MILLBURN ACADEMY
MATHS DEPARTMENT
S4 HOMEWORK BOOKLET
National 5
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
(15)
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
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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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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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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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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
7
3
4. A function is given by f x   4 x .Calculate f   .
3
5. (a) Express 150  24  54 as a surd in its simplest form.
5
2
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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