MILLBURN ACADEMY
MATHS DEPARTMENT
S5 HOMEWORK BOOKLET
National 5
Pythagoras Theorem
1.
A circular table cloth has to be designed so that it
completely covers a square table with side 2 metres.
The designer wishes to use the minimum amount of
material. The diagram below shows the designers initial
plan.
What is the diameter of the smallest possible tablecloth
which the designer could make?
Round your answer to 1 decimal place.
2.
4
In a switch mechanism lever AB rotates around A
until it rests against the rod CD.
Point B touches rod CD at E.
AB = 11cm and AC = 8.4cm as shown.
For the switch to work the distance from C to E
must be more than 7cm.
Will this switch mechanism work?
3.
4.
Your answer must be accompanied by appropriate working and explanation.
4
A spotlight is to be positioned in the
centre of a hall by suspending it on
a cable fixed to two opposite walls,
as shown in the diagram.
The hall is 21 metres wide and the
cable is 22 metres long. Both wall
fixings are the same height above
the floor.
Find the height of the wall fixings
above the floor if the spotlight has
to be 5 metres above the floor.
4
A triangular paving slab has measurements as shown.
Is the slab in the shape of a right-angled triangle?
Give a reason for your answer.
110 cm
60 cm
90 cm
3
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Percentages
1.
John bought his house 2 years ago for £165,000.
He estimates that its value appreciated by 2% after 1 year and by 5%
over the course of the 2nd year.
Calculate its current value.
2.
Kirsty invests £12,500 with the Highland Bank at 3% for 4 years.
Calculate how much interest she earned over that period assuming she withdrew
no money.
3.
4
The Marks & Spencer share price has fallen 6% in the last week.
If a share can now be bought for £4.23, calculate last week's share price.
6.
3
Mr Smith buys a concert ticket for £45 and sells it on for £58.
Calculate his profit as a percentage of the cost price giving your answer correct
to 2 significant figures.
5.
3
The value of a classic car bought 6 years ago for £19,500 has appreciated
by 2% per annum.
Calculate its current value.
4.
3
3
After VAT at 17.5% is added on, a pair of designer jeans is sold for £141.
Calculate the price before VAT.
3
(19)
Factorisation
Go on to www.thatquiz.org on your PC.
Your teacher will give you a set of codes.
Enter the first code into box at the bottom right-hand corner of the page.
Select your name from the pull down list of students.
Complete the exercise online.
Now do the same for all of the homework codes.
Do all necessary working in your homework jotter and then enter your answers online.
How to enter answers:
 There should be no spaces between brackets or between terms in a bracket
 All letters are lower case
 To enter “squared” use ^ e.g. 2x2 is entered as 2x^2
 For a Difference of 2 Squares: enter the bracket ( - ) before ( + ) e.g. (y-3)(y+3)
 For a quadratic with a=1, the first bracket should contain the smaller digit e.g. (g+3)(g-7)
 For a quadratic with a>1, the first bracket should contain the larger coefficient of
the variable e.g. (3c -5)(2c+2)
Arcs & Sectors
1.
A sector of a circle, centre O, is shown.
The radius of the circle is 2.3 metres.
Angle AOB is 65º.
Find the length of the arc AB.
2.
A fan has four identical plastic blades.
Each blade is a sector of a circle of radius
5 centimetres.
The angle at the centre of each sector is 64⁰.
Calculate the total area of plastic required to
make the blades.
3.
3
Contestants in a quiz have 25 seconds to
answer a question.
Time is indicated on the clock.
The tip of the clock hand moves through
the arc AB as shown.
a) Calculate the size of angle AOB.
b) The length of arc AB is 120 centimetres.
Calculate the length of the clock hand.
4.
3
1
4
A cone is formed from a paper circle with a sector removed as shown.
The radius of the paper circle is 30cm.
Angle AOB is 100⁰.
Calculate the area of paper used to make the cone.
Calculate the circumference of the base of the cone.
3
3
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Fractions Review
Do NOT use a calculator for this exercise.
1.
2.
3.
4.
5.
Evaluate:
3
1
b)
43 − 28
4
1
d)
5 10 ÷ 2 3
a)
25 + 14
c)
19 × 12
1
2
1
7
2
12
1
In a year group election, 2 of the students voted for Amy, 4 voted for Sophie
1
and 5 voted for Ryan. The remaining votes were for Andrew.
a) What fraction of the votes did Andrew get?
b) One hundred students voted. How many votes did each person get?
2
2
In 2002 the Highland Building Society offer saving accounts with a rate of
7
interest at 7¼ % per annum. In 2012 the interest rate is 8 % per annum.
Calculate the drop in the percentage interest rate.
3
A map has a scale of 1 inch to 3¼ miles. Calculate the actual distance between
2 towns that are 9 inches apart on the map.
3
7
A fully loaded truck can carry 4 8 tonnes.
a) Calculate the least number of loads needed to move 25 tonnes of earth.
b) The truck is filled to its maximum capacity whenever possible.
What is the weight of earth in the final load.
3
2
(27)
Spread of Data
1.
The ages of the ten people on a roller-coaster are:
15
20
15
20
16
24
18
27
19
49
a) Calculate the range of the ages.
b) Calculate the inter-quartile range of the ages.
c) Which value gives a better measure of spread?
Explain your answer.
2.
91
84
71
79
75
Calculate the mean and standard deviation of these times.
4
The pulse rates, in beats per minute, of 6 adults in a hospital waiting room are:
66
b.
1
The running times, in minutes, of 6 television programmes are:
77
3a.
1
1
73
86
72
82
78
Calculate the mean and standard deviation of this data.
4
Six children in the same waiting room have a mean pulse rate of
89·6 beats per minute and a standard deviation of 5·4.
Make two valid comparisons between the children’s pulse rates
and those of the adults.
2
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Volume
1.
2.
3.
4.
5.
A tin of tuna is in the shape of a cylinder.
It has diameter 10 centimetres and height 4 centimetres.
Calculate its volume.
2
The diagram opposite represents a sphere.
The sphere has a diameter of 6 centimetres.
Calculate its volume.
2
A glass ornament in the shape of a
cone is partly filled with coloured water.
The cone is 24 centimetres high and has a
base of diameter 30 centimetres.
The water is 16 centimetres deep and
measures 10 centimetres across the top.
What is the volume of the water?
Give your answer correct to 2 significant figures.
5
A company manufactures aluminium tubes.
The cross-section of one of the tubes is
shown in the diagram below.
The inner diameter is 74 millimetres.
The outer diameter is 82 millimetres.
The tube is 900 millimetres long.
Calculate the volume of aluminium used to
make the tube.
Give your answer correct to three significant figures.
3
A health food shop produces cod liver oil capsules
for its customers.
Each capsule is in the shape of a cylinder with
hemispherical ends as shown in the diagram below.
The total length of the capsule is 23 millimetres and the length of the cylinder is 15
millimetres.
Calculate the volume of one cod liver oil capsule.
6
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Algebraic Fractions
Simplify the following:
1)
4)
7)
6𝑎
2)
8𝑎2
9𝑥 − 6
5)
15𝑥 − 10
2𝑚2 − 5𝑚 − 3
3𝑚 − 9
8)
12𝑎𝑏
3)
9𝑏𝑐
𝑥2 − 4
6)
𝑥+2
5
10𝑥 + 15
𝑥 2 + 5𝑥 + 4
𝑥 2 − 3𝑥 − 4
𝑎2 − 1
8
𝑎4 −1
Express as a single fraction in its simplest form:
9)
12)
15)
𝑥
5
× 𝑥2
2
𝑝
10)
𝑝
− 𝑞2
𝑞
2
13)
1
+ 𝑥−2
𝑥+5
16)
4
10
÷ 𝑦3
𝑦
4
11)
3
+𝑦
𝑥
1
14)
3
− 𝑥+2
𝑥
𝑚
3
𝑛
+5
𝑥+1
2
+
1,1,2
𝑥+3
5
2,2,2
2,2
Solve the following equations:
17)
3𝑥 + 1 =
𝑥−5
2
18)
𝑚
3
=
1− 𝑚
5
3,3
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Equation and Gradient of a Straight Line
1.
Find the equation of the line AB in the diagram.
2.
A taxi fare consists of a £2 “call-out” charge
plus a fixed amount per kilometre.
The graph shows the fare, f pounds for a
journey of d kilometres.
The taxi fare for a 5 kilometres journey is £6.
3.
3
Find the equation of the straight line in terms of d and f.
4
The straight line with equation
4x + 3y = 36 cuts the y-axis at A.
a)
Find the coordinates of A.
1
This line meets the line through B(0,8),
parallel to the x-axis, at C as shown above.
b)
Find the coordinates of C.
2
4.
Find the equation of the line through the points A(1,0) and B(-2,-6).
5.
The results for a group of students who sat tests in mathematics and physics are shown
below.
a)
b)
Calculate the standard deviation for the mathematics test.
The standard deviation for physics was 6.8.
Make an appropriate comment on the distribution of the marks in the
two tests.
These marks are shown on the scattergraph.
A line of best fit has been drawn.
c)
Find the equation of the line of best fit.
d)
Another pupil scored 76% in the
mathematical test but was absent from
the physics test.
Use your answer to part (c) to predict his
physics mark.
6.
A straight line is represented by the equation y = mx + c.
Sketch a possible straight line graph to illustrate this equation when m > 0 and c < 0.
3
4
1
3
1
2
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Similarity
1. Each of the diagrams below contains a pair of similar triangles.
Calculate the labelled lengths.
3,4
4,3
2. On the outside of a new building there are two similar glass elevators.
The first is 1·5 m wide and has a volume of 4 m3.
The other is 2 m wide. What is its volume?
4
3. A Cessna aeroplane has a wingspan of 1020 cm.
A scale model of the same plane has a wingspan of 120 cm and a wing area of 0·4 m2.
Calculate the wing area of the actual Cessna.
4
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Sine Rule, Cosine Rule and Area of a Triangle
1.
2.
Brunton is 30 kilometres due North
of Appleton.
From Appleton, the bearing of Carlton
is 065°.
From Brunton, the bearing of Carlton
is 153°.
Calculate the distance between Brunton
and Carlton.
4
A telegraph pole is 6.2metres high.
The wind blows the pole over into the position as shown below.
AB is 2.9metres and angle ABC is 130°.
Calculate the length of AC.
3.
Paving stones are in the shape of a rhombus.
The side of each rhombus is 40centimetres long.
The obtuse angle is 110°.
Find the area of one paving stone.
4.
4
4
Triangle DEF is shown below.
It has sides of length 10.4metres, 13.2metres and 19.6metres.
Calculate the size of angle EDF.
3
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Simultaneous Equations
1.
2.
3.
Draw the lines with equation x + y = 6 and 2x + y = 8.
Find the point of intersection of these lines.
5
The graph below shows two straight lines.
• y = 2x – 3
• x + 2y = 14
The lines intersect at the point P.
Use the substitution method to find the coordinates of P.
4
(a) Brian, Molly and their four children visit Waterworld.
The total cost of their tickets is £56.
Let a pounds be the cost of an adult’s ticket and c pounds the cost of a child’s
ticket.
Write down an equation in terms of a and c to illustrate this information.
1
(b) Sarah and her three children visit Waterworld.
The total cost of their tickets is £36.
Write down another equation in terms of a and c to illustrate this information.
1
(c) (i) Calculate the cost of a child’s ticket.
(ii) Calculate the cost of an adult’s ticket.
2
1
(14)
Vectors
1.
2.
3.
Write down the components of
⃗⃗⃗⃗⃗
a)
𝐴𝐵
⃗⃗⃗⃗⃗
b)
𝑃𝑄
c)
u
1
1
1
PQRSTU is a regular hexagon.
Find a vector equal to:
⃗⃗⃗⃗⃗ + 𝑄𝑅
⃗⃗⃗⃗⃗ + ⃗⃗⃗⃗⃗
a)
𝑃𝑄
𝑅𝑆
⃗⃗⃗⃗
⃗⃗⃗⃗⃗ + 𝑇𝑆
b)
𝑈𝑃
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗
c)
𝑇𝑈 − 𝑃𝑈
1
2
3
Use the diagram to name a vector equal to:
a)
a + 2b
b)
c–b
c)
a–d
1
1
2
4.
A boat sets off north east at 7.5 km/h but meets a current of 4 km/h from the north west.
a)
Draw a diagram to show the resultant velocity of the boat.
2
b)
Calculate the boat’s resultant speed and its bearing.
7
5.
OABCD is a rectangular based pyramid
of height 7 units.
The point D is vertically above the point
of intersection of the diagonals of rectangle
OABC.
State the coordinates of the points A, B, C and D.
6.
7.
The diagram shows cuboid PQRSTUVW.
State the components of:
⃗⃗⃗⃗⃗
a)
𝑃𝑅
⃗⃗⃗⃗⃗
b)
𝑆𝑉
⃗⃗⃗⃗⃗
c)
𝑉𝑄
⃗⃗⃗⃗⃗
d)
𝑇𝑅
⃗⃗⃗⃗⃗⃗⃗
e)
𝑄𝑊
1
1
1
1
1
−2
1
If u = ( 3 ) and v = (−2), express in component form:
5
4
a)
8.
5
u+v
b)
u–v
Calculate the magnitude of each of these vectors:
6
−4
⃗⃗⃗⃗⃗
a)
𝐴𝐵 = ( )
b)
w = (−2)
8
−3
c)
2u + 3v
4
6
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Scientific Notation
1.
Write the following numbers in standard form:
a)
2.
5.
0.000 000 483
4
4.2 x 106
b)
5.39 x 10-4
2
Calculate, giving your answer in scientific notation:
a)
b)
4.
b)
Write the following numbers in full:
a)
3.
56 000 000 000
( 3.5 x 107 ) x ( 4.9 x 10-3 )
( 1.248 x 1012 ) ÷ ( 4.8 x 105 )
2
2
Macro Computers make an average profit of £20 000 per hour.
Calculate their annual profit in 2011.
Give your answer in scientific notation.
3
There are 6.023 x 10²³ atom of silver in 108 grams of silver.
How many atoms of silver are there in 1 gram of silver?
3
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Functions, Polynomials & Graphs
x2 – 10x + 18 = (x – a)2 + b, find the values of a and b.
1.
Given that
2.
Two functions are given below.
f(x) = x2 – 4x,
g(x) = 2x + 7
2
(a) If f(x) = g(x), show that x – 6x – 7 = 0.
(b) Hence find algebraically the values of x for which f(x) = g(x).
2
2
Solve the equation
3x2 – 2x – 10 = 0.
Give your answer correct to 2 significant figures.
4
Find the range of values of p such that the equation px2 – 2x + 3 = 0, p ≠ 0,
has no real roots.
4
3.
4.
5.
6.
3
The profit made by a publishing company of a magazine is calculated by the
formula y = 4x (140 – x), where y is the profit (in pounds) and x is the selling
price (in pence) of the magazine.
The graph below represents the profit y against the selling price x.
Find the maximum profit the company can make from the sale of the magazine.
4
The equation x2 – 6x + 8 = 0 can also be written as (x – 2)(x – 4) = 0.
(a) Write down the roots of the equation x2 – 6x + 8 = 0.
1
Part of the graph of y = x2 – 6x + 8 is shown below.
(b)
(c)
State the coordinates of the points A, B and C.
What is the equation of the axis of symmetry of this graph?
3
1
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Trigonometric Graphs & Equations
1.
Part of the graph of y = a sin bx ° is shown in the diagram.
State the values of a and b.
2.
2
The graph shown below has an equation of the form y = cos(x – a)°.
Write down the value of a.

1
3.
Sketch the graph of
y = 4 cos 2x °,
4.
Solve the equation
5 tan x ° – 6 = 2,

5.
Given that
6.
If sin x ° =
7.
Simplify
0 ≤ x ≤ 360.
0 ≤ x < 360.
cos 60° = 0.5, what is the value of cos 240 °?
4
5
and cos x ° =
cos3 𝑥°
1− sin2 𝑥°
3
5
, calculate the value of tan x °.
3
3
1
2
2
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Indices & Surds
1. Simplify the following, giving your answer with positive indices:(a)
2b 
3 2
6 p 3  2 p 4
4 p 1
(b)
b 2
8
2. Expand the brackets and simplify, giving your answer with positive indices:(a) u 3 / 2 u 1 / 2  u 1 / 2 
(b)
a
1/ 2

 a 1 / 2 a 1 / 2  a 1 / 2

(c)
1

v  v 
v

14
3. Evaluate the following expressions.
(a)
2x 1 / 2 , when x  9
(b)
1 3 / 4
x
, when x  81
2
3
4. A function is given by f x   4 x .Calculate f   .
2
7
3
5. (a) Express 150  24  54 as a surd in its simplest form.
5
96
(b) Simplify
8
.
3
(c) Simplify 18  6
6. (a) Express
(b) Express
7
3
as a fraction with a rational denominator.
15
3 1
as a fraction with a rational denominator.
3
2
3
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