Shaded questions are from non-calculator papers
Expanding brackets and simplifying
|
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 |
|
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
|
|
5. |
Expand and simplify (x + 5)(x - 6) [2] |
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6. |
Expand and simplify 4(2y - 3) - 3(y + 5) [2] |
|
7. |
Expand the following expression,
simplifying your answer as far as possible
|
Transforming formulae
|
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] |
Factorising
|
1. |
Factorise 6x² - 8xy [2] |
|
2. |
Factorise 3a² - 6ac [2] |
|
3. |
Factorise 3xy² - 6xy [2] |
|
4. |
Factorise 4x²y - 6xy² [2] |
Simultaneous Equations
|
1. |
Solve the following simultaneous equations by an algebraic (not graphical) method. [4]
|
3x - 2y = 16 |
|
2. |
Solve the following simultaneous equations by an algebraic (not graphical) method. [4]
|
4x + 5y = -5 |
|
3. |
Solve the following simultaneous equations by an algebraic (not graphical) method. Show all your working. [4]
|
3x - 4y
= 22 |
|
4. |
Solve the following simultaneous equations by an algebraic (not graphical) method. Show all your working. [4]
|
4x + 2y = 17 |
|
5. |
Solve the following simultaneous equations by an algebraic (not graphical) method. Show all your working. [4]
|
4x - 3y = 20 |
|
6. |
Solve the following simultaneous equations by an algebraic (not graphical) method. Show all your working. [4]
|
5x + 2y = 10 |
Quadratics Factorising
|
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] |
Loci
|
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 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] |
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3. |
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. |
· 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.
|
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.
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
points are further from B than the distance BC. [2]Straight line graphs
|
1. |
Find the equation of the line that passes through the points (0, 8) and (-2, 2) [4] |
Forming expressions (Higher tier)
|
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] |
Scatter Diagrams
|
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] |
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|
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 c) Use your line of best fit to estimate the acceleration time for a car with an engine capacity of 1750 cc. [1] |
Percentages
|
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 1st Marcus owed £250 on his credit card account. The credit card company requires Marcus to pay at least 10% of the balance on the 20th of each month. The company charges interest at 2% on what the balance is on the 28th of every month. Marcus pays the minimum payment on time every month. Write down the full details of his account up to May 31st. [3] April 1st £250 |
|
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] |
Ratio
|
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. |
|
3. |
Arwyn, Betty and Clive share £3600 in the ratio 4:5:9. How much do they each get ? [3] |
Upper and Lower Bounds
|
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.
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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] |
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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
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|
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] |
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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] |
Find
correct to three significant figures, the greatest possible area of triangle
ABC. [3]
Find
the greatest length of diagonal PQ.
[3]Proportion
|
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] |
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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] |
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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] |
Right-Angled Trigonometry
|
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]
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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]
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|
3. |
Calculate the size of ÐBDE. [5]
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|
4. |
The diagram shows a cuboid. AB = 5 cm, BC = 7 cm and
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5. |
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|
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) |
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|
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
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 
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 
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.
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|
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.
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|
10. |
The diameter of a circle, AB, is of length 8.7 cm. BC has length 5.4 cm and ÐACB = 90°.
|
Calculate
the size of 
Calculate
the length of AC. [3]Area and Volume
|
1. |
Calculate the mass, in kilograms, of |
|
2. |
a) Calculate the length of PQ. [3] b) The
density of the material from which the cuboid is made is
|
|
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)
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|
4. |
|
|
5. |
In quadrilateral PQRS, the line PQ is parallel to SR, with PQ = 16 cm
|
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. 
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.
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]
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]
and
SR = 18 cm. The perpendicular distance between PQ and SR is 8 cm. Calculate the
area of the quadrilateral PQRS. [2]Estimating Mean and Standard Deviation
|
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] |
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|
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] |
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|
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] |
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|
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 1st 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 1st November 2004. [1] |
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|
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] |
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|
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] |
Frequency Polygons
|
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] |
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|
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] |
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|
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]
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Cumulative Frequency
|
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] |
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|
2. |
The times taken by 160 pupils to travel to school were measured and the results are summarised in the following table.
a) Copy
and complete the following cumulative frequency table. [1]
b) On graph paper, draw a cumulative frequency diagram to show this information. [3] c) Use your cumulative frequency diagram to find the interquartile range. [2] d) Use
your cumulative frequency diagram to complete the following statement. |
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|
3. |
A group of 200 pupils sat an examination. The table gives a grouped frequency distribution of their marks in the examination.
a) Copy
and complete the following cumulative frequency table. [1]
b) On graph paper draw a cumulative frequency table (x-axis, 2 cm = 10 units , y-axis 2 cm = 20 units) [3] c) Use your cumulative frequency diagram to find the median. [1] d) The minimum mark for the top grade A was 58. Use your cumulative frequency diagram to estimate how many pupils achieved grade A. [2] |
Transformations
|
1. |
a) |
|
|
|
b) |
i) Rotate the shape C through 90° anti-clockwise about the point (1, -2). Label the image D. [2] ii) Draw the image of shape C after a translation of –5 units in the x direction and –3 units in the y direction. [2]
|
|
2.
|
|
Find the single transformation
which takes triangle ABC to |
|
3. |
a) |
|
|
|
b) |
|
Reflect
the shape A in the line y = 
The
diagram shows triangles ABC and A1B1C1 drawn
to scale.
Draw
the image when the triangle ABC is reflected in the line y = 
Draw
the image when the triangle marked D is rotated through 90° anticlockwise
about the Standard Form
|
1. |
Write each of the following in standard form a) 3895584 [1] b) 0.0000002567 [1] |
|
2. |
Find, in standard form, the value of: a) (4 ´ 10-4) ´ (1.2 ´ 10-5) [1] b) [2] |
|
3. |
a) Write the following number in standard form 0.0026 [1] b) Find,
in standard form, the value of |
|
4. |
a) The
following numbers have been written in standard form. Write each in
decimal form. b) Find
in standard form, the value of |
|
5. |
Find, in standard form, the value of |
|
6. |
a) Write
each of the following numbers in standard form. b) Find,
in standard form, the value of |
Probability
|
1. |
A survey of cars was carried out. It was noted whether the cars were up to 3 years old inclusive or over 3 years old. It was also noted whether the cars had a diesel engine or a petrol engine. The results of the survey were as follows.
Use this information to estimate how many cars with diesel engines you would expect to find in a country known to have 40 000 cars. [3] |
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2. |
A bag contains 7 banana, 6 cherry, 5 lemon and 2 pineapple flavoured sweets. Two sweets are selected at random without replacement from the bag. Calculate the probability that a) The two sweets are both banana. [2] b) The two sweets are either both lemon or both pineapple. [2] c) Exactly one of the sweets is pineapple. [3] d) At least one of the sweets is lemon. [3] |
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|
3. |
a) Copy
and complete the following tree diagram.
[2] b) Calculate the probability that both dice show 4. [2] c) Calculate the probability that exactly one dice shows 4. [2] |
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4.
|
At a certain driving test centre a record was kept of the gender and age of each driving test candidate. On the basis of these records, the probability of a randomly selected driving test candidate being a male under 25 is estimated to be 0.6. It was also estimated that the probability of a randomly selected driving test candidate being a female under 25 is 0.3. a) Using these estimates, calculate the probability that a randomly selected driving test candidate is not a male under 25. [1] b) Consider the next two driving test candidates. Calculate the probability that i) Both are female under 25, [2] ii) Only one of them is a male under 25. [3] |
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5. |
A bag contains 7 blue balls and 5 green balls. Another bag contains 4 blue balls and 6 red balls. A ball is drawn at random from the first bag and its colour is noted. A ball is then drawn at random from the second bag and its colour is noted. a) b) Calculate the probability that both balls are blue. [2] c) Calculate the probability that at least one ball is blue. [2] |
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6. |
a) Copy and complete the following tree diagram by entering all the probabilities on the branches. [2] b) Calculate the probability that both letters chosen are vowels. [2] c) Calculate the probability that only one letter is a vowel. [2] |
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7. |
Two bags contains some coloured balls, which are identical
except for their colour. One ball is taken at random from each bag and their
colours noted. The probability of the selected ball from the first bag being a) Draw a tree diagram to show this information. [2] b) Calculate the probability that both balls are red. [2] c) Calculate the probability that only one ball is red. [3] |
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8. |
A sample of boys and girls at a school yielded the following results for their eye colour. There are 930 boys and 720 girls at the school. Use the results of the sample and these totals to find an estimate for the total number of pupils in the school with brown eyes. [4]
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9. |
A candidate sits a multiple choice examination. For each question in the examination, five possible answers are given but only one of these answers is correct. The candidate knows 70% of the facts tested in the examination and for each question based on these facts she selects the correct answer. On all other questions she selects at random one of the five possible answers. a) A question is selected at random from the paper. Calculate the probability that the candidate correctly answers the question. [4] b) The examination has 150 questions. Calculate how many questions you might expect the candidate to answer correctly. [2]
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10. |
a) Copy and complete this tree diagram. [2] b) Calculate the probability that an item chosen at random is accepted. [2]
|
Vivienne
has a fair cubical dice with its faces numbered from 1 to 6 and a biased dice
for which the probability of throwing a 4 is .
She throws the two dice and notes whether or not a 4 is obtained on each
dice.
Copy
and complete the following tree diagram.
[2]
Paul
selects a letter at random from a passage of text, he then selects a second
letter at random from the text. The probability that any letter is a vowel is
.
A
factory has two machines, A and B, which it uses to make large numbers of a
certain item. Machine A is used to make 60% of the factory’s total output and
Machine B is used for the remainder. The probability that an item made on
Machine A is rejected is 0.1. The probability that an item made on Machine B
is rejected is 0.2.
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11. |
A clown has seven pairs of shoes, one pair in each of the colours of the rainbow. The shoes are kept in a trunk in a dark room. The clown selects two shoes at random. a) What
is the probability that the clown selects one left shoe and one b) What is the probability of selecting a matching pair of shoes ? [2] |
|
12. |
A bag contains coloured counters, 3 green, 4 blue, 1
yellow and 2 red. a) The two counters are both red. [2] b) Exactly one of the counters is red. [3] |
|
13. |
A professional navigation assessment consists of two elements. The first element is a practical test and the second element is a written examination. Past records show that
Calculate the probability that a candidate a) Passes both the written examination and the practical test. [2] b) Passes only one of the two assessment elements. [3] c) Fails both the written examination and the practical test. [2] |
|
14. |
In a small pack of nine cards, the cards are numbered 1, 2, 3, 4, 5, 6, 7, 8 and 9 respectively. A fair cubical dice has faces numbered 1, 2, 3, 4, 5 and 6 respectively. Terry draws a card at random from the pack and rolls the dice. Calculate the probability that the number on the card is even and that the dice shows 5. [3] |
|
15. |
Peter and Rees regularly play chess matches against each other, each match consisting of two games. The probability that Peter will win the first game of the match is 0.3. If Peter wins the first game of a match, the probability that he will win the second game is 0.6. If Peter does not win the first game of a match, the probability that he will win the second game is 0.1. a) Calculate the probability that Peter will win both the first and second games of a match. [2] b) Calculate the probability that Peter will win at least one game in a match. [3] |
|
16. |
The letters of the word MAESTEG are written on seven cards, one letter per card and placed in a box. Similarly, the ten letters of CAERNARFON are written on ten cards and placed in another box. A person selects one card at random from each of the two boxes. What is the probability that the person has the letter E on both cards ? [3] |
Inequalities
|
1. |
Find all integer values of n that satisfy the inequality -10 < 5n £ 17. [3] |
|
2. |
On graph paper using a scale of 2 cm for 1 unit on both axes, and a range of x from -3 to 4 and of y from -4 to 6, draw the reqion which satisfies all of the following inequalities. x ³
-3 Make sure that you clearly indicate the region that represents your answer. [4] |
|
3. |
a) Rearrange
the inequality 35 -
3n > 2n + 7 into the form b) Given that n also satisfies the inequality 3n > 1, write down all the integer values of n that satisfy both inequalities. [2] |
|
4. |
On graph paper and using a scale of 2 cm = 1 unit on both axes, draw the region, which satisfies all of the following inequalities. Draw your axes from -5 to 5. x < 4 Make sure that you clearly indicate the region that represents your answer. [3] |
Indices
|
1. |
a) Simplify each of the following b) Given that 0 < x < 1 write in ascending order. [2] |
|
2. |
Simplify . [2] |
|
3. |
Simplify |
|
4. |
Simplify 2a5b² ´ 3a³b. [2] |
|
5. |
Simplify the expression [2] |
|
6. |
Simplify each of the following |
|
7. |
Simplify |
|
8. |
Simplify leaving your answer in fractional form. [2] |
|
9. |
Simplify
|
|
10. |
Simplify each of the following |
|
11. |
a) Simplify [2] b) Simplify [3] |
Curved graphs
|
1. |
a) Copy and complete the table for the equation y = 2x² - 3x
b) Draw the graph of y = 2x² - 3x. c) Use your graph to find the value of y when x = 2.3 d) Use your graph to find the value of y when x = -1.5 e) Use your graph to solve 2x² - 3x = 5 f) Use your graph to solve 2x² - 3x = 8 |
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|
2. |
The table shows some of the values of y = 2x² - 5x
- 8
for values of x
a) Copy and complete the table by finding the value of y for x = 3. [1] b) On graph paper, draw the graph of y = 2x² - 5x - 8 for values of x between -2 and 4. Use a scale of 2cm = 1 unit on both axes. The x axis should extend from -2 to 5 and the y axis from -12 to 10. [2] c) Draw the line y = 3 on your graph and write down the x-values of the points of intersection of your line with y = 2x² - 5x - 8. [2] d) Write
down and simplify the equation in x whose solution you found
|
|
3. |
The graph of y = x³ - 5x² - 12x + 36 for values of x between x = -4 and x = 7, has been drawn below.
a) Use the graph to solve the equation x³ - 5x² - 12x + 36 = 0. [1] b) Using
the graph, estimate the gradient of the curve c) By
drawing an appropriate line on the graph, solve the equation d) Use the trapezium rule with 5 strips to estimate the area enclosed by the x – axis and the curve between x = -3 and x = 2. [4] |
|
4. |
The diagram shows a sketch of y = x². a) On a copy of the diagram, sketch the curves i) y = 4x² [1] ii) y = 4x² - 1 [1] b) |
|
5. |
The graphs of y = 9 - x² and y = x² - 3x - 4 have been drawn below.
a) Write down the gradient of the graph y = 9 - x² at x = 0. [1] b) Use
the graph of y = x² -
3x - 4
and the graph of y = 1 to solve c) Write
down and simplify the equation in x satisfied by the |
|
5. |
a) Using
a scale of 2 cm = 1 unit for both axes, draw the graph of b) Use your graph to estimate the gradient of y = x² - x - 5 at the point where i) x = ½, [1] ii) x = 1 [3] c) On the same axes, plot the graph of y = -x² for -2 £ x £ 2. [2] d) Use the graphs of y = x² - x - 5 and y = -x² to solve 2x² - x - 5 = 0 [2] |
Write
down the coordinates of the points where the curve y = 4x² 
|
6. |
a) Copy
and complete the table which gives the values of y = 2x² + 4x - 5
for values of x ranging from -4
to 3. [2] b) On graph paper, using a scale of 2 cm = 5 units on the y –axis and 2 cm = 1 unit on the x – axis, draw the graph of y = 2x² + 4x - 5 for values of x ranging from -4 to 3. [2] c) Draw the line y = 8 on the same graph paper and write down the x-values of the points where your two graphs intersect. [2] d) Write
down the equation in x whose solutions are the x-values you |
Estimation
|
1. |
ESTIMATE the value of , giving your answer as a decimal. Show clearly how you obtain your answer. [3] |
Trial and Improvement
|
1. |
A solution to the equation x³ - 5x - 66
= 0 lies between 4.4 and 4.5. |
|
2. |
A solution to the equation x³ - 6x - 3
= 0 lies between 2.6 and 2.7. |
|
3. |
A solution to the equation x³ +4x - 8 = 0 lies between 1.3
and 1.4. |
Calculator Skills
|
1. |
Use your calculator to find the value of correct to 2 decimal places. [2] |
Solving linear equations
|
1. |
Solve the equation 9x - 1 = 4(x + 5). [3] |
|
2. |
Solve the equation 6(x - 5) = 4x + 1 [3] |
|
3. |
Solve the equation 16x - 5 = 3(4x + 7) [3] |
|
4. |
Solve the following equation [3] |
|
5. |
Solve the following equation [3] |
|
6. |
Solve the following equation [3] |
|
7. |
Solve the following equation [3] |
|
8. |
a) Write down an equation that x satisfies. [2] b) Solve the equation and hence find the length of a side of the pentagon. [3] |
|
9. |
Solve the equation 5x - 6 = 3(10 - x) [3] |
|
10. |
Solve the following equation [3] |
|
11. |
Solve the equation 4(2x - 5) = 2x + 1. [3] |
|
12. |
a) Given that the length of the fence is 74 metres, write down an equation that x satisfies. [2] b) Solve the equation to find the value of x. [2] |
|
13. |
Solve the following equation [3] |
The
sides of a regular octagon are x cm long. Each side of a regular pentagon is
6 cm longer than each side of the octagon. The perimeter of the octagon is 3
cm longer than the perimeter of the pentagon. 
The
diagram represents a rectangular lawn measuring 15 metres by 12 metres,
surrounded by a path of width x metres. There is a fence all around the
outside of the path.Prime Factors
|
1. |
a) Write 600 as the product of its prime factors in index form. [2] b) What is the smallest number that 600 must be multiplied by so that the answer is a square number ? [1] |
|
2. |
Express 504 as a product of prime numbers in index form.[3] |
|
3. |
Express 882 as a product of prime numbers in index form.[3] |
|
4. |
a) Write 700 as the product of its prime factors in index form. [3] b) Use your result in part (a) to write down the smallest multiple of 700 which is a perfect square. [1] |
|
5. |
Express 108 as a product of prime numbers in index form. [3] |
Dimensions
|
1.
|
The diagram shows a solid. The lengths D, R and H are as shown.
One of the following formulae may be used to estimate V, the volume of the solid. V = 3H + 2R + 5D V = 3R + 5DR V = 3R²H + 2R²D V = 3R(4D + 5H) a) Explain why the formula V = 3H + 2R + 5D cannot be used to estimate the volume of the solid. [1] b) State, with a reason, which of the above formulae may be used to estimate the volume of the solid. [2] |
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|
2. |
In each of the following formulae, every letter stands for the measurement of a length. By considering the dimensions implied by each formula, write down, for each case, whether the formula could be for a length, an area, a volume or none of these. The first one has been done for you. [2]
|
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|
3. |
In each of the following formulae, every letter stands for the measurement of a length. By considering the dimensions implied by each formula, write down, for each case, whether the formula could be for a length, an area, a volume or none of these. The first one has been done for you. [2] |
|
4. |
Each of the following quantities has a particular number of dimensions. Give the number of dimensions of each quantity. The first has been done for you. [2]
|
Similarity
|
1. |
The diagram shows two similar
triangles in which ,
AC = 15, BC = 20, XY = 6
Calculate the length of a) YZ [2] b) AB [2] |
|
2. |
a) Calculate the volume of the smaller cone. [3] b) The smaller cone is melted down and recast as 20 identical cylinders. The length of each cylinder is 1.8 cm. Calculate the radius of each cylinder, giving your answer to an appropriate degree of accuracy. [4]
|
|
3. |
In the diagram, BC is parallel to DE, and the triangles
ABC and ADE are similar. AB = 9 cm, AC = 6 cm, Showing all your working, find the length of
|
|
4. |
The diagram shows two similar cylinders. The radius of the smaller cylinder is half the radius of the larger cylinder. The volume of the smaller cylinder is 200 cm³.
|
|
5. |
a) |
|
|
b) |
|
6. |
In the diagram, AB is parallel to DE, and the triangles ABC and EDC are similar. AB = 6 cm, AC = 8 cm, DE = 7.2 cm and CD = 10.8 cm. Showing all your working, find the length of
|
|
7. |
Two cones are similar in shape and have surface areas of 60 cm² and 135 cm². The height of the smaller cone is 5 cm. Find the height of the larger cone. [2] |
A
solid metal cone has a height of 80 cm and radius of 30 cm. A smaller cone
has height 20 cm is obtained by cutting off the top of the original cone.
a) BC
[2] b) AE
[2]
Find
the volume of the larger cylinder.
[2]
Explain
clearly why triangles ABC and
Explain
clearly why triangles PQR and XYZ are not similar.
a) CE
[2] b) BC
[2]
|
8. |
a. Every square is similar to every other square. Name another geometrical figure that has this property. [1] b. Every cube is similar to every other cube. Name another 3 dimensional object that has this property. [1] |
|
9. |
The diagram shows a special edition postage stamp and an enlargement of the stamp.
|
The
height of the original stamp is 1.8 cm, and the height of the enlarged stamp
is 3.6 cm. The area of the original stamp is 6.4 cm². Find the area of the
enlarged stamp. [2]Rational, Irrational Numbers and Surds
|
1. |
Expand and state whether the result is rational or irrational. [2] |
|
2. |
a) Express as a fraction. [2] b) Show that [2] |
|
3. |
a) Express as a fraction. [2] b) Write down a value of x for which is rational. [1] c) Give an example of an irrational number whose square is rational. [1] d) Give an example of an irrational number whose square is irrational. [1] e) Find the value of [2] |
|
4. |
Express as a fraction. [2] |
|
5. |
Given that , simplify each of the following, indicating in each case whether your answer is rational or irrational. [5] |
|
6. |
Express as a fraction. [2] |
|
7. |
Show that [2] |
|
8. |
a) Express as a fraction. [2] b) Simplify and state whether your answer is rational or irrational.[3] |
Advanced Algebraic Fractions
|
1. |
Express as a single fraction, simplifying your answer. [4] |
|
2. |
a) Factorise x² - 16. [1] b) Simplify [3] |
|
3. |
Simplify [4] |
|
4. |
Simplify the expression [3] |
|
5. |
Express the following as a single fraction in its simplest form, [4] |
Sectors. Arc Length
|
1. |
Calculate the length of arc GH, giving your answer to an appropriate degree of accuracy. [3] |
|||
|
2. |
OA and OP are radii of the smaller circle. OB and OQ are radii of the larger circle. The radii of the smaller circle is 3.8 cm and ÐBOQ = 75°. a) The area of the sector BOQ is 48.2 cm². Calculate the radius of this sector. [3] b) Calculate the area of the triangle AOP. [3] |
The
diagram shows two concentric circles with centre O.Transforming graphs
|
1.
|
a) |
The diagram shows a sketch of y = f(x). On the same diagram, sketch the curve of y = f(x) - 2. Mark clearly the coordinates of the point where the curve crosses the y-axis. [2]
|
|
|
b) |
The diagram shows a sketch of y = g(x). On the same diagram, sketch the curve of y = -g(x). Mark clearly the coordinates of the point where the curve crosses the y-axis. [2]
|
|
|
c) |
The diagram shows a sketch of y = h(x). On the same diagram, sketch the curve of y = h(x - 3). Mark clearly the coordinates of the point where the curve crosses the x-axis. [2]
|
|
2. |
a) |
|
|
|
b) |
|
|
|
c) |
|
The
diagram shows a sketch of y = f(x). On the same diagram, sketch the curve of
y = 
The
diagram shows a sketch of y = g(x). On the same diagram, sketch the curve of
y = g(x + 2). Mark clearly the coordinates of the point where the curve
crosses the x-axis. [2] 
The
diagram shows a sketch of y = h(x). On the same diagram, sketch the curve of
y = h(x) + 3. Mark clearly the coordinates of the point where the curve
crosses the y-axis. [2] Histograms
|
1. |
a) Use the histogram to calculate the number of driveways measured. [3] b) Find the length exceeded by 75% of the driveways measured. Give your answer to 2 decimal places. [3] |
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|
2. |
The heights of a group of children are summarised in the grouped frequency distribution. a) Copy and complete the frequency density column and draw a histogram. [3] b) Calculate an estimate for the number of children in the group whose heights are at least 142 cm. [3] |
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A survey of time wasted by pupils at a school in one hour of supervised study time was carried out. The lengths of times wasted were noted by an observer. The results are summarised in the grouped frequency distribution. a) Copy and complete the frequency density column and draw a histogram. [3] b) |
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|
4. |
a) Use
the histogram to complete the grouped frequency table below. [1]
b) Calculate
an estimate of the number of cars with speeds exceeding c) A
further study was carried out after the placement of a “SLOW DOWN” warning
sign. The results are summarised in the grouped frequency distribution below.
Complete the frequency density row in the table and draw a histogram to illustrate the results of this survey. [3] d) Compare the two histograms. Do you consider the “SLOW DOWN” warning sign to have been effective ? Give a reason for your answer. [1] |
A
survey was carried out to measure the lengths of the driveways to a number of
houses. The histogram shows the results of the survey.
Vectors
|
1. |
ABCD is a parallelogram. BAF is a straight line. FE is parallel to AD. AF = 2BA and FE = 3AD.
Diagram not drawn to scaleGiven that AB = p and AD = q, express each of the following in terms of a) AC [1] b) FE [1] c) BF [1] d) CE [2] |
|
2. |
Given that AD = x, AB = y and DH = z, express each of the following in terms of x and y. Give your answers in the simplest form. [3] a) AC b) AM c) MH |
The
diagram shows a cuboid ABCDHGFE with M the mid-point of BF.
|
3. |
The diagram shows a parallelogram OXYZ. The point P is on OX such that OP : PX = 1 : 2. The point R is on OY such that OR : RY = 1 : 5. a) b) Show that ZX = 6RP [3] c) Describe fully the geometrical relationship between ZX and RP. [2] |
|
4. |
PQRS is a parallelogram. The mid – point of QR is J and the mid-point of RS is K.
a) Given
that PQ = a, PS = b, express each of the following in terms of a
and b. b) Describe fully the geometrical relationships between QS and JK. [2] |
|
5. |
You are given that OP = a + b, OQ = 4a
+ 3b and a) Express
each of the following in terms of a and b in their simplest
form. b) Show that PR = kPQ where k is a constant. [1] c) Explain fully the geometrical implication of your answer in part b). [2] |
|
6. |
The points P and Q are the midpoints of OZ and XY respectively. Given that OX = x, OY = y and ZY = x, a) Express,
in their simplest form, each of the following in terms of x and y. b) Show that PQ is parallel to OX [2] |
Given
that OX = x and OY = y,
express each of the following in terms of x and y.
Vectors
OP, OQ and OR are shown
in the diagram.
The
diagram shows a trapezium OXYZ. Angles in a Circle
|
1. |
Diagram not drawn to scale
Given that ÐAOD = 62°, find each of the following angles. Give reasons for your answers. a) ÐABD [2] b) ÐADC [2] c) ÐCAD [2] d) ÐCDP [2] |
|
2. |
Given that ÐDBA = 42°, find each of the following angles. Give reasons for your answers. a) ÐOAQ [1] b) ÐACD [1] c) ÐDEA [2]
|
|
3. |
The four points A, B, C and D lie on the circumference of a circle centre O. The tangent PQ touches the circle at C. The point X on the tangent PQ is such that OAX is a straight line.
a) ÐAOC b) ÐOXC c) ÐADC [4]
|
|
4. |
Find each of the following angles, giving reasons for your answers. a) ÐAED b) ÐAOD c) ÐBDC [4]
|
|
5. |
The tangent RT touches the circle at A.
Given that ÐACD = 30° and ÐCAD = 26°, find each of the following angles. a) ÐDAT [1] b) ÐBCA [1] c) ÐBPC [2] d) ÐDOC [1]
|
A,
B, C and D are four points on the circumference of a circle centre O. AC is a
straight line passing through the centre of the circle. The tangent PT meets
the circle at D.
The
five points A, B, C, D and E lie on the circumference of a circle centre O.
The tangent PQ meets the circle at A.
Given
that 
Five
points A, B, C, D and E lie on the circumference of the circle centre O with
BOD a straight line. The tangent RT touches the circle at C. 
The
diagram shows A, B, C and D are four points on the circumference of a circle
centre O. The diameter AOB is extended to P, so that BP = BC. Forming and Solving Quadratic Equations
|
1.
|
Triangle PQR is right-angled at Q. The length of QR is x cm and PQ is 3.6 cm longer than QR. The area of the triangle is 64.8 cm².
a) Show that x satisfies the equation x² + 3.6x - 129.6 = 0. [2] b) |
Solve
the equation x² + 3.6x
|
2. |
ABCD is a square of side 10 cm. The points P, Q, R and S lie on the sides of the square ABCD. AS = BP = CQ = DR = x cm.
The area of the square PQRS is 75 cm². a) a) Solve the equation 2x² - 20x + 25 = 0. [4] |
|
3. |
Solve the equation [7] |
|
4. |
The volume of a cuboid with height 8 cm, length (x + 2) cm and width (x - 5) cm is 20.6 cm³. a) Show that x satisfies the equation 8x² - 24x - 100.6 = 0. [4] b) Use
the formula method to solve the equation c) Hence write down the dimensions of the cuboid. [1] |
|
5. |
Use the formula to solve 2x² + 3x - 6 = 0, giving your answers correct to 2 decimal places. Show your working. [3] |
|
6. |
The surface area of a cuboid with length x cm, width (x - 1) cm and height 3 cm is 63 cm³. a) Show that x satisfies the equation 2x² + 10x - 69 = 0. [3] b) Solve the equation 2x² + 10x - 69 = 0, giving solutions to two decimal places. [3] c) Hence write down the dimensions of the cuboid. [1] |
|
7. |
a) Show
clearly that the equation maybe written as b) Factorise 6x² - 7x - 20 = 0 [2] c) Hence solve the equation [2] |
Show
that x satisfies the equation 2x² Random Samples
|
1. |
Castle High School has 1800 pupils. The table shows the number of pupils in each year group.
a) Explain why selecting the first pupil on each class register will not give a random sample. [2] b) Calculate the number of pupils from each year group who would be selected for a stratified random sample of exactly 110 pupils. [4]
|
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|
2. |
The table shows the age and gender distribution of members of a tennis club.
A stratified sample of 10 members is required. The sample is to be stratified with respect to age and gender. a) Calculate how many men under 30 should be in the sample. [2] b) Use the following extract from a table of random digits to select 2 men under 30 and 2 men aged 30 or over for the stratified random sample. Start from the first number and explain your method. [3] 25 79 46 25 02 93 68 58 13 71 46 04 |
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|
3. |
Some of the people visiting a historic site in Wales signed the visitors’ book and left their addresses. The table below shows the frequency distribution of the country of origin of the group of visitors.
a) Advertising material is to be sent to some visitors. A random sample of size 20 stratified on the basis of country of origin is to be selected from the above group of visitors for this purpose. Find the number of people from each of the five countries that should be selected for the sample.[5] b) Use the following extract from a table of random digits to show how you would select 8 persons from a list of the 92 visitors from Wales for the sample. Explain your method. [3] 34 45 98 78 13 45 03 65 72 39 92 57 06 34 39 08 99 62 29 81 47 11
|
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|
4. |
Eleri and Mick wish to survey pupils’ opinion concerning the proposed new sports facility in school. The table below gives the distributions of boys and girls within the Year Groups in the school.
a) Mick decides to select a stratified random sample of size 60, stratified with respect to Year Group and gender. i) Calculate the number of Year 7 girls that should be selected. [1] ii) Calculate the number of Year 11 boys that should be selected. [1] b) As the Year Group 11 pupils will not be in school next year, Eleri decides to select pupils only from Year Groups 7 to 10. She selects a stratified sample of size 48 by choosing the first 6 boys and 6 girls on each of these Year Group lists. i) Explain why this method of sampling is not stratified random sampling. [2] ii) Give one advantage of this method of sampling over stratified random sampling. [1] |
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|
5. |
The Headteacher of a school wants to investigate opinions of pupils about meals sold in the canteen. She asks the first 20 pupils in the queue for dinner their opinions. a) Explain why this is not a satisfactory method of selecting a sample of pupils to ask about school meals. [1] b) Explain how the Headteacher could select a random sample of 20 pupils. [2] |
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|
6. |
The table shows the details of the departments in a computer company.
|
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|
7. |
A survey is to be carried out on a Monday afternoon by the cinema management to ascertain views of the public on the choice of films screened at a local cinema. Give reasons why asking every tenth person outside the cinema will not give a) A representative sample [1] b) A random sample. [1] |
Sine and Cosine Rules
|
1. |
The diagram shows a trapezium DEFG. DE is parallel to GF. DE = 8.6 cm, GF = 14.8 cm, DF = 13.2 cm and ÐDEF = 103°. Calculate a) ÐEFD [3] b) ÐDFG [1] c) the length of DG [3]
|
|
2. |
Two ships A and B sail from port P.
Calculate the bearing of ship A from ship B at this time. [3] |
|
3. |
The diagram shows triangle ABC. The point D is on the side BC of the triangle.
|
|
4. |
Calculate ÐTSR, the angle the cable makes with the ground. [5] |
|
5. |
The diagram shows quadrilateral PQRS.
|
Ship
A sails out of the port on a bearing of 034° (N34°E) and ship B sails out of
port on a bearing of 126° (S54°E). When ship A is 10 km from 
Given
that 
A
vertical pole TR, 10.8 metres tall stands on sloping ground RS. A cable is
attached to the top of the pole at T and to the ground at S. The distance
Given
that Graphs of symbolic relationships
|
1. |
The graph shows the relationship y = pqx.
Given that y = pqx, use the graph above to find the values of p and q. [4] |
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|
2. |
The data in the table was recorder during an experiment. Results were recorded for two variables x and y. a) On graph paper, plot the values of y against the values of x³. [2] b) Before starting the experiment it was already known that y is approximately equal to px³ + q. Use your graph to estimate p and q. [3] |
Velocity-Time Graphs
|
1. |
a) Use
the trapezium rule with three strips of equal width to find the approximate
area of the shaded region between the curve and the t-axis from t = 0 to t =
3. [4] b) The
area found in part a) represents one of the following. |
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|
2. |
The graph below shows the speed of a train, in m/s, over a
period of a) Estimate the acceleration of the train at time t = 25 seconds. [3] b) The table below gives the speed of the train between t = 0 and t = 30.
c) Use
the trapezium rule with values taken from the table to estimate the distance,
in kilometres, travelled by the train between t = 0 and d) Hence estimate the total distance travelled during the 60 seconds. [1] |
The
velocity v metres per second, of a particle at time t seconds is given by the
equation v = t³ + 4t² + 1. A table of values of v for values of t between t =
0 and t = 3 is given here.
Trigonometric Graphs
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1. |
a) Sketch the graph of y = tan x for values of x from -180° to 180°. [3] b) Find all solutions of the following equation in the range -180° to 180°. tan x = -14.3 [2] |
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2. |
a) Sketch the graph of y = sin x for values of x from -180° to 180°. You must show the values -180° and 180° on the x-axis. [1] b) Write down the minimum and maximum values of y. [2] |
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3. |
The graph of y = sin x for x values between 0° and 360° is
given below.
Use the graph of y = sin x to find all the solutions of the following equations in the range 0° to 360°. a) sin x = -0.5 [2] b) 10 sin x = 6.5 [2] |
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4. |
a) Sketch the graph of y = cos x in the range x = -180° to x = 180°. b) Find
all the solutions of the following equations in the range -180°
to 180° |
