Year 10 Higher Revision
Package II
Name:
Taken from AQA New GCSE Higher Revision Packages
Essential Skills
1 Geoff calculated that the mean age of the members of his badminton club was
16 years 8 months, and the range of their ages was 2 years 1 month.
A new member, aged 14 years 10 months, joins the club.
a Will the mean age of the members increase, decrease, stay the same, or is it
impossible to tell?
Explain your answer.
b Will the range of ages increase, decrease, stay the same, or is it impossible to tell?
Explain your answer.
9 The diagram shows a circle with diameters
AC and BD.
Prove that triangle ABD is congruent to triangle DCA.
Explain your method clearly.
2 The circumference of the circle and the
perimeter of the square are equal.
Calculate the radius of the circle.
Show your method.
10 The difference between two numbers is 4.
The difference between the squares of these two numbers is also 4.
Use an algebraic method to find a pair of numbers for which these statements are true.
11 a Each side of a square is increased by 5%.
By what percentage is the area of the square increased?
b The length of a rectangle is increased by 10%.
The width is decreased by 10%.
By what percentage is the area of the rectangle changed?
c A 20% increase followed by another 20% increase is not the same as a total
increase of 40%.
What is the total percentage increase?
Show your working.
3 Show that 32 + 23 = (32)2 о 43
4 Look at these expressions.
What value of y makes the first expression twice as great as the second expression?
Show your working.
12 The lowest of four consecutive numbers is n.
a Prove that there are only two sets of four consecutive numbers where the sum of the
four numbers is equal to the product of the highest and lowest numbers.
b Write down the two sets of four consecutive numbers.
5 Holly wrote the following:
1 1
p q
Essential Skills
1
pq
13 The diagram shows a prism.
Show that Holly’s statement is not correct.
6 I fill a glass with orange juice and lemonade in the ratio 1 : 5.
The volume of the prism is 12x3 о 2x2y о 2xy2
I drink 1 of the contents of the glass, then I fill the glass using orange juice.
5
Show that the depth of the prism is 2x о y
Now what is the ratio of orange juice to lemonade in the glass?
Show you working and write the ratio in its simplest form.,
7 A farmer keeps sheep and hens.
He has 84 creatures altogether.
Between them they have 288 legs.
Work out how many sheep and how many hens the farmer has.
Show your method.
8 A student wrote ‘For all numbers, (p + q)2 = p2 + q2’
Show that the student is wrong.
Could (p + q)2 ever be the same as p2 + q2?
Explain your answer.
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New AQA GCSE Mathematics © Nelson Thornes Ltd 2010
Number
1 The audience in a theatre is made up of the following ratio:
Number
8 a The cost of 5 metres of wire is £4.
What is the cost of 8 metres of the same wire?
b It takes 3 men 4 days to build a wall.
How long would it take 2 men to build the same wall?
men : women : children = 3 : 4 : 5
a
b
c
d
4
There are 348 people in the audience. Calculate the number of men.
What fraction of the audience are women?
What percentage are children?
Another night the audience was made up of the following ratio:
9 a Write down any irrational number.
b 30 x 40
x has a rational value. Write down a possible value for x.
c 2<y<3
y has an irrational value. Write down a possible value for y.
men : women : children = 2 : 5 : 6
One of the officials recorded that there were 310 people in the audience.
He made a mistake in writing this figure down. Explain how you know this.
10 a Express 5 as a recurring decimal.
11
2 a Rhian measures the height of one of her tomato plants as 20 cm.
The next week it is 15% taller. What is its new height?
b Another tomato plant grows from 240 cm to 312 cm.
Calculate the percentage change in height.
b Which of the following fractions are recurring decimals?
7
18
13
20
2
35
19
25
11
16
11 a Write each of the following in standard index form
3 Geoff filled the petrol tank in his car with unleaded petrol.
i 27 300 000
The petrol cost him £52.65.
a How many litres did he buy?
b How much more would it have cost Geoff if he had filled his petrol tank with super
unleaded instead?
ii 0.00000000006
b Find, in standard index form, the value of each of the following
i (1.25 × 10о4) × (9.4 × 10о5)
4
ii 8.88 u 10
1.2 u 10 3
12 Luke buys a new car for £35 000.
By the end of each year the car has lost 20% of its value at the beginning of that year.
a How much is the car worth when it is one year old?
b How much is the car worth when it is four years old?
13 A dealer buys items from auctions and sells them via the internet.
a He buys a painting for £56 and makes a profit of 65% when he sells it.
What does he sell it for?
b Another time he makes a profit of 40% on a table which he sells for £112.
What did he buy the table for?
c Once he made a loss of 55% when he sold a bureau for £162.
What had he paid for the bureau?
4 a Write 48, 180 and 108 each as a product of its prime factors.
b Find the highest common factor of 48, 180 and 108.
c What is the lowest common multiple of 48, 180 and 108.
5 Show clearly how you would obtain an estimate for this calculation:
607 u 4.97
0.214
14 The power, P, of a car is proportional to the velocity, v.
When P = 3000 watts, v = 8 metres per second.
a Find a formula for P in terms of v.
b Find the power, P, when v = 5.2 metres per second.
6 Work out each of the following:
a 73 21 32
8
2
b The reciprocal of 5 divided by the square root of 1 .
3
4
7 Use the rules of indices to simplify the following.
Give your answers in index form.
a 43 × 4 5
b 38 ÷ 32
c (t4)3
d
15 The length and width of a rectangle are 8 cm and 5 cm, each measured to the nearest
centimetre.
a Write down the upper and lower bounds of the length of the rectangle.
b Write down the upper and lower bounds of the width of the rectangle.
c Find the difference between the maximum and minimum possible areas.
m9
m2 u m 4
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New AQA GCSE Mathematics © Nelson Thornes Ltd 2010
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Number
1 Factorise:
a 12a + 8b
1
16 Calculate the value of ( 4.41 u 102 )2
b 32
1
2
c 252
d 273
e 625
2
4
b 5 3u 3
d 3pq2 – 12p2r
3 Remove the brackets and then simplify:
a 3(5x + y) + 2(3y о 2x)
b 5(2m + 3) о 3(4 о m)
18 Simplify each of these expressions containing surds:
a 3u 5
c 2k2 + 6k
b xy – 2x
2 a Write down all of the whole number values of x, such that 5 x d 3
b Represent the inequality 5 x d 3 on a number line.
17 Simplify:
a 80
Algebra
4 Here are the first five numbers of a sequence.
e 6
d 21
c 28
4
2
3,
19 Write the recurring decimal 0.4444… as a fraction.
a
b
c
d
9,
15,
21,
27
Write down the next two numbers in the sequence.
Write down, in words, the term-to-term rule to continue this sequence.
Write down an expression for the nth term of this sequence.
What will the 20th term of the sequence be?
5 Solve the inequality 6 y 4 d 2 y 7
6 Solve the following equations:
a 4(a + 3) = 6(a о 1)
b
x 1 2x 3
=1
2
5
7 Starting with x = 4, use a trial and improvement method to find, correct to one decimal
place, a solution to the equation x3 + x = 84
Show all your working.
8 a Copy and complete the table for y = x2 о 2x о 2
x
о2
y
6
о1
0
1
2
о2
3
4
1
b Draw x- and y-axes with the x-axis from о2 to 4 and the y-axis from о4 to 6.
On the axes, draw the graph of y = x2 о 2x о 2 for values of x from о2 to 4.
c Write down the equation of the line of symmetry of the graph.
d Write down the coordinates of the minimum point on the graph.
e Use your graph find the values of x when y = 0
9 In each of the following, make a the subject of the formula.
a 2q
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p at
bv
3a
5
2
c d
d 4a m
3a c
n aq
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Algebra
10 M is the point (о2, 4) and N is the point (6, о8).
a Find the coordinates of the midpoint of the line MN.
b Find the gradient of the line MN.
c Find the equation of this line.
d Another line PQ is parallel to MN and passes through the point (1, 5).
Find the equation of PQ.
Algebra
16 Solve each of these quadratic equations:
b x2 + 6x = 0
c x2 о 64 = 0
a x2 + 2x о 24 = 0
d 2x2 + 5x о 12 = 0
17 Solve these equations, giving each answer correct to two decimal places.
a x2 + 7x + 8 = 0
b x2 о 3x + 1 = 0 c 2x2 + 10x о 3 = 0
18 a Sketch the graph of y = cos x for values of x from 0º to 360º.
b Use your sketch, together with your calculator, to solve the equation cos x = о0.4
Find all the solutions for x that lie between 0º and 360º.
Give your solutions correct to 1 decimal place.
11 The diagram shows a cuboid.
The coordinates of point A are (6, 0, 0).
The coordinates of point B are (6, 0, 5).
The coordinates of point C are (4, 3, 0).
Write down the coordinates of point D.
19 Solve these quadratic equations by completing the square.
a x2 о 8x + 1 = 0
b x2 + 4x + 2 = 0
c x2 + 2x о 4 = 0
20 Each of the equations in the table represents one of the graphs A to F.
Copy and complete the table, writing the letter of each graph alongside the correct equation.
Equation
Graph
y = x2 + 2x о 1
y = x3 о 3x
y = 3x о x3
12 Solve these pairs of simultaneous equations:
a 3x + 2y = 12
b
5x о 2y = 7
x + 2y = 2
x + 2y = 11
y = 4x о x2
c
7x о 3y = 48
2x + y = 10
d
3x о 4y = 14
5x + 3y = о54
2
y=x о2
13 a Copy and complete the table of values for y = x3 о 2x2 о 4x
x
о3
y
о33
о2
о1
0
1
4
x
y
2
0
3
4
о3
b Draw the graph of y = x3 о 2x2 о 4x for values of x from о3 to 4.
c Use your graph to solve the equations
i x3 о 2x2 о 4x = 1
ii x3 о 2x2 о 4x = о5
14 a Multiply out and simplify:
i (x + 4)(x + 7)
ii (x о 6)(x + 3)
b Factorise:
i x2 + 3x о 18
iii (x + 5)(x о 5)
iv (3x + 2)(5x о 4)
iii 2x2 о 5x о 3
iv 6x2 о 27x + 30
ii x2 о 9x + 20
15 Simplify:
a
x2 9
3x 9
b
2 x2 5 x 3
8x 4
c
x2 6 x 8
x2 4
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New AQA GCSE Mathematics © Nelson Thornes Ltd 2010
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Geometry
1 The lines VW and XY are parallel. Find the values of angles a, b and c.
Geometry
5 Calculate:
a the exterior angle of a regular octagon
b the sum of the interior angles of a decagon
c the interior angle of a regular 15-sided polygon.
6 The diagram shows a prism with a cross-section in an L-shape.
Find:
a the area of the L-shaped cross-section
b the volume of the prism
c the surface area of the prism.
2 The diagram shows three airports U, T and V.
V is due east of U.
Angle VUT is 25º and angle UTV is 92º.
a What is the bearing of T from U?
b Calculate the angle UVT. Show your working.
c Calculate the bearing of T from V.
3 Find the area of each of these shapes.
7 Describe fully the transformation that maps shape:
a A on to F
b B on to C
c E on to D
d E on to A
e A on to C.
4 The diagram shows a model made with nine cubes.
Five of the cubes are grey. The other four cubes are white.
Draw each of the following, shading the correct cubes:
a the side elevation A
b the side elevation B
c the plan view.
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New AQA GCSE Mathematics © Nelson Thornes Ltd 2010
Geometry
8 a
b
c
d
Construct a triangle ABC with AB = 11 cm, AC = 8 cm and BC = 9.5 cm
Construct the locus of points 5 cm from A.
Construct the locus of points equidistant from BA and BC.
Shade the area inside triangle ABC that is less than 5 cm from A and nearer to
AB than BC.
12
Geometry
13 A ship, A, is 11.3 km from a port P, on a bearing of 038º. Another ship, B, is 4.8 km from P
a bearing of 119º.
a Calculate the distance, AB, between the two ships.
b Calculate the bearing of ship A from ship B.
Give your answer to the nearest degree.
on
9 The diagram shows a park PQRS.
PQ is 47 m long.
QR is 21 m long.
RS is 35 m long.
Angles PQR and RSP are right angles.
There is a path PR running across the park.
a Calculate the length of the path, PR.
b Calculate the length of the side of the park, PS.
14 PQRS is a trapezium with PQ parallel to SR.
PQRT is a parallelogram.
TR is twice the length of ST.
10 In the diagram, PQ is a diameter of the circle and O is the centre.
Calculate the size of angles
a ROQ
b RPQ
Given that PQ = a and QR = b,
express each of the following in terms of vectors a and b.
c PRQ
a PT
Give a reason for each of your answers.
b PR
c RT
d SR
e PS
11 The base of a pyramid, vertex V, is a rectangle ABCD.
The rectangle measures 6 cm × 5 cm
The length of a slant edge of the pyramid is 7 cm.
Calculate:
a the height of the pyramid
b the volume of the pyramid
c the angle which the slant edge AV makes with the base.
12 The diagram shows a church clock at 12:20.
The hour hand is 0.8 m long and the minute hand is 1.3 m long.
a Calculate the angle that the hour hand has moved through
since 12:00.
b Calculate the length of the arc, in cm, swept out by the tip of the
hour hand since 12:00.
c Calculate the area, in cm2, swept out by the hour hand since 12:00.
d Calculate the distance, in m, between the tips of the hour hand and
the minute hand at 12:20.
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New AQA GCSE Mathematics © Nelson Thornes Ltd 2010
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Statistics
1 Shelley is doing a survey to find out how many people eat five portions of fruit or
every day.
She decides to ask 10 people as they come out of a local gym.
Give two different reasons why Shelley’s method might not give very good data.
vegetables
2 Asmat conducted a survey about how accurately people could guess the length of a piece
string to the nearest centimetre.
The results of the survey are given in the stem-and-leaf diagram.
Statistics
Age
21
16
20
29
22
16
23
15
22
25
No. of hours
sleep
7
10.5
8
5.5
7.5
9.5
7
10
6.5
10
a Plot this information on a scatter graph and draw the line of best fit.
b Use your line of best fit to estimate how many hours sleep a person of 26 years of age
would need.
c Comment on the relationship between age and the number of hours of sleep needed.
of
5 James is an unemployed engineer.
He is planning to move to the North East, North West or West Midlands region of the UK
find work.
to
He downloads from the internet details of 25 vacancies for engineers in each region.
A summary of the salaries in the North East and North West regions is illustrated in the
plots below.
box
a How many people took part in Asmat’s survey?
b Find the median length guessed.
c Find the range of lengths guessed.
d The actual length of the string was 44 cm. What percentage of the people surveyed
were more than 10 cm out on their guess? Give your answer to one decimal place.
3 The annual salaries of the workers in a factory are:
20 apprentices
15 semi-skilled workers
10 skilled workers
2 foremen
1 manager
£8500 each
£15 000 each
£18 500 each
£23 000 each
£37 000
a What is the range of salaries in the North West?
b Find the inter-quartile range of salaries in the North West region.
c James summarises his results from the West Midlands region.
The workers were discussing the average earnings in the factory.
The median salary is £25 500, the lower quartile salary is £23 800 and the inter-quartile
range is £2900.
The highest salary is £27 200 and the lowest £22 600.
Use these results to draw a box plot for the West Midlands salaries, using the same
scale for the horizontal axis as used on the diagram above.
a The union representative quoted the modal salary as the average.
What is the modal salary?
b One of the foremen quoted the median as the average.
What is the median?
c Calculate the mean salary of all the people who work in the factory.
d Why is there a large difference between the modal and mean salaries?
d Comment on the salaries for engineers in the three regions that James is looking at.
4 Twenty people were asked their age and the number of hours of sleep they felt they needed.
The results are shown in the table.
Age
11
17
20
28
27
25
17
19
24
19
No. of hours
sleep
10.5
9
8.5
6
5
6.5
8.5
8
6
7.5
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New AQA GCSE Mathematics © Nelson Thornes Ltd 2010
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Statistics
6 Katie has a biased dice. The probability of throwing a six with the dice is 0.4.
Katie throws the dice twice.
Statistics
9 The two-way table shows the number of males and females in a school.
a Copy and complete the following probability tree diagram.
Staff
Students
Male
57
509
Female
42
592
The headteacher conducts a survey to find the reaction of the school to the introduction of a new
uniform.
He decides to ask 200 people in total using stratified sampling.
How many of the sample will be:
a male staff
b What is the probability that Katie does not throw a six in her two throws?
c What is the probability that Katie throws exactly one six in her two throws?
b female staff
c male students
The results are shown in the grouped frequency table below.
7 Ewan wants to find out the length of time cars are left in a car park.
His results, to the nearest minute, are given in the table
Number of eggs, f
30 d x 50
5
50 d x 60
15
Number of cars
(frequency)
0 t d 15
0
60 d x 90
30
15 t d 30
21
90 d x 110
10
30 t d 45
36
45 t d 60
42
60 t d 75
62
75 t d 90
22
11
120 t d 135
6
Cumulative
frequency
Weight, x grams
Length of stay
(minutes)
90 t d 120
d female students?
10 A farmer weighs all 55 eggs collected one day.
Frequency density
a Copy the table and complete the frequency density column.
b Draw a histogram for the data.
c Estimate the mean weight of the eggs collected on that day.
a Copy and complete the table.
b Draw a cumulative frequency diagram for the data.
c Use your diagram to estimate the inter-quartile range.
d The owners of the car park think that about two-thirds of the cars are parked for
between 40 and 80 minutes.
Do Ewan’s results support this?
Give a reason for your answer.
8 A bag contains 4 red balls, 5 blue balls and 3 yellow balls.
One ball is selected at random and not replaced. A second ball is then selected at
a Calculate the probability that both balls are blue.
b Calculate the probability that the two balls are different colours.
Show your method.
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random.
17
New AQA GCSE Mathematics © Nelson Thornes Ltd 2010
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Answers
Statistics
Number
1
c 33 %
3
12
d Because the parts of the ratio add up to 13, and 310 is not exactly divisible by 13.
The numbers of men, women and children must be whole numbers.
b 5
1 a 87
2 a 23 cm
1 Answers may vary. Suggestions are:
10 is too small a sample to get a true analysis of the situation.
People coming out of a gym are likely to be health conscious anyway, so the sample is
biased.
b 30% increase
3 a 45 litres
2 a 35
b £2.25
5 607 u 4.97 is approximately the same as 600 u 5 = 600 u 5 u 5 = 15 000
1
0.2
0.214
b 46 cm
c 42 cm
d 25.7%
3 a £8500
b £15 000
c £13 812.50
d The modal salary is that of the lowest-paid workers. The mean is distorted upwards by
the high salaries of the smaller groups of skilled workers, foremen and especially the
manager.
4 a 48 = 2 × 2 × 2 × 2 × 3; 180 = 2 × 2 × 3 × 3 × 5; 108 = 2 × 2 × 3 × 3 × 3
b HCF = 12
c LCM = 6480
6 a65
Answers
4 a
b2
24
5
7 a 48
b 36
8 a £6.40
b 6 days
c t12
d m3
9 a Any irrational number; examples are
2, S ,
3
b Answers may vary; examples are 6, 5 1 , 6 1
2
4
c Answers may vary; examples are 5 , 2 2 , 7
10 a 0.45
11 a i
b i
b 7
18
2
35
2.73 u 107
ii
6 u 1011
1.175 u 10 8
ii
7.4 u 101
12 a £28 000
b £14 336
13 a £92.40
b £80
14 a P = 375v
b 1950 watts
15 a Upper bound = 8.5 cm
b Upper bound = 5.5 cm
c 13 cm2
b 6.05 hours (this may vary depending on the position of the line of best fit).
c The graph shows negative correlation. In general, the higher the age of a person, the
lower the number of hours sleep that is thought to be needed.
c £360
Lower bound = 7.5 cm
Lower bound = 4.5 cm
16 0.21
17 a 1
b1
18 a 15
b 15
9
c5
c2 7
d9
e 1
25
d3
e3 2
2
19 4
9
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New AQA GCSE Mathematics © Nelson Thornes Ltd 2010
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Answers
5 a £3300
c
b £1200
Answers
b
d The range of salaries is much greater in the West Midlands than in either of the other
two areas, but the median salary is also higher in the West Midlands.
There is not a lot of difference between the North West and the North East, the range
of salaries being similar, but the salaries are on average slightly higher in the North
East.
6 a
c Inter-quartile range = 72 о 42 = 30
d Ewan’s results show that the owners were nearly correct.
An estimated 123 cars were left in the car park for between 40 and 80 minutes, and
b 0.36
c 0.48
7 a
this is 61.5%, which is not much less than 66.66% or 2 .
3
Length of stay
(minutes)
Number of cars
(frequency)
Cumulative
frequency
8 a 5 u 4
12
0 t d 15
0
0
15 t d 30
21
21
30 t d 45
36
57
45 t d 60
42
99
60 t d 75
62
161
75 t d 90
22
183
90 t d 120
11
194
120 t d 135
6
200
New AQA GCSE Mathematics © Nelson Thornes Ltd 2010
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11
20
132
b
P(red and blue) = 4 u 5
20
11 132
4
3
12
P(red and yellow) = u
12 11 132
5
4
20
P(blue and red) = u
12 11 132
15
P(blue and yellow) = 5 u 3
12 11 132
3
4
12
P(yellow and red) = u
12 11 132
15
P(yellow and blue) = 3 u 5
12 11 132
20
12
20 15 12 15
P(different colours) =
132
12
New AQA GCSE Mathematics © Nelson Thornes Ltd 2010
94
132
47
66
22
Answers
9 a9
b7
c 85
Answers
Algebra
d 99
10 a
1 a 4(3a + 2b)
Weight, x grams
Number of eggs, f
Frequency density
30 d x 50
5
0.25
50 d x 60
15
1.5
60 d x 90
30
1
90 d x 110
10
0.5
b x(y – 2)
d 3p(q2 – 4pr)
c 2k(k + 3)
2 a о4, о3, о2, о1, 0, 1, 2, 3
b
3 a 11x + 9y
b 13m + 3
4 a 33, 39
b add 6
c 6n о 3
d 117
5 y чϮ͘ϳϱϭϲ
b
6 a a=9
b x = о1
7 x = 4.3
8 a
c 70.9 grams
x
о2
о1
0
1
2
3
4
y
6
1
о2
о3
о2
1
6
b
c x=1
9 a a
d (1, о3)
2q p
t
10 a (2, о2)
e x = о 0.7 or 2.7
b a
2( v 5)
3
b 3 c 2y + 3x = 2
2
c a
d2 c
3
d a
mn
4q
d 2y + 3x = 13
11 D (6, 3, 5)
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New AQA GCSE Mathematics © Nelson Thornes Ltd 2010
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Answers
12 a x = 5, y
1
1
2
b x = 3, y = 4
c x = 6, y = о2
Answers
18 a
d x = о6, y = о8
13 a
x
о3
о2
о1
0
1
2
3
4
y
о33
о8
1
0
о5
о8
о3
16
b
b x = 92.3º and 267.7º
19 a x = 7.87 and 0.13
b x = о0.52 and о3.68
c x = 1.24 and о2.62
20
Equation
Graph
y = x2 + 2x о 1
f
y = x3 о 3x
e
y = 3x о x3
b
y = 4x о x2
d
4
x
a
y
y = x2 о 2
c i x = о1, о0.3 and 3.3
14 a i x2 + 11x + 28
b i (x + 6)(x – 3)
ii x = о1.7, 1 and 2.8
ii x2 о 3x о 18
ii (x о 5)(x – 4)
iii x2 о 25
iii (2x + 1)(x – 3)
15 a x 3
b x3
c x4
16 a x = 4 and о6
b x = 0 and о6
c x = 8 and о8
3
iv 15x2 о 2x о 8
iv (2x о 5)(3x – 6)
x2
4
17 a x = о1.44 and о5.56
c
b x = 2.62 and 0.38
New AQA GCSE Mathematics © Nelson Thornes Ltd 2010
d x
1
1 and о4
2
c x = 0.38 and о5.19
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New AQA GCSE Mathematics © Nelson Thornes Ltd 2010
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Answers
Geometry and measure
Answers
11 a 5.81 cm
12 a 10º
1 a = 116º
b 58.09 cm3
b 13.96 cm
c 56.1º
c 558.51 cm2
d 1.74 m (2 d.p.)
b = 69º c = 133º
2 a 065º
b 63º
c 333º
3 a 23.56 cm2 b 66.36 cm2
c 10.675 m2
d 15.26 cm2
13 a 11.57 km
b 014º
14 a PT = b
b PR = a + b
c RT = оa
e 5.31 m2 (2 d.p.)
d SR =
3a
2
e PS = b a
2
4
5 a 45º
b 1440º
c 156º
6 a 32 cm2
b 384 cm3
c 424 cm2
7 a A rotation of 90º anticlockwise about the origin.
§ 10·
b A translation on the vector ¨
¸
© 4 ¹
c An enlargement, scale factor 2, centre of enlargement is the point (о1, о4).
d A reflection in the line y = x
e A reflection in the line x = 1
8 Student’s accurate drawing.
Not drawn accurately
9 a PR = 51.48 m (2 d.p.)
b PS = 37.75 m (2 d.p.)
10 a 50º The angle subtended by an arc at the centre is twice the angle subtended at the
circumference.
b 25º Angles subtended by the same arc are equal.
c 90º The angle in a semicircle is a right angle.
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New AQA GCSE Mathematics © Nelson Thornes Ltd 2010
Answers
Essential skills
28
Answers
The farmer has 60 sheep and 24 hens.
1 a The mean will decrease because the age of the new member is less than the original
mean.
b The range cannot decrease because the difference between the lowest and highest
ages cannot get smaller. It is not possible to tell whether the range will increase or stay
the same, because it depends whether the age of the new member is lower than the
lowest age already in the club.
2 The perimeter of the square = 16 cm
S × r = 16
S ×r=8
2×
So
r=
8
S
= 2.55 cm (3 s.f.)
3 32 + 23 = 9 + 8 = 17
(32)2 о 43 = 92 о 64 = 81 о 64 = 17
So 32 + 23 = (32)2 о 43
4 Let
6y о 4
6y о 4
6y о 4y
2y
y
= 2(2y + 3)
= 4y + 6
=6+4
= 10
=5
5 Let p = 3 and q = 4, so
1 1
p q
1 1
3 4
The common denominator for 3 and 4 is 12 (3 × 4 not 3 + 4).
So
1 1
3 4
43
12
7
1 1
and 12
p q
q p
1
not
pq
pq
6 1:2
7 Let x represent the number of sheep.
Let y represent the number of hens.
Write down two simultaneous equations, and solve for x and y.
x + y = 84
4x + 2y = 288
Equation 1
Equation 2
2x + 2y = 168
2x = 120
x = 60
4 × 60 + 2y = 288
2y = 48
y = 24
60 + 24 = 84D
multiply Equation 1 by 2 to form Equation 3
subtract Equation 3 from Equation 2 to eliminate y
solve for x
substitute this value for x in Equation 2
simplify
solve for y
check values for x and y by substitution in Equation 1
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New AQA GCSE Mathematics © Nelson Thornes Ltd 2010
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Answers
8 (p + q)2 = (p + q) (p + q) = p2 + 2pq + q2
(p + q)2 could be the same as p2 + q2 if pq = 0, so p or q or both would have to be zero.
9 Angle ADC = angle BAD = 90º
BD = AC
Angle ABD = angle ACD
an angle in a semi-circle is a right angle
both are diameters of the circle
angles subtended by the same arc are equal
So triangles ABD and DCA are congruent.
10 2 1 and 11
2
2
11 a 10.25%
b Decreased by 1%
c 44%
12 a Sum of consecutive numbers = 4n + 6
Product of highest and lowest = n2 + 3n
n2 + 3n = 4n + 6
n2 о n о 6 = 0
(n + 2)(n о 3) = 0
n = о 2 or 3
So there are only two sets of four consecutive numbers where the sum of the
numbers is equal to the product of the highest and the lowest numbers.
b о2, о1, 0, 1 and 3, 4, 5, 6
13 The area of cross-section is (3x × x) + (2x × y) + (3x × x)
= 6x2 + 2xy
= 2x (3x + y)
The volume of the prism is
12x3 о 2x2y о 2xy2
= 2x (6x2 о xy о y2)
= 2x (2x о y)(3x + y)
So the depth of the prism
= 2x (2x о y)(3x + y)
2x (3x + y)
= 2x о y
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