1.

a)     Expand 5(3x - 1)  [1]

b)     Expand 3x(2x + 4)  [1]

c)     Expand and simplify 2(3x + 5) - 3(4x - 2)  [2]

d)     Expand and simplify 5(2y - 3) + 2(y - 1)  [2]

e)     Expand and simplify (3x - 4)(2x - 1)  [2]

2.

Expand the following expression, simplifying your answer as far as possible
(x + 7)(x - 3).  [2]

3.

a)     Expand 2x(x² + 3)   [2]

b)     Expand and simplify 4(3x - 1) + 3(x - 5).    [2]

4.

Expand the following expression, simplifying your answer as far as possible 
(x - 4)(x + 7)  [2]

5.

Expand and simplify (x + 5)(x - 6)   [2]

6.

Expand and simplify 4(2y - 3) - 3(y + 5)  [2]

7.

Expand the following expression, simplifying your answer as far as possible 
(x - 2)(x - 6)  [2]

1.

Make t the subject of the formula w(5 - 3t) = 2(t + 5)  [4]

2.

Make d the subject of the formula 4(d - 2e) = 7 + 3e   [3]

3.

Make e the subject of the formula    [4]

4.

Make q the subject of the formula  8qt = 3a (q + t) + d   [4]

5.

Make p the subject of the formula  4(t + 3p) = 6 - 3t   [3]

6.

Make d the subject of the formula  .  [2]

7.

Make p the subject of the formula  a(3t - 2p) = p(3b - w) - w   [4]

8.

Make t the subject of the formula  2n + 5 = 3(8 - 3t)   [3]

9.

Make c the subject of the formula. Simplify your answer.

c(5b - a) = a(c - 3b)   [4]

1.

Factorise 6x² - 8xy  [2]

2.

Factorise 3a² - 6ac   [2]

3.

Factorise 3xy² - 6xy  [2]

4.

Factorise 4x²y - 6xy²  [2]

1.

Solve the following simultaneous equations by an algebraic (not graphical) method.  [4]

3x - 2y = 16
x + 3y = - 2

2.

Solve the following simultaneous equations by an algebraic (not graphical) method.  [4]

4x + 5y = -5
6x + 4y = 3

3.

Solve the following simultaneous equations by an algebraic (not graphical) method. Show all your working.  [4]

3x - 4y = 22
2x + 3y = - 8

4.

Solve the following simultaneous equations by an algebraic (not graphical) method. Show all your working.  [4]

4x + 2y = 17
5x + 3y = 23  

5.

Solve the following simultaneous equations by an algebraic (not graphical) method. Show all your working.  [4]

4x - 3y = 20
6x - 5y = 22  

6.

Solve the following simultaneous equations by an algebraic (not graphical) method. Show all your working.  [4]

5x + 2y = 10
2x + 3y = -7  

1.

Factorise x² - 2x - 15  [2]

2.

Factorise 16x² - 1  [2]

3.

a)     Factorise x² - 5x - 14  [2]

b)     Hence solve the equation x² - 5x - 14 = 0  [1]

4.

Factorise x² - 16.  [1]

5.

Factorise x² + 2x - 8.  [2]

6.

a)     Factorise x² +7x - 18  [2]

b)     Hence solve the equation x² + 7x - 18 = 0  [1]

7.

Factorise the expression 4x² - 81 and hence solve the equations 4x² - 81 = 0.[3]

8.

Factorise x² - 4x - 12  [2]

9.

Factorise       a)    2a² - 50b²  [2]               b)   5x² + 17x - 12   [2]

10.

Show that the expression 4x² - 20x + 25 can be written in the form (ax + b)². Write down the values of a and b.   [3]

11.

a)     Factorise   (i)  9q² - 100       [2]          (ii)  3q² + 4q - 20    [2]

b)     Simplify   [1]

1.

ABCD is a rectangle in which AB is 8 cm and AD is 13 cm.

a)     Draw ABCD.  [1]

b)     Draw the locus of all points inside the rectangle whose distance from AB is the same as their distance from AD. [1]

c)     Draw the locus of all the points inside the rectangle which are 6 cm
from DC.  [1]

d)     Draw the locus of all the points inside the rectangle whose distance from A is the same as the length of AB.  [1]

2.

Draw a line 12 cm long. Label the end points A and B.

Find and shade the region of points that satisfy both of the following conditions.

a)     The points are nearer to A than to B.  [1]

b)     They are not further than 7 cm from B.  [2]

3.

A dog is tied to a 4 m length of rope, at the other end of which is a ring. This can slide over a rod AB of length 6 m, which is attached to a long brick wall. The ring cannot slide off the rod due to stoppers at A and B.

Using a scale of 1 cm to represent 1 m, draw a diagram to show the region in which the dog can move.  [4]

4.

Enid and George hide a box in their garden which measures 12m by 8 m. They make a map of the garden using a scale of 1 cm to represent 1 m. The diagram shows a sketch of this garden. They give the map to some friends together with the following clues.

·         The box is nearer the end A of the hedge than to the end C.

·         The box is less than 6 m away from the tree marked T.

·         The box is nearer the garden wall AB than the hedge AC.

Draw an accurate diagram of the garden and shade the region in which the box is hidden.  [4]

5.

Shade in the region of points inside the triangle ABC which satisfy both of the following conditions.

• The points are nearer the point A than the point B.   [1]
• The points are further from B than the distance BC.  [2]

1.

Find the equation of the line that passes through the points (0, 8) and (-2, 2)  [4]

1.

A circle of radius 2x cm is cut out of a rectangular piece of paper with length 6x cm and width 5x cm. Find, in terms of x and p, the area of paper remaining after the circle has been removed.  [4]

2.

A cone with base of radius x cm and height h cm, has the same volume as a cuboid with length x cm, width (x - 2) cm and height (x + 2) cm.

Find an expression for h in terms of x and p.  [5]

1.

The assessment for a mathematics examination consists of two parts, namely coursework marked out of 50, and written papers, marked out of 200. The marks for ten pupils are given in the table.

a)     Draw a scatter diagram to show these results.  [2]

b)     The mean coursework mark for the pupils is 25 and the mean mark of the written papers is 98. Draw a line of best fit on your scatter diagram.  [2]

c)     Another pupil completed the coursework and was given a mark of 19, but was absent from the written papers examination. Use your line of best fit to estimate the mark on the written papers for this pupil.  [1]

2.

The engine capacity, measured in cubic centimetres (c.c.) and the time (in seconds), taken to accelerate to a certain speed, for each of 8 cars, are given in the table.

a)     Draw a scatter graph to show these results.  [2]

b)     The mean engine capacity is 1425 cc and the mean acceleration time is
11 seconds. Draw a line of best fit on your scatter diagram.  [2]

c)     Use your line of best fit to estimate the acceleration time for a car with an engine capacity of 1750 cc.   [1]

1.

Three years ago a car was bought for £8000. Each year the car’s value depreciates by 12% of its value at the start of the year. Calculate how much the car is worth today.  [3]

2.

On April 1^st Marcus owed £250 on his credit card account. The credit card company requires Marcus to pay at least 10% of the balance on the 20^th of each month.

The company charges interest at 2% on what the balance is on the 28^th of every month. Marcus pays the minimum payment on time every month.

Write down the full details of his account up to May 31^st.   [3]

April 1^st      £250
April 20^th

3.

Find, to the nearest penny, the compound interest when £2000 is invested at 6% per annum for 3 years.  [4]

4.

A bill comes to £89.30 inclusive of VAT at 17½ %. How much as the bill before VAT was added ?  [3]

5.

Every year, an item of furniture depreciates by 15% of its value at the start of that year. An item of furniture is bought for £3000. How much will it be worth in 3 years time?  [3]

6.

a)     The population of a country increased from 56 000 000 to 59 500 000. What percentage increase is this ?  [3]

b)     What will be the amount if £5000 is invested for 3 years at the rate of 4% compound interest per annum?  [3]

1.

Daniel, Richard and Tina share £200 in the ratio of 4:5:7. Calculate how much each one receives.  [3]

2.

Jill and Alan invest some money and share any profit made in the ratio 5:4.

a)     How much does Jill get when they make a profit of £270 ?  [2]

b)     On another occasion, Alan receives £136.
How much profit were they sharing ?  [2]

3.

Arwyn, Betty and Clive share £3600 in the ratio 4:5:9.

How much do they each get ?  [3]

1.

In triangle ABC, ÐBAC = 135° measured correct to the nearest degree. AC = 4.9 cm and AB = 3.8 cm both measured correct to the nearest mm.

Find correct to three significant figures, the greatest possible area of triangle ABC. [3]

2.

A time of 24.4 seconds, measured to the nearest tenth of a second, was recorded for the winner of a 200 metres race. The race track had been marked out to within an accuracy of ±0.1%. Explaining clearly your reasoning, calculate the greatest and least possible values of the average speed of the winner, giving your answers in metres per second.  [5]

3.

A rod is 18.5 cm measured to the nearest mm.

a)     Write down the least possible length and the greatest possible length of the rod.  [2]

b)     Keri places 30 of these rods end to end in a straight line. Write down the least possible length and the greatest possible length of this straight line of 30 rods.  [2]

4.

The diagram shows a cuboid with dimensions 2.6 cm, 3.4 cm and
4.2 cm measured correct to the nearest mm. Two vertices are labelled on the diagram, P and Q.

Find the greatest length of diagonal PQ.  [3]

5.

The capacity of a jug is 250 ml, measured to the nearest 10 ml.

a)     Write down the least and greatest value of the capacity of the jug.  [2]

b)     The capacity of a bucket is 5.1 litres, measured correct to the nearest ¹/₁₀ of a litre. The jug is filled with water and then the water is poured into the bucket. This is done 20 times in all. Explain, showing all your calculations, why it is not always possible for the bucket to hold all this water.  [5]

6.

Sacks are filled with 50 kg of sand measured correct to the nearest kg.

a)     Write down the least and greatest amounts of sand there could be in the sack.  [2]

b)     A person buys 20 sacks of sand. Write down the least and greatest amounts of sand he could receive.  [2]

1.

a)     A pebble is dropped from rest and falls a distance d metres in t seconds, The distance d is proportional to the square of the time t. Given that the pebble falls 1¼ metres in the first ½ second, find an expression for d in terms of t.   [3]

b)     Calculate the distance that the pebble falls in the first 3 seconds.  [1]

c)     Calculate the time taken in seconds for the pebble to fall 405 metres from rest.  [2]

2.

A pebble is thrown vertically upwards with a speed of s metres per second. The pebble reaches a maximum height of h metres, before falling vertically downwards.

It is known that h is directly proportional to the square of s.

a)     Given that a pebble thrown with a speed of 10 ms^-1 reaches a maximum height of 5 metres, find an expression for h in terms of s.  [3]

b)     Calculate the maximum height reached when a pebble is thrown with a speed of 5 metres per second.  [2]

c)     The pebble reaches a maximum height of 0.45 metres. Calculate the speed at which the pebble was thrown.  [2]

3.

Spheres are made of a particular metal. The mass, m grams, of such a sphere is directly proportional to the cube of the radius, r centimetres.

a)     Given that the mass of a sphere with radius 2 cm is 80 g, find an expression for m in terms of r.  [3]

b)     Calculate the mass of a sphere with radius 4 cm.  [1]

c)     Calculate the radius of a sphere of mass 270 g.  [2]

4.

Given that y is inversely proportional to x², and that y = 2 when x = 5.

a)     Find an expression for y in terms of x.  [3]

b)     Calculate the value of y when x = 2.  [1]

c)     Calculate a value of x when y = 0.5.  [2]

5.

An alloy is made by using copper and zinc in the ratio 17:3.

a)     How much zinc is used to make 4 kg of the alloy ?  [2]

b)     There is only 1.5 kg of zinc available, but plenty of copper. What is the greatest amount of the alloy that can be made ?  [2]

1.

A ladder which is 7.6 m long is placed against a vertical wall. The foot of the ladder rests on a horizontal floor and is 2.4 m away from the bottom of the wall. Calculate how far the top of the ladder is above the floor.  [3]

2.

PQRS is a rectangle in which its diagonal SQ is 23 cm long. The line ST is drawn 10 cm long to meet the line TQ so that ÐTSR = 36° and ÐSTQ = 90°. Find ÐTSQ and the length of RQ. [5]

3.

In the diagram below, ÐABC = 90°, ÐBED = 90°, ÐBAC = 37°,
AB = 17.8 m, CD = 23.6 m and BE = 21.4 m.

Calculate the size of ÐBDE.   [5]

4.

The diagram shows a cuboid.

AB = 5 cm, BC = 7 cm and
CG = 15 cm.

Calculate ÐAGD, giving your answer to an appropriate degree of accuracy.  [4]

5.

PQRS represents the symmetrical cross-section of the roof of a house, where SQ is perpendicular to PR and Q is the mid-point of PR. The width of the house, PR is 7.2 m and the length of the rafter, PS, is
4.3 m. Calculate the height SQ.  [3]

6.

a)     The angle of elevation of the top of a building from a point 75 m horizontally from the foot of the building is 48°. Calculate the height of the building, giving your answer to an appropriate degree of accuracy.  [3]

b)     Triangle PQR is right-angled at R. The length of PQ is 35 m and the length of QR is 12 m. Calculate the size of ÐQPR.  [3]

7.

AB and CD represent the vertical walls of two buildings that are 8.3 m apart on level ground AC. The points B and D are 10.6 m and 15.2 m vertically above the ground respectively.

Calculate the distance BD.  [4]

8.

ACD and BCE are two triangles right-angled at C. The point D lies on CE at a distance of 23.7 cm from C and B lies on AC such that AB is 5.7 cm. The side BE has length 63 cm.

Calculate the size of ÐBEC.   [6]

9.

A vertical flagpole, BDC stands on horizontal ground ABE. It is supported by two ropes AC and DE. The length of AC is 13.5 m, and the distance CD is 4.7 m. The rope AC makes an angle of 62° with the ground and the rope DE is fixed to the ground at E such that BE is 8.4 m.

Calculate the size of ÐBDE.   [6]

10.

The diameter of a circle, AB, is of length 8.7 cm. BC has length 5.4 cm and ÐACB = 90°.

Calculate the length of AC.  [3]

1.

The diagram represents a solid metal bar with a uniform cross-section in the form of the trapezium ABCD, in which AB = 9.3 cm and DC = 5.8 cm. The height of the bar is 3.5 cm and the length of the bar is 14.7 cm. The density of the metal is 5.6 g/cm³.

Calculate the mass, in kilograms, of
the bar.   [4]

2.

The diagram shows a cuboid of length 53.1 cm. The cross-section, PQRS, is such that PR = 24.7 cm and QR = 16.3 cm.

a)     Calculate the length of PQ.   [3]

b)     The density of the material from which the cuboid is made is
4.3 g/cm³. Calculate the mass of the cuboid in kilograms.  [3]

3.

ABCFED represents the uniform cross-section of a solid block of material. ABCD is a rectangle in which AB = 6.1 cm and BC = 4.2 cm. EF is of length 2.3 cm and is parallel to AB. The distance between EF and AB is 8.5 cm.

a)     Calculate the area of cross-section of the block.  [3]

b)     The block has this uniform cross-section along its length of 12.6 cm and has a mass of 2 kg. Calculate the density, in g/cm³, of the material from which the block is made.  [4]

4.

The diagram represents a prism with a uniform cross-section of area 78 cm³. The prism is 54 cm long and has a mass of 19.6 kg. Find the density, in g/cm³, of the material from which the prism has been made.  [3]

5.

In quadrilateral PQRS, the line PQ is parallel to SR, with PQ = 16 cm

and SR = 18 cm. The perpendicular distance between PQ and SR is 8 cm. Calculate the area of the quadrilateral PQRS.  [2]

1.

The speeds of 120 cars on a stretch of motorway were measured and the following results were obtained,

Find an estimate for the mean speed of the cars.  [4]

2.

The weekly expenses of the people employed by a travel company are summarised in the grouped frequency distribution below.

a)     Use the distribution to calculate an estimate of the mean and standard deviation of the weekly expenses of these employees.  [5]

b)     The weekly expenses of all employees are reduced by £5. Use your answers for the mean and standard deviation from part a) to calculate estimates of the mean and standard deviation for the new reduced weekly expenses. [2]

3.

A grouped frequency distribution of the marks scored by 90 girls in an English examination is given in the table below.

An estimate for the mean marks scored by the girls is 57.5.

a)     Calculate an estimate for the standard deviation of the marks.  [4]

b)     Each pupil had her mark for the examination increased by 3 marks for good spelling, punctuation and grammar. State estimate for the mean and standard deviation of the increased marks.  [2]

4.

The following table shows the distribution of the ages, in years on their last birthday, of 140 members of a youth sports club on the 1^st November 2003.

a)     Calculate, correct to two decimal places, estimates for the mean and standard deviation of these ages.  [5]

b)     The membership of the youth sports club does not change over the next 12 months. Write down estimates for the mean and standard deviation of the ages on the 1^st November 2004.  [1]

5.

Calculate the mean and standard deviation of the following set of 12 numbers.

34, 23, 35, 64, 56, 52, 48, 32, 40, 57, 36, 45     [3]

6.

A survey of the cost per litre of unleaded petrol at garages in a particular area was carried out. It was calculated that the mean was 76.8p and the standard deviation was 2.8p. Following a price cut all the garages then reduced the price of unleaded petrol by 1p per litre. Write down the mean and standard deviation of the unleaded petrol following the price cut.  [2]

1.

The table below shows a grouped frequency distribution for the number of empty seats on 90 flights from Heathrow to Florida.

a)     On graph paper, draw a frequency polygon for this data. Use the scale 2cm = 5 units on the x-axis and 2 cm = 4 units on the y-axis  [2]

b)      Calculate an estimate for the mean number of empty seats on each flight. [4]

2.

The following table shows the value of a selection of 90 houses on one estate agency’s books.

a)     On graph paper, draw a frequency polygon for this data. Use a scale of 2 cm = 20 units of the x-axis and 2 cm = 10 units on the y-axis.  [2]

b)     Find an estimate for the mean price for these houses.  [4]

3.

The batting scores of 100 cricketers were recorded and the results are summarised in the following table.

a)     Draw a frequency polygon for the data.  [2]

b)     Find an estimate for the mean of the batting scores.  [4]

1.

The times taken, in minutes, by 120 people to complete a task were recorded. Below is a cumulative frequency polygon of the results.

Use the cumulative frequency polygon to find

a)     The median time taken to complete the task.  [1]

b)     How many people took more than 38 minutes to complete the task.   [2]

2.

The times taken by 160 pupils to travel to school were measured and the results are summarised in the following table.