square of side x centimetres has a diagonal 6 centimetres long.
Calculate the value of x, giving your answer
as a surd in its simplest form.
3 RE
2006 – Paper 2
4.
(b) Expand
m
1
2
(2 + m )
2
2 KU
2 20 - 3 5
(c) Simplify, leaving your answer as a surd.
2 KU
2005 – Paper 1
11.
f ( x) = 4 x + 2
(a) Find the value of f (72) as a surd in its simplest form.
3 KU
(b) Find the value of t, given that f (t ) = 3 2 .
3 RE
2004 – Paper 1
11.
(a) Simplify
2 75
2 KU
(b) Evaluate
20 + 3-1
2 KU
-8-
2003 – Paper 1
12.
a) Evaluate
8
2
3
2 KU
24
2
b) Simplify
2 KU
2002 – Paper 1
27 + 2 3
10.
Simplify
11.
Express in its simplest form
2 KU
y8 ´ ( y3 )
-2
2 KU
2001 – Paper 1
10.
Simplify
3
24
Express your answer as a fraction with a rational denominator
-9-
3 KU
Data Handling
2007– Paper 1
3.
There are 400 people in a studio audience.
The probability that a person chosen at random from this audience is male is
5
.
8
2 KU
How many males are in this audience?
2005 – Paper 1
4.
Two identical dice are rolled simultaneously.
Find the probability that the total score on adding both numbers
will be greater than 7 but less than 10
2 KU
2004 – Paper 1
7.
John’s school sells 1200 tickets for a raffle.
John buys 15 tickets.
John’s church sells 1800 tickets for a raffle.
John buys 20 tickets.
3 RE
In which raffle has he a better chance of winning the first prize ?
Show clearly all your working.
2003 – Paper 1
8.
A mini lottery game uses red, green, blue and yellow balls.
There are 10 of each colour, numbered from 1 to 10.
The balls are placed in a drum and one is drawn out.
What is the probability that it is a 6 ?
What is the probability it is a yellow 6 ?
a)
b)
1 KU
1 KU
2001 – Paper 1
7.
A garage carried out a survey on 600 cars.
The results are shown in the table below:
Engine size (cc)
Less than 3
years
3 years or
more
Age
0 – 1000
1001 – 1500
1501 – 2000
2001 +
50
80
160
20
60
100
120
10
a)
What is the probability that a car chosen at random, is less than 3 years old?
1 KU
b)
In a sample of 4200 cars, how many would be expected to have an
engine size greater than 2000cc and be 3 or more years old.
2 KU
- 10 -
Statistics – Statistical Diagrams
2008 – Paper 2
2.
In a class, 30 pupils sat a test.
The marks are illustrated by the stem and leaf diagram below.
(a)
(b)
Write down the median and the modal mark.
Find the probability that a pupil selected at random scored at least 40 marks.
2 KU
1 KU
2005 – Paper 1
7.
The speeds (measured to the nearest 10 kilometres per hour) of 200 vehicles
are recorded as shown.
Construct a cumulative frequency table and hence find the median for this data.
3 KU
2004 – Paper 1
5.
The average monthly temperature in a holiday resort was recorded in
degrees Celsius (°C).
Draw a suitable statistical diagram to illustrate the median and the quartiles
of this data.
4 RE
2003 – Paper 1
9.
A random check is carried out on the contents of a number of matchboxes.
A summary of the results is shown in the boxplot below.
47
50
51
53
54
What percentage of matchboxes contains fewer than 50 matches.
- 11 -
1 RE
2002 – Paper 1
8.
Fifteen medical centres each handed out a questionnaire to fifty patients.
The numbers who replied to each centre are shown below.
11
19
22
25
25
29
31
34
36
38
40
46
49
50
50
Also, they each posted the questionnaires to another fifty patients.
The numbers who replied to each centre are shown below.
15
15
21
22
23
25
26
31
33
34
37
39
41
46
46
Draw an appropriate statistical diagram to compare these two sets of data.
3 RE
Standard Deviation
2007 – Paper 2
3.
(a)
During his lunch hour, Luke records the number of birds that visit
his bird-table.
The numbers recorded last week were:
28 32 14 19 18 26
31
Find the mean and standard deviation for this data.
(b)
4 KU
Over the same period, Luke’s friend, Erin also recorded the number
of birds visiting her bird-table.
Erin’s recordings have a mean of 25 and a standard deviation of 5.
Make two valid comparisons between the friends’ recordings.
2 RE
The pulse rates, in beats per minute, of 6 adults in a hospital waiting
area are:
68 73
86 72 82 78
Calculate the mean and standard deviation of this data.
3 KU
2006 – Paper 2
2.
(a)
(b)
6 children in the same waiting area have a mean pulse rate
of 89.6 beats per minute and a standard deviation of 5.4.
Make two valid comparisons between the children’s pulse rates and those
of the adults.
- 12 -
2 RE
2005 – Paper 2
2.
The running time in minutes, of 6 television programmes are:
77
91
84
71
79
75
4 KU
Calculate the mean and standard deviation of these times.
2004 – Paper 2
3.
Bottles of juice should contain 50 millilitres.
The contents of 7 bottles are checked in a random sample.
The actual volume in millilitres are as shown below:
52,
50,
51,
49,
52,
53,
50
4 KU
Calculate the mean and standard deviation of the sample.
2003 – Paper 2
2.
Fiona checks out the price of a litre of milk in several shops.
The prices in pence are:
49 44 41 52 47 43
a)
Find the mean price of a litre of milk.
1 KU
b)
Find the standard deviation of the prices.
2 KU
c)
Fiona also checks out the price of a kilogram of sugar in the same
shops and finds that the standard deviation of the prices is 2.6.
Make one valid comparison between the two sets of prices.
1 RE
2001 – Paper 2
2.
The price, in pence per litre, of petrol at 10 city garages is shown below:
84.2
84.4
85.1
83.9
81.0
84.2
85.6
85.2
84.9
84.8
3 KU
a) Calculate the mean and standard deviation of these prices.
b)
In 10 rural garages, the petrol prices had a mean of 88.8 and a standard deviation of 2.4
How do the rural prices compare with the city prices ?
2 RE
2001 – Paper 1
5.
A furniture maker investigates the delivery times, in days, of two local wood
companies and obtains the following data.
Company
Timberplan
Allwoods
a)
b)
Minimum
16
18
Maximum
56
53
Lower
Quartile
34
22
Median
38
36
Upper
Quartile
45
49
Draw an appropriate statistical diagram to illustrate these two sets of data.
Given that consistency of delivery is the most important factor, which
company should the furniture maker use? Give a reason for your answer.
- 13 -
3 RE
1 RE
Area & Volume
2007 – Paper 2
12.
(a)
A cylindrical paperweight of radius 3 centimetres and height 4 centimetres is filled with sand.
2 KU
Calculate the volume of sand in the paperweight.
(b)
Another paperweight, in the shape of a hemisphere, is filled with sand.
It contains the same volume of sand as the first paperweight.
Calculate the radius of the hemisphere.
[The volume of a hemisphere with radius r is given by the formula, V = p r 3 ]
2
3
3 RE
2006 – Paper 2
7.
(a)
A block of copper 18 centimetres long
is prism shaped as shown.
The area of its cross section
is 28 square centimetres.
1 RE
Find the volume of the block.
(b)
The block is melted down to make a cylindrical cable
of diameter 14 millimetres.
4 KU
Calculate the length of the cable.
- 14 -
2006 – Paper 1
10.
Triangle ABC is right-angled at B.
The dimensions are as shown.
(a)
Calculate the area of triangle ABC.
1 KU
(b)
BD, the height of triangle ABC, is drawn as shown.
Use your answer to part (a) to calculate the height BD.
3 RE
2004 – Paper 2
9.
A gift box, 8 centimetres high, is prism shaped.
The uniform cross section is a regular pentagon.
Each vertex of the pentagon is 10 centimetres from the centre O.
5 KU
Calculate the volume of the box.
- 15 -
2003 – Paper 2
4.
A mug is in the shape of a cylinder with
diameter 10 centimetres and height 14 centimetres.
2 KU
a)
Calculate the volume of the mug.
b)
600 millilitres of coffee are poured in.
Calculate the depth of the coffee in the cup.
3 RE
2002 – Paper 2
5.
A feeding trough, 4 metres long, is prism shaped.
The uniform cross-section is made up of a rectangle and semi-circle
as shown below.
Find the volume of the trough, correct to 2 significant figures.
5 KU
2001 – Paper 2
5.
A cylindrical soft drinks can is 15 centimetres in height and
6.5 centimetres in diameter.
A new cylindrical can holds the same volume but has a reduced height
of 12 centimetres.
What is the height of the new can ?
Give your answer to 1 decimal place.
4 RE
2001 – Paper 2
8.
A metal doorstop is prism shaped,
as shown in Figure 1
Figure 1.
The uniform cross-section
as shown in Figure 2:
Figure 2.
Find the volume of metal required to make the doorstop.
- 16 -
4 KU
Similar Shapes & Similar Triangles
2007 – Paper 1
8.
Mick needs an ironing board.
He sees one in a catalogue with measurements
as shown in the diagram below.
When the ironing board is set up, two similar triangles are formed.
Mick wants an ironing board which is at least 80 centimetres in length.
Does this ironing board meet Mick’s requirements ?
Show all your working.
3 RE
2006 – Paper 2
11.
In triangle ABC
BC = 8 centimetres
AC = 6 centimetres
PQ is parallel to BC
M is the mid-point of AC
Q lies on AC, x centimetres from M, as shown on the diagram.
(a) Write down an expression for the length of AQ.
(
(b) Show that PQ = 4 +
4
x
3
)
1 RE
3 RE
2006 – Paper 1
7.
Coffee is sold in regular cups and large cups.
The two cups are mathematically similar in shape
The regular cup is 14 centimetres high and holds 160 millilitres.
The large cup is 21 centimetres high.
Calculate how many millilitres the large cup holds.
- 17 -
4 KU
2005 – Paper 2
6.
A vertical flagpole 12 metres high stands
at the centre of the roof of a tower.
The tower is cuboid shaped with a square
base of side 10 metres.
At a point P on the ground, 20 metres
from the base of the tower, the top of the
flagpole is just visible, as shown.
4 RE
Calulate the height of the tower.
2003 – Paper 2
9.
Two perfume bottles are mathematically similar in shape.
The smaller one is 6 centimetres high and
holds 30 millilitres of perfume.
The larger one is 9 centimetres high.
What volume of perfume will the larger one hold.
3 KU
2002 – Paper 2
12.
A metal beam, AB, is 6 metres long.
It is hinged at the top, P, of a vertical post 1 metre high.
When B touches the ground, A is 1.5 metres above the ground, as shown
In Figure 1.
When A comes down to the ground, B rises, as shown in Figure 2.
By calculating the length of AP, or otherwise, find the height of B above the ground.
Do not use a scale drawing.
- 18 -
5 RE
Pythagoras
2008 – Paper 1
11.
A right angled triangle has dimensions as shown.
Calculate the length of AC,
leaving your answer as a surd in its simplest form.
3 KU
2008 – Paper 2
5.
A circle, centre the origin, is shown.
P is the point (8, 1).
2 RE
(a) Calculate the length of OP.
The diagram also shows a tangent from P
which touches the circle at T.
The radius of the circle is 5 units.
(b)
2 RE
Calculate the length of PT.
2007 – Paper 1
9.
A square of side x centimetres has a diagonal 6 centimetres long.
Calculate the value of x,
giving your answer as a surd in its simplest form.
3 RE
2007 – Paper 1
12.
The diagram shows water lying in a length of roof guttering.
The cross-section of the guttering is a semi-circle
with diameter 10 centimetres.
The water surface is 8 centimetres wide.
Calculate the depth, d, of water in the guttering.
4 RE
- 19 -
2005 – Paper 2
5.
A triangular paving slab has measurements as shown.
Is the slab in the shape of a right angled triangle ?
Show your working.
3 RE
2005 – Paper 1
10.
A badge is made from a circle of radius 5 centimetres.
Segments are taken off the top and the bottom of the circle as shown.
The straight edges are parallel.
The badge measures 7 centimetres from the top to the bottom.
The top is 8 centimetres wide.
Calculate the width of the base.
5 RE
2004 – Paper 2
8.
The curved part of a doorway is an arc of a circle with radius
500 millimetres and centre C.
The height of the doorway to the top of the arc is 2000 millimetres.
The vertical edge of the doorway is 1800 millimetres.
5 RE
Calculate the width of the doorway.
- 20 -
2003 – Paper 2
10.
A sheep shelter is part of a cylinder as shown in Figure 1.
It is 6 metres wide and 2 metres high.
The cross-section of the shelter is a segment of a circle
with centre O, as shown in Figure 2.
OB is the radius of the circle.
4 RE
Calculate the length of OB.
2002 – Paper 2
6.
An oil tank has a circular cross section of radius 2.1 metres.
It is filled to a depth of 3.4 metres.
a)
Calculate x , the width in metres of the oil surface.
3 KU
b)
What other depth of oil would give the same surface width.
1 RE
- 21 -
Circle - Arcs and Sectors
2007 – Paper 2
7.
A fan has four identical plastic blades.
Each blade is a sector of a circle of radius 5 centimetres.
The angle at the centre of each sector is 64°
Calculate the total area of plastic required to make the blades.
3 KU
2006 – Paper 2
8.
A set of scales has a circular dial.
The pointer is 9 centimetres long.
The tip of the pointer moves through
an arc of 2 centimetres for each
100 grams of weight on the scales.
A parcel, placed on the scales, moves the pointer through an angle of 284°
Calculate the weight of the parcel.
- 22 -
4 RE
2005 – Paper 2
10.
The chain of a demolition ball is 12.5 metres long.
When vertical, the end of the chain is 1.5 metres from the ground.
It swings to a maximum height of 2.5 metres above the ground on both
sides.
(a) At this maximum height, show that the angle x°, which the chain
makes with the vertical, is approximately 23°
4 RE
(b) Calculate the maximum length of the arc through which the end of the
chain swings. Give your answer to 3 significant figures.
4 RE
2008 – Paper 2
9.
Contestants in a quiz have 25 seconds to answer a question.
This time is indicated on the clock.
The tip of the clock hand moves through the arc AB as shown.
(a)
1 KU
Calculate the size of angle AOB.
(b) The length of arc AB is 120 centimetres.
4 RE
Calculate the length of the clock hand.
- 23 -
Trigonometry – SOH-CAH-TOA
2006 – Paper 2
5.
ST, a vertical pole 2 metres high,
is situated at the corner of a rectangular
garden, PQRS.
RS is 8 metres long and QR is 12 metres long.
The pole casts a shadow over the garden.
The shadow reaches M, the midpoint of QR.
Calculate the size of the shaded angle TMS
4 KU
2004 – Paper 2
6.
The diagram below shows a spotlight at point S, mounted 10 metres directly
above a point P at the front edge of a stage.
The spotlight swings 45° from the vertical to illuminate another point Q,
also at the front edge of the stage.
Through how many more degrees must the spotlight swing to illuminate
a point B, where Q is the mid-point of PB ?
5 RE
2003 – Paper 2
6.
In the diagram
· Angle STV = 34°
· Angle VSW = 25°
· Angle SVT = Angle SWV = 90°
· ST = 13.1 centimetres
4 KU
Calculate the length of SW
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Trigonometry – Non-right angled triangles
2008 – Paper 2
7.
A telegraph pole is 6.2 metres high.
The wind blows the pole over into the position
as shown below.
AB is 2.9 metres and angle ABC is 130 °.
Calculate the length of AC.
4 RE
2008 – Paper 2
8.
A farmer builds a sheep-pen
using two lengths of fencing and a wall.
The two lengths of fencing are
15 metres and 18 metres long.
(a)
(b)
Calculate the area of the sheep-pen, when the angle between the
fencing is 70 °.
3 KU
What angle between the fencing would give the farmer the largest
possible area?
1 RE
2007 – Paper 2
6.
Brunton is 30 kilometres due North of Appleton.
From Appleton, the bearing of Carlton is 065°
From Brunton, the bearing of Carlton is 153°
Calculate the distance between Brunton and Carlton.
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4 RE
2007 – Paper 2
8.
In triangle PQR
· QR = 6 centimetres
· Angle PQR = 30°
· Area of triangle PQR = 15 square centimetres.
3 RE
Calculate the length of PQ.
2006 – Paper 2
6.
(a) There are three mooring points
A, B and C on Lake Sorling.
(b)
From A, the bearing of B is 074°.
From C, the bearing of B is 310°
Calculate the size of angle ABC.
2 RE
B is 230 metres from A and 110 metres from C.
Calculate the direct distance from A to C.
Give your answer to 3 significant figures.
4 RE
2005 – Paper 2
3.
4 KU
Calculate the area of triangle PQR.
2005 – Paper 2
7.
David walks on a bearing of 050°
from hostel A to a viewpoint V,
5 kilometres away.
Hostel B is due east of hostel A.
Susie walks on a bearing of 294°
from hostel B to the same viewpoint.
Calculate the length of AB, the distance between the two hostels.
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5 KU
2004 – Paper 2
5.
A helicopter, at point H, hovers between
two aircraft carriers at points A
and B which are 1500 metres apart.
From carrier A, the angle of elevation of the helicopter is 50°.
From carrier B, the angle of elevation of the helicopter is 55°.
Calculate the distance from the helicopter to the nearer carrier.
4 RE
2004 – Paper 2
7.
A square trapdoor of side 80 centimetres is held open by a rod as shown.
The rod is attached to the corner of the trapdoor and placed 40 centimetres
along the edge of the opening.
The angle between the trapdoor and the opening is 76°.
Calculate the length of the rod to 2 significant figures.
4 KU
2004 – Paper 2
9.
A gift box, 8 centimetres high, is prism shaped.
The uniform cross section is a regular pentagon.
Each vertex of the pentagon is 10 centimetres from the centre O.
Calculate the volume of the box.
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5 KU
2003 – Paper 2
3.
Two yachts leave from harbour H.
Yacht A sails on a bearing of 072° for 30 kilometres and stops.
Yacht B sails on a bearing of 140° for 50 kilometres and stops.
How far apart are the two yachts when they have both stopped?
Do not use a scale drawing.
4 RE
2003 – Paper 2
7.
The area of triangle is 38 square centimeters.
AB is 9 centimetres and BC is 14 centimetres.
3 RE
Calculate the size of the acute angle ABC
2002 – Paper 1
7.
In triangle ABC
AB = 4 units
AC = 5 units
BC = 6 units
Show that cos A =
1
8
3 KU
2002 – Paper 2
4.
A TV signal is sent from a transmitter T, via a satellite S, to a village V, as
shown in the diagram. The village is 500 kilometres from the transmitter.
The signal is sent out at an angle of 35° and is received in the village at an angle of 40°.
Calculate the height of the satellite above the ground.
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5 RE
2001 – Paper 2
6.
The radio masts, Kangaroo (K),
Wallaby (W) and Possum (P) are situated
in the Australian outback.
Kangaroo is 250 kilometres due south of Wallaby.
Wallaby is 410 kilometres from Possum
Possum is on a bearing of 130° from Kangaroo.
Calculate the bearing of Possum from Wallaby.
Do not use a scale drawing.
4 RE
2001 – Paper 2
10.
Each leg of a folding table is prevented from opening too far by a metal bar.
The metal bar is 21 centimetres long.
It is fixed to the table top 14 centimetres from the hinge and to the table leg
12 centimetres from the hinge.
a) Calculate the size of the obtuse angle which the table top makes with the leg.
3 KU
b) Given that the table leg is 70 centimetres long, calculate the height of the table.
3 RE
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Gradients and the Straight Line
2008 – Paper 1
4.
A straight line cuts the x-axis at the point (9, 0)
and the y-axis at the point (0, 18) as shown.
3 KU
Find the equation of the line.
2007 – Paper 1
6.
A taxi fare consists of a £2 “call-out” charge plus a fixed amount per kilometre.
The graph shows the fare, f pounds for a journey of d kilometres.
The taxi fare for a 5 kilometre journey is £6.
Find the equation of the straight line in terms of d and f.
4 KU
2006 – Paper 1
4.
Find the equation
of the given straight line.
3 KU
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2005 – Paper 2
9.
The monthly bill for a mobile phone is made up
of a fixed rental plus call charges.
Call charges vary as the time used.
The relationship between
the monthly bill, y (pounds), and the time used,
x (minutes) is represented in the graph.
(a) Write down the fixed rental.
1 RE
(b) Find the call charge per minute.
3 RE
2005 – Paper 1
5.
In an experiment involving two variables, the following values for x and y
were recorded.
The results were plotted and a straight line was drawn through the points.
Find the gradient of the line and write down its equation.
3 KU
2004 – Paper 2
2.
A tank which holds 100 litres of water has a leak.
After 150 minutes there is no water left in the tank.
The above graph represents the volume of water (v litres) against
time (t minutes).
(a)
Find the equation of the line in terms of v and t.
3 KU
(b)
How many minutes does it take for the container to lose 30 litres of water?
3 RE
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2004 – Paper 1
10.
Two variables x and y are connected by the relationship, y = ax + b .
Sketch a possible graph of y against x to illustrate this relationship
when a and b are each less than zero.
3 RE
2003 – Paper 1
6.
In the diagram, A is the point (-1, 7)
and B is the point (4, 3).
y
B(4, 3)
1 KU
a) Find the gradient of the line AB.
O
b) AB cuts the y-axis at the point (0, -5).
Write down the equation of the line AB
c) The point (3k, k) lies on AB
Find the value of k.
A(-1, -7)
x
1 KU
2 RE
2002 – Paper 1
12.
The graph shows the relationship
between the history and geography
marks of a class of students
A best fitting straight line, AB has been drawn.
Point A represents 0 marks for history and 12 marks for geography.
Point B represents 90 marks for history and 82 marks for geography.
Find the equation of the straight line AB in terms of h and g.
4 RE
2001 – Paper 1
6.
A is the point (a2, a)
T is the point (t2, t)
a≠t
Find the gradient of the line AT
3 KU
Give your answer in its simplest form.
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2001 – Paper 2 (also in simultaneous equations)
4.
A water pipe runs between two buildings.
These are represented by the points A and B in the diagram below.
a) Using the information in the diagram, show that the equation
of the line AB is 3 y - x = 6 .
b)
3 KU
An emergency outlet pipe has to be built across the main pipe.
The line representing this outlet pipe has equation 4 y + 5 x = 46
Calculate the coordinates of the point on the diagram at which
the outlet pipe will cut across the main water pipe.
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4 RE
Simultaneous Equations
2008 – Paper 2
4.
Aaron saves 50 pence and 20 pence coins in his piggy bank.
Let x be the number of 50 pence coins in his bank.
Let y be the number of 20 pence coins in his bank.
(a)
(b)
(c)
There are 60 coins in his bank.
Write down an equation in x and y to illustrate this information.
1 KU
The total value of the coins is £17.40.
Write down another equation in x and y to illustrate this information.
1 KU
Hence find algebraically the number of 50 pence coins Aaron has in
his piggy bank.
3 RE
2007 – Paper 1
11.
(a)
(b)
(c)
A cinema has 300 seats which are either standard or deluxe.
Let x be the number of standard seats and y be the number of deluxe seats.
Write down an algebraic expression to illustrate this information.
1 KU
A standard seat costs £4 and a deluxe seat costs £6.
When all the seats are sold the ticket sales are £1380.
Write down an algebraic expression to illustrate this information.
2 KU
How many standard seats and how many deluxe seats are in the cinema?
3 RE
2006 – Paper 1
9.
Euan plays in a snooker tournament which consists of 20 games.
He wins x games and loses y games
(a)
Write down an equation in x and y to illustrate this information.
1 KU
(b)
He is paid £5 for each game he wins and £2 for each game he loses.
He is paid a total of £79.
Write down another equation in x and y to illustrate this information.
2 RE
How many games did Euan win
3 RE
(c)
2004 – Paper 1
8.
7,
-2,
5,
3,
8
In the sequence above, each term after the first two terms is the sum of
the previous two terms. For example: 3rd term = 5 = 7 + (-2)
(a)
A sequence follows the above rule.
The first term is x and the second term is y.
The fifth term is 5.
x, y, -, -, 5
Show that 2 x + 3 y = 5
2 RE
This question is continued over the page
- 34 -
2004 – Paper 1 (continued)
8.
(b)
(c)
Using the same x and y, another sequence follows the above rule.
The first term is y and the second term is x.
The sixth term is 17.
y, x, -, -, -, 17
Write down another equation in x and y.
Find the values of x and y.
2 RE
3 RE
2003 – Paper 1
7.
Andrew and Doreen each book in at the Sleepwell Lodge.
a)
b)
c)
Andrew stays for 3 nights and has breakfast on 2 mornings.
His bill is £145
Write down an algebraic equation to illustrate this.
1 KU
Doreen stays for 5 nights and has breakfast on 3 mornings.
Her bill is £240.
Write down an equation to illustrate this.
1 KU
Find the cost of one breakfast.
3 KU
2002 – Paper 1
13.
a)
b)
c)
4 peaches and 3 grapefruit cost £1.30
Write down an algebraic equation to illustrate this.
1 KU
2 peaches and 4 grapefruit cost £1.20.
Write down an algebraic equation to illustrate this.
1 KU
Find the cost of 3 peaches and 2 grapefruit.
4 RE
2001 – Paper 2 (also in gradients and the straight line)
4.
A water pipe runs between two buildings.
These are represented by the points A and B in the diagram below.
a)
b)
Using the information in the diagram, show that the equation
of the line AB is 3 y - x = 6 .
3 KU
An emergency outlet pipe has to be built across the main pipe.
The line representing this outlet pipe has equation 4 y + 5 x = 46
Calculate the coordinates of the point on the diagram at which
the outlet pipe will cut across the main water pipe.
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4 RE
Functions and the Parabola (Quadratic)
2008 – Paper 1
8.
The curved part of the letter A in the Artwork logo is in the shape of a parabola.
The equation of this parabola is y = ( x - 8 )( 2 - x )
(a)
Write down the coordinates of Q and R.
2 KU
(b)
Calculate the height, h, of the letter A.
3 RE
2008 – Paper 1
10.
Part of the graph of y = a x , where a > 0, is shown below
The graph cuts the y-axis at C.
(a)
1 KU
Write down the coordinates of C.
B is the point (2, 16)
(b)
Calculate the value of a.
2 KU
2008 – Paper 1
12.
Given that
x 2 - 10 x + 18 = ( x - a ) + b
2
Find the values of a and b.
3 KU
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2007 – Paper 2
13.
The profit made by a publishing company
of a magazine is calculated by the formula
y = 4 x (140 - x ) ,
where y is the profit (in pounds) and x is
the selling price (in pence) of the magazine.
The graph represents the profit y against
the selling price x.
Find the maximum profit the company can make from the sale of the magazine.
4 RE
2006 – Paper 1
8.
The graph of y = x 2 has been moved
to the position shown in Figure 1.
The equation of this graph
is y = ( x - 2 ) + 3
2
The graph of y = x 2 has now been moved
to the position shown in Figure 2.
Write down the equation of
the graph in Figure 2.
2 RE
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2003 – Paper 2
8.
The diagram shows part of the graph of a
quadratic function, with equation of the form
y = k ( x - a )( x - b)
The graph cuts the y-axis at (0, -6) and
the x-axis at (-1, 0) and (3, 0)
a)
Write down the values of a and b.
2 KU
b)
Calculate the value of k.
2 KU
c)
Find the coordinates of the minimum
turning point of the function
2 RE
2001 – Paper 1
8.
The diagram below shows part of the graph of y = 4 x 2 + 4 x - 3
The graph cuts the y-axis at A and the x-axis at B and C.
a)
Write down the coordinates of A
1 KU
b)
Find the co-ordinates of B and C.
3 KU
c)
Calculate the minimum value of 4 x 2 + 4 x - 3
2 RE
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Making and Using Formulae
2008 – Paper 1
6.
Jane enters a two-part race.
(a)
(b)
(c)
She cycles for 2 hours at a speed of (x + 8) kilometres per hour.
Write down an expression in x for the distance cycled.
1 KU
She then runs for 30 minutes at a speed of x kilometres per hour.
Write down an expression in x for the distance run.
1 KU
The total distance of the race is 46 kilometres.
Calculate Jane’s running speed.
3 RE
2008 – Paper 1
7.
The 4th term of each number pattern below is the mean of the
previous three terms.
1 KU
(a)
When the first three terms are 1, 6 and 8, calculate the 4th term.
(b)
When the first three terms are x, (x + 7) and (x + 11), calculate the 4 term.
(c)
When the first, second and fourth terms are
th
-2x,
(x + 5), _______ ,
1 RE
(2x + 4),
2 RE
rd
Calculate the 3 term.
2008 – Paper 2
10.
To hire a car costs £25 per day
plus a mileage charge.
The first 200 miles are free with each
additional mile charged at 12 pence.
(a)
(b)
Calculate the cost of hiring a car for 4 days
when the mileage is 640 miles.
1 KU
A car is hired for d days and the mileage is m miles where m > 200.
Write down a formula for the cost £C of hiring the car.
3 RE
2008 – Paper 2
11.
The minimum number of roads joining 4 towns
to each other is 6 as shown.
The minimum number of roads, r, joining n towns
to each other is given by the formula
r = n ( n - 1)
1
2
(a)
State the minimum number of roads needed to join 7 towns to each other.
1 KU
(b)
When r = 55, show that n 2 - n - 110 = 0 .
2 RE
(c)
Hence find algebraically the value of n.
3 RE
- 39 -
2007 – Paper 1
12.
The diagram shows water lying in a length of roof guttering.
The cross-section of the guttering is a semi-circle with diameter 10 centimetres.
The water surface is 8 centimetres wide.
Calculate the depth, d, of water in the guttering.
4 RE
2007– Paper 2
11.
(a)
A decorator’s logo is rectangular and measures 10 centimetres by 6 centimetres.
It consists of three rectangles: one red, one yellow and one blue.
The yellow rectangle measures 10 centimetres by x centimetres.
The width of the red rectangle is x centimetres.
Show that the area, A, of the blue rectangle is given by the expression
A = x 2 - 16 x + 60
(b)
The area of the blue rectangle is equal to
2 RE
1
of the total area of the logo.
5
Calculate the value of x.
4 RE
- 40 -
2006 – Paper 2
9.
The number of diagonals, d, in a polygon of n sides is given by the formula
d = n ( n - 3)
1
2
(a)
How many diagonals does a polygon of 7 sides have ?
(b)
A polygon has 65 diagonals.
n 2 - 3n - 130 = 0
2 RE
Hence find the number of sides in this polygon.
3 RE
Show that for this polygon,
(c)
2 KU
2006 – Paper 1
11.
(a)
One session at the Leisure Centre costs £3.
Write down an algebraic expression for
the cost of x sessions
(b)
1 RE
The Leisure Centre also offers a
monthly card costing £20.
The first 6 sessions are then free,
with each additional session costing £2.
(c)
(i) Find the total cost of a monthly card and 15 sessions.
1 KU
(ii) Write down an algebraic expression for the total cost of a monthly
card and x sessions where x is greater than 6.
2 RE
Find the minimum number of sessions required for the monthly card
to be the cheaper option.
Show all working.
3 RE
2005 –Paper 2
8.
The side length of a cube is
2x centimetres.
The expression for the volume in cubic centimetres is equal to the
expression for the surface area in square centimetres.
5 RE
Calculate the side length of the cube.
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2005 – Paper 2
9.
The monthly bill for a mobile phone is made up of a fixed rental plus call charges.
Call charges vary as the time used.
The relationship between the monthly bill, y (pounds), and the time used,
x (minutes) is represented in the graph below.
(a)
Write down the fixed rental.
1 RE
(b)
Find the call charge per minute.
3 RE
2005 – Paper 1
9.
(a)
Emma puts £30 worth of petrol into the empty fuel tank of her car.
Petrol costs 75 pence per litre.
Her car uses 5 litres of petrol per hour, when she drives at a particular
constant speed.
At this constant speed, how many litres of petrol will remain in the car
after 3 hours?
(b)
2 KU
The next week, Emma puts £20 worth of petrol into the empty fuel tank of her car.
Petrol costs c pence per litre.
Her car uses k litres of petrol per hour, when she drives at another constant speed.
Find a formula for R, the amount of petrol remaining in the car after t hours.
3 RE
2005 – Paper 1
12.
The height of a triangle is ( 2 x - 5) centimetres and the base is 2x centimetres.
The area of the triangle is 7 square centimetres.
Calculate the value of x.
5 RE
- 42 -
2004 – Paper 2
11.
A rectangular lawn has a path,
1 metre wide, on 3 sides as shown.
The breadth of the lawn is x metres.
The length of the lawn is three times its breadth.
The area of the lawn equals the area of the path.
(a)
Show that 3 x 2 - 5 x - 2 = 0
3 RE
(b)
Hence find the length of the lawn.
4 RE
2004 – Paper 1
12.
A piece of gold wire 10 centimetres long
is made into a circle.
The circumference of the wire is equal
to the length of the wire.
Show that the area of the circle is exactly
25
p
square centimetres.
4 RE
2003 – Paper 1
13.
A rectangular clipboard has a triangular
plastic pocket attached as shown in Figure 1.
The pocket is attached along edges TD
And DB as shown in Figure 2.
B is x centimetres from the corner C.
The length of the clipboard is 4x centimetres
and the breadth is 3x centimetres.
The area of the pocket is a quarter of the area of the clipboard.
Find in terms of x , the length of TD.
4 RE
2003 – Paper 2
5.
The number of diagonals, d, in a polygon with n sides is given by the formula:
d=
n(n - 3)
2
A polygon has 20 diagonals.
How many sides does it have?
4 RE
- 43 -
2003 – Paper 2
11.
(a)
A driver travels from A to B, a distance of x miles at a constant speed
of 75 kilometres per hour.
Find the time taken for the journey in terms of x.
(b)
The time for the journey from B to A is
1 KU
x
hours
50
Hence calculate the driver’s average speed for the whole journey.
4 RE
2002 – Paper 2
9.
Esther has a new mobile phone and considers the following daily rates.
Easy Call
Green Call
25 pence per minute for
the first 3 minutes
40 pence per minute for
the first 2 minutes
5 pence per minute after
the first three minutes.
2 pence per minute after
the first two minutes.
a)
For Easy Call, find the cost of ten minutes in a day.
1 KU
b)
For Easy Call, find a formula for the cost of “m” minutes in a day, m > 3
1 RE
c)
For Green Call, find a formula for the cost of “m” minutes in a day, m > 2
1 RE
d)
Green Call claims that its system is cheaper.
Find algebraically, the least number of minutes (to the nearest minute)
which must be used each day for this claim to be true.
3 RE
2001 – Paper 1
11.
The intensity of light, I, emerging after passing through a liquid
with concentration, c, is given by the equation
I=
20
2c
c³0
a)
Find the intensity of light when the concentration is 3.
1 KU
b)
Find the concentration of the liquid when the intensity is 10
2 KU
c)
What is the maximum possible intensity ?
3 RE
2001 – Paper 2
11.
A rectangular wall vent is 30 centimetres long and 20 centimetres wide.
It is to be enlarged by increasing both the length and the width by x centimetres.
1 RE
a)
Write down the length of the new vent.
b)
Show that the Area, A, square centimetres, of the new vent is given by
A = x 2 + 50 x + 600
c)
2 RE
The area of the new vent must be at least 40% more than the original area.
Find the minimum dimensions to the nearest centimetres, of the new vent.
- 44 -
5 RE
Trigonometry – Graphs and Equations
2008 – Paper 2
12.
The diagram shows part of the graph of y =tan x °.
The line y =5 is drawn and intersects the graph of y = tan x ° at P and Q.
(a)
Find the x-coordinates of P and Q.
3 RE
(b) Write down the x-coordinate of the point R, where the line y = 5
next intersects the graph of y = tan x °
1 RE
2007 – Paper 2
10.
Solve algebraically the equation
5cos x° + 4 = 0,
0 £ x < 360
3 KU
2007 – Paper 1
13.
Part of the graph of y = cos bx° + c is shown below.
Write down the values of b and c.
2 KU
2006 – Paper 2
10.
Emma goes on the “Big Eye”.
Her height, h metres, above the ground
is given by the formula
h = -31cos t ° + 33
where t is the number of seconds after the start.
(a) Calculate Emma’s height above the ground 20 seconds after the start.
2 KU
(b) When will Emma first reach a height of 60 metres above the ground ?
3 RE
(c) When will she next be at a height of 60 metres above the ground.
1 RE
- 45 -
2005 – Paper 2
11.
(a) Solve algebraically the equation
3 sin x° - 1 = 0
0 £ x £ 360
3 KU
0 £ x £ 90
1 RE
(b) Hence write down the solution of the equation
3 sin 2 x° - 1 = 0
2004 – Paper 2
10.
Solve algebraically the equation
4sin x° + 1 = - 2
0 £ x < 360
3 KU
2004 – Paper 1
9.
The graph of y = a cos bx°, 0 £ x £ 90° is shown below.
Write down the value of a and b
2 KU
2002 – Paper 2
8.
The diagram shows part of the graph of y = sin x.
The line y = 0.4 is drawn and cuts the graph of y = sin x at A and B.
Find the x-coordinates of A and B.
3 RE
2001 – Paper 2
7.
Solve algebraically the equation:
tan 40° = 2sin x° + 1
- 46 -
0 £ x £ 360
3KU
Ratio & Proportion
2007 – Paper 2
9.
To make “14 carat” gold, copper and pure gold are mixed in the ratio 5:7.
A jeweller has 160 grams of copper and 245 grams of pure gold.
What is the maximum weight of “14 carat” gold that the jeweller can make?
3 RE
2003 – Paper 1
10.
School theatre visits are arranged for parents, teachers and pupils.
The ratio of parents to teachers to pupils must be 1 : 3 : 15.
a)
b)
45 pupils want to go to the theatre.
How many teachers must accompany them?
1 KU
The theatre gives the school 100 tickets for a play.
What is the maximum number of pupils who can go to the play?
3 RE
2002 – Paper 2
7.
A coffee shop blends its own coffee and sells it in one-kilogram tins.
One blend consists of two kinds of coffee, Brazilian and Columbian, in the ratio 2 : 3.
The shop has 20 kilograms of Brazilian and 25 kilograms of Columbian in stock.
What is the maximum number of one-kilogram tins of this blend
which can be made.
- 47 -
3 RE
Variation & Proportion
2008 – Paper 2
6.
The distance, d kilometres, to the horizon, when viewed from a cliff top, varies directly
as the square root of the height, h metres, of the cliff top above sea level.
From a cliff top 16 metres above sea level, the distance to the horizon is 14 kilometres.
A boat is 20 kilometres from a cliff whose top is 40 metres above sea level.
Is the boat beyond the horizon?
Justify your answer.
5 RE
2007 – Paper 1
10.
A relationship between T and L is given by the formula, T =
k
where k is a constant.
L3
When L is doubled, what is the effect on T?
2 RE
2002 – Paper 2
10.
A weight on the end of a string is spun
in a circle on a smooth table.
The tension, T, in the string varies directly
as the square of the speed, v,
and inversely as the radius, r, of the circle.
a)
Write down a formula for T in terms of v and r.
b)
The speed of the weight is multiplied by 3 and
the radius of the string is halved.
1 KU
2 RE
What happens to the tension in the string.
2001 – Paper 2
9.
The electrical resistance, R, of copper wire varies directly as its length, L metres,
and inversely as the square of its diameter, d millimetres .
Two lengths of copper wire, A and B, have the same resistance.
Wire A has a diameter of 2 millimetres and a length of 3 metres.
Wire B has a diameter of 3 millimetres
4 RE
What is the length of wire B.
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Distance, Speed & Time
2004 – Paper 2
1.
Radio signals travel at a speed of 3 ´ 108 metres per second.
A radio signal from Earth to a space probe takes 8 hours.
What is the distance from Earth to the probe?
Give your answer in scientific notation.
4 KU
2003 – Paper 2
11.
a)
A driver travels from A to B, a distance of x miles at a constant speed
of 75 kilometres per hour.
Find the time taken for the journey in terms of x.
b)
The time for the journey from B to A is
1 KU
x
hours
50
Hence calculate the driver’s average speed for the whole journey.
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4 RE
Sequences
2007 – Paper 1
14. The sum S n of the first n terms of a sequence, is given by the formula
S n = 3n - 1
(a)
Find the sum of the first 2 terms.
1 RE
(b)
When S n = 80 , calculate the value of n.
2 RE
2005 – Paper 1
8.
A number pattern is given below.
(a)
(b)
1st term:
22 - 02
2nd term:
32 - 12
3rd term:
42 - 22
Write down a similar expression for the 4th term.
th
Hence or otherwise find the n term in its simplest form.
1 RE
3 RE
2003 – Paper 1
11.
Using the sequence
1, 3, 5, 7, 9, …..
a)
Find S3, the sum of the first 3 numbers.
1 RE
b)
Find Sn, the sum of the first n numbers.
2 RE
c)
th
Hence or otherwise, find the (n + 1) term of the sequence
2 RE
2002 – Paper 2
11
2n = 32
1 KU
a)
Solve the equation
b)
A sequence of numbers can be grouped and added together as shown.
The sum of 2 numbers:
The sum of 3 numbers:
The sum of 4 numbers:
c)
(1 + 2) = 4 – 1
(1 + 2 + 4) = 8 – 1
(1 + 2 + 4 + 8) = 16 – 1
Find a similar expression for the sum of 5 numbers.
1 KU
Find a formula for the sum of the first n numbers of this sequence.
2 RE
2001 – Paper 1
9.
13 + 1 = (1 + 1)( 12 - 1 + 1)
A number pattern is shown here:
23 + 1 = (2 + 1)(2 2 - 2 + 1)
33 + 1 = (3 + 1)(32 - 3 + 1)
a)
Write down a similar expression for 73 + 1
1 RE
b)
Hence write down an expression for n3 + 1
1 RE
c)
2 RE
3
Hence find an expression for 8p + 1
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