MILLBURNACADEMY
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.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
2.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. 4
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.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.Draw the lines with equation x + y = 6 and 2x + y = 8.
Find the point of intersection of these lines. 5
2.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
3. (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 andc 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 andc to illustrate this information. 1
(c) (i) Calculate the cost of a child’s ticket. 2
(ii) Calculate the cost of an adult’s ticket. 1
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Vectors
1.Write down the components of
a)1
b)1
c)u1
2.PQRSTU is a regular hexagon.
Find a vector equal to:
a)1
b)2
c)3
3.Use the diagram to name a vector equal to:
a)a+ 2b1
b)c – b1
c)a – d 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.5
6.The diagram shows cuboid PQRSTUVW.
State the components of:
a) 1
b)1
c)1
d)1
e)1
7.If u = and v= , express in component form:
a)u + vb)u – v c)2u + 3v4
8.Calculate the magnitude of each of these vectors:6
a) b)w = (42)
Trigonometric Graphs & Equations
1.Part of the graph of y = a sin bx° is shown in the diagram.
State the values of a andb. 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 °, 0 ≤ x ≤ 360. 3
4.Solve the equation 5 tan x ° – 6 = 2, 0 ≤ x < 360. 3
5.Given that cos 60° = 0.5, what is the value of cos 240 °? 1
6.If sin x ° = and cos x ° = , calculate the value of tan x °. 2
7.Simplify 2
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Functions, Polynomials & Graphs
1.Given that x2 – 10x + 18 = (x – a)2 + b, find the values of a and b. 3
2.Two functions are given below.
f(x) = x2 – 4x, g(x) = 2x + 7
(a) If f(x) = g(x), show that x2 – 6x – 7 = 0.2
(b) Hence find algebraically the values of x for which f(x) = g(x).2
3.Solve the equation3x2 – 2x – 10 = 0.
Give your answer correct to 2 significant figures.4
4.Find the range of values of p such that the equation px2– 2x + 3 = 0, p 0,
has no real roots.4
5.The profit made by a publishing company of a magazine is calculated bythe
formula y= 4x (140 – x),where y is the profit (in pounds) and x is the selling
price (in pence) of themagazine.
The graph below represents the profit y against the selling price x.
Find the maximum profit the company can make from the sale of themagazine.4
6.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) State the coordinates of the points A, B and C.3
(c) What is the equation of the axis of symmetry of this graph?1
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Indices & Surds
- Simplify the following, giving your answer with positive indices:-
(a)(b) 8
- Expand the brackets and simplify, giving your answer with positive indices:-
(a)(b)(c) 14
- Evaluate the following expressions.
(a), when (b), when 7
- A function is given by .Calculate . 3
5.(a)Express as a surd in its simplest form. 5
(b)Simplify . 3
(c)Simplify 3
6.(a)Express as a fraction with a rational denominator. 2
(b)Express as a fraction with a rational denominator. 3
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