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BIGGAR HIGH SCHOOL
HOMEWORK BOOKLET
NATIONAL 5
Brackets & Equations
1. Remove the brackets in the following expressions:
a) 3(x + 4) b) 5(y – 8) c) 6(2g + 5)
d) -4(3 – 7x) e) x(4x + 3) f) a(2b + 3c)
g) 3k(6k² - 5k) h) 12(3a – 7b + 2c) 8
2. Remove the brackets and simplify where possible:
a) (x + 3)(x + 5) b) (y – 4)(y + 7) c) (x – 1)(x – 1)
d) (6t + 5)(2t – 1) e) (f + 7)² f) (3h – 4)² 12
3. Solve the following equations:
a) 3(2x – 1) = 21 b) 21 – 2(x + 3) = 3x + 5 2,3
c) 7(3y + 5) – 4(2y – 3) = 5(y – 1) 4
d) (m + 3)(m – 4) = (m – 2)(m + 5) 4
4. The triangle and rectangle below have the same area.
Form an equation and solve it to find the value of x. 4
(37)
2x
x + 10
x + 6
x + 2
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. 4
2. 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?
Your answer must be accompanied by appropriate working and explanation. 4
3. 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
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.
3
(15)
90 cm
110 cm
60 cm
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. 3
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
3. The value of a classic car bought 6 years ago for £19,500 has appreciated
by 2% per annum.
Calculate its current value. 3
4. 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. 4
5. 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. 3
6. After VAT at 17.5% is added on, a pair of designer jeans is sold for £141.
Calculate the price before VAT. 3
(19)
Algebra Skills 1
1. Multiply the brackets and simplify where possible:
a) 2x(x + 4) b) (y + 3)(y – 2) c) x(6x – 2) – x(x – 1) 1,2,3
d) (a + 3)2 e) w(w2 + 3w – 7) f) (x – 3)(x2 – 4x + 1) 2,1,3
2. Solve the following equations:
a) 5x – 2 = 17 b) 45 = 24 – 7x 2,2
c) 5 + 3a = a – 15 d) 5 – 2(1 + 3x) = 27 3,3
3. Solve the inequalities:
a) 5 – x > 2(x + 1) b) 2y < 3 – (y + 6) 3,3
c) 2 + 5x > 8x – 16 3
4. Given that a = 4, b = -5 and c = -8, evaluate the following:
a) c2 – 2a b) 𝑏𝑐
𝑎 2,2
c) √𝑏𝑐 − 𝑎 d) 3
4𝑐 −
2
5𝑏 2,2
(39)
Factorisation - 1
1. Factorise by finding the common factor
a. 2x + 4 b. 3d + 9 c. 6s + 3 d. 12x + 4
e. 6 + 9a f. 2b + 8 g. 5y + 10 h. 10 + 15c
i. 12x + 16 j. 18m + 24 k. 30 + 36a l. 14y + 21
12
2. Factorise by finding the common factor
a. 3x 6 b. 4y 8 c. 16 8a d. 10c 15
e. 9s 12 f. 2b 14 g. 12x 20 h. 22m 33
i. 15x 10 j. 18 12y k. 25b 20 l. 18d 30
12
3. Factorise by finding the common factor
a. 2a + 4b b. 10x 12y c. 18m + 24n d. 10c + 15d
e. 6a 9x f. 18s 12t g. 12x + 15y h. 14a 7b
i. 25c + 10d j. 9b 15y k. 18x + 24y l. 6a + 28b
12
4. Factorise by finding the common factor
a. ax + ay b. xy2 + xa2 c. pqr + pst
d. xay bac e. pq + p f. y2 + y
g. a2 ab h. ab bc i. n2 3n
j. xy + y2 k. abc abd l. fgh efg
12
5. Factorise by finding the highest common factor
a. 2ax + 6a b. 3y + 9y2 c. 24a 16ab
d. pq2 pq e. 12xy 9xz f. 6b2 4b
g. 3a2 + 27ah h. 15abc + 20abd i. 3s3 9s2
j. 14x 12xyz k. 10b2c 15bcd l. 2r 2 + 2rh
12
6. Factorise
a. ap + aq ar b. 2a + 2b + 2c c. 6e 2f + 4g
d. p2 + pq + xp e. 3ab 6bc 9bd f. ½ ah + ½ bh + ½ ch
g. 5x2 8xy + 5x h. 4ac + 6ad 10a2 i. 15p2 + 10pq + 20ps
9
7. Factorise
a. ab2c a2bd b. a3 a2 a c. 2x2 50x + 12xy
d. x6 + x4 + x2 e. 25p2 + 15pq + 10p f. x2yz + axy + bxy2
g. 3a4 + 9a3 6a h. abx + bcx bcy i. ½ gtT ½ gt2
9 (66)
Factorisation – 2
1. Factorise the following expressions, which contain a difference of squares
a. a2 b2 b. x2 y2 c. p2 q2 d. s2 t2
e. a2 32 f. x2 22 g. p2 92 h. c2 52
i. b2 1 j. y2 16 k. m2 25 l. a2 9
m. 36 d2 n. 4 q2 o. 49 w2 p. x2 64
16
2. Factorise the following expressions, which contain a difference of squares
a. a2 4b2 b. x2 25y2 c. p2 64q2 d. 16c2 d2
e. 81 4g2 f. 36w2 y2 g. 4a2 1 h. g2 81h2
i. 49x2 y2 j. 9c2 16d2 k. 4p2 9q2 l. b2 100c2
m. 25 16a2 n. 4d2 121 o. 225 49k2 p. 9x2 0.25
16
3. Factorise the following expressions
a. 2a2 2b2 b. 5p2 5 c. 45 5x2 d. 4d2 36
e. 2y2 50 f. 4b2 100 g. 3q2 27 h. 8a2 32b2
i. ab2 64a j. xy2 25x k. abc2 ab l. 8p2 50q2
m. 2x2 2.88 n. ak2 121a o. 10s2 2.5 p. ½ y2 450
16
(48)
Factorisation - 3
1. Factorise the following quadratic expressions
a. x2 + 3x + 2 b. a2 + 2a + 1 c. y2 + 5y + 4
d. c2 + 8x + 7 e. x2 + 6x + 9 f. b2 + 8b + 12
g. a2 + 9a + 14 h. w2 + 10w + 9 i. d2 + 7d + 10
j. x2 + 10x + 21 k. p2 + 9p + 20 l. c2 + 10c + 24
m. s2 + 12s + 36 n. x2 +11x + 28 o. y2 + 10y + 25
15
2. Factorise the following quadratic expressions
a. a2 8a + 15 b. x2 9x + 8 c. c2 9c + 18
d. y2 4y + 4 e. b2 6b + 5 f. x2 15x + 14
g. c2 10c + 16 h. x2 7x + 6 i. y2 12n + 32
j. p2 11p + 24 k. a2 13a + 36 l. x2 15x + 36
m. b2 4b + 3 n. q2 11q + 10 o. a2 7y + 12
15
3. Factorise the following expressions
a. b2 + 3b 10 b. x2 + 6x 7 c. y2 y 6
d. a2 a 20 e. q2 + 2q 8 f. x2 8x 20
g. d2 + 4d 21 h. c2 + 9c 36 i. p2 5p 24
j. y2 7y 8 k. a2 + 5a 6 l. x2 5x + 36
m. b2 4b 5 n. s2 + 2s 24 o. d2 + 6d 16
15
4. Factorise the following expressions
a. 3x2 + 7x + 2 b. 2a2 + 5a + 2 c. 3c2 + 8c + 5
d. 2p2 + 11p + 9 e. 2y2 + 11y + 5 f. 3d2 + 11d + 6
g. 5q2 + 9q + 4 h. 4b2 + 8b + 3 i. 6x2 + 13x + 6
j. 3a2 + 14a + 15 k. 10x2 + 17x + 3 l. 9c2 + 6c + 1
m. 6y2 + 11y + 3 n. 3b2 + 5b + 2 o. 8x2 + 14x + 3
15
5. Factorise the following expressions
a. 2x2 7x + 3 b. 2a2 5a + 3 c. 5p2 17p + 6
d. 5b2 7b + 2 e. 6x2 7x + 2 f. 4y2 11y + 6
g. 7c2 29c + 4 h. 4m2 9m + 2 i. 16a2 10a + 1
j. 8y2 22y + 5 k. 3p2 37p + 12 l. 4x2 25x + 6
m. 15a2 16a + 4 n. 24c2 22c + 3 o. 6b2 35b + 36
15
6. Factorise the following expressions
a. 3x2 2x 1 b. 2a2 a 3 c. 4p2 p 3
d. 2c2 + 7c 4 e. 6y2 11y 2 f. 3w2 + 10w 8
g. 3m2 + 2m 5 h. 4q2 + 5q 6 i. 6b2 + 7b 20
j. 4t2 4t 3 k. 12z2 + 16z 3 l. 4d2 4d 15
m. 7s2 27s 4 n. 15x2 + 16x 15 o. 36v2 + v 2
15 (90)
Factorisation – 4
1. Fully factorise these expressions
a. 3x2 3 b. 2p2 + 12p + 10 c. 9x2 36
d. 5x2 + 25x + 30 e. ax2 + 5ax + 6a f. 3y2 12y 15
g. 15c2 + 27c + 12 h. 16b2 + 28b + 6 i. 9q2 + 33q + 18
j. 10s2 35s + 15 k. 8m2 20m + 12 l. 8a2 36a + 36
m. 4t2 + 2t 56 n. 90d2 60d 80 o. 400x2 4
(15)
Scientific Notation
1. Write the following numbers in standard form:
a) 56 000 000 000 b) 0.000 000 483 4
2. Write the following numbers in full:
a) 4.2 x 106 b) 5.39 x 10-4 2
3. Calculate, giving your answer in scientific notation:
a) ( 3.5 x 107 ) x ( 4.9 x 10-3 ) 2
b) ( 1.248 x 1012 ) ÷ ( 4.8 x 105 ) 2
4. Macro Computers make an average profit of £20 000 per hour.
Calculate their annual profit in 2011.
Give your answer in scientific notation. 3
5. 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
(16)
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. 3
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. 1
b) The length of arc AB is 120 centimetres.
Calculate the length of the clock hand. 4
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. 3
Calculate the circumference of the base of the cone. 3
(17)
Fractions Review
Do NOT use a calculator for this exercise.
1. Evaluate:
a) 23
5+ 1
1
4 b) 4
2
3− 2
7
8
c) 14
9× 1
1
2 d) 5
1
10÷ 2
2
3 12
2. In a year group election, 1
2 of the students voted for Amy,
1
4 voted for Sophie
and 1
5 voted for Ryan. The remaining votes were for Andrew.
a) What fraction of the votes did Andrew get? 2
b) One hundred students voted. How many votes did each person get? 2
3. In 2002 the Highland Building Society offer saving accounts with a rate of
interest at 7¼ % per annum. In 2012 the interest rate is 7
8 % per annum.
Calculate the drop in the percentage interest rate. 3
4. 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
5. A fully loaded truck can carry 4 7
8 tonnes.
a) Calculate the least number of loads needed to move 25 tonnes of earth. 3
b) The truck is filled to its maximum capacity whenever possible.
What is the weight of earth in the final load. 2
(27)
35km
Braley
Cannich Aldwich
N
A
B C
D
18m
Trigonometry
1. A shelf bracket is constructed using three steel rods.
The sloping side makes an angle of 28˚ with the horizontal and is 24cm long.
Calculate the size of the horizontal rod marked x, giving your answer to the
nearest centimetre. 4
2. A radio mast AD is held in place by two cables DC and DB.
AC = 15 metres and CB = 18 metres, and the angle of elevation of the top of the mast
from C is 65˚.
Calculate a) The height of the mast AD 3
b) The angle marked x˚. 3
3. The diagram below shows the positions of three towns. Braley is 35 km from Cannich.
Cannich is due east of Aldwich. Braley is due south of Cannich.
Braley is on a bearing of 130˚ from Aldwich.
Calculate the distance from Aldwich to Braley. 4
(14)
15m 65˚ x˚
28˚
24cm
x
Changing the Subject of a Formula
The following formulae appear in the work of departments across the school.
1. In Technological Studies they study Pneumatic/Hydraulic Systems.
The terms force and pressure are used extensively in the study of fluid
power. Force means a total push or pull. It is push or pull exerted
against the total area of a particular surface and is expressed in pounds
or grams. Pressure means the amount of push or pull (force) applied to
each unit area of the surface and is expressed in pounds per square inch
(lb/in2) or grams per square centimetre (gm/cm2). Pressure may be exerted
in one direction, in several directions, or in all directions.
A formula is used in computing force, pressure, and area in hydraulic
systems. In this formula, P refers to pressure, F indicates force, and A represents area.
P = F
A
P = F
A
Change the subject to A. 2
2. In Physics they study Kinetic Energy.
The kinetic energy of an object is the energy which it possesses due to its
motion. It is defined as the work needed to accelerate a body of a given
mass from rest to its stated velocity. The formula is 𝐸 = 1
2𝑚𝑣2
where m is the mass and v is the speed (or the velocity) of the body. Mass is usually
measured in kilograms, speed in metres per second, and the resulting kinetic energy is in
joules.
Change the subject of this formula to v. 3
3. In Business Studies/Accounting they study Profit and Loss.
The formula for calculating gross percentage profit is
P =G
S 100
P =G
S 100 , where G is the gross profit and S the total sales.
Change the subject to G. 2
4. Change the subject of the following formulae to y.
a)
2(m + y) = 3
2(m + y) = 3 b)
y – 3 = B
y – 3 = B c)
Q = 2 +y
3
Q = 2 +y
3 d)
by = ay + 4
by = ay + 4 10
(17)
Spread of Data
1. The ages of the ten people on a roller-coaster are:
15 15 16 18 19
20 20 24 27 49
a) Calculate the range of the ages. 1
b) Calculate the inter-quartile range of the ages. 1
c) Which value gives a better measure of spread?
Explain your answer. 1
2. The running times, in minutes, of 6 television programmes are:
77 91 84 71 79 75
Calculate the mean and standard deviation of these times. 4
3a. The pulse rates, in beats per minute, of 6 adults in a hospital waiting room are:
66 73 86 72 82 78
Calculate the mean and standard deviation of this data. 4
b. 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
(13)
Algebraic Fractions
Simplify the following:
1) 6𝑎
8𝑎2 2) 12𝑎𝑏
9𝑏𝑐 3)
5
10𝑥 + 15
4) 9𝑥 − 6
15𝑥 − 10 5)
𝑥2 − 4
𝑥 + 2 6)
𝑥2+ 5𝑥 + 4
𝑥2 − 3𝑥 − 4
7) 2𝑚2 − 5𝑚 − 3
3𝑚 − 9 8)
𝑎2 − 1
𝑎4−1 8
Express as a single fraction in its simplest form:
9) 𝑥
2×
5
𝑥2 10) 4
𝑦÷
10
𝑦3 11) 𝑚
3+
𝑛
5 1,1,2
12) 𝑝
𝑞−
𝑝
𝑞2 13) 4
𝑥+
3
𝑦 14)
𝑥 + 1
2+
𝑥 + 3
5 2,2,2
15) 2
𝑥 + 5+
1
𝑥 − 2 16)
1
𝑥−
3
𝑥 + 2 2,2
Solve the following equations:
17) 3𝑥 + 1 =𝑥 − 5
2 18)
𝑚
3=
1− 𝑚
5 3,3
(28)
Algebraic Patterns
1. Find an expression for the nth term in the following sequences:
a) 2, 5, 8, 11, 14 1
b) 4, 12, 20, 28, 36 1
c) 3, 6, 11, 18, 27 1
2. A sequence of terms, starting with 1, is
1, 5, 9, 13, 17, …….
Consecutive terms in this sequence are formed by adding 4 to the previous term.
The total of consecutive terms of this sequence can be found using the following pattern.
Total of the first 2 terms: 1 + 5 = 2 x 3
Total of the first 3 terms: 1 + 5 + 9 = 3 x 5
Total of the first 4 terms: 1 + 5 + 9 + 13 = 4 x 7
Total of the first 5 terms: 1 + 5 + 9 + 13 + 17 = 5 x 9
a) Express the total of the first 9 terms of this sequence in the same way. 2
b) The first n terms of this sequence are added.
Write down an expression, in n, for the total. 3
(8)
Volume
1. A tin of tuna is in the shape of a cylinder.
It has diameter 10 centimetres and height 4 centimetres.
Calculate its volume. 2
2. The diagram opposite represents a sphere.
The sphere has a diameter of 6 centimetres.
Calculate its volume. 2
3. 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
4. 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
5. 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
(18)
Equation and Gradient of a Straight Line
1. Find the equation of the line AB in the diagram. 3
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.
Find the equation of the straight line in terms of d and f. 4
3. 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). 3
5. The results for a group of students who sat tests in mathematics and physics are shown
below.
a) Calculate the standard deviation for the mathematics test. 4
b) The standard deviation for physics was 6.8.
Make an appropriate comment on the distribution of the marks in the
two tests. 1
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. 3
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. 1
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. 2
(24)
