Firrhill High School

Mathematics Department

National 5 Mathematics
Relationships Homework
Booklet
National 5 Homework Relationships

DETERMINING the EQUATION of a STRAIGHT LINE

1. Calculate the gradients of the lines AB and CD shown below. (2)

y C A

B x
0

2. A line passes through the points A(2, 4) and B(8, 1).

(a) Find the gradient of the line AB. (2)

(b) Find the equation of the line AB. (2)

3. Find the equation of the line passing through P(4, 6) which is parallel to the line
with equation 4x 2y + 6 = 0. (4)

4. A straight line has equation 3y 2x = 6.

Find the gradient and y-intercept of the line. (3)

5. Find the equation of the straight line joining the points P(4, 1) and Q(2, 3). (3)

16 marks
National 5 Homework Relationships

FUNCTIONAL NOTATION

1. A function is defined as f (x) = x2 4. Evaluate

(a) f (1) (b) f (0) (c) f (9) (4)

2. A function is defined by the formula g(x) = 12 5x

(a) Calculate the value of g(5) + g(2) (3)

(b) If g(k) = 14, find k. (3)

3. A function is defined as f (x) = x2 + 3

Find a simplified expression for f (a + 2) f (a 5) (6)

16 marks
National 5 Homework Relationships

EQUATIONS and INEQUATIONS

1. Solve these equations

(a) 2x 12 = 3 (b) 5z + 9 = 4 (c) 6y 9 = 2y + 5 (6)

2. Solve these equations by first multiplying out the brackets

(a) 3(2x 4) = 6 (b) 6(a 1) = 4(a + 2) (5)

3. Solve these inequalities

(a) 7x > 42 (b) 3x 2 > 11 (3)

4. Solve these inequalities

(a) 9x + 2 6x + 11 (b) 5(y 2) > 2(y + 4) (5)

5. Solve these inequalities, giving your answer from the set {3, 2, 1, 0, 1, 2, 3, 4, 5, 6}

(a) 7x 3 > 2x 23 (b) 9(y + 2) 7(y + 4) (5)

24 marks
National 5 Homework Relationships

WORKING with SIMULTANEOUS EQUATIONS

1. Two lines have equations 2x + 3y = 12 and x + y = 5.

By drawing graphs of the two lines, find the point of intersection of the 2 lines. (3)

2. Solve, by substitution, the equations 3a + 12b = 144

a = 05b + 3 (4)

3. Solve, by elimination, the equations 3p 2q = 4

p 3q = 13 (3)

4. Mr. Martini is ordering tea and coffee for his cafe. He spends exactly 108 on these each month.

In March he orders 4kg of tea and 6kg of coffee. In April he changes his order to 8kg of tea and 3
kg of coffee.

How much do the tea and coffee cost each per kilogram? (6)

5. An electrical goods warehouse charges a fixed price per item for goods delivered plus a fixed rate
per mile.

The total cost to a customer 40 miles from the warehouse for the delivery of 5 items was 30.

A customer who lived 100 miles away paid 54 for the delivery of 2 items.

Find the cost to a customer who bought 3 items and lives 70 miles away. (5)

6. A straight line with equation y = ax + b passes through the points (2, 4) and (2, 2).

Find the equation of the line. (4)

25 marks
National 5 Homework Relationships

CHANGING the SUBJECT of a FORMULA

5
1. The formula for changing from oC to oF is C= F 32
9

Change the subject of the formula to F. (3)

50
2. H w Change the subject of the formula to m. (4)
m2

3. Change the subject of the formula to x: A = 5 + 4x (3)

bc
4. Given that A = , express b in terms of A and c. (4)
b

14 marks
National 5 Homework Relationships

QUADRATIC GRAPHS

1. (a) This graph has equation in the form y = kx. Find the value of k.

(2, 8)

O x (2)

(b) This graph has equation of the form y = (x + p) + q. Write down its equation.

(3, 2)
x (2)
O

2. Sketch the graphs of the following showing clearly any intercepts with the axes and the turning
point.

(a) y = (x 4)(x + 2) (b) y = (x 5) + 3 (7)

3. For the quadratic function y = 3 (x + )2, write down

(a) its turning point and the nature of it. (3)

(b) the equation of the axis of symmetry of the parabola. (1)

15 marks
National 5 Homework Relationships

WORKING with QUADRATIC EQUATIONS

1. Draw a suitable sketch to solve these quadratic equations.

(a) x(x 4) = 0 (b) x2 + 8x + 12 = 0 (5)

2. Solve these quadratic equations algebraically.

(a) 5x2 15x = 0 (b) 6x2 7x 3 (5)

3. Solve the equation 3x2 3x 5 = 0, giving your answer correct to 2 decimal places. (4)

4. Solve the equation 4x(x 2) = 7, giving your answer correct to 1 decimal place. (5)

y
5.
D
The graph shows the parabola y = 16 + 6x x2.

Find the coordinates of A, B, C and D.

C
(6)

A 0 B x

6. Use the discriminant to determine the nature of the roots of these quadratic equations.

(a) x 6x + 8 = 0 (b) 4x + x + 3 = 0 (5)

30 marks
National 5 Homework Relationships

APPLYING the THEOREM of PYTHAGORAS

1. A square snakes & ladders board has

100 squares and a diagonal of length
35 cm.

Find the length of side of one of the

small squares. (4)

2.
The figure shows the cross section of a tunnel
with a horizontal floor AB which is 24 metres
Height of
O wide.
. the tunnel
25m
The radius OA of the cross section is 25
metres.
A B
(4)
Find the height of the tunnel.

A B
3. Calum is making a picture frame, ABCD .

It is 25 cm high and 215 cm wide.

25cm d
To check whether the frame is rectangular, he
measures the diagonal, d.
C
D
It is 315 cm long. 215cm
(4)

Is the frame rectangular?

4. Calculate the perimeter of this field, which is made up

of a rectangle and a right angled triangle. 80 m (4)

40 m

16 marks 70 m
National 5 Homework Relationships

APPLYING PROPERTIES of SHAPES (1)

1. Find the missing angles in each of these diagrams. Each circle has centre C. (7)

(a) (b) co
(c)
ho
o
d jo
30o ao go
ko
C C io
C lo
25o
bo
eo fo mo

2. Use symmetry in the circle to find the missing angles in the circles (centre C) below. (4)

(a) (b) (c)

go
eo ho

C do C C
o
o
a b o 61 o
i 25o
40o co fo

3. Calculate the sizes of the missing angles in each diagram. (4)

(a) (b)
ho
40o
20o
io
55o
co
go
o
b
ao do eo fo

4. PR is a tangent to the circle, centre O, at T. O

(4)

13cm
5cm
Calculate the length of the line marked x.
P x T R

20 marks
National 5 Homework Relationships

APPLYING PROPERTIES of SHAPES (2)

1. Find the area of each shape below.

(a) (b) (4)

7 cm 9 cm

18 cm 9 cm

2. Find each shaded area below.

85 m
(a) (b) (6)

7m
7m

35 m

15 m 7m

3. A window is in the shape of a rectangle 4m by 2m

with a semicircle of diameter 4m on top.

Find the area of glass in the window. (3)

2m

4m
4. By dividing the pentagon into triangles or otherwise,
find the size of angle ABC. (2)

B C 15 marks
National 5 Homework Relationships

SIMILARITY (1)

1. Calculate the value of x and y in the diagrams below. (7)

(a) (b)
720cm A
J K

y cm
640cm 600cm
B 15cm C
L
x cm 5 5 cm
N M D E
270cm 20cm

2. P
The diagram shows a system of roads
which are represented below as similar
25 km 2 km triangles.

M R A man driving from P to S, reaches R

3 km before discovering that the road between R
and S is blocked.

T 36 km S He takes the detour P RMTS.

PM = 25 km, MR = 3 km, PR = 2 km and ST = 36 km.

25 2

M How much greater was his journey than going directly from P to S?
R
3

(5)
T S
36

12 marks
National 5 Homework Relationships

SIMILARITY (2)

1. These two rugs are mathematically similar.

3m
2m

The area of the larger one is 45m. What is the area of the smaller one? (3)

2. I have two triangular plots in my garden which I have had turfed.

The diagrams below show plans of both areas. Equal angles are marked with the same shape.
51m

11m
31m
17m

The cost depends on the area being tiled.

It cost 16.75 to buy turf for the smaller area. How much did it cost for the larger one if the
triangles are mathematically similar? (3)

3. These two parcels are mathematically similar.

The smaller one has dimensions which are half

those of the larger.

If the smaller one has volume 150cm3, calculate

the volume of the larger. (3)

4. These two perfume bottles are mathematically similar.

The cost depends on the volume of perfume in them.

4cm 25cm
The larger bottle costs 62.

Find the cost of the smaller bottle correct to the nearest penny. (3)

12 marks
National 5 Homework Relationships

TRIGONOMETRY (1)

1. Write down the equations of the following graphs. (6)

(a) (b)
y y
4 3

360o
0 180o 360o 0 o x
x 180

-4 -3

2. Write down the equation of each graph shown below: (5)

(a) y
8

0 xo
90 180 270 360

xo
0 90 180 270 360
(b)

3. Make a neat sketch of the function y = 3 sin 2xo, 0 x 360, showing the important values. (3)

4. Make a neat sketch of each of the following for 0 x 360, showing all important points.

(a) y = 4sin(x 45)o (b) y = 2cos x + 1 (6)

20 marks
National 5 Homework Relationships

TRIGONOMETRY (2)

1. Write down the exact values of :

(a) sin 60o (b) tan 225o (c) cos 300o (d) sin 315o (4)

2. Write down the period of the following

(a) y = 3 cos 2xo (b) y = 2 sin 5xo (c) y = 4 cos xo (3)

3. Solve for 0 x 360, giving your answer correct to 3 significant figures.

(a) sin x = 0839 (b) 4cos x + 7 = 6 (c) tan 2 x = 25 (11)

4. Prove the following identities:

1
(a) (sin x + cos x)2 = 1 + 2 sin xcos x (b) tanx sinx = cos x cos xo (6)

24 marks