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 + 3v**4

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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