Transcript (a) [2]

1.(a) Given that 20 kg is approximately 44 lb (pounds), complete the
statement below.
lb (pounds)
[1]
(b) The label on a pack of cheese reads:
10 litres of milk make 1lb (pound) of cheese
HIGHER Paper 1
GCSE MATHEMATICS - LINEAR
1 kg =
Calculate how many litres of milk are needed to make 15 kg of cheese.
[3]
Part c
1.(a) Given that 20 kg is approximately 44 lb (pounds), complete the
statement below.
2.2
lb (pounds)
[1]
This is 44 ÷ 20
(b) The label on a pack of cheese reads:
10 litres of milk make 1lb (pound) of cheese
HIGHER Paper 1
GCSE MATHEMATICS - LINEAR
1 kg =
Calculate how many litres of milk are needed to make 15 kg of cheese.
15kg = 15 × 2.2
= 33 lb
This is a 2-step calculation.
33 × 10 = 330 litres
[3]
Part c
Reveal
1 (c) The label on the pack of cheese also states:
Typical value per 100g:
Energy
Protein
Carbohydrate
Fat
HIGHER Paper 1
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[4]
1700kj
25·0g
0·1g
34·4g
410kcal
Calculate the amount of protein in 1·5 kg of cheese. Give your answer in grams.
[4]
Part a & b
1 (c) The label on the pack of cheese also states:
Typical value per 100g:
Energy
Protein
Carbohydrate
Fat
HIGHER Paper 1
GCSE MATHEMATICS - LINEAR
[4]
1700kj
25·0g
0·1g
34·4g
410kcal
Calculate the amount of protein in 1·5 kg of cheese. Give your answer in grams.
1.5kg = 1500g
Cheese  protein
100g  25.0g
× 15
× 15
1500g  375g
Answer = 375g
[4]
Part a & b
Reveal
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GCSE MATHEMATICS - LINEAR
2. Calculate the size of each of the angles marked x, y and z in the diagram
below.
Diagram not drawn to scale.
x=
°,
y=
°,
z=
°
[3]
Alternate
angles
35°
HIGHER Paper 1
GCSE MATHEMATICS - LINEAR
2. Calculate the size of each of the angles marked x, y and z in the diagram
Opposite
below.
angles
Diagram not drawn to scale.
x = 35° (opposite angles)
y = 35° (alternate angles)
This angle is 35° too.
z = 180° – 35° (angles on a straight line)
z = 145°
x=
35 °,
y=
35 °,
z = 145 °
[3]
Reveal
(a) On the graph paper below, draw a scatter diagram of these results.
(b) Describe the correlation
between the average
percentage of cloud cover
and the amount of rainfall.
HIGHER Paper 1
GCSE MATHEMATICS - LINEAR
3. Every Saturday for 5 weeks in the Autumn the number of centimetres of rainfall and the
average percentage of cloud cover were recorded by a group of students. The table below
shows the results.
[1]
(c) Find an estimate of the
average percentage of cloud
cover on a day with 0·6cm of
rainfall clearly showing your
method.
[2]
Each little square
is worth 0.02 cm
(a) On the graph paper below, draw a scatter diagram of these results.
(b) Describe the correlation
between the average
percentage of cloud cover
and the amount of rainfall.
HIGHER Paper 1
GCSE MATHEMATICS - LINEAR
3. Every Saturday for 5 weeks in the Autumn the number of centimetres of rainfall and the
average percentage of cloud cover were recorded by a group of students. The table below
shows the results.
Positive
In order to arrive at an estimate
you need to draw a line of best
fit on the graph – this does not
have to pass through the origin
[1]
(c) Find an estimate of the
average percentage of cloud
cover on a day with 0·6cm of
rainfall clearly showing your
method.
64%
Each little square is worth 2%
64%
Reveal[2]
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5. (a) Draw an enlargement of the shape shown below using
a scale factor of 2.
Use the point A as the centre of the enlargement.
Part b
[3]
This is the correct size but
drawn in the wrong place.
[3]
Make sure you use the
centre of enlargement, A
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GCSE MATHEMATICS - LINEAR
5. (a) Draw an enlargement of the shape shown below using
a scale factor of 2.
Use the point A as the centre of the enlargement.
Part b
Reveal
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GCSE MATHEMATICS - LINEAR
5 (b) Rotate the shape shown below through 90° anticlockwise about the
point (2, 1).
Part a
[2]
[2]
Remember the three key
facts:
Angle: 90°
Centre: (2, 1)
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5 (b) Rotate the shape shown below through 90° anticlockwise about the
point (2, 1).
Direction: anticlockwise
Part a
Reveal
Mrs. Roberts is travelling to Hong Kong on business.
(a) There is a time difference between the UK and Hong Kong.
When the time is 6 a.m. in the UK the time is 2 p.m. on the same day in Hong Kong.
(i) When it is 10 a.m. in the UK what time is it in Hong Kong?
[1]
(ii) Mr. Roberts stays in the UK and has given his wife his time schedule, shown below.
Mrs. Roberts will be in meetings most of
the day in Hong Kong from 8 a.m. until 11
a.m., then from 12 noon to 6 p.m.
She plans to telephone her husband at a
convenient time during the day.
During which time period should Mrs.
Roberts telephone her husband?
Give your answer in UK and Hong Kong
times.
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GCSE MATHEMATICS - LINEAR
6. You will be assessed on the quality of your written communication in part (b) of this
question.
Part b
Part c
[2]
6. You will be assessed on the quality of your written communication in part (b) of this
question.
(i) When it is 10 a.m. in the UK what time is it in Hong Kong?
6am (UK) is 2pm (HK) (+8hrs)
so
10am (UK) is 6pm (HK)
[1]
(ii) Mr. Roberts stays in the UK and has given his wife his time schedule, shown below.
HIGHER Paper 1
GCSE MATHEMATICS - LINEAR
Hong Kong is 8
Mrs. Roberts is travelling to Hong Kong on business.
hours ahead of UK
(a) There is a time difference between the UK and Hong Kong.
When the time is 6 a.m. in the UK the time is 2 p.m. on the same day in Hong Kong.
pm
meeting
am
Mrs. Roberts will be in meetings most of
the day in Hong Kong from 8 a.m. until 11
a.m., then from 12 noon to 6 p.m.
She plans to telephone her husband at a
convenient time during the day.
During which time period should Mrs.
Roberts telephone her husband?
Give your answer in UK and Hong Kong
times.
HK
2:00
2:30
3:30
7:30
8:30
10:00
0:00
2:00
5:00
6:00
Between
8:30pm – 10:00pm HK time [Mrs Roberts has finished work]
12:30pm – 2:00pm UK time [Mr Roberts is having lunch]
So they are both free to talk.
Part b
Part c
Reveal
[2]
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6 (b) Mrs. Roberts is going to be in Hong Kong for 4 nights.
She finds two suitable hotels on the internet.
Which hotel should Mrs. Roberts choose? You must show your working and give a
reason for your answer.
Part a
Part c
[5]
HIGHER Paper 1
GCSE MATHEMATICS - LINEAR
6 (b) Mrs. Roberts is going to be in Hong Kong for 4 nights.
She finds two suitable hotels on the internet.
Which hotel should Mrs. Roberts choose? You must show your working and give a
reason for your answer.
Hotel Bear
Hotel Gelton
£107 × 3 = £321 B&B + dinner
Remember to include a
valid reason for your choice
Choose Hotel Bear as you also get dinner for 4
nights for an extra £1
£80 × 4 = £320
Part a
Part c
B&B
Reveal
[5]
(i) How much of the £400 did Mrs. Roberts spend when in Hong Kong? Give
your answer in dollars.
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6 (c) The currency in Hong Kong is dollars, ($).
Mrs. Roberts changes £400 into dollars. She
returns from Hong Kong with $1500. The
bank gives the exchange rates shown below.
[3]
(ii) On return from her business trip Mrs. Roberts exchanges $1500
for pounds. Will she receive more or less than £100? You must
give a reason for your answer.
Part a
Part b
[2]
(i) How much of the £400 did Mrs. Roberts spend when in Hong Kong? Give
your answer in dollars.
400  15 = $6000
HIGHER Paper 1
GCSE MATHEMATICS - LINEAR
6 (c) The currency in Hong Kong is dollars, ($).
Mrs. Roberts changes £400 into dollars. She
returns from Hong Kong with $1500. The
bank gives the exchange rates shown below.
6000 – 1500 = $4500
[3]
(ii) On return from her business trip Mrs. Roberts exchanges $1500
for pounds. Will she receive more or less than £100? You must
give a reason for your answer.
She will receive less than £100 because
1500 ÷ 17 = £88.24
Part a
Part b
[2]
Reveal
(a) Fill in the numbers on these houses.
[1]
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7. The houses on one side of a long street have odd numbers and the houses on the other
side of the street have even numbers.
(b) The numbers on five houses next to each other on one side of the street total 65.
What are the numbers on these five houses?
[3]
(c) The product of the numbers on two houses which are directly opposite each other is 380.
What are the numbers on these two houses?
[1]
7. The houses on one side of a long street have odd numbers and the houses on the other
side of the street have even numbers.
+2 +2
(a) Fill in the numbers on these houses.
+2 +2
97 99 101
105
98 100 102 104 106
HIGHER Paper 1
GCSE MATHEMATICS - LINEAR
+1
[1]
(b) The numbers on five houses next to each other on one side of the street total 65.
What are the numbers on these five houses?
65 ÷ 5 = 13
[3]
This must be the middle house number.
So, the solution is:
9
11
13
15
17
(c) The product of the numbers on two houses which are directly opposite each other is 380.
What are the numbers on these two houses?
The numbers will be consecutive i.e.
(number) × (number + 1) = 380
The numbers are 19 and 20.
[1]
Product means
“multiply”
Reveal
9.
y (4y 3+ 1)
[2]
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(a) Expand
(b) Simplify
t6
t2
[1]
9.
y (4y 3+ 1)
= 4y 4 + y
Remember – everything
in the bracket is
multiplied by the y
[2]
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(a) Expand
(b) Simplify
t6
t2
subtract the powers
= t4
[1]
Reveal
10.(a) A teacher recorded the time taken by each of 30 pupils in her class to
complete a task. The table below shows a summary of her results.
(i) On the graph paper below draw a frequency polygon for this data.
HIGHER Paper 1
GCSE MATHEMATICS - LINEAR
[2]
(ii) Using the table, give the class
interval which contains the
median time taken.
[1]
Part b
10.(a) A teacher recorded the time taken by each of 30 pupils in her class to
complete a task. The table below shows a summary of her results.
[2]
Mid point
Remember:
15
25
Plot the mid-points
35
(i) On the graph paper below draw a frequency polygon for this data.
The frequency polygon starts
at the first point and ends at
the last point
HIGHER Paper 1
GCSE MATHEMATICS - LINEAR
5
(ii) Using the table, give the class
interval which contains the
median time taken.
30 pupils  median is between
15th and 16th pupil, both of
whom are in the interval
10 < t ≤ 20
[1]
Part b
Reveal
Use the cumulative frequency diagram
to find an estimate for the interquartile
range.
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10 (b) Another teacher recorded the times taken to complete the same task for his
class of 32 pupils and he drew the following cumulative frequency diagram.
[2]
(c )Is it possible for the median times
of the two classes to be the same?
Give a reason for your answer.
[2]
Part a
A quarter of 32 is 8.
Use the cumulative frequency diagram
to find an estimate for the interquartile
range.
Interquartile range = UQ – LQ
Interquartile range = 19.5 – 11.5
UQ: at 24
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GCSE MATHEMATICS - LINEAR
10 (b) Another teacher recorded the times taken to complete the same task for his
class of 32 pupils and he drew the following cumulative frequency diagram.
Interquartile range = 8
[2]
Median:
at 16
(c )Is it possible for the median times
of the two classes to be the same?
Give a reason for your answer.
[2]
Median time for this class ≈ 15.5
LQ: at 8
Part a
Your answer
Both medians are in the range 10 < t ≤ 20,
must be justified so yes it is possible that they could be the
using results
same.
15.5
11.5 19.5
Reveal
On the same diagram, draw a graph to
show the volume of the cuboid for values
of x from 0 to 1 using the values in the
following table.
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11 (c) On the diagram below, the graph drawn shows the volume of the pyramid for
values of x from 0 to 1.
[3]
(d) Explain what the intersection of the two graphs tells you.
Part a
[1]
Scales
Vertical: 1 small square = 0.2
Horizontal: 1 small square = 0.02
On the same diagram, draw a graph to
show the volume of the cuboid for values
of x from 0 to 1 using the values in the
following table.
HIGHER Paper 1
GCSE MATHEMATICS - LINEAR
11 (c) On the diagram below, the graph drawn shows the volume of the pyramid for
values of x from 0 to 1.
0
0.4 1.6 3.6 6.4 10
Substitute each value of x from the
table into 10x2 to calculate the
volume
(e.g. 10 × 0.2² = 10 × 0.04 = 0.4)
[3]
(d) Explain what the intersection of the two graphs tells you.
At the point of intersection the cuboid and the pyramid have the
same volume.
Part a
Reveal[1]
12. When Dylan has lunch the probability that he has a dessert is 1 .
4
Whether or not he has a dessert the probability that he has coffee is 2 .
5
HIGHER Paper 1
GCSE MATHEMATICS - LINEAR
(a) Complete the following tree diagram.
(b) Calculate the probability that Dylan has a dessert or coffee, but not both.
[2]
[2]
Remember each set of
branches total 1
HIGHER Paper 1
GCSE MATHEMATICS - LINEAR
12. When Dylan has lunch the probability that he has a dessert is 1 .
4
Whether or not he has a dessert the probability that he has coffee is 2 .
2 5
5
(a) Complete the following tree diagram.
3
5
2
5
3
4
3
5
[2]
(b) Calculate the probability that Dylan has a dessert or coffee, but not both.
P(dessert and NOT coffee) or P(NOT dessert and coffee)
1 3
3 2
= 4× 5 + 4 ×5
6
3
= 20 + 20
9
= 20
Reveal
[2]
13. A bag contains 16 red beads, 4 green beads and 1 yellow bead. Two beads are drawn at
random without replacement from the bag.
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GCSE MATHEMATICS - LINEAR
(a) Calculate the probability that the two beads are of the same colour.
(b) Calculate the probability that one of the two beads selected is yellow.
[3]
[2]
13. A bag contains 16 red beads, 4 green beads and 1 yellow bead. Two beads are drawn at
random without replacement from the bag.
Remember there is NO replacement
(a) Calculate the probability that the two beads are of the same colour.
P(r)
= 16
21
P(g) =
4
21
= 16
20
P(g) = 3
[3]
P(r,r) + P(g,g) + P(y,y)
15
= 16 ×
+
21 20
P(r)
HIGHER Paper 1
GCSE MATHEMATICS - LINEAR
P(r)
= 15
20
4
× 3
21 20
+
1
× 0
21
12
= 240 +
420
420
20
= 252
420
P(y)
=
1
21
=
3
5
P(y) = 0
(b) Calculate the probability that one of the two beads selected is yellow.
P(r,y) + P(g,y) + P(y,r) + P(y,g)
1
= 16 ×
+ 4 × 1 + 1 × 16
21
20
21 20
21
20
4
4
= 16 +
+ 16 +
420
420
420
420
= 40
420
= 2
21
+
1 × 4
21 20
[2]
Remember to consider
all options.
e.g. ‘red, yellow’ is
different to ‘yellow, red’
Reveal
14.
(a) Simplify
x2 + 5x + 6
[3]
(b) Simplify
HIGHER Paper 1
GCSE MATHEMATICS - LINEAR
3x + 6
(3ab 7)3
(c) Make d the subject of the following formula.
de – c
2d + g
[2]
=5
[4]
14.
(a) Simplify
x2 + 5x + 6
=
(x + 3) (x + 2)
3 (x + 2)
=
(x + 3)
3
[3]
(b) Simplify
HIGHER Paper 1
GCSE MATHEMATICS - LINEAR
3x + 6
Numerator and denominator both need to
be factorised before you can cancel
(3ab 7)3
The 3, the a and the b7 need to be cubed
27a 3b 21
(c) Make d the subject of the following formula.
de – c
2d + g
de – c
de – c
de – 10d
d(e – 10)
=
d
=
factorise
=
=
=
[2]
=5
Multiply by denominator
5(2d + g)
10d + 5g
5g + c
Collect d terms together on left hand side
5g + c
[4]
5g + c
(e – 10)
Reveal
The points A, B, C and D are on the
circumference of a circle with
centre O and BOD = 6x. Find the
size of BCD in terms of x.
HIGHER Paper 1
GCSE MATHEMATICS - LINEAR
15.(a)
Diagram not drawn to scale.
Part b
[2]
15.(a)
The points A, B, C and D are on the
circumference of a circle with
centre O and BOD = 6x. Find the
size of BCD in terms of x.
3x
HIGHER Paper 1
GCSE MATHEMATICS - LINEAR
Remember, the angle at circumference is half the angle at the centre
Diagram not drawn to scale.
BAD = 3x
(angle at circumference is half angle at centre)
BAD + BCD = 180° (opposite angles of a cyclic quadrilateral)
3x + BCD = 180°
BCD = 180° – 3x
Part b
Reveal
[2]
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GCSE MATHEMATICS - LINEAR
16.(a) The diagram shows a sketch of
y = – x 3. On the same diagram,
sketch the curve y = – 2x 3.
[1]
(b) The diagram shows a
sketch of y = f (x). On the
same diagram, sketch the
curve y = f (x + 5).Indicate
the coordinates of one
point on the curve.
[2]
Part c
Graph stretches this way:
Every point will end up
being twice as far from
the x-axis than it was.
HIGHER Paper 1
GCSE MATHEMATICS - LINEAR
16.(a) The diagram shows a sketch of
y = – x 3. On the same diagram,
sketch the curve y = – 2x 3.
[1]
(b) The diagram shows a
sketch of y = f (x). On the
same diagram, sketch the
curve y = f (x + 5).Indicate
the coordinates of one
point on the curve.
As the + 5 is inside the bracket,
you move the graph to the LEFT 5
Part c
(– 5, 0)
[2]
Reveal
16 (c)
On the same diagram, sketch
the curve y = f (x) – 3.
Indicate the coordinates of one
point on the curve.
HIGHER Paper 1
GCSE MATHEMATICS - LINEAR
The diagram shows a sketch of
y = f (x).
Part a&b
[2]
16 (c)
On the same diagram, sketch
the curve y = f (x) – 3.
Indicate the coordinates of one
point on the curve.
HIGHER Paper 1
GCSE MATHEMATICS - LINEAR
The diagram shows a sketch of
y = f (x).
As the – 3 is outside the bracket,
you move the graph DOWN 3
(0, – 3)
Part a&b
[2]
Reveal
HIGHER Paper 1
GCSE MATHEMATICS - LINEAR
17. Solve
[7]
17. Solve
Write the two fractions as one on left hand side
20(n + 3) + 5n(n + 1) = 6
(n + 1)(n + 3)
Multiply throughout by denominator
20(n + 3) + 5n(n + 1) = 6(n + 1)(n + 3)
20n + 60 + 5n2 + 5n = 6n2 + 24n + 18
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GCSE MATHEMATICS - LINEAR
See mark scheme for alternative
start to this question
0 = n2 – n – 42
n2 – n – 42 = 0
(n – 7)(n + 6) = 0
Either
n –7=0
n=7
Expand all brackets
Collect all terms
on right hand
side (more n2)
Rearrange so that
right hand side is
equal to zero
or n + 6 = 0
or
n=–6
[7]
Reveal
Pattern 1
Pattern 2
Pattern 3
Pattern 4
Diagrams not drawn to scale.
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GCSE MATHEMATICS - LINEAR
18. Patterns are generated as shown in the diagram.
Find the perimeter of Pattern 6 in the form a + √b , where a and b are whole numbers.
Show your working.
[4]
Pattern 1
Pattern 2
Pattern 3
Pattern 4
Diagrams not drawn to scale.
HIGHER Paper 1
GCSE MATHEMATICS - LINEAR
18. Patterns are generated as shown in the diagram.
Find the perimeter of Pattern 6 in the form a + √b , where a and b are whole numbers.
Show your working.
P1 = 1 + 1 + √2
= 2 + √2
P2 = 1 + 1 + 1 + √3 = 3 + √3
Note: Pn = (n+1) + √(n+1)
P3 = ……………
= 4 + √4
P4 = ……………
= 5 + √5
P6 = 7 + √7
Reveal
[4]
1. The numbers on opposite faces of a dice add up to 7.
HIGHER Paper 2
GCSE MATHEMATICS - LINEAR
Complete the following diagrams for nets of dice.
[4]
1. The numbers on opposite faces of a dice add up to 7.
These faces are opposite each
other and must add up to 7
These faces are opposite each
other and must add up to 7
5 or 2
5 or 2
HIGHER Paper 2
GCSE MATHEMATICS - LINEAR
Complete the following diagrams for nets of dice.
2 or 5
2 or 5
[4]
These faces are opposite each
other and must add up to 7
These faces are opposite each
other and must add up to 7
Reveal
2. The owner of a takeaway coffee shop uses two types of paper cups.
Hi-rim cup
Base-stay cup
Hi-rim cup
Base-stay cup
Diagrams are not drawn to scale.
HIGHER Paper 2
GCSE MATHEMATICS - LINEAR
They can be stacked like this...
(a) How high is a stack of 25 Hi-rim cups?
[2]
(b) A stack of Base-stay cups is 18.6 cm high.
How many Base-stay cups are in the stack?
[2]
Part (c)
2. The owner of a takeaway coffee shop uses two types of paper cups.
Hi-rim cup
Hi-rim cup
Base-stay cup
Base-stay cup
Diagrams are not drawn to scale.
HIGHER Paper 2
GCSE MATHEMATICS - LINEAR
They can be stacked like this...
(a) How high is a stack of 25 Hi-rim cups?
Remember there are 24 cups
above the bottom cup
Therefore 24 × 0.5 NOT 25 × 0.5
Height of cup 1 = 14 cm
Height of cups 2 to 25 = 24 × 0.5 = 12 cm
Total height = 14 + 12 = 26 cm
Add height of bottom cup
[2]
(b) A stack of Base-stay cups is 18.6 cm high.
How many Base-stay cups are in the stack?
Remove cup 1: 18.6 – 9 = 9.6
Number of cups 9.6 ÷ 1.2 = 8
8 cups + 1 cup = 9 cups
[2]
Part (c)
Reveal
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GCSE MATHEMATICS - LINEAR
2 (c) A stack of Hi-rim cups is the same height as a stack of Base-stay cups.
There are 21 Base-stay cups in the stack.
How many cups are there in the stack of Hi-rim cups?
[3]
Part (a) & (b)
Height of Base-stay
20 × 1.2 + 9 = 33 cm
To find number of Hi-rim
33 – 14 = 19
19 ÷ 0.5 = 38 cups
38 + 1 = 39 cups
Don’t forget the bottom cup
HIGHER Paper 2
GCSE MATHEMATICS - LINEAR
2 (c) A stack of Hi-rim cups is the same height as a stack of Base-stay cups.
There are 21 Base-stay cups in the stack.
How many cups are there in the stack of Hi-rim cups?
[3]
Part (a) & (b)
Reveal
HIGHER Paper 2
GCSE MATHEMATICS - LINEAR
3. Mrs Evans received an electricity bill from Wales Electricity Company.
The bill, with some of the entries removed, is shown below.
Use the information given on the bill to complete all of the missing entries and calculate
the total amount that Mrs Evans has to pay.
[6]
Change
to £
HIGHER Paper 2
GCSE MATHEMATICS - LINEAR
3. Mrs Evans received an electricity bill from Wales Electricity Company.
The bill, with some of the entries removed, is shown below.
Use the information given on the bill to complete all of the missing entries and calculate
the total amount that Mrs Evans has to pay.
[6]
1100
156.20
Add on
191.08
9.55
Check your
answer makes
sense, an
electricity bill isn’t
usually thousands
of pounds
200.63
This
means
that they
paid too
much on
their last
bill.
188.63
Reveal
(a) Are there any balls of another colour in the bag?
Give a reason for your answer.
HIGHER Paper 2
GCSE MATHEMATICS - LINEAR
4. The table below shows the probabilities of selecting one ball at random
from a bag of coloured balls.
[2]
(b) What is the probability of selecting either a yellow or a purple ball?
[2]
(a) Are there any balls of another colour in the bag?
Give a reason for your answer.
0.25 + 0.14 + 0.06 + 0.15 + 0.40 = 1
HIGHER Paper 2
GCSE MATHEMATICS - LINEAR
4. The table below shows the probabilities of selecting one ball at random
from a bag of coloured balls.
There are no balls of any other colour because the
probabilities add up to 1.
[2]
(b) What is the probability of selecting either a yellow or a purple ball?
P(yellow or purple)
= P(yellow) + P(purple)
= 0.06 + 0.40
= 0.46
The ball can’t be yellow and purple at
the same time, so the rule
P(A or B) = P(A) + P(B) works.
[2]
Reveal
[2]
(b) Solve 8x + 7 = 2x + 10.
HIGHER Paper 2
GCSE MATHEMATICS - LINEAR
5. (a) Write down the first three terms of the sequence with an nth term of n 2 + 10.
[3]
12 + 10 = 11
22 + 10 = 14
Start with n = 1
32 + 10 = 19
[2]
(b) Solve 8x + 7 = 2x + 10.
8x – 2x = 10 – 7
HIGHER Paper 2
GCSE MATHEMATICS - LINEAR
5. (a) Write down the first three terms of the sequence with an nth term of n 2 + 10.
6x = 3
Be careful with signs
x =3
6
x =1
2
[3]
Reveal
[2]
(b) Calculate 5.6 × 3.4
8.1 – 2.7
giving your answer correct to two decimal places.
HIGHER Paper 2
GCSE MATHEMATICS - LINEAR
6. (a) Express 104 as a percentage of 260.
[2]
(c) Two friends share £280 in the ratio 3:4.
Find how much each friend receives.
[2]
6. (a) Express 104 as a percentage of 260.
= 40%
(b) Calculate 5.6 × 3.4
( 8.1 – 2.7 )
Put brackets around the denominator
when you put it into your calculator
[2]
giving your answer correct to two decimal places.
HIGHER Paper 2
GCSE MATHEMATICS - LINEAR
104
× 100 %
260
3.53
As this number is 5 or higher,
3.52 rounds up to 3.53
3.52592592….
[2]
(c) Two friends share £280 in the ratio 3:4.
Find how much each friend receives.
3+4=7
280 = 40
7
Total number of parts
3 × 40 = £120
4 × 40 = £160
How much each
part is worth
[2]
Reveal
HIGHER Paper 2
GCSE MATHEMATICS - LINEAR
7. The test scores of 20 people were recorded and the results are
summarised in the following table.
Calculate an estimate for the mean of the test scores.
[3]
7. The test scores of 20 people were recorded and the results are
summarised in the following table.
4.5
0 + 9 = 4.5
2
14.5
24.5
Total = 20
HIGHER Paper 2
GCSE MATHEMATICS - LINEAR
Mid-point
Calculate an estimate for the mean of the test scores.
mean = total of ‘mid-point × frequency’
total frequency
= 4.5 × 7 + 14.5 ×11 + 24.5 × 2
20
= 240
20
= 12
[3]
Reveal
[2]
(b) On the graph paper below, draw the graph of y = 2x 2 – 5 for values of x between – 2 and 2.
[2]
HIGHER Paper 2
GCSE MATHEMATICS - LINEAR
8. The table shows some of the values of y = 2x 2 – 5 for values of x from –2 to 2.
(a) Complete the table by finding the value of y for x = –2 and x = 1.
(c) Write down the x-coordinates of
the points of intersection of
y = 2x 2 – 5 with the x-axis.
[2]
(d) Write down the minimum value of y.
[1]
3
–3
[2]
(b) On the graph paper below, draw the graph of y = 2x 2 – 5 for values of x between – 2 and 2.
[2]
(– 2)2 = 4,
So, 2(4) – 5 = 3
HIGHER Paper 2
GCSE MATHEMATICS - LINEAR
8. The table shows some of the values of y = 2x 2 – 5 for values of x from –2 to 2.
(a) Complete the table by finding the value of y for x = –2 and x = 1.
(c) Write down the x-coordinates of
the points of intersection of
y = 2x 2 – 5 with the x-axis.
x ≈ 1.6 and x ≈ – 1.6
[2]
(d) Write down the minimum value of y.
–5
Reveal
[1]
A number is written on each of five cards.
The cards are arranged in ascending order.
It is known that the mean of the five numbers is 9.6, the range is 12, the median is 10,
the greatest number is 16 and the fourth number is twice the second number.
Explaining your reasoning, find the five numbers written on the cards.
HIGHER Paper 2
GCSE MATHEMATICS - LINEAR
9. You will be assessed on the quality of your written communication in this question.
[3]
4
6 10 12 16
A number is written on each of five cards.
The cards are arranged in ascending order.
It is known that the mean of the five numbers is 9.6, the range is 12, the median is 10,
the greatest number is 16 and the fourth number is twice the second number.
Explaining your reasoning, find the five numbers written on the cards.
HIGHER Paper 2
GCSE MATHEMATICS - LINEAR
9. You will be assessed on the quality of your written communication in this question.
[3]
The median is 10. Therefore, the number on the middle card is 10.
The largest number is 16.
The range is 12. Therefore, the smallest number is 16 – 12 = 4
Now, mean × number of cards = total
So, 9.6 × 5 = 48
total of 5 cards – total of 3 cards = 48 – (4 + 10 + 16)
= 18
The fourth number is twice the second, and the two add up to 18.
The fourth number is 12, the second is 6.
Reveal
Make sure that you clearly indicate the region that represents your answer.
[4]
HIGHER Paper 2
GCSE MATHEMATICS - LINEAR
13. On the graph paper below, draw the region which satisfies all of the
following inequalities.
[4]
Draw:
HIGHER Paper 2
GCSE MATHEMATICS - LINEAR
13. On the graph paper below, draw the region which satisfies all of the
following inequalities.
5 ≤0 +7 
5 ≥1 –0 
5 ≥3 
0 ≤4 
Make sure that you clearly indicate the region that represents your answer.
y =x +7
x
y
–5
2
y = 1 – 2x
x
y
–5 0 5
11 1 – 9
0 5
7 12
y =3
x =4
Check :
Consider point within region
(0, 5)
Reveal
14. (a) Factorise 6x 2 + 18xy.
(b) Factorise x 2 – 25.
[1]
(c) Solve 4n – 5 < n + 22.
HIGHER Paper 2
GCSE MATHEMATICS - LINEAR
[2]
[2]
(d) Solve the equation
decimal places.
3x 2
+ 19x + 11 = 0, giving your solutions correct to two
[3]
14. (a) Factorise 6x 2 + 18xy.
Difference of
two squares
[2]
(b) Factorise x 2 – 25.
(x + 5)(x – 5)
(c) Solve 4n – 5 < n + 22.
HIGHER Paper 2
GCSE MATHEMATICS - LINEAR
6x (x + 3y)
4n – n < 22 + 5
3n < 27
n < 27
3
(d) Solve the equation
decimal places.
3x 2
[1]
Use inequality
sign
throughout
[2]
+ 19x + 11 = 0, giving your solutions correct to two
x = – b ± √ – 4ac
2a
b2
[3]
with a = 3, b = 19, c = 11
2
x = – 19 ± √ 19 – (4 × 3 × 11)
2×3
x = – 19 ± √ 229
6
x = – 0.64, x = – 5.69
If the question asks for
solutions to a given
number of decimal places,
don’t try and factorise,
use the formula.
Reveal
(a) Find an estimate for the median of this distribution.
[1]
(b) Draw a histogram to illustrate the distribution on the graph below.
HIGHER Paper 2
GCSE MATHEMATICS - LINEAR
15. There are 100 pupils in Year 10. The time taken by each pupil to answer a question
was recorded. The following grouped frequency distribution was obtained.
[2]
15. There are 100 pupils in Year 10. The time taken by each pupil to answer a question
was recorded. The following grouped frequency distribution was obtained.
10
10
10
10
50
20
50
(a) Find an estimate for the median of this distribution.Median
30
[1]
(b) Draw a histogram to illustrate the distribution on the graph below.
HIGHER Paper 2
GCSE MATHEMATICS - LINEAR
Interval width
Frequency
density
4
3
2
1
Areas of bars show the
frequency (no. of pupils)
[2]
frequency
frequency
=
density
interval width
6 = 0.6
10
14 = 0.7
20
0.6, 1.9, 2.5, 3.6, 0.7
Use these values to decide
on a scale
Reveal
HIGHER Paper 2
GCSE MATHEMATICS - LINEAR
17. (a) Using the axes below, sketch the graph of y = sin x
for values of x from – 180° to 360°.
[2]
(b) Find all solutions of the following equation in the range – 180° to 360°.
sin x = – 0.788
[3]
17. (a) Using the axes below, sketch the
of y =1sin
Plotgraph
– 1 and
onxthe
for values of x from – 180° to 360°.
[2]
1
– 180
HIGHER Paper 2
GCSE MATHEMATICS - LINEAR
y-axis and
– 180º, – 90º, 0º, 90º, 180º, 270º,
360º at even intervals on the x-axis.
– 90
90
180
270
360
– 0.788
–1
Use your
calculator to find
firstallvalue
of x of the following equation in the range – 180° to 360°.
(b) Find
solutions
sin x = – 0.788
[3]
x = sin (– 0.788)
x = – 52º (to the nearest º)
Using graph … x = –180º + 52º, x = 180º + 52º, x = 360º – 52º
x = –128º,
x = 232º,
x = 308º
Reveal