Year 11
Maths
Booklet
1
Year 11 Scheme of Work
Content
• Sine and Cosine Rule (if not covered in year 10)
• 3D Pythagoras and Trigonometry
• Surds
• Recurring Decimals
Test 1 - Revision resources at www.adamsmaths.uk
• Accuracy
• Graph Transformations (not in booklet, printed worksheets provided)
• Length of Arc, Area of Sector and Area of Segment
• Similar Solids
• Volumes and Surface Areas of Cones, Spheres and Cylinders
Mock Exams – Revision resources at www.adamsmaths.uk
• Circle Theorems
• Congruent Triangles
Christmas
• Algebraic Proof
• Cumulative Percentage Change
• Equations involving Indices and Surds
Test 2 - Revision resources at www.adamsmaths.uk
• Vectors (not in booklet, printed worksheets provided)
• Linear and Quadratic Inequalities
• Linear, Quadratic, Fibonacci and Geometric Sequences
• Samples and Counting Problems
• Density and Pressure
Easter
• Revision
Half Term
What’s on www.adamsmaths.uk?
Half Term Test Revision Resources
Mock Exam Revision Resources
Revision Questions by Topic
All the Past Papers
2
Sine and Cosine Rule
1) Find the missing side or angle using the sine or the cosine rule
a)
b)
c)
d)
𝑥
56𝑜
68
8
𝑥
𝑜
38𝑜
8
5
4
e)
12
𝑥
𝑥
38
19
73𝑜
𝑜
12
𝑥
35𝑜
8
7
f)
g)
h)
8
26
18
8
𝑥
𝑥
l)
68𝑜
17
112𝑜 44𝑜
𝑥
𝑥
8
𝑥
m)
16
n)
9
5
32
o)
6
26𝑜
23
𝑥
j)
7
38𝑜
17
𝑥
9
𝑜
38𝑜
14
k)
i)
11
7
39𝑜
𝑥
8
12
𝑥
106𝑜
𝑥
13
5
p)
q)
12
r)
17
𝑥
𝑜 𝑥
112
17
7
s)
𝑜
12 26
29
t)
𝑥 6
112𝑜
𝑥
35
17
25
𝑥
5
39𝑜
𝑜
Area of a Triangle
1) Work out the areas of the triangles.
a)
b)
c)
d)
e)
6
5
81𝑜
5
125𝑜
78
9
11 22
𝑜
𝑜
11
38𝑜
21
13
8
11
2) Work out the areas of the triangles (you may need to use other triangle rules first).
a)
b)
c)
d)
e)
17
9
14
11
9
67𝑜
f)
36
8
g)
14
15
68𝑜
10
39𝑜
h)
72𝑜
17
11
𝑜
4
14 49𝑜
4
37𝑜
i)
j)
8
41𝑜
5
67𝑜
67𝑜
12
21
7
15
3
3D Pythagoras and Trigonometry
1) The following shape
is a cube (all sides equal).
𝐴𝐵 = 6cm
𝐸
a) Calculate 𝐴𝐶
b) Calculate 𝐴𝐺
c) Calculate angle 𝐺𝐴𝐶
𝐻
2) In the following shape
𝐺
𝐸𝐹 = 6cm, 𝐸𝐻 = 5cm, and
𝐴𝐸 = 10cm.
a) Calculate 𝐴𝐶
b) Calculate 𝐴𝐺
c) Calculate angle 𝐺𝐴𝐶
𝐻
𝐹
𝐷
𝐸
𝐹
𝐶
𝐴
𝐷
𝐵
𝐴
𝐵
𝐹
𝐷
𝐴
5) a) Calculate the length 𝐴𝐶.
b) Calculate the vertical height
of the shape.
c) Calculate the angle 𝐸𝐶𝐴
22cm
14cm
𝐸
14cm
𝐶
𝐷
𝐶
15cm
𝐴
7cm
𝐴
𝐵
7) Calculate the angle 𝐴𝐶𝐸
8cm
8) Calculate the angle 𝐸𝐵𝐷
𝐵
𝐸
𝐸
17cm
18cm
10cm
𝐴
12cm
𝐷
𝐺
𝐹
12cm 𝐸
𝐴
𝐵
9) In the following shape, calculate:
a) Length AF
b) Angle 𝐴𝐷𝐸
𝐵
c) Angle 𝐵𝐸𝐺
15cm
𝐴
d) Angle 𝐵𝐸𝐹
𝐶
𝐷
𝐶
𝐷
𝐶
𝐵
6) a) Calculate the length 𝐴𝐶
b) Find the height of the
following pyramid.
c) Calculate the angle 𝐸𝐶𝐴
𝐸
𝐷
𝐺
𝐸
𝐶
𝐵
𝐶
𝐻
4) In the following shape
3) In the following shape 𝐴𝐵 = 7cm, 𝐴𝐸 = 6cm
𝐸𝐹 = 8cm, 𝐸𝐻 = 7cm, and
and 𝐴𝐷 = 9cm.
𝐻
𝐺
𝐴𝐸 = 13cm.
a) Calculate 𝐵𝐷
𝐸
𝐹
a) Calculate 𝐸𝐺
b) Calculate 𝐵𝐻
b) Calculate 𝐸𝐶
c) Calculate angle 𝐻𝐵𝐷
𝐷
c) Calculate angle 𝐶𝐸𝐺
𝐴
𝐺
9cm
7cm
𝐵
10) In the following symmetric shape 𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻,
calculate:
a) Length 𝐶𝐷
𝐺 28cm
𝐹
b) Angle 𝐶𝐷𝐴
c) Angle 𝐶𝐸𝐷
𝐶
𝐵
10cm
𝐻
20cm
𝐻
𝐶
𝐸
24cm
12cm
𝐴
4
37cm
𝐷
3D Pythagoras and Trigonometry (Hard Exam Questions)
1) In the triangular prism, 𝑀 is the midpoint of 𝐴𝐶
such that 𝐴𝐵𝐶 is an isosceles triangle.
Calculate to 3 s.f.:
𝐸
a) Length 𝐵𝑀
b) Length EM
2) 𝐼𝐽𝐾𝐿𝑀𝑁 is a triangular prism.
Calculate to 3 s.f.:
a) Length 𝐽𝐾
b) Area 𝐼𝐽𝐾
c) Volume of the triangular prism.
𝑀
𝐵
10cm
𝐷
𝐶
𝑀
8m
5m
25cm
59𝑜
𝐴
𝐽
𝐹
𝐿
𝐼
3) The diagram shows the triangle 𝐶𝐸𝐻 inside the
cuboid 𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻.
Calculate to 3 s.f.:
a) Length 𝐸𝐻
𝐸
𝐹
b) Length 𝐶𝐸
4m
c) Length 𝐶𝐻 𝐴
𝐵
d) Angle 𝐸𝐶𝐻
𝐺
𝐻
4) 𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻 is a cuboid.
𝐴𝐵 = 7.3cm , 𝐶𝐻 = 8.1cm , Angle 𝐵𝐶𝐴 = 48𝑜
Find the size of the
𝐻
𝐺
angle between 𝐴𝐺
𝐸
and the plane 𝐴𝐵𝐶𝐷,
𝐹
to 1 d.p.
𝐷
𝐷
8m
9m
𝐾
11m
6m
𝐶
𝑁
𝐶
𝐴
𝐵
6) 𝐴𝐷 = 15cm , 𝐴𝐵 = 15cm , 𝐵𝐴𝐸 = 35𝑜
Calculate, to 3 s.f.:
a) Length 𝐵𝐸
b) Length 𝐴𝐹
𝐹
c) Angle 𝐶𝐴𝐹
5) 𝐴𝐵𝐶𝐷𝐸 is a square based pyramid.
𝐴𝐵 = 5m.
𝐸
The vertex 𝐸 is 12m vertically
above the midpoint of 𝐴𝐶.
𝐸
Calculate the angle 𝐸𝐴𝐶.
𝐶
𝐷
𝐶
35𝑜
𝐴
𝐵
𝐷
15cm
𝐵
15cm
𝐴
8) Angle EBC = 60o, Angle ECB = 50o, and ABCD
is a rectangle.
𝐷
Work out the length AB.
7) In the pyramid ABCDP. O is the centre of the
square ABCD, and P is vertically above O.
PA = 11cm, and angle PBA = 72o.
Work out the height OP to 1 d.p. 𝑃
𝐸
11cm
𝐴
𝐷
𝐶
60𝑜
𝑂
72𝑜
𝐴
50𝑜
𝐵
𝐵
5
12cm
𝐶
Surds
1) Simplifying Surds – Simplify the following surds:
a) √18
c) √28
b) √50
d) √45
e) √150
f) √12
g) √72
h) √125
i) √90
j) √63
k) √80
l) √160
m) √98
n) √54
o) √32
p) √48
q) √75
r) √147
s) √27
t) √40
u) √250
v) √96
w) √112
x) √128
2) Adding and Subtracting Surds – Calculate the following:
a) √8 + √18
b) √27 + √75
c) √50 − √18
d) √24 + √54
e) √48 − √27
f) √32 + √72
g) √160 − √90
h) √180 − √45
i) 5√48 − 2√75
j) 9√8 + 4√32
k) 7√45 − √80
l) 6√108 + 7√48
m) 9√8 + 3√50
n) 12√45 − 2√125
o) 8√48 − √108
p) 8√54 + 3√96
3) Multiplying Surds – Calculate the following, leave your answer in simplified form:
b) √6 × √10
a) √7 × √14
c) √10 × √15
d) √8 × √14
e) √12 × √15
f) √14 × √2
g) √18 × √2
h) √20 × √10
i) √22 × √6
j) √12 × √14
k) √3 × √21
l) √5 × √30
m) 2√3 × √6
n) 5√6 × √10
o) 3√8 × √6
p) 2√10 × 3√15
q) 5√3 × √12
r) 8√6 × 2√2
s) 9√3 × √15
t) 8√2 × 3√10
4) Expanding Brackets – Calculate the following, simplifying each surd:
a) (2 + √2)(3 + √2)
b) (4 − √2)(5 + √2)
c) (3 + √5)(2 + 2√5)
d) (4 − √3)(5 + 2√3)
e) (5 + 2√3)(2 + √6)
f) (7 + √6)(3 + √10)
g) (5 + 3√3)(2 − √2)
h) (5 − 3√6)(5 − 3√6)
i) (5 + 2√2)
j) (5 − 2√3)
2
k) (2√5 − 1)
2
l) (3 − √2)
5) Rationalising the Denominator – Simplify the following expressions:
7
1
3
5
b)
d)
c)
a)
√2
√5
√7
√3
14
18
10
15
i)
j)
h)
g)
√5
√12
√6
√3
11
10
14
5
n)
o)
p)
m)
2√3
3√2
5 √6
2√2
s)
√5
3√2
t)
√7
4√3
u)
√5
2√2
v)
6
3√10
8√6
e)
k)
5
√6
12
q)
w)
√8
10
7√2
2+√3
√5
2
3
f)
l)
r)
x)
4
√2
15
√10
9
4√6
8−√2
√3
6) Rationalising the Denominator – Simplify the following expressions:
3
9
5
a)
c)
b)
4+√3
√11+2
√7−2
6
8
12
f)
g)
e)
2−√3
√5+√2
√3+1
2√7
4√3
√5
i)
j)
k)
5−√2
2+√3
√7−2
6
3+√2
4−√7
n)
m)
o)
3−2√2
4−√2
2+√7
1
10
9√3
r)
s)
q)
2√5−3
√7−1
√6−√3
d)
h)
l)
p)
t)
6
3−√5
12
√5−2
2√5
3−√5
6√3
7−2√3
3+√2
5−2√2
7) Mixed Questions – Calculate the following, leaving your answer as a simplified surd:
9
2
c)
a) √20
d) √27 + √75
b) (1 + √3)
√3
e)
8
f) 7√8 − √18
√11+3
i) 4 +
m)
3
j)
√2
22
10
k) (2 − √2)
√14−2
r) (3 + √2)
2
l) 6√5 × 2√10
o) (3 + √8)(2 + 3√6)
n) √98
2√3−1
√5−2
q)
2√3
h) (4 − √2)(8 − √8)
g) √128
3
s) 2 −
p)
3
√5
t)
18
3−√5
3+√2
2+2√2
8) More Mixed Questions – Calculate the following, leaving your answer as a simplified surd:
12
14
15
a) 3√27 −
+ 2√12
b) √50 + 4√8 −
c) 2√20 +
−2√45
√3
√2
√5
6
44
√2
e)
+ √63 − √49
f)
− 3√12 + 2√147
d)
+√90 + 4√18
5−√3
√7−2
√5+2
10
21
21
i) 5√8 + 3+√2 −√72
g) √50 +
−2√98
h) 3√12 +
− 2√75
√2
√3
20
55
6
26
24
l)
+ − √147
j) 2√45 +
− √20
k) 5√44 +
−
√3+4
√3
√5
√11 √11−3
9) Simplifying Surds with Algebra – Simplify the following surds:
a) √45𝑎2
b) √50𝑎2 𝑏 2
c) √28𝑥 3
d) √12𝑎5
g) √𝑥 4 𝑦 3
h) √54𝑥 4 𝑦 5
j) √75𝑎3 𝑏 8
i) √40𝑥 7 𝑦 6
e) √63𝑥 7
f) √32𝑎3 𝑏 2
k) √𝑥 5 𝑦10
l) √150𝑥 3
10) Mixed Questions with Algebra – Calculate the following, leaving your answer as a simplified surd:
a) √18𝑎3
b) √8𝑎5
c) √90𝑥 3
d) √12𝑎5
e) √3𝑎 × √6𝑎3
f) √10𝑎3 × √15𝑎2
4𝑎
j)
√𝑎
√𝑎−√2
n)
√2𝑎
g) √75𝑎 − √27𝑎
h) (2 + √𝑎)(3 − √𝑎)
6𝑎2
l)
√2𝑎
i) (√𝑎 + √𝑏)(2√𝑏 − √𝑎)
m) (√𝑎 + √2𝑎)
2
k)
o)
7
4𝑎
3+√𝑎
12
√5−√𝑎
p) (3√𝑎3 − √𝑎)
2
Trigonometry with Standard Results (Non-Calculator)
Calculate the value for 𝑥 in each of these questions:
8
Surds – Application Questions
1) A rectangle has a width of (3 + 4√2)cm and a height of (4 − √2)cm. Calculate the perimeter and area of
the rectangle.
𝑥 m
2) The following diagram (on the right) shows a patio
area of 5√2 − √3m2. One of the dimensions of the
patio area is (3 + √2)m. Calculate the length of the
other dimension.
Area = 𝟓√𝟐 − √𝟑 m2
3) The following diagram shows a right-angled triangle.
Calculate the value of the missing side, leaving your
answer in exact terms.
3 + √2 m
12 − 2√3
8 − 3√3
4) The following diagram shows a shaded region T formed by
removing an equilateral triangle 𝑃𝑄𝑅 from a regular hexagon
𝐴𝐵𝐶𝐷𝐸𝐹.
The points 𝑃 and 𝑄 lie on 𝐴𝐵 such that 𝐴𝐵 = 1.5 × 𝑃𝑄
Given that the area of 𝐓 is 72√3cm2.
Work out the length of 𝑃𝑄.
5) Given that (8 − √𝑥)(5 + √𝑥) = 𝑦√𝑥 + 21 where 𝑥 is a prime
number and 𝑦 is an integer, find the values of 𝑥 and 𝑦.
6) Given that 𝑦 is a prime number, express
3
2−√𝑦
in the form
𝑎+𝑏√𝑦
𝑐−𝑦
where 𝑎, 𝑏 and 𝑐 are integers.
7) The area of a rectangle is 18cm2. The length of the rectangle is (√7 + 1)cm. Find the width of the
rectangle in the form 𝑎√𝑏 + 𝑐 where 𝑎, 𝑏 and 𝑐 are integers.
8) Solve √3(𝑥 − 2√3) = 𝑥 + 2√3. Give your answer in the form 𝑎 + 𝑏√3 where 𝑎 and 𝑏 are integers.
9) Martin wants to complete the question: rationalise the denominator
Here is how he answered the question:
Find Martin’s mistake.
14
2+√3
=
=
=
14
2+√3
14×(2−√3)
(2+√3)(2−√3)
28−14√3
4+2√3−2√2+3
28−14√3
7
9
= 4 − 2√3
Accuracy
Error Intervals
1) The number 𝑛 is rounded to 4.76 to 2 d.p. Using inequalities, write down the error interval for 𝑛.
2) Jess rounds a number, 𝑥, to 1 d.p.. The result is 9.8. Write down the error interval for 𝑥.
3) Martin truncates the number 𝑁 to 1 digit. The result is 7. Write down the error interval for 𝑁.
4) Sally used her calculator to work out the value of a number 𝑦. The answer on the calculator display began
8.3. Write down an error interval for 𝑦.
5) The length of a pencil is 128mm correct to the nearest millimetre. Using the letter 𝑝, write an error
interval for the length of the pencil.
6) The value of 𝑥 is calculated to be 4,000 to 2 significant figures. Write down an error interval for 𝑥.
7) The value of 𝑥 is calculated to be 375 rounded the nearest multiple of 5. Write an error interval for 𝑥.
Upper and Lower Bound Calculations
1) 𝐷 = 𝑥 2 + 𝑎
𝑥 = 13.6 correct to 3 significant figures.
𝑎 = 4.2 correct to 2 significant figures.
Calculate the upper and lower bounds for 𝐷.
2) 𝑇 = 4𝑥 − 2𝑦
𝑥 = 83.2 correct to 1 decimal place.
𝑦 = 9.3 correct to 1 decimal place.
Calculate the upper and lower bounds for 𝑇.
1
3) 𝐴 = 2 𝑏 × ℎ
𝑏 = 7.24 correct to 3 significant figures.
ℎ = 1.5 correct to 2 significant figures.
Calculate the upper and lower bounds for 𝐴.
3𝑟
4) 𝑅 = 2𝑎
𝑟 = 54 correct to the nearest whole number.
𝑎 = 5.6 correct to 1 decimal place.
Calculate the upper and lower bounds for 𝑅.
5) 𝑌 = 5𝑥 − 2𝑎
𝑥 = 820 correct to 2 significant figures.
𝑎 = 115 correct to the nearest multiple of 5.
Calculate the upper and lower bounds for 𝑌.
𝑣−𝑢
8) 𝑎 = 𝑡
𝑣 = 40 to the nearest whole number.
𝑢 = 15 to the nearest whole number.
𝑡 = 12 to the nearest whole number.
Calculate the upper and lower bounds of 𝑎
𝑑
9) 𝑆 =
𝑡
𝑑 = 110 miles to the nearest mile.
𝑡 = 1 hour and 20 minutes, correct the nearest
10 minutes.
Calculate the upper and lower bounds of the
speed, 𝑆.
10) A train travelled along a track in 110 minutes,
to the nearest 5 minutes.
Jake finds out that the track is 270km long.
He assumes that the track has been measured
correct to the nearest 10km.
Could the average speed of the train have been
greater than 160km/h?
11) In the following diagram, all values are correct
to the nearest whole number. Calculate the upper
and lower bound of the angle 𝐶𝐴𝐵.
𝑉
6) 𝐼 = 𝑅
𝑉 = 65 measured to the nearest multiple of 5.
𝑅 = 9 measured to the nearest whole number.
Calculate the upper and lower bounds for 𝐼.
𝐴
9cm
7) 𝑇 = 𝑎2 − 𝑏 2
𝑎 = 6.3 correct to 1 d.p.
𝑏 = 2.6 correct to 1 d.p.
Calculate the upper and lower bounds for 𝑇.
𝐶
10
35𝑜
11cm
𝐵
Suitable degree of accuracy
𝑢
5) 𝑋 = 6𝑎 − 𝑏 2
𝑎 = 35.46 correct to 2 decimal places.
𝑏 = 5.23 correct to 2 decimal places.
By considering bounds, calculate the value of 𝑋
to a suitable degree of accuracy.
1) 𝑓 =
𝑣
𝑢 = 5.34 correct to 3 significant figures.
𝑣 = 0.52 correct to 2 decimal places.
By considering bounds, work out the value of 𝑓,
giving your answer to a suitable degree of
accuracy.
𝐼
6) 𝑉 = 𝑅
𝐼 = 75.34 correct to 2 decimal places.
𝑅 = 5.23 correct to 2 decimal places.
By considering bounds, calculate the value of 𝑉
to a suitable degree of accuracy.
1
2) 𝑑 = 8 𝑐 3
𝑐 = 10.9 correct to 3 significant figures.
By considering bounds, work out the value of 𝑑
to a suitable degree of accuracy.
𝑎+𝑏
7) 𝐷 = 𝑐
𝑎 = 14.56 correct to 2 decimal places.
𝑏 = 12.933 correct to 3 decimal places.
𝑐 = 3.12 correct to 2 decimal places.
By considering bounds, calculate the value of 𝐷
to a suitable degree of accuracy.
3) 𝑐 = 𝑎2 − 𝑏
𝑎 = 6.043 correct to 4 significant figures.
𝑏 = 2.95 correct to 3 significant figures.
By considering bounds, work out the value of 𝑐
to a suitable degree of accuracy.
𝑢2
4) 𝐷 = 2𝑎
𝑢 = 26.2 correct to 3 significant figures.
𝑎 = 4.3 correct to 2 significant figures.
By considering bounds, work out the value of 𝐷
to a suitable degree of accuracy.
8) A race is measured to have a distance of 10.6km,
correct to the nearest 0.1km. Sam runs the race in a
time of 31 minutes 48 seconds, to the nearest
second.
Sam’s average speed was 𝑉 km/hour.
By considering bounds, work out the value of 𝑉 to
a suitable degree of accuracy.
Recurring Decimals
1) Without a calculator, convert the following fractions into recurring decimals:
a)
3
7
b)
5
9
c)
4
13
d)
7
e)
11
8
11
Convert the following into fractions:
2) a) 0.44̇
b) 0.33̇
c) 0.22̇
d) 0.88̇
e) 0.99̇
3) a) 0. 1̇2̇
b) 0. 3̇4̇
c) 0. 5̇6̇
d) 0. 7̇8̇
e) 0. 0̇9̇
4) a) 0.46̇
b) 0.02̇
c) 0.67̇
d) 0.52̇
e) 0.39̇
5) a) 0.81̇2̇
b) 0.34̇1̇
c) 0.54̇8̇
d) 0.13̇2̇
e) 0.57̇3̇
6) a) 0. 3̇12̇
b) 0. 5̇02̇
c) 0.12̇35̇
d) 0. 9̇237̇
e) 0. 1̇62̇
7) a) 2. 3̇4̇
b) 4.18̇
c) 3. 9̇23̇
d) 4. 5̇
e) 2.95̇
8) Evaluate 0. 4̇ + 0. 3̇5̇ + 0.01̇3̇
11
f)
9
28
Length of Arc & Area of Sector and Segment
Simple Calculations of Arc Length and Area of a Sector
1)
2)
3)
15cm
6cm
130
40o
3cm
Calculate to 3 s.f.:
a) Arc length
b) Area of Sector
5)
4)
𝑜
290o
10o
Calculate to 3 s.f.:
a) Arc length
b) Area of Sector
Calculate to 3 s.f.:
a) Arc length
b) Area of Sector
Calculate to 3 s.f.:
a) Arc length
b) Area of Sector
6)
7)
8)
7cm
o
85
2cm
Calculate in terms of 𝜋:
a) Arc length
b) Area of Sector
5cm
230o
o
110o
4cm
25
10cm
Calculate in terms of 𝜋:
a) Arc length
b) Area of Sector
Calculate in terms of 𝜋:
a) Arc length
b) Area of Sector
Calculate in terms of 𝜋:
a) Arc length
b) Area of Sector
2)
3)
4)
Finding a Missing Value
1)
95o
30o
40cm
15cm
260o
Area = 40cm2
16cm
Calculate to 3 s.f.
a) Radius
b) Area of Sector
Area = 56cm2
Calculate to 3 s.f.
a) Radius
b) Length of Arc
Calculate to 3 s.f.
a) Radius
b) Area of Sector
5)
6)
7)
Area =
300cm2
2cm
225o
30cm
8)
4cm
3cm
120o
Calculate to 3 s.f.
a) Radius
b) Perimeter of shape
Calculate to 3 s.f.
a) Angle
b) Length of Arc
Calculate to 3 s.f.
a) Angle
b) Area of Sector
Calculate to 3 s.f.
a) Radius
b) Area of Sector
Area = 10cm2
Calculate to 3 s.f.
a) Angle
b) Arc Length
3)
4)
Calculating the Area of a Segment
1)
2)
40o
9cm
120o
70o 4cm
5cm
55
o
3cm
Calculate the shaded area
Calculate the shaded area
Calculate the area of the
shaded segment
12
Calculate the area of the
shaded segment
Length of Arc & Area of Sector and Segment
Past Paper Questions – from easy to hard
1) Calculate the area of the shaded segment to 3 s.f.
2) Calculate the perimeter of the shaded segment to
3 s.f.
𝑂
𝐶
72𝑜
5.4cm
𝐵
5.4cm
8cm
𝐵
𝐴
120𝑜
𝑂
𝑃
𝐴
3) The sector area = 20𝜋cm2
4) The arc length 𝐴𝐵𝐶 = 3𝜋 cm
𝐴
𝐵
Calculate the perimeter of
the sector 𝑂𝐴𝐵𝐶
Calculate the area of the
sector 𝑂𝐴𝐵𝐶
𝐵
𝐶
𝑂
𝑂
5) The length of the arc = 3𝜋 cm
𝐴
𝐴
𝐶
50𝑜
Calculate the area of the
shaded segment to 1 d.p.
8cm
12cm
6) The area of the sector 𝑂𝐴𝐵𝐶 is 5𝜋cm2.
𝐶
𝐵
𝐵
15cm
Calculate the perimeter
of the shaded segment
to 3 s.f.
𝐴
𝐶
6cm
𝑂
𝑂
7) The following shape shows a pentagon drawn
inside a circle. The radius is 6.8cm.
Calculate the area of the shaded region, correct to 3
s.f.
6cm
8) In the following diagram, the radius is 9cm, and
the angle 𝐶𝐷𝑂 is 35𝑜 .
Calculate the area
of the shaded
region to 3 s.f.
13
9) In the following shape 𝐷𝑂 = 9cm.
𝐴
Work out the length of the arc 𝐴𝐵𝐶 to 3 s.f.
5cm
𝐵
𝑂
𝐷
5cm
𝐶
10) Using the following diagram:
𝐵
a) Calculate the length of 𝐵𝐶 to 3 s.f.
b) Calculate the total area of the shape 3 s.f.
65𝑜
𝐴
8cm
11) In the following diagram:
𝑂𝐴 = 6cm.
𝐴𝐷 = 14cm.
Angle 𝐴𝑂𝐷 = 140𝑜
Angle 𝑂𝐴𝐷 = 24𝑜
Calculate the perimeter of the shape.
12) In the following diagram, 𝐴𝐵 and 𝐵𝐶 is
a tangent to the sector 𝑂𝐴𝐶.
Calculate the area of the shaded region.
14
35𝑜
𝑂
𝐶
13) The diagram shows two circles with centre 𝑂 and a regular
pentagon 𝐴𝐵𝐶𝐷𝐸.
The pentagon has sides of length 8cm.
The diagram is shaded such that
𝑆ℎ𝑎𝑑𝑒𝑑 𝐴𝑟𝑒𝑎 = 𝑈𝑛𝑠ℎ𝑎𝑑𝑒𝑑 𝐴𝑟𝑒𝑎
Work out the radius of the smaller circle, to 3 s.f.
14) The following diagram shows the cross section of a
circular water pipe.
The shaded region in the diagram represents the water
flowing in the pipe.
The water flows at 14cm/s in the pipe.
Work out the volume of water that flowed through the pipe in
3 minutes. Give your answer to 3 s.f.
15) In the following diagram 𝑂𝐴𝐵 is sector 𝐒 of a circle with centre
𝑂 and radius (𝑟 + 7) metres.
A different circle 𝐂 has radius (𝑟 − 2) metres.
Given that the area of sector 𝐒 is twice the area of circle 𝐂.
Find the value of 𝑟.
16) The following diagram shows the overlap between two circles.
The radius of one circle is 5cm and the radius of
another circle is 4cm.
Calculate the area of region 𝐑. Give your answer to 3
s.f.
15
Similar Solids
1) The following two shapes 𝐴 and 𝐵 are similar.
The volume of 𝐴 is 70cm2.
Calculate the volume of 𝐵.
2) The following two shapes 𝐴 and 𝐵 are similar.
The surface area of 𝐵 is 125cm2.
Calculate the surface area of 𝐴.
𝐵
6cm
12cm
Radius = 3cm
3) The following two shapes 𝐴 and 𝐵 are similar.
The volume of 𝐴 is 200cm3.
Calculate the volume of shape 𝐵.
𝐴
𝐵
𝐴
𝐴
Radius = 5cm
4) The following two shapes 𝐴 and 𝐵 are similar.
The surface area of 𝐴 is 16cm2.
Calculate the surface area of 𝐵.
𝐵
𝐵
𝐴
3cm
Radius = 7cm
10cm
Radius = 2cm
5) The following two shapes 𝐴 and 𝐵 are similar.
The surface area of 𝐵 is 200cm2
Calculate the surface area of 𝐴.
6) The following two shapes 𝐴 and 𝐵 are similar.
The volume of 𝐵 is 50cm3.
Calculate the volume of 𝐴.
𝐵
𝐴
7cm
10cm
7) The following two shapes 𝐴 and 𝐵 are similar.
The volume of 𝐴 is 128cm3.
Calculate the volume of shape 𝐵.
𝐴
𝐴
𝐵
Radius = 8cm
Radius = 5cm
8) The following two shapes 𝐴 and 𝐵 are similar.
The surface area of 𝐵 is 324cm2.
Calculate the surface area of 𝐴.
𝐵
𝐵
𝐴
Radius = 9cm
9cm
Radius = 5cm
12cm
9) The radius of a sphere increases by 30%. Find
the percentage increase in the surface area.
10) The side length of a dice (cube shaped) is
increased by 25%. Find the percentage increase in
the volume.
16
Harder Similar Solids
1) The following two shapes 𝐴 and 𝐵 are similar.
Volume 𝐴 = 15cm3. Volume 𝐵 = 405cm3.
Surface area 𝐴 = 20cm2.
Calculate the surface area of 𝐵.
2) The following two shapes 𝐴 and 𝐵 are similar.
Surface area 𝐴 = 50cm2. Surface area 𝐵 = 312.5cm2
Volume 𝐴 = 40cm3.
Calculate the volume of 𝐵.
𝐵
𝐴
𝐵
𝐴
3) The following two shapes 𝐴 and 𝐵 are similar.
Volume 𝐴 = 135cm3. Volume 𝐵 = 40cm3.
Surface area 𝐴 = 67.5cm2.
Calculate the surface area of 𝐵.
𝐴
4) The following two shapes 𝐴 and 𝐵 are similar.
Surface area 𝐴 = 20cm2. Surface area 𝐵 = 245cm2
Volume 𝐵 = 1029cm3.
Calculate the volume of 𝐴.
𝐵
𝐵
𝐴
5) The following two shapes 𝐴 and 𝐵 are similar.
Volume 𝐴 = 250cm3. Volume 𝐵 = 1024cm3.
Surface area 𝐴 = 300cm2.
Calculate the surface area of 𝐵.
6) The following two shapes 𝐴 and 𝐵 are similar.
Surface area 𝐴 = 193.6cm2. Surface area 𝐵= 40cm2
Volume 𝐴 = 532.4cm3.
Calculate the volume of shape 𝐵.
𝐵
𝐴
𝐴
7) The following two shapes 𝐴 and 𝐵 are similar.
Surface area 𝐴=187.5cm2. Surface area 𝐵 = 30cm2
Volume of 𝐴 = 312.5cm3.
Calculate the volume of 𝐵.
𝐴
𝐵
8) The following two shapes 𝐴 and 𝐵 are similar.
Volume of 𝐴 = 60cm3. Volume of 𝐵 = 349.92cm3
Surface area of 𝐵 = 162cm2.
Calculate the surface area of 𝐴.
𝐵
𝐵
𝐴
17
Harder Similar Solids with Ratios
1) The following two shapes 𝐴 and 𝐵 are similar.
Surface area 𝐴 : Surface area 𝐵 = 25: 49
Volume 𝐵 = 82.32cm3.
Calculate the volume of 𝐴.
2) The following two shapes 𝐴 and 𝐵 are similar.
Volume 𝐴 : Volume 𝐵 = 27: 8.
Surface area 𝐴 = 90cm2
Calculate the surface area of 𝐵.
𝐵
𝐴
𝐴
3) The following two shapes 𝐴 and 𝐵 are similar.
Surface area 𝐴 : Surface area 𝐵 = 16: 9
Volume 𝐴 = 192cm3.
Calculate the volume of 𝐵.
𝐴
𝐵
4) The following two shapes 𝐴 and 𝐵 are similar.
Volume 𝐴 : Volume 𝐵 = 8: 125
Surface area 𝐵 = 500cm2
Calculate the surface area of 𝐴.
𝐵
𝐵
𝐴
5) The following two shapes 𝐴 and 𝐵 are similar.
Surface area 𝐴 : Surface area 𝐵 = 4: 9
The volume of 𝐴 is 60cm3.
Calculate the volume of 𝐵.
6) The following two shapes 𝐴 and 𝐵 are similar.
Volume 𝐴 : Volume 𝐵 = 27: 8
The surface area of 𝐴 is 180cm2.
Calculate the surface area of 𝐵.
𝐵
𝐴
𝐴
7) The following two shapes 𝐴 and 𝐵 are similar.
Surface Area 𝐴 : Surface Area 𝐵 = 𝑛 ∶ 4
Volume 𝐴 = 625cm3. Volume 𝐵 = 40cm3
Calculate the value of 𝑛.
𝐴
𝐵
8) The following two shapes 𝐴 and 𝐵 are similar.
Volume 𝐴 : Volume 𝐵 = 27: 𝑛
Surface area 𝐴 = 36cm2. Surface area 𝐵 = 64cm2.
Calculate the value of 𝑛.
𝐵
𝐵
𝐴
18
Volumes and Surface Areas of Cones, Spheres and Cylinders
Formulae you don’t get in the exam
Formulae you get in the exam
1
Volume of Cylinder = 𝜋𝑟 2 ℎ
Volume of Cone = 3 𝜋𝑟 2 ℎ
𝑙
ℎ
Surface area = 2𝜋𝑟 2 + 2𝜋𝑟ℎ
Curved Surface area of
Cone = 𝜋𝑟𝑙
𝑟
Area of Circle = 𝜋𝑟 2
Circumference = 2𝜋𝑟
4
Volume of Sphere = 3 𝜋𝑟 3
Surface area of Sphere = 4𝜋𝑟
ℎ
𝑟
𝑟
2
Simple Calculations of Volume and Surface Area
1) A sphere has a radius of 5cm.
Calculate the:
a) Surface area to 3 s.f.
𝑟
b) Volume to 3 s.f.
2) In the following cone, the
radius is 6cm and the height is
8cm. Calculate
a) Volume to 3 s.f.
b) Surface area to 3 s.f.
𝑙
ℎ
𝑟
3) In the following cylinder,
the radius is 8cm and the
height is 12cm.
Calculate:
a) Volume to 3 s.f.
b) Surface area to 3 s.f.
ℎ
5) A sphere has a radius of 4cm.
Calculate the:
a) Surface area in terms of 𝜋
b) Volume in terms of 𝜋
7) In the following cylinder,
the radius is 10cm and the
height is 14cm.
Calculate:
ℎ
a) Volume in terms of 𝜋
b) Surface area in terms of 𝜋.
4) In the following hemisphere,
the radius is 7cm.
Calculate the:
a) Surface area to 3 s.f.
b) Volume to 3 s.f.
𝑟
6) In the following cone, the
radius is 5cm and the height is
12cm. Calculate
a) Volume in terms of 𝜋.
b) Surface area in terms of
𝜋.
𝑟
𝑙
ℎ
𝑟
8) In the following hemisphere, the radius is 7cm.
Calculate the:
a) Surface area in terms of 𝜋
b) Volume in terms of 𝜋.
𝑟
19
Finding a Missing Value to Solve a Question
1) The following sphere has a surface area of
36𝜋cm2.
Calculate:
a) the radius.
𝑟
b) the volume in terms of 𝜋.
3) In the following cone the
height is 9cm, and the volume
is 192𝜋cm3.
Calculate:
a) the radius.
b) the curved surface area
to 3 s.f.
𝑙
2) The following cylinder has a
height of 8cm and a volume of
288𝜋cm3.
Calculate:
ℎ
a) the radius
b) the surface area in terms of
𝜋.
ℎ
𝑟
5) The following sphere has a volume of 2000cm3.
Calculate:
a) the radius to 2 d.p.
b) the surface area to 3 s.f.
𝑟
7) In the cylinder, the ratio of
the radius to the height is 2: 5.
The volume of the cylinder
is 540𝜋cm3.
ℎ
Calculate:
a) The length of the radius
b) The total surface area.
9) In the following cone, the ratio
of the height of the cone to the
radius of the cone is 4: 3.
𝑙
The volume of the cone is
96𝜋cm3.
Calculate:
a) the radius of the cone.
b) the curved surface area to
3 s.f.
4) In the following hemisphere, the volume is
144𝜋cm3.
Calculate:
a) the radius
b) the surface area in terms of
𝜋.
6) The following cone has a
height of 10cm and volume of
500cm3.
Calculate:
a) the radius to 2 d.p.
b) the slanted height to 2
d.p.
𝑙
ℎ
𝑟
8) In the following hemisphere, the total surface
area is 90𝜋cm2.
Calculate:
a) the radius in simplified surd
form.
b) the volume of the
hemisphere in terms of 𝜋.
𝑟
ℎ
𝑟
𝑟
10) In the following sphere, the total
surface area is 588𝜋cm2.
Calculate:
a) the radius in simplified surd
form.
b) the volume of the sphere in
the form 𝑎√3𝜋, where 𝑎 is a
constant to be found.
20
𝑟
Composite Solids
1) The following shape is a cone on top of a
hemisphere.
The height of the shape is 13cm.
The diameter of the
hemisphere is 6cm.
2) The following shape is a cone on top of a
cylinder.
The height of the cone is 10cm.
The height of the cylinder is 10cm.
The radius of the cone is 10cm.
Calculate the total
volume in terms of 𝜋.
Show that the surface area
is (300 + 100√2)𝜋cm2.
13cm
10cm
6cm
10cm
10cm
3) The following diagram shows a hemisphere
connected to a cylinder.
The radius of the cylinder is 3.4cm, and the height
of the cylinder is 8.3cm.
4) The following diagram shows a hemisphere on
top of a cylinder.
The radius of the hemisphere is 10cm. The height
of the cylinder is ℎcm.
Calculate the
total surface
area to 3 s.f.
The total surface area is
1000𝜋cm2.
3.4 cm
Calculate ℎ.
ℎ cm
8.3cm
10 cm
5) The following diagram shows a cone on top of a
hemisphere.
The overall height of the shape is ℎ cm.
The diameter of the cone and hemisphere is 6cm.
The volume of the shape is 54𝜋cm3.
6) The following diagram shows a cone to top of a
hemisphere.
The height of the cone is 4𝑟, and the radius is 3𝑟.
The total volume of the shape is 330𝜋cm3.
3
Find the value of 𝑟 in the form √𝑛 where 𝑛 is an
integer.
Calculate the height of the
overall shape.
4𝑟
ℎ cm
3𝑟
6cm
21
Frustums
1) The diagram shows a frustum is made by
removing a cone with height 5cm from a solid cone
with height 10cm and base
diameter of 12cm.
2) The diagram shows a frustum made by removing
a cone with height 8cm from a solid cone with
height 12cm and base radius
9cm.
5cm
Calculate the volume of the
frustum in terms of 𝜋.
8cm
Calculate the volume of the
frustum in terms of 𝜋.
5cm
4cm
12cm
9cm
3) The diagram shows a frustum made by removing
a cone with height 𝑥 cm from a
solid cone to leave a frustum of
𝑥
height 6cm. The top diameter is
4cm and the bottom diameter
is 12cm.
4cm
6cm
a) Calculate the value of 𝑥.
4) The diagram shows a frustum made by removing
a cone with height 8cm from a
solid cone with height 12cm and
base radius 9cm.
8cm
Calculate the surface area of
the frustum in terms of 𝜋.
4cm
12cm
9cm
b) Calculate the volume of
the frustum to 3 s.f.
5) The diagram shows a frustum made by removing
a cone with height 𝑥 cm from a solid cone to leave
a frustum of height 12cm. The top radius is 3cm
and the bottom radius is 12cm.
𝑥
a) Show that 𝑥 = 4
b) Calculate the surface
area of the frustum in
terms of 𝜋.
3cm
12cm
6) The diagram shows a frustum made by removing
a cone with height 12cm
from a solid cone to leave a
frustum of height 4cm.
12cm
The base radius is 14cm.
Calculate the volume
of the frustum in
terms of 𝜋.
4cm
14cm
12cm
7) The diagram shows a frustum made by removing
a cone with height 𝑥 cm from a solid cone to leave
a frustum of height 15cm.
𝑥
a) Calculate 𝑥.
b) Given that the volume of
the frustum is 1040𝜋, find
the value of 𝑟.
𝑟 cm
15cm
3𝑟 cm
8) The diagram shows a frustum made by removing
a cone with height 15cm from a
solid cone to leave a frustum of
height 20cm. The top radius is
15cm
3𝑟.
a) Find an expression for
the bottom radius.
b) Given that the volume of
the frustum is 555𝜋, find
the value of 𝑟.
22
3𝑟
5cm
Comparing Two Solids
1) In the following diagrams, the cylinder has
a height ℎ and radius 𝑟.
The sphere has a radius 2𝑟.
The volume of the two shapes are equal.
ℎ
2𝑟
Find an expression for ℎ in terms of 𝑟.
𝑟
2) The following two diagrams have
the same volume.
Find an expression for ℎ in terms of 𝑥.
5𝑥
ℎ
𝑥
2𝑥
3) In the following two shapes the base radius
of the cone is three times the radius of the sphere.
Given that the volume of the cone is equal
to the volume of the sphere, find an expression
for the radius of the sphere in terms of ℎ.
ℎ
4) In the following diagrams it is given that
Total surface area of cylinder
=2
surface area of sphere
𝑟
ℎ
Find the value of
𝑟
Volume of cylinder
Volume of sphere
23
Solids - Exam Questions
1) The solid is made from a cone and a hemisphere.
The radius of the cone and hemisphere are both 20cm.
The curved surface area of the cone is 580𝜋cm2.
The volume of the solid is 𝑘𝜋cm3.
Work out the exact value of 𝑘.
2) The following solid shows a hemisphere and a of radius 𝑥, and a
cylinder with radius 𝑥 and height 3𝑥.
The total surface area is 81𝜋cm2.
Find the value of 𝑥.
3) The following diagram shows a sphere with diameter 𝑥, and a
rectangular based pyramid as shown below.
The volume of the sphere is
288𝜋cm3.
Calculate the total surface
area of the pyramid. Give
your answer to the nearest
cm2.
4) The diagram shows a solid cone and a solid sphere.
The base radius of the cone is equal to the radius of
the sphere.
𝑙
ℎ
Given that
𝑟
𝑘 × volume of the cone = volume of the sphere
Show that the total surface area of the cone can be written in the form
𝜋𝑟 2 (
𝑘 + √𝑘 2 + 𝑎
)
𝑘
24
𝑟
Revision of Circle Theorems from Year 9
1) Calculate:
𝐷
2) Calculate:
𝑄
𝑅
a) 𝑃𝑄𝑆
a) 𝐺𝐷𝐸
41𝑜
𝑇
b) 𝑄𝑆𝑅
b) 𝐸𝐹𝐺
𝑂
98𝑜
34𝑜
c) 𝑄𝑇𝑃
𝑃
𝐺
𝐸
𝐹
𝑆
3) Calculate:
4) Calculate:
𝑄
a) 𝑃𝑂𝑅
a) 𝐴𝑂𝐵
b) 𝑃𝑄𝑅
b) 𝐵𝐴𝐷
𝑂
𝑅
𝐶
106𝑜
𝐷
57𝑜
𝐵
𝑂
c) 𝑂𝐴𝐷
36𝑜
𝑃
𝐴
𝐶
5) Calculate:
6) Calculate:
112𝑜
a) 𝐵𝐴𝐷
𝐷
b) 𝐴𝐷𝐵
𝑀
a) 𝐿𝑂𝑃
48𝑜
b) 𝑂𝑃𝐿
𝑂
𝐵
𝑂
𝐿
𝐴
𝑃
7) Calculate:
8) Calculate:
𝐵
𝐶
a) 𝐴𝐷𝐵
b) 𝐷𝐶𝐵
𝐴
a) 𝐷𝑂𝐵
𝐴
75𝑜
𝐷
b) 𝑂𝐷𝐶
58𝑜
𝑂
𝐷
𝑂
𝐶
27𝑜
𝐵
25
Circle Theorems – Angles on Tangents
1) Calculate:
𝐶
2) Calculate:
𝐵
a) 𝐴𝑂𝐶
b) 𝐴𝐵𝐶
b) 𝐴𝐷𝐵
𝑂
𝐴
𝐵
103𝑜
a) 𝐴𝐵𝐷
𝐷
76𝑜
𝐶
39𝑜
𝑃
𝐴
𝐷
𝑇
3) Calculate the value of 𝑥.
𝐵
4) Calculate:
𝑥𝑜
𝐴
a) 𝐴𝐶𝐷
𝐴
𝐵
𝑃
b) 𝐷𝐴𝐶
𝐶
c) 𝐴𝐵𝐶
𝑂
29𝑜
𝐶
53𝑜
61𝑜
𝐸
5) Calculate angle 𝑆𝑇𝐴
𝐹
𝐷
6) Calculate angle 𝐶𝑃𝐴
𝐶
𝑅
73𝑜
𝑄
𝑂
100𝑜
𝑆
𝐵
𝑃
26𝑜
𝐴
𝐴
𝑇
7) Calculate:
8) Calculate:
𝐵
𝐸
a) 𝐴𝐶𝐵
35𝑜
b) 𝐶𝐴𝑂
𝐷
a) 𝐴𝐶𝐵
𝐴
b) 𝐶𝐴𝐷
𝑂
56𝑜
𝐶
𝐴
260𝑜
𝑂
𝐸
𝐶
30𝑜
𝐵
𝐷
26
Mixed Circle Theorem Questions
1) Calculate angle 𝐶𝐴𝐵.
𝐵
2) Calculate angle 𝐴𝐷𝐸.
40𝑜
𝐵
𝐶
𝑂
𝐷
𝐴
32
𝐶
𝑜
𝐷
𝐴
3) Calculate angle 𝐴𝐵𝐷.
𝐵
𝐴
4) Calculate angle 𝐷𝐹𝐸
𝐶
𝑂
𝐹
40𝑜
𝐸
𝐸
𝐹
𝐷
100𝑜
𝐷
48𝑜
𝐴
𝐸
5) Calculate angle 𝑂𝑃𝑄
𝐵
𝐶
6) Calculate 𝐵𝐷𝐸
𝑄
𝐸
𝑉
18𝑜
238𝑜
𝑂
𝐹
𝐷
60𝑜
𝑅
𝑃
39𝑜
𝐴
𝑇
7) Show that the line 𝐴𝐶 is parallel to 𝐸𝐹.
𝐸
𝐶
𝐵
8) Given that 𝐵𝐴𝐷: 𝐵𝐶𝐷 = 3: 1
Calculate 𝑆𝐵𝐴.
𝐵
𝑆
𝐹
40𝑜
𝑇
20𝑜
𝐴
𝐷
70𝑜
𝐴
𝐵
𝐶
𝐷
𝐶
27
Mixed Circle Theorem Questions
1) Calculate in terms of 𝑥
2) Calculate:
𝐶
a) 𝐵𝐷𝐴
𝑃
a) 𝑃𝑄𝑅
2𝑥
𝐵
b) 𝐴𝑂𝐷
b) 𝑆𝑃𝑂
𝑂
c) 𝐴𝐵𝐷
𝑂
𝑆 70𝑜
c) Explain why
𝑃𝑄𝑅𝑆 is not a
cyclic quadrilateral.
𝑥
𝐷
𝑄
𝑅
𝐴
3) Calculate:
4) Calculate:
𝐵
a) 𝐴𝑃𝑇
a) 𝐶𝑂𝑇
b) 𝐵𝑇𝑆
𝐶
b) 𝐷𝐶𝑇
𝐵
𝐶
𝑂
c) 𝐴𝐵𝐶
25𝑜
42𝑜
𝐷
𝑇
𝐴
𝑃
𝑆
𝐴
𝑇
5) Calculate:
6) Calculate:
a) Show that 𝑥 2 − 6𝑥 − 39 = 0
a) 𝐴𝐶𝐵
𝑇
b) Hence, find
the value of
the radius to
3 s.f.
𝐸
𝐴
b) 𝐵𝐴𝐶
𝑥+5
𝑥
𝐵
𝑂
𝑂
𝑥+8
63𝑜
𝐴
𝐶
𝐹
1
7) 𝑃𝑄𝑅𝑆𝑇 is a regular pentagon.
Prove that 𝑆𝑈 = 𝑈𝑇.
𝑄
8) Show that 𝐴𝐵𝐶 = 90 − 2 𝑥
𝑅
𝐴
𝐶
𝑆
x
𝑂
𝑈
𝑇
𝑥
𝑃
𝐵
𝑇
28
𝑃
9) Calculate the radius of the circle.
10) Calculate the radius of the circle.
𝑆
𝑇
𝐶
54
20cm
𝑂
30𝑜
𝐴
𝑃
𝐵
14cm
11) Calculate:
𝑅
𝑄
12) Calculate:
𝐷
a) 𝐵𝐴𝑂
𝑇 a) Angle 𝑃𝑂𝑇
𝐶
b) 𝐷𝐴𝑇
30𝑜
108𝑜
b) Length 𝑃𝐶
𝑂
𝑂
𝐶
5.8cm
𝐴
𝑃
13) Calculate:
𝑇
12.5cm
𝐵
14) Calculate:
𝐴
𝐵
a) Length 𝐴𝐵
a) 𝐵𝐶𝐴
b) Area of circle.
b) 𝐷𝐵𝐶
𝑂
16cm
c) 𝐵𝑂𝐴
𝐴
35𝑜
𝑂
𝐷
𝐶
12cm
𝐶
𝐵
15) Angle 𝐵𝐶𝐸 = 63𝑜
𝐵
𝐸
a) 𝐴𝐶𝐵
b) 𝐵𝐴𝐶
16) 𝐴𝐵 = 7cm , 𝐵𝑋 = 5cm, 𝐶𝑋 = 6cm.
Calculate:
a) Length 𝐴𝐶
b) Length 𝐷𝐶
𝐴
𝐴
𝑂
7cm
63𝑜
𝐵
𝐶
5cm
𝐷
𝐹
29
𝐶
28𝑜
6cm
𝑋
Congruent Triangles
1) Given that 𝐴𝐵𝐶𝐷 forms a rectangle, prove that
triangles 𝐵𝑂𝐶 and 𝐴𝑂𝐷 are congruent.
𝐴
2) Given that 𝐴𝐵𝐶𝐷 is a parallelogram, prove that
triangles 𝐴𝐵𝐷 and 𝐵𝐶𝐷 are congruent.
𝐵
𝐴
𝐵
𝑂
𝐷
𝐶
𝐷
3) Given that 𝐴𝐵𝐶𝐷 is a parallelogram, prove that
𝐴𝑂𝐵 is congruent to 𝐷𝑂𝐶.
𝐴
𝐵
𝐶
4) Given that lengths 𝐵𝑌 and 𝐷𝑋 are equal, and that
𝐴𝐵𝐶𝐷 is a square, prove that 𝐴𝐵𝑌 and 𝐴𝐷𝑋 are
congruent.
𝐵
𝐴
𝑌
𝑂
𝐶
𝐷
𝐷
5) In the following diagram 𝐴𝐵𝐶𝐷 and 𝐿𝑀𝑁𝑂 are
squares. Angle 𝐿𝐵𝐶 = 𝑥o. Prove that triangles 𝐴𝐵𝑂
and 𝐶𝐵𝐿 are congruent.
𝐴
𝐵
𝐿
𝑥
𝑋
6) In the following diagram, 𝑂
is the centre of the
circle, and 𝐴𝐵 and 𝐴𝐶
are tangents to the circle.
Prove that triangles 𝑂𝐵𝐴
𝐶
and 𝑂𝐶𝐴 are congruent.
𝐶
𝑂
𝐵
𝑂
𝑀
𝐴
𝑁
𝐷
𝐶
7) In the following diagram 𝐴𝐵𝐶 is an isosceles
triangle and 𝑁𝐶𝐵 = 𝐶𝐵𝑀.
𝐴
Prove that triangles 𝐶𝐵𝑁
and 𝑀𝐵𝐶 are congruent.
8) Given that triangle 𝑃𝑄𝑅 is equilateral, and
𝑃𝑋 = 𝑍𝑅 = 𝑄𝑌, use congruent triangles to prove
that 𝑋𝑌𝑍 is also an equilateral
𝑃
triangle.
𝑋
𝑁
𝑀
𝑍
𝑅
𝐵
𝐶
30
𝑌
𝑄
9) In the following diagram 𝑃𝑄𝑅𝑆 and 𝑃𝑇𝑈𝑉 are
squares, and 𝑃𝑆𝑇 is an isosceles triangle.
Show that triangles 𝑄𝑃𝑇 and 𝑆𝑃𝑉 are congruent.
𝑄
10) In the following diagram, 𝐴𝐵𝐶𝐷 and 𝐷𝐸𝐹𝐺 are
both squares. Prove that 𝐴𝐷𝐸 and 𝐶𝐷𝐺 are
congruent triangles.
𝐺
𝐹
𝑉
𝑃
𝐴
𝑅
𝑈
𝑆
𝐸
𝐷
𝐵
𝐶
𝑇
11) Given that 𝑃𝑄𝑅𝑆 is a triangle an 𝐴𝑄𝑅 is an
equilateral triangle, prove that 𝐴𝑃𝑄 and 𝐴𝑆𝑅 are
congruent.
𝑄
𝑃
12) Given that 𝐴𝐵𝐶𝐷 and 𝐶𝑃𝑄𝑅 are squares, prove
that 𝐶𝐷𝑃 is congruent to 𝐵𝐶𝑅.
𝐵
𝐴
𝑄
𝐴
𝑅
𝑃
𝑅
𝑆
𝐷
13) Given that triangle 𝐴𝐷𝐸 is an equilateral
triangle, show that triangle 𝐴𝐵𝐶 is congruent to
𝐴
triangle 𝐷𝐶𝐵.
𝐶
14) Given that 𝐴𝐷 and 𝐵𝐶 are parallel, 𝐴𝑃 and 𝑄𝐶
are parallel and 𝐶𝐵𝑄 = 𝑃𝐷𝐴 = 90o , prove that
triangle 𝐴𝐷𝑃 and triangle 𝑄𝐵𝐶 are congruent.
𝐴
𝑃
𝐵
𝐷
𝐵
𝐸
𝑄
𝐶
𝐷
𝐶
15) State which two of the following triangles are congruent, and state the reason.
45𝑜
55𝑜
10cm
45
55𝑜
45𝑜
10cm
Triangle 𝐀
8cm
10cm
𝑜
80𝑜
8cm
10cm
Triangle 𝐁
Triangle 𝐂
31
Triangle 𝐃
Algebraic Proof
1) Prove that the sum of two even integers is
always even.
18) Prove that the difference between two
consecutive square numbers is always odd.
2) Prove that the sum of two odd integers is always
even.
19) Prove that the sum of two consecutive odd
square numbers is always 2 more than a multiple of
4.
3) Prove that the sum of an odd integer and an even
integer is always odd.
4) Prove that the sum of three odd integers is
always odd.
5) Prove that the multiplication of two even
numbers is always even.
6) Prove that the product of two odd numbers is
always odd.
20) Prove that the sum of two consecutive even
square numbers is always a multiple of 4.
Algebraic Questions
21) Prove that (3𝑛 + 1)2 − (3𝑛 − 1)2 is always a
multiple of 12.
22) Prove that (2𝑛 + 3)2 − (2𝑛 − 3)2 is always a
multiple of 8.
23) Prove that (5𝑛 + 1)2 − (5𝑛 − 1)2 is always a
7) Prove that the product of one odd integer and one multiple of 5.
even integer is always even.
24) Prove that:
8) Prove that the difference between two even
(2𝑎 − 1)2 − (2𝑏 − 1)2 = 4(𝑎 − 𝑏)(𝑎 + 𝑏 − 1)
numbers is always even.
25) Prove that (3𝑛 + 1)2 + (3𝑛 + 2)2 is 1 less than
9) Prove that the difference between two odd
a multiple of 3.
numbers is always even.
26) Prove that (4𝑛 + 1)2 + (4𝑛 + 2)2 is always
10) Prove that the difference between an odd
one more than a multiple of 4.
number and an even number is always odd.
1
11) Prove that the square of an odd number is
always odd.
12) Prove that the square of an even number is
always even.
27) Prove that the sum of 2 𝑛(𝑛 + 1) and
1
2
(𝑛 + 1)(𝑛 + 2) is always a square number.
Splitting into odd and even cases
27) Prove that for all integers 𝑛, 𝑛2 + 𝑛 is always
even.
Consecutive Terms
13) Prove that the sum of two consecutive integers
is always odd.
28) Prove that for all integers 𝑛, 𝑛2 − 3𝑛 is always
even.
14) Prove that the sum of three consecutive integers
is divisible by 3.
29) Prove that for all integers 𝑛, 𝑛2 + 2𝑛 + 41 is
always odd.
15) Prove that the sum of four consecutive integers
is always 2 more than a multiple of 4.
30) Prove that for all integers 𝑛, 𝑛3 + 3𝑛 is always
even.
16) Prove that the sum of two consecutive odd
integers is always a multiple of 4.
Extension: By considering odd and even cases,
prove that the sum of two squares is never equal to
a value 1 less than a multiple of 4.
17) Prove that the sum of two consecutive square
numbers is always odd.
32
Percentage Change
Revision – Percentage Change:
1) A bike costs £400. It is then in the sale at a price
of £300. Find the percentage discount of the bike.
13) The initial number of bacteria in a petri dish is
200. This increases by 40% per hour. Find the
number of bacteria in the petri dish after 8 hours.
2) The value of a vintage pen is £350. It then
increases in value to £490. Find the percentage
increase in the value of the pen.
14) Dominic invests £3,000 in bitcoin. The value of
his investment increases by 20% per year in year 1
and 2, but then decreases in value by 15% in year 3.
Find the value of his investment at the end of year 3
3) A dress is priced at £60. In the sale it is priced at
£42. Calculate the percentage discount in the sale.
4) The value of a house in 2020 is £210,000. The
following year it is worth £218,400. Calculate the
percentage increase in the value of the house.
5) A car originally is priced at £14,000. The
following year it is worth £11,900. Calculate the
percentage decrease in the value of the car.
Revision – Reverse Percentages:
6) A dress is in a 15% discounted sale for £42.50.
Calculate the original price of the dress.
7) The value of a vintage watch increases in value
by 35% to the value of £67,500 in one year.
Calculate the original price of the watch.
8) The value of a car decreases by 18% in one year.
The value in 2021 is £11,480. Calculate the value
of the car in 2020.
9) The value of a house increases by 4% in one
year. The value in 2020 is £249,600. Calculate the
value of the house in 2019.
10) The sale price for a pair of shoes is £33.80.
Given that the pair of shoes is in a 35% sale,
calculate the original price for the shoes.
Compound Interest:
11) The value of a vintage painting increases in
value by 5% per year. In 2020 the value of the
painting is £2,000,000. Calculate the value of the
painting in 2025.
12) Simon invests £6,000 in a bank that pays 4%
interest per year. Calculate the value of the
investment 6 years afterwards.
15) The value of a rare musket increases in value by
15% per year, over three years. Calculate the
overall percentage increase in the value of the
musket over the three years.
16) Mary invests in a gold piece of jewelry. The
value of gold increases by 5% per year in year 1
and year 2, and then decreases by 7% in year 3.
Calculate the percentage change in the value of the
gold piece of jewelry over the three years.
Compound interest – finding an unknown
17) The value of a vintage necklace increases in
value by 𝑥% each year over 4 years. In total, the
piece of jewelry increases in value by 15%.
Calculate the value of 𝑥.
18) Emily invests in Apple stocks. The value of the
stocks increase by 𝑥% each year over 5 years. In
total, the value of the stocks increases by 60%.
Calculate the value of 𝑥.
19) The value of a car decreases by 40% over 3
years. Given that the value of the car decreases by
the same percentage each year, calculate the yearly
percentage decrease in the value of the car.
20) Oliver buys shares in twitter.
Initially, they increase in value by 20% in the first
year. Over the next four years they decrease by the
same percentage each year.
In total, over the 5 years, the shares decrease in
value by 10%. Find the percentage the shares
decrease by in each of the four years.
21) Kirsty buys a house in 2010. In 2010 and 2011
the house decreases by 15% per year. In 2012, 2013
and 2014, the house increases in value by the same
percentage each year. At the end of 2014, the house
price is 5% more than it was worth in 2010.
Find the percentage increase in 2012, 2013 and
2014.
33
Exponential Equations and Graphs
1) Match the following graph to the equation at the bottom
𝑦 = 𝑥2 − 7
𝑦 = 2−𝑥
𝑦 = −𝑥 2
𝑦 = 2𝑥
𝑦 = cos(𝑥)
𝑦 = 𝑥3
34
1
𝑥
1
𝑦=−
𝑥
𝑦=
𝑦 = sin(𝑥)
2) The following graph has equation 𝑦 = 𝑎 𝑥 .
Calculate the value of 𝑎.
(2,9)
3) The following graph has equation 𝑦 = 2𝑥 + 𝑏.
a) Calculate the value of 𝑏.
b) Find the equation of the asymptote.
(2,7)
(1,3)
4) The following graph has equation 𝑦 = 𝑎 × 2−𝑥
Calculate the value of 𝑎.
(0,4)
5) The following graph has equation 𝑦 = 𝑎 × 𝑏 𝑥
Calculate the values of 𝑎 and 𝑏.
(2,18)
(0,6)
(1,6)
6) The following graph has equation 𝑦 = 2𝑥 + 𝑏.
a) Find the value of 𝑏.
b) Find the value of 𝑘.
7) The following graph has equation 𝑦 = 1.5−𝑥 + 𝑏
a) Find the value of 𝑏.
b) Find the value of 𝑘.
(2, 𝑘)
(−1, 𝑘)
൬2,
(−1,5.5)
35
40
൰
9
Modelling with Trigonometric Equations
1) The following graph shows
𝑦 = 2cos(𝑥).
𝐴
Write down coordinates:
a) 𝐴
𝐵
b) 𝐵
𝑂
c) 𝐶
d) Write down the value of
2cos(630𝑜 )
𝐶
2) The following graph shows
𝑦 = 2 sin(𝑥) + 1.
Write down the coordinates:
𝐴
𝐵
a) 𝐴
b) 𝐵
c) 𝐶
𝑂
d) Write down the value of
2 sin(−540) + 1
𝐶
3) The following graph shows
𝑦 = cos(𝑥 − 𝑎) + 𝑏. The
coordinate (45,3) is a maximum
point labelled on the graph.
(45,3)
a) Find the values of 𝑎 and 𝑏.
𝐵
Write down the coordinates of
𝐴
b) 𝐴
𝑂
c) 𝐵
4) The following graph shows 𝑦 = 𝑎 sin(𝑥)
and 𝑦 = cos(𝑥) + 𝑏.
𝑦 = 𝑎sin(𝑥)
a) Find the values of 𝑎 and 𝑏
b) Use the graph to evaluate
𝑎 sin(𝑥) − (cos(𝑥) + 𝑏) when 𝑥 = 270.
c) Estimate the solution to
𝑎 sin(𝑥) = cos(𝑥) + 𝑏
where 0 < 𝑥 < 360.
𝑦 = cos(𝑥) + 𝑏
36
5) The following two graphs labelled Graph
𝐴 and Graph 𝐵 are transformations
of 𝑦 = cos(𝑥).
Graph 𝐴
a) Find the equation of Graph 𝐴.
b) Find the equation of Graph 𝐵.
Graph 𝐵
6) The following graph shows a
transformation of 𝑦 = tan(𝑥).
Find an equation for the graph
shown.
7) The following graph shows 𝑦 = cos(2𝑥)
Write down coordinates:
𝐶
a) 𝐴
b) 𝐵
𝐴
c) 𝐶
𝐵
8) The following graph shows:
𝑦 = − sin(𝑥) + 3
𝐵
𝐴
Write down coordinates:
a) A
b) 𝐵
37
9) The following graph shows 𝑦 = cos(2𝑥) − 1
Write down coordinates:
a) 𝐴
b) 𝐵
c) Write down the value of cos(540) − 1
𝐴
𝐵
10) The following two graphs labelled
Graph 𝐴 and Graph 𝐵 are
transformations of 𝑦 = sin(𝑥)
Graph 𝐴
a) Find the equation of Graph 𝐴
b) Find the equation of Graph 𝐵
Graph 𝐵
11) The following graph is a transformation of
𝑦 = tan(𝑥).
Write down two possible equations that could
represent this graph.
12) The following graph shows a
graph transformation of 𝑦 = cos(𝑥).
Write down the equation of the graph.
38
13) The following graph is 𝑦 = cos(𝑥)
Use the graph to find two solutions between
0 to 360:
a) cos(𝑥) = 0.4
b) cos(𝑥) = −0.7
14) The following graph is 𝑦 = sin(𝑥)
Use the graph to find two solutions
between 0 to 360:
a) sin(𝑥) = 0.8
b) sin(𝑥) = −0.3
Context Questions
1) The depth, 𝑑, in metres, of water at the end of a jetty 𝑡 hours after noon is modelled by the formula
𝑑 = 4 + 2.5cos(30𝑡)
a) Find the depth of water at
i) noon
ii) 2pm
iii) 3pm
iv) 6pm
v) midnight
b) Find the first time, correct to the nearest minute, when the depth of the water is 6 metres.
c) Sketch the graph of 𝑑 against 𝑡 for 0 ≤ 𝑡 ≤ 12.
2) One end of a spring is fixed to a wall at point P. A mass M, which lies on the table is attached to the other
end. PM is horizontal. Damien pushes the mass towards P and releases it. He models the distance 𝑦cm, of
the mass from P at time 𝑡 seconds after releasing it by the formula
𝑦 = 15 − 5cos(45𝑡)
a) Find the distance of the mass from the wall when i) 𝑡 = 2
ii) 𝑡 = 4 iii) the mass is released.
b) Sketch the graph of 𝑦 against 𝑡 for 0 ≤ 𝑡 ≤ 8.
3) 𝑡 hours after midnight, the depth of water, 𝑑 metres, at the entrance of a harbour is modelled by the
formula
𝑑 = 6 + 3sin(30𝑡)
a) What is the depth of water at i) 1 am
ii) noon?
b) What is the depth of water at low tide?
c) Find the times of high tide during a complete day.
d) Sketch the graph of 𝑑 against 𝑡 for 0 ≤ 𝑡 ≤ 24.
4) The diameter of a big wheel is 16m. Its centre is 9m above the ground. The wheel rotates clockwise.
Mandy rides on the big wheel and starts to time it when her chair reaches the highest point. The wheel
rotates once every 20 seconds.
a) Find the constants 𝑝 and 𝑞 so that 𝑦 = 𝑝 + 𝑞𝑐𝑜𝑠(18𝑡) is a suitable model for the height of the chair, 𝑦
metres, above the ground 𝑡 seconds after timings start.
b) Find the times during the first half minute when the chair is 13m above the ground.
c) Find the times during the first half minute when the chair is 5m above the ground.
d) Sketch the graph of 𝑦 against 𝑡 for 0 ≤ 𝑡 ≤ 30.
39
Equations with Indices
1) Revision – Numerical Indices – Evaluate the following values
1
a) 5−2
1
4
3
f) 64
g) 81
3 −2
−
k) ( )
3
2
8 −3
m) ( )
27
l) ( )
4
4
1
4 2
i) ( )
25
2 −2
h) ( )
1 −3
e) 252
3
2 2
3
4
3
d) 3−3
c) 49−2
b) 362
j) 36−2
3
o) 16−4
n) ( )
5
2) Revision – Rules of Indices with Algebra – Simplify the following pieces of algebra
a) 4𝑥 5 × 3𝑥 3
b) 3𝑥 2 𝑦 4 × 4𝑥 3 𝑦
c) 6𝑥 3 𝑦 5 × 2𝑦 2 𝑧 4
d) 7𝑥 3 × 2𝑦 4
e) (3𝑥 4 )2
f) (2𝑥 5 )3
g) (4𝑥 2 )3
h) √16𝑥 6
i)
14𝑥 3 𝑦 8
j)
7𝑥 2 𝑦3
60𝑥 7 𝑦 2
k)
40𝑥 3 𝑦
50𝑥 4 𝑦 2
20𝑥 3 𝑦 9
l)
32𝑥 7 𝑦 4
16𝑥 4 𝑦 2
3) Easy – Equations involving Surds – Solve the following equations:
a) 9√3 = 3𝑥
e)
9
√3
b) 8√2 = 2𝑥
= 3𝑥
f)
3
i) 2𝑥 = √2 × √2
16
c) 25√5 = 5𝑥
= 2𝑥
√2
j) 3𝑥 =
g)
1
5
√5
= 5𝑥
k) 2𝑥 =
√3
1
√32
d) √27 = 3𝑥
h) 3𝑥 =
81
√3
1
l) 5𝑥 = 125
4) Medium – Changing the base of an Indices Equation – Solve the following equations:
a) 4𝑥 = 8
e)
b) 9𝑥 = √3
1
1
f) 25𝑥 = 125
= √8
4𝑥
3
i) 5√5 = 25𝑥
j) 82 =
1
d) √4 = 8𝑥
g) 9𝑥 = √27
h) 4√2 =
1
k) 84 = 4𝑥 × √2
4𝑥
3
c) 27𝑥 = 9
1
8𝑥
l) 25𝑥 = 5 ×
25
√5
5) Hard – Equations including Indices – Solve the following equations:
3
a) √5 = 25𝑥 ÷ 52
3
5𝑥
e) √25 =
125
1
1
i)
= 274 ÷ 3𝑥+1
√9
b) √27 = 9𝑥 ÷ 3
f)
1
√8
1
=
32
4𝑥
3
j) 165 × 2𝑥 = 84
1
c) 324 = 4𝑥 ÷ 8
1
g) 27−𝑥 =
9√3
27
k) 9𝑥−1 =
√3
d) 27𝑥 =
1
93
h) 25√5 = 125 ÷ 25𝑥
l) 8√2 =
1
4 𝑥−1
6) Rearrange Formulae including Indices – Write an expression for 𝑦 in terms of 𝑥.
8
1
9
9𝑥
b) 27𝑦 = 9𝑥 × √3
a) 4𝑦 = 𝑥
c) 𝑦 = 𝑥
d) 27𝑦+1 =
2
9
27
√3
1
√32
g) 8𝑦 = 4 ÷ 32𝑥
f) 9−𝑦 = √3𝑥
h) 8𝑦−1 = 64𝑥 √32
e) 𝑦 = 𝑥
8
4
1
1
i) 9−𝑦 = √27 ÷ 3𝑥+1
l) 9𝑦+1 = 27𝑥 ÷ √243
j) 25−𝑦 =
k) 25𝑦 × 125 = 𝑥
𝑥+1
125
5
40
Column Vectors
1) Basics – Write the following translations as vectors:
a) 4 right, 5 up
b) 3left, 8 up
d) 8 left, 6 up
e) 2 left
g) 8 left, 8 down
h) 2 right, 7 up
j) 5 down
k) 6 left, 2 up
c) 6 right, 9 down
f) 7 right, 3 up
i) 3 right, 9 down
l) 8 right, 4 up
2) Describe the following vectors as translations:
5
a) ( )
3
−4
)
5
f) (
−7
)
5
3
)
−4
6
d) ( )
2
e) (
9
)
−5
i) (
−3
)
−6
j) (
b) (
c) (
2
g) ( )
0
h) (
−5
)
−4
0
)
−7
3) a) Write down vector 𝐚
b) Write down vector 𝐛
c) Write down the vector from coordinate (2,5) to (6,2).
d) Write down the vector from coordinate (8,4) to (5,6)
4) Adding and Subtracting Vectors – Calculate the following:
8
7
a) ( ) + ( )
5
6
b) (
3
11
)−( )
9
4
c) (
−4
8
e) ( ) − ( )
9
4
7
9
f) ( ) + ( )
6
−9
g) (
4
−5
)+( )
−3
4
6
−5
d) ( ) − ( )
4
4
−7
3
)+( )
4
−8
h) (
−6
−8
)−( )
−3
4
6
−3
−7
5) If 𝐚 = ( ) , 𝐛 = ( ) and 𝐜 = ( ), calculate:
5
5
7
a) 𝐚 + 𝐛
b) 3𝐚
c) 𝐛 − 𝐚
d) 𝐜 − 𝐛
f) 5𝐚 + 3𝐜
g) 3𝐚 − 5𝐛
h) 6𝐛 − 4𝐚
i) 2 𝐚
2
21
6) If 𝐚 = ( ) and 3𝐚 + 5𝐛 = ( ), find the vector 𝐛.
3
14
7
3
7) If 𝐚 = ( ) and 5𝐚 + 4𝐛 = ( ), find the vector 𝐛.
32
4
13
3
8) If 𝐚 = ( ) and 7𝐚 − 2𝐛 = (
), find the vector 𝐛.
−13
−1
3
−24
9) If 𝐛 = ( ) and 6𝐚 − 4𝐛 = (
), find the vector 𝐚.
5
−38
−26
−5
10) If 𝐛 = ( ) and 4𝐚 − 3𝐛 = (
), find the vector 𝐚.
45
6
41
1
e) 3𝐛 − 𝐜
1
j) 2 (𝐛 + 𝐜)
Inequalities
1) Linear Inequalities – Solve the following inequalities
a) 4𝑥 − 7 ≥ 29
b) 23 ≥ 3𝑥 − 1
d) 9𝑥 − 13 < 23
e) 18 − 2𝑥 > 10
g) 17 − 5𝑥 ≤ 4𝑥 − 19
j) 2(𝑥 − 3) > 43 − 5𝑥
h)
5(𝑥−4)
𝑥+1
≤3
8
f) 13 ≥ 28 − 3𝑥
c)
i) 1 −
𝑥
< −6
2
𝑥
l) 32 − 2𝑥 ≥ 7 −
> 10
8
𝑥−3
k) 11 − 𝑥 ≤
3
7
2) Two Linear Inequalities – Find the region of values for 𝑥 that satisfies both these inequalities
a) 2𝑥 + 5 > 11 and 4 ≤ 19 − 3𝑥
b) 17 − 𝑥 ≤ 15 and 16 < 30 − 2𝑥
c) 2𝑥 + 6 ≥ 0 and 3𝑥 + 3 < 2𝑥 + 7
d) 𝑥 − 16 ≤ 4𝑥 − 1 and 10 + 𝑥 ≤ 28 − 2𝑥
e) 5 − 3𝑥 < 10 − 2𝑥 and 2𝑥 + 5 ≤ 41 − 2𝑥
f) 8 − 3𝑥 ≤ 14 − 𝑥 and 3𝑥 − 10 ≤ 23
3) Triple Linear Inequalities – Solve the following inequalities
a) 3 < 2𝑥 + 7 ≤ 13
b) 8 ≤ 3𝑥 − 4 < 23
c) −6 ≤ 4 − 2𝑥 < 10
d) 1 < 5 − 𝑥 ≤ 12
e) 2 ≤ 9 − 𝑥 < 6
f) −3 < 7 − 2𝑥 ≤ 11
g) −7 < 5 − 3𝑥 ≤ 14
h) −11 < 4 − 3𝑥 ≤ 13
i) −9 ≤ 9 − 2𝑥 < 1
4) Quadratic Inequalities – Splitting into two parts – Solve the following inequalities
a) 𝑥 2 ≤ 16
b) 𝑥 2 ≥ 9
c) 𝑥 2 < 25
d) 𝑥 2 ≥ 49
e) (𝑥 − 4)2 ≤ 36
f) (𝑥 − 5)2 > 9
g) (𝑥 + 2)2 < 100
h) (4 − 𝑥)2 ≤ 49
i) (7 − 𝑥)2 ≥ 81
j) (3 − 𝑥)2 > 64
k) (3 − 2𝑥)2 ≤ 49
l) (5 − 3𝑥)2 > 169
5) Quadratic Inequalities – Solve the following inequalities
a) 𝑥 2 − 6𝑥 + 5 ≥ 0
b) 𝑥 2 + 𝑥 − 12 < 0
c) 𝑥 2 − 9𝑥 + 20 > 0
d) 𝑥 2 + 2𝑥 − 15 ≤ 0
e) 𝑥 2 − 11𝑥 + 24 < 0
f) 𝑥 2 + 5𝑥 − 24 ≥ 0
g) 12𝑥 ≥ 𝑥 2 + 35
h) 7𝑥 < 𝑥 2 + 12
i) 𝑥 2 < 7𝑥 − 6
j) 5𝑥 − 77 ≤ 9𝑥 − 𝑥 2
k) 3𝑥 − 14 ≥ 40 − 𝑥 2
l) 3𝑥 − 𝑥 2 < 8𝑥 − 24
m) 15 − 𝑥 2 ≥ 2(𝑥 − 10)
n) 6𝑥 − 12 < 15 − 𝑥 2
o) 100 − 𝑥 2 > 3𝑥 − 80
6) Harder Quadratic Inequalities – Solve the following inequalities
a) 3𝑥 2 + 2𝑥 − 16 ≥ 0
b) 2𝑥 2 − 𝑥 − 21 < 0
c) 7𝑥 2 + 23𝑥 + 6 > 0
d) 2𝑥 2 + 9𝑥 + 10 ≤ 0
e) 6𝑥 2 − 𝑥 − 22 ≥ 0
f) 12𝑥 2 − 29𝑥 − 21 < 0
g) 25𝑥 + 35 ≤ 7 − 3𝑥 2
h) 45 − 7𝑥 < 12𝑥 − 2𝑥 2
i) 10𝑥 2 > 13𝑥 + 14
j) 14𝑥 2 ≤ 27𝑥 + 20
k) 3 − 13𝑥 < 18 − 6𝑥 2
l) 27𝑥 − 6𝑥 2 > 50𝑥 + 15
42
7) One Linear and One Quadratic Inequality – Find the region of values for 𝑥 that satisfy both inequalities
a) 𝑥 2 < 16 and 3𝑥 + 5 > 11
b) 𝑥 2 − 2𝑥 − 15 ≤ 0 and 3𝑥 + 11 < 17
c) 𝑥 2 + 5𝑥 − 14 ≤ 0 and 10 ≥ 7 − 2𝑥
d) 𝑥 2 − 2𝑥 − 24 ≤ 0 and 13 − 2𝑥 > 12 − 3𝑥
e) 𝑥 2 + 9𝑥 + 14 ≤ 0 and 13 > 3 − 2𝑥
f) 𝑥 2 − 3𝑥 − 10 ≤ 0 and 𝑥 + 4 > 10 − 2𝑥
g) 𝑥 2 < 3𝑥 + 28 and 𝑥 − 4 ≤ 8 − 2𝑥
h) 𝑥 2 < 5𝑥 + 24 and 16 + 3𝑥 ≥ 6 − 2𝑥
8) In the following diagram, the area of the
rectangle is greater than the area of the triangle.
3𝑥 − 2
2
a) Show that 2𝑥 − 5𝑥 + 2 > 0
b) Find the possible set of value of 𝑥.
2𝑥
𝑥−1
𝑥
9) In the following diagram, the area of the
triangle is greater than the are of the rectangle.
a) Show that 7𝑥 2 − 35𝑥 > 0
3𝑥 + 4
b) Find the possible set of value of 𝑥.
5𝑥 − 6
𝑥+3
4𝑥 − 4
2
10) The area of the triangle is less than 7cm .
a) Show that 2𝑥 2 − 5𝑥 − 25 < 0
2𝑥 − 3
b) Given that all sides of the triangle are positive,
find the range of possible values of 𝑥.
30𝑜
𝑥−1
11) The area of the parallelogram is greater than 15cm2.
2𝑥 − 1
a) Show that 2𝑥 2 − 21𝑥 + 40 < 0
150𝑜
b) Find the range of possible values of 𝑥.
10 − 𝑥
43
Sequences
1) Revision - Linear Sequence – Find an expression for the nth term of the following sequences
a) 5 , 9 , 13 , 17, 21 , …
b) 2 , 9 , 16 , 23 , 30 , …
c) −6 , 3 , 12 , 21 , 30 , … d) 19 , 17, 15 , 13 , 11 , ..
e) 14 , 11 , 8 , 5 , 2 , …
f) −9, −5, −1, 3, …
g) −2, 1, 4, 7, …
i) −10, −25, −40, −55,
j) −41, −81, −121, …
k) 5 , 1 , 5 , 5 , …
3
7
9
h) 40, 37, 34, 31, …
l) −5, −8, −11, −14, …
2) Revision - Iterative Sequence – Find the value of 𝑥2 , 𝑥3 , 𝑥4 and 𝑥5 in the following sequences
1
a) 𝑥𝑛+1 = 3𝑥𝑛 − 4 , 𝑥1 = 3
c) 𝑥𝑛+1 = 1.2𝑥𝑛 + 2 , 𝑥1 = 3.2
b) 𝑥𝑛+1 = 𝑥𝑛 + 3 , 𝑥1 = 4
2
d) 𝑥𝑛+1 = √𝑥𝑛 + 2 , 𝑥1 = 3
g) 𝑥𝑛+1 = 9 −
4
𝑥𝑛
, 𝑥1 = 2
e) 𝑥𝑛+1 = √7 − 𝑥𝑛 , 𝑥1 = 2
h) 𝑥𝑛+1 =
𝑥𝑛
, 𝑥1 = 3
√𝑥𝑛 +3
f) 𝑥𝑛+1 =
3𝑥𝑛
𝑥𝑛 +2
, 𝑥1 = 2
1
i) 𝑥𝑛+1 = 3 𝑥𝑛 + √𝑥𝑛 , 𝑥1 = 5
3) Fibonacci Type Sequences
a) In the following Fibonacci type sequence, the 3rd
term is 12, and the 5th term is 31. Calculate the 7th
term in the sequence.
b) In the following Fibonacci type sequence, the 4th
term is 14 and the 7th term is 60. Calculate the 8th
term in the sequence.
c) In the following Fibonacci type sequence, the 3rd
term is 6 and the 8th term is 68. Calculate the 10th
term in the sequence.
d) In the following Fibonacci sequence, the 4th term
is 11 and then 7th term is 45. Calculate the 5th term
in the sequence.
e) In the following Fibonacci type sequence, the 3rd
term is 17 and the 6th term is 69. Calculate the 8th
term in the sequence.
f) In the following Fibonacci type sequence, the 4th
term is 12 and the 8th term is 81. Calculate the 6th
term in the sequence.
4) Quadratic Sequences – Find an expression for the nth term of the following sequences
a) 3, 14, 29, 48, 71,…
b) 8, 15, 28, 47, 72, …
c) −1, 5, 13, 23, 35, …
d) 0, 3, 10, 21, 36, …
e) −1, 9, 25, 47, 75
f) 8, 18, 30, 44, 60, …
g) 7, 12, 21, 34, 51, …
h) 1, 10, 27, 52, 85, …
i) −7, −1, 9, 23, 41, …
j) 1, 11, 27, 49, 77, …
k) 5, 7, 13, 23, 37, …
l) −5, 0, 7, 16, 27, …
5) Geometric Sequences – Find the next three terms in the following geometric sequences
3
a) 5, 10, 20, 40, …
b) 4, 12, 36, 108, …
c) 12, 6, 3, , …
d) 3, 15, 75, 375, …
2
5
g) 1.4 , 2.8 , 5.6 , 11.2 , …
e) 45, 15, 5, 3 , …
h) 5 , 5.5 , 6.05 , 6.655, …
f) 48, 24, 12, 6, …
j) 5 , 5√2 , 10 , 10√2 , …
k) 2 , 2√3 , 6 , 6√3 , …
l) 4√5, 20, 20√5 , 100, …
i) 8 , 9.6 , 11.52 , 13.824, …
6) The first three terms in a geometric sequence are 𝑥 − 1 , 𝑥 + 4 , 3𝑥 + 2. Calculate the values of 𝑥.
7) The first three terms in a geometric sequence are 𝑥 − 5 , 𝑥 + 3 , 4𝑥. Given that 𝑥 > 0, calculate the 5th
term in the sequence
8) The first three terms in a geometric sequence are 𝑥 − 2 , 𝑥 + 1 , 3𝑥 − 3. Calculate the values of 𝑥.
9) The first three terms in a geometric sequence are √𝑥 − 1 , 1 , √𝑥 + 1.
a) Find the value of 𝑥.
b) Show that the 5th term is 7 + 5√2
44
Peterson Capture-Recapture Method
𝑁𝑢𝑚𝑏𝑒𝑟 𝑐𝑎𝑝𝑡𝑢𝑟𝑒𝑑 𝑎𝑛𝑑 𝑡𝑎𝑔𝑔𝑒𝑑 𝑖𝑛 𝑟𝑜𝑢𝑛𝑑 1 𝑁𝑢𝑚𝑏𝑒𝑟 𝑐𝑎𝑝𝑡𝑢𝑟𝑒𝑑 𝑤𝑖𝑡ℎ 𝑡𝑎𝑔 𝑜𝑛 𝑖𝑛 𝑟𝑜𝑢𝑛𝑑 2
=
𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑 𝑡𝑜𝑡𝑎𝑙 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛
𝑇𝑜𝑡𝑎𝑙 𝑐𝑎𝑢𝑔ℎ𝑡 𝑖𝑛 𝑟𝑜𝑢𝑛𝑑 2
1) A scientist wishes to estimate the number of rabbits on an island. The scientist initially captures 40
rabbits, tags them, and releases them. 30 days later, the scientist captures 50 rabbits, of which 5 of them have
tags on. Use the following information to estimate the population of rabbits on the island.
2) Shirley wants to find an estimate for the number of bees in her hive. On Monday she catches 90 of the
bees, places a mark on each of these bee’s, and returns it to the hive. On Tuesday she catches 120 of the
bees, and she finds that 20 of these bees have been marked. Work out an estimate for the total number of
bees in her hive.
3) Jeremy wishes to calculate the number of bats in a cave. He captures 40 bats and places a mark on each of
these bats, before returning it to the cave. 30 days later, he captures 30 bats, of which 10 of them have a
mark on. Work out an estimate for the number of bats in the cave.
4) Julie wants to estimate the number of butterfly’s in a field. On Monday she captures 50 butterfly’s and
places a mark on them. On Saturday she captures 36 butterfly’s and finds that 9 have a mark on them. Work
out an estimate for the number of butterfly’s in the field.
5) Stuart wants to estimate the number of puffin’s on Puffin island (in Wales). On Monday he captures 20
puffins, places a mark on them, and releases them. On Wednesday he captures 35 puffins, of which 2 have a
mark on them. Estimate the number of puffin’s on Puffin island.
Sampling
1) In a company there are 40 office staff, 50 warehouse staff, and 20 managers. A survey is to be conducted
about the staff Christmas party by taking a sample of 44 members of staff from the company. Calculate the
number of each type of staff member that should be included in the survey.
2) Each person in a fitness club is going to get a free gift. Stan conducts a survey from a sample of 50 people
at the fitness club. 17 people choose sports bag, 7 choose towels, 11 choose headphones and 15 choose
vouchers. In total there are 700 people at the fitness club. Work out the number of sports bags Stan should
order to give away as a free gift.
3) Hannah is planning a day trip for 195 students. She asks a sample of 30 students where they want to go.
Each student chooses one place. 10 choose a theme park, 5 choose the theatre, 8 choose a sports centre and 7
choose the seaside. Work out an estimate for the number of students who will want to go to the theme park.
4) Stuart is planning a tea party. He surveys 20 people who are attending whether they would like tea,
coffee, wine, or fruit juice. 3 people choose tea, 5 people choose coffee, 8 people choose wine, and 4 people
choose fruit juice. There are 110 people attending the tea party. Calculate an estimate for the number of
people who will be drinking fruit juice at the party.
5) Fred is having a biscuit party. He surveys 30 people who are attending the biscuit party which biscuits are
their favourite. 13 choose digestives, 8 choose custard crème, and 9 choose jammy dodgers. There are 150
people coming to the biscuit party. Calculate an estimate for the number of people whose favourite biscuit is
a jammy dodger.
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Counting Problems
1) In a restaurant, there are three options for starter, 6 options for main course, and 4 options for dessert.
Work out the number of different ways of choosing a starter, main course and dessert.
2) There are 16 hockey teams in a league. Each team plays two matches against each of the other teams.
Work out the total number of matches played.
3) On a padlock, each dial can be set to one of the numbers 1, 2, 3, 4, 5. Work out the
number of different combinations that can be set for the combination lock.
4) Sadia is going to buy a new car. For the car, she can choose one body colour, one
roof colour and one wheel type. She can choose from
19 different body colours
25 different wheel colours
The total number of ways Sadia can choose the body colour and the roof colour and
the wheel type is 3325. Work out the number of different roof colours that Sadia can
choose from.
5) Jack is in a restaurant. There are 5 starters, 8 main courses and some desserts on the menu. Jack is going
to choose one starter, one main course and one dessert. He says there are 240 ways that he can choose his
starter, his main course and his dessert. Could Jack be correct? Justify your answer.
6) Julie is in a restaurant. There are 6 starters, 7 main courses and 5 desserts. Julie is only going to choose
two options, one from either
• Starter and main course or
• Main course and dessert.
Calculate the number of different ways Julie can choose from these options.
7) A switch board has 32 switches that can either be turned on or off. Write down the number of possible
combinations the switches can be in at any one time. Write your answer to 3 s.f. and in standard form.
Pressure
𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 =
𝐹𝑜𝑟𝑐𝑒
𝐴𝑟𝑒𝑎
1) A cuboid is placed on a table. The cuboid has dimensions 5cm × 5cm × 5cm. The force on the table is
80N. Calculate the pressure the cuboid placed on the table in N/cm2.
2) A coin is placed on a table. The radius of the coin is 6mm. The force of the coin on the table is 2N.
Calculate the pressure the coin placed on the table in N/mm2.
3) A box in the shape of a cuboid is placed on a horizontal floor. The box exerts a force of 180 newtons on
the floor. The box exerts a pressure of 187.5 N/m2 on the floor. The face in contact with the floor is a
rectangle of length 1.2 metres and width 𝑥 metres. Work out the value of 𝑥.
4) A coin is placed on a table. The radius of the coin is 𝑥. The coin exerts a force of 10N on the table. The
coin exerts a pressure of 0.2N/mm2 on the table. Work out the value of 𝑥 to 3 s.f.
5) A force of 70N acts on an area of 20cm2. The force is increased by 10N. The area is increased by 10cm2.
Helen says
“the pressure decreases by less than 20%”
Is Helen correct?
6) The diagram shows a prism placed on a horizontal floor. The prism has a
height of 3m. The volume of the prism is 18m3. The pressure on the floor
due to the prism is 75N/m2.
Work out the force exerted by the prism on the floor.
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Density
𝐷𝑒𝑛𝑠𝑖𝑡𝑦 =
𝑀𝑎𝑠𝑠
𝑉𝑜𝑙𝑢𝑚𝑒
1) 25cm3 of metal 𝐴 and 30cm3 of metal 𝐵 are going to be smelted together.
• Metal 𝐴 has a density of 6.8g/cm3
• Metal 𝐵 has a density of 7.2g/cm3
Calculate the density of the metal after the two metals have been smelted together.
2) Liquid 𝐴 and liquid 𝐵 are mixed to make liquid 𝐶.
• Liquid 𝐴 has a density of 70kg/m3 and a mass of 1400kg
• Liquid 𝐵 has a density of 280kg/m3 and a volume of 30m3.
Work out the density of liquid 𝐶.
3) Metal 𝐴 and metal 𝐵 are going to be smelted together in the ratio 2:5 by volume to make metal 𝐶.
• Metal 𝐴 has a density of 8.2g/cm3
• Metal 𝐵 has a density of 6.9g/cm3
Calculate the density of metal 𝐶.
4) The density of ethanol is 1.09g/cm3. The density of propylene is 0.97g/cm3. 60 litres of ethanol are mixed
with 128 litres of propylene to make 188 litres of antifreeze. Work out the density of antifreeze to 2 d.p.
5) Julie is making a drink for a party containing apple juice, fruit syrup and carbonated water.
• The density of apple juice is 1.05g/cm3
• The density of fruit syrup is 1.4g/cm3
• The density of carbonated water is 0.99g/cm3
She uses 25cm3 of apple juice, 15cm3 of fruit syrup and 280cm3 of carbonated water to make a drink with a
volume 320cm3. Work out the density of the drink.
6) Jackson is trying to find the density, in g/cm3, of a block of wood. The block of wood is in the shape of a
cuboid. He measures
• The length is 13.2cm, correct to the nearest mm
• The width is 16.0cm, correct to the nearest mm
• The height as 21.7cm, correct to the nearest mm.
He measures the mass as 1970g, correct to the nearest 5g.
By considering bounds, calculate the density of the wood. Give your answer to a suitable degree of
accuracy.
7) Liquid 𝐴 and liquid 𝐵 are mixed together in the ratio 2:13 by volume to make
liquid 𝐶.
• Liquid 𝐴 has density 1.21g/cm3
25 cm
• Liquid 𝐵 has density 1.02g/cm3
A cylinder container is filled completely with liquid 𝐶. The cylinder has radius 3cm
and height 25cm. Work out the mass of the liquid in the container to 3 s.f.
3cm
8) The diagram shows that the frustum is made by removing a cone with height 3.2cm
from a solid cone with height 6.4cm and base diameter of 7.2cm.
The frustum is joined to a solid hemisphere of diameter 7.2cm to form the solid 𝑆 shown below.
The density of the frustum is 2.4g/cm3
The density of the hemisphere is 4.8g/cm3
3.2cm
Calculate the average density of the solid 𝑆.
7.2cm
3.2cm
7.2cm
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3.2cm