_MATHEMATICS_
MATHS PARTS I and II are comprehensive packages for pupils undertaking
match courses in the 13-17 year age range.
The package consists of :
1. Basic problems in maths with an almost infinite variation
in the data provided.
2. Hints on revision.
3. A set of notes.
_MATHS_PART_1_
SIDE A Program names
"ALG 1"
"ALG 2"
"ALG 3"
"REVISION"
SIDE B
"PLANE"
"SOLIT"
"GEN"
_"ALG 1"___"ALG 2"___"ALG 3"_
Three areas are covered - functions, algebra and equations.
You are expected to have revisied these areas thoroughly before attempting
the problems. As a guide you should have covered all aspects of functions,
mapping, reversed functions, solutions of equations, relationships, linear
graphs, simultaneous equations, algebraic expressions, quadratic factors,
factorisation, quadratic graphs, completing the square, algebraic
manipulation and subjects of formulae.
Each of the three programs contains five different basic problem
types giving a total of 15 problems all similar to those from past
examination papers. Within each random numbers are generated to give a
wide variety. Each program will go through each problem type twice to
give ten questions. If your first answer is incorrect then you will be
told the answer - there is no second chance in the exams - so check your
working carefully. If you are wrong find out why. At the end you will be
given a score out of ten. You should not feel happy about the
thoroughness of your revision or your accuracy until you can
consistently score ten on all three programs.
_"REVISION"_
This program contains advice gained from many years of teaching
experience. There are comments for those who start early on a program
of revision and for those panicking at the last minute.
_"PLANE"___"SOLID"___"GEN"_
Three programs are provided to give you experience in dealing
with basic problems involving perimeters, areas, surface areas and
volumes.
"PLANE" gives questions on the perimeters and areas of plane
shapes. Similarly "SOLID" gives questions on the surface areas and
volumes of all the common solids that you will meet. The third
program "GEN" provides ten more general problems on the above topics.
Within each program random numbers are again generated to give a wide
variety. Each program will give you ten randomly selected problems.
You need give the answer to one decimal place and it will be considered
correct if within 0.1 of the actual answer. If your first answer is
incorrect then you will be told the answer so please check your working
carefully. If you are wrong again find out why. At the end you will be
given a score out of ten. You should not feel happy until you can
consistently obtain ten out of ten with this series of three programs.
_REVISION_NOTES_
You will gain far more benefit by producing your own revision
notes tailormade to your requirements, however much of this time is often
spent extracting sections straight from books. A minimum of time being
actually spent in learning and understanding the work.
To help you make your work more time effective we have produced
a set of revision notes. In the text we have noted the pertinent points;
we do expect you to add to this booklet and thus build up a set of
notes to _suit_your_ needs.
All rights of the producer and of the owner of the work(s) being
produced, are reserved. Unauthorised copying, lending, hiring,
broadcasing and public performances of this cassette are
prohibited. The publisher assumes no responsibility for errors
nor liability for damage arising from its use.
_LOADING_INSTRUCTIONS_
SPECTRUM 48K LOAD " "
DRAGON 32 CLOAD
BBC CHAIN " "
Details of other programs in this series and of other educational
programs can be obtained from your software supplier or from:
SCISOFT 5 Minster Gardens, Newthorpe, Eastwood, NOTTS. NG16 2AT
_FUNCTIONS_ 3 ——__ __—> 3
__X__
1 —— —> 9
In this diagram of two sets X and Y
a relationship is shown by the arrows 2 ————————> 6
4 ————————>12
1 ———> 3 2 ———> 6
3 ———> 9 4 ———>12 X Y
The number 1 in X _maps_onto_ the number 3 in Y
The number 3 in Y is the _image_ of 1 in X
More generally:-
If x ε X then x ——> x x 3 or x ——> 3x
x ———>3x is a _rule_ by which the image of X can be found
A rule can produce
(i) an image in Y for every element of X
(ii) an image in Y for some values of X
(iii) more than one image in Y for some values of X
(iv) no image in Y for some values of X
A rule which produces one and only one image in a set Y for each element
of a set X is called a _functions_.
Example: X Y X Y
2 ————————> 4 2 ————————> 4
\___ \__
3 ————————> 5 3 ————\———>10
——\
5 ————————> 7 5 ————————>12
8 ————————>10
x —> x+2 x —> multiple of x
(function) (not a function)
A function is normally referred to by a letter, the most common being
f (and also g, h).
Thus
f:x———>4x means f is the rule x——>4x
Using this rule, the image of 3 is 12 so f:3 ——>12
Another way of writing this is f(3) = 12
or more generally f(x) = 4x
1 1
As another example if g:x ——>————— then g(1) = —
(x+4) 5
The normal rules of arithmetic are followed in calculating images.
Thus
g: x ——>(x+2) 3 is the same as
g: x ——>(3x+6)
Similarly
g: x ——>(9x-81) is the same as
g: x ——>9(x-9)
A _reversed_ function can be written as
2x+5 ——> x
In this we have to find what value of x needs to be input to produce a
specified output. Thus if we require an image of 19, we can write
2x+5 = 19
This is an _equation._
The value of x that fits the equation is called the _solution._
Finding the correct value for x is _solving_ the equation.
Thus solving the equation 2x+5 = 19 gives a solution of x = 7.
Solving the following equations gives the solutions stated:-
Equations Solutions
(i) 1
— x-6 = 3 x = 18
2
(ii) 3x+2
———— = 4 x = 6
5
(iii) 3(y+7)
—————— = 9 y = -1
2
(iv) 2
— c-6 = 0 c = 15
5
A functions such as f:x —>x+3 can also be represented as a set of
ordered pairs.
Thus L = {(4,7), (2,5), (-1,2)...} are some of the ordered pairs
belonging to L.
If (x,y) ε L then y = x+3
y /
L is known as the _solution_set_ of the equation y = x+3. ┼ 3
/ │
/ │
The ordered pairs can be represented graphically. / ┼ 2
/ │
/ │
The graph of y = x+3 cuts / ┼ 1
the x-axis at (-3,0) / │
and the y-axis at (0,3) / │
-3 ─┴─────┴─────┴─────┼─────┴─────┴─x
/ -2 -1 │0 1 2
More generally a straight line is │
represented by the equation ┼ -1
y = ax+b
(0,b) is the point at which the line cuts the y-axis (_ordinate_).
a is the _slope_ or _gradient_ of the line.
b
(- —,0) is the point at which the line cuts the x-axis (_abscissae_).
a
Thus two lines with the same b will intersect at (0,b)
and two lines with the same a will be parallel.
Plot y = 2x+8 y = 3x-6
y = 2x+1 y = 2x-6
y y
│ ∕ │
│8 ∕ │
┼∕ 2┼ /
∕ ┼ ┼ 1 2 / 3 ∕
∕ ┼6 ───┼────┴────┴────∕───
∕ ┼ 0┼ / ∕
∕ ┼4 ∕ -2┼ / ∕
∕ ┼ ∕ ┼ / ∕
┼2 ∕ -4┼ /∕
│∕ ┼/∕
────┴────┴──∕─┼────┴────┴────── x -6┼
-2 -1∕ ┤0 1 2 ∕/┼
∕ ┼-2 ∕ -8┼
┼ ∕ / │
Many processes can be described in terms of an equation. For example,
the length of time that it takes to discuss an item in a meeting is
estimated as three times the number of people present, plus ten. This
can be written as
t = 3n+10
This shows the _relationship_ between n and t.
An equation can be used to describe the balance between two expressions.
For example □
□ o □ □ o
□ □ /5\ □ □ /2\
═════════────────────────────────────═════════
/\
shows us that if t is the weight of a tin then
3t+5 = 5t+2
The balance of the equation is preserved if the same arithmetical
operation is carried out to both sides. Thus
5 = 2t+2 (subtract 3t)
3 = 2t (subtract 2)
3
———= t (divide by 2)
2
All equations can be solved using the balancing idea.
Remember to simplify expressions before solving. Hence
6(x+4) + 2(3-x) - 5(2-3x) = 58
becomes 6x+24+6-2x-10+15x = 58
collecting terms
19x+20 = 58
19x = 38
x = 2
_SIMULTANEOUS_EQUATIONS_
Two equations which are true statements at the same time are called a
pair of simultaneous equations.
An ordered pair can make two equations true at the same time. Thus the
ordered paid (6,2) makes the equations T = 3D and D = T-4 true at the
same time. It is the point where the graphs of the two equations intersect
i.e. where T = 6 and D = 2.
Parallel lines have no point of intersection.
y = x+5
y = 2x
These two equations represent simultaneous equations. At the point of
intersection of the two lines, the values of x and y are the same for
each equation. This occurs when
x+5 = 2x
i.e. x = 5
When x = 5, y = 10 (checked in both simultaneous
equations by substituting back)
Simultaneous equations can be solved in a different way. If we have
5a + 3b = 35
4a + 3b = 31
a + 0 = 4
then a = 4 by subtraction.
By substituting back b = 5.
Often one equation has to be multiplied by a suitable number. Thus
suppose we have the simultaneous equations 9x+2y = 41 and x+y = 10.
The second equation is equivalent to 2x+2y = 20
and we have
9x + 2y = 41
x + 2y = 20
————————————
7x + 0 = 21
x = 3 and y = 7
Further, it may be necessary to alter both equations in order to eliminate
one of the letters
e.g. 5x + 3y = 19 .....Eq(1)
6x + 5y = 27 .....Eq(2)
can be written 25x + 15y = 95 Eq(1) x 5
18x + 15y = 81 Eq(2) x 3
————————————————
7x + 0 = 14 Subtract
x = 2, y = 3
_FACTORS_IN_ALGEBRA_
Algebraic expressions follow the normal arithmetical rules
Thus (a+b)c = ac + bc
a(x+y) + b(x-y) = ax + ay + bx - by (removing brackets)
= ax + bx + ay - by (collecting like terms)
= (a+b)x + (a-b)y (simplifying)
_EXPRESSIONS_TO_KNOW:_
(a+b)² = a² + 2ab + b² ^┌───────────┬───┐
- - - - - - - - - - - b ││ (2) │(3)│
v├───────────┼───┤
Proof: Total area = (a+b) x (a+b) ^│ │ │
││ │ │
a ││ (1) │(4)│
Total area = area (1) + area (2) ││ │ │
v└───────────┴───┘
+ area (3) + area (4) <———a——————><—b—>
= a x a + a x b + b x b + a x b
= a² + 2ab + b²
(a-b)² = a² - 2ab + b² ^┌───────────┬───┐^
- - - - - - - - - - - b ││ (2) │ ││
v├───────────┤ ││
area (1) = (a-b)² ^│ │ ││
││ │ ││a
a-b ││ (1) │(3)││
area (1) = Total area - area (2) - area (3) ││ │ ││
v└───────────┴───┘v
= a x a - b(a-b) - a x b <———a-b————><—b—>
= a² - ab + b² - ab
= a² - 2ab + b²
a² - b² = (a+b)(a-b) ┌───────────┬───┐^
- - - - - - - - - - │ (2) │(3)││
^├───────────┼───┤│
a² - b² = total area - area (1) ││ │ ││a
││ (1) │(4)││
= area (2) + area (3) + area (4) b ││ │ ││
││ │ ││
= b(a-b) + (a-b)(a-b) + b(a-b) v└───────────┴───┘v
<————b—————>
= (a-b)(b + a-b + b)
= (a-b)(a+b)
This expression can be used to find the difference between two squares.
For example
(1.6)² - (1.4)² = 3.0 x 0.2
= 0.6
_QUADRATIC_FACTORS_
An expression which contains a squared term is called a _quadratic_expression_
e.g. x² + 7x + 10
This can be written as
(x+5)(x+2) (check this)
and the expression is then said to be _factorised_
(x+5) is one factor, (x+2) is the other
The _solution_set_ for a quadratic equation are the values of x which
satisfy the equation. Thus
x² + 7x + 10 = 0
has a solution set -5 and -2
Practise simplifying quadratic expressions
e.g. (3x-5)(7x+4) = 21x² - 23x - 20
and also factorizing quadratic expressions
e.g. 3x² + 7x + 4
(3x + 4)(x+1)
_QUADRATIC_GRAPHS_
Quadratic graphs are represented by quadratic expressions
e.g. f(x) = x² - 5x + 4
Factorising gives
f(x) = (x-4)(x-1)
This will help in sketching the curve. By finding where f(x)=0, i.e.
where the curve crosses the x-axis, it becomes easy to fix the general
position of the U-shaped curve.
y
3 ┼ |
│ | |
2 ┼ \ /
│ | |
1 ┼ \ /
│ |1 2 3 |4
──────┼────\────┴────┴────/───────── x
0 │ \ /
-1 ┼ -- --
│ \ /
-2 ┼ ----
│
The curve for a quadratic expression with a positive x² term is U-shaped
The curve for a quadratic expression with a negative x² term is ∩-shaped.
Sometimes a quadratic expression can be written in the form
f(x) = (x-a)² + b
The associated curve can be obtained by translating f(x) = x² 'a' units
to the right and 'b' units upwards.
This method can also be used to find the solution to a quadratic equation
e.g. x² - 6x + 5 = 0
x² - 6x = -5
(x-3)² - 9 = -5
(x-3)² = 4
x-3 = +2 or -2
x = 5 or 1
This is known as _completing_the_square_.
A quadratic equation may have two, one or no elements in its solution set
depending on whether the associated quadratic curve cuts, touches or does
not cut the x-axis
| |
│ \ / | |
│ \__/ \ /
│ | | \__/
──┼─────────────\────/────────────────────
│ \__/
│
For the general quadratic equation
ax² + bx + c = 0
the solution set is given by
_________ __________
x = -b + √ b² - 4ac or -b -√ b² - 4ac
———————————————— ————————————————
2a 2a
_ALGEBRAIC_MANIPULATION_
Manipulating symbols is an improtant aspect of mathematics.
In order to add algebraic fractions it is necessary to find the
equivalent fractions which have a common denominator. These can then be
added together.
a b ay bx ay + bx
e.g. 1. — + — = —— + —— = ———————
x y xy xy xy
2 3 2(x-2) 3(x-1)
e.g. 2. ——— + ——— = —————————— + ——————————
x-1 x-2 (x-1)(x-2) (x-1)(x-2)
2(x-2) + 3(x-1)
= ———————————————
(x-1)(x-2)
5x-7
= ——————————
(x-1)(x-2)
When an equation has one letter expressed in terms of other letters, the
first letter is called the _subject_of_the_formula_
e.g. V = πr²h V is the subject
If the equation is rewritten then another letter can become the subject
V
e.g. h = ——— h is the subject
πr²
Manipulations can be complex and need care. They are sometimes necessary
in solving equations.
For example find the solution set of
5 4
————— + ————— = 2
(x+2) (x-3)
Answer: 5(x-3) + 4(x-2) = 2(x-2)(x-3)
9x - 23 = 2x² + 12 - 10x
2x² - 19x + 35 = 0
(2x-5)(x-7) = 0
5
x = — or 7
2
Remember, always check your work carefully.
_PERIMETERS_AND_AREAS_OF_PLANE_SHAPES_
The _perimeter_ of a plane shape is the length of its boundary. For
example, if the shape is a polygon (a clsoed plane shape with a
finite number of straight edges), its perimeter is the sum of the
lengths of its sides. The area of a plane shape is measured in
square units.
When calculating areas, it is very important to make sure that all
the linear dimensions are measured in the same units.
If the linear units are in cm, the area is given in sq.cm (or cm²).
Similarly if in mm, the area is given in sq.mm (or mm²).
You will be required to know the following basic plane shapes but
problems may combine two or more of these shapes.
(a) _SQUARE_ ┌┬───┬┐
├┘ └┤
For a square of side a: a │ │
perimeter = 4a ├┐ ┌┤
area = a² └┴───┴┘
a
e.g For a square of side 3 cm,
the perimeter = 12 cm and the area = 9 cm²
(b) _RECTANGLE_
┌┬─────────────┬┐
For a square of length a ├┘ └┤
and breadth b: b│ │
perimeter = 2b+2a = 2(b+a) ├┐ ┌┤
area = ba └┴─────────────┴┘
a
e.g For a rectangle of length 4 m and breadth 2 m,
the perimeter = 12 m and the area = 8 m²
B
__/¦\_
(c) _TRIANGLE_ a__/ ¦ \_c
__/ ¦h \_
For a triangle with sides a,b,c and __/ ¦ \_
perpendicular height (as shown) h: C/____________¦□_______\A
perimeter = a+b+c b
area = ½bxh (½ base x perpendicular height)
e.g. For a triangle with base 4 cm and perpendicular
height 2 cm, the area = 4 sq.cm.
If the perpendicular height is not known, but
two sides (a,b) and the included angle (C) are,
then the area of the triangle = ½ab sin C
If the lengths of all three sides are known then the area
________________
of the triangle = √S(S-a)(S-b)(S-c) where S = ½(a+b+c)
e.g. For a triangle with sides measuring 6 cm, 5 cm and
3 cm the perimeter = 14 cm and the area
________________ __
= √7(7-6)(7-5)(7-3) = √56 = 7.5 sq.cm
(d) _PARALLELOGRAM_
_______________
A parallelogram is a quadrilateral /¦ /
with opposite sides parallel. b/ ¦ /
/ ¦h /
If the sides are of lengths a and b /___¦□_________/
and the perpendicular height is h then: a
perimeter = 2 (a+b)
area = ah (i.e. base x perpendicular height)
e.g. For example if the base is 10 cm and the
perpendicular height is 5 cm then the area
= 50 sq.cm
(e) _RHOMBUS_ ________
/\ ∕/
A rhombus is just a special case a / \□∕ /
of a parallelogram in which a = b / ∕ \ /
i.e. all sides are equal. /∕_____\/
a
(f) _TRAPEZIUM_
A trapezium is a quadrilateral
with one pair of parallel sides.
_______a____
If the lengths of the sides are ∕¦ \
a,b,c and d and h is the c ∕ ¦ \ d
perpendicular distance between ∕ ¦h \
the parallel sides then: ∕______¦□_____________\
perimeter = a+b+c+d b
area = ½(1+b)h
e.g. For example if the parallel sides of a trapezium
have lengths 5 cm and 7 cm and the perpendicular
distance between them is 2 cm then the area = 12 sq.cm
(g) _CIRCLE_
______
For a circle with radius r _/ \_
(diameter d = 2r) then: / \
perimeter = 2πr = πd |_____d______|
area = πr² | \ |
\_ \r _/
The value of P will normally be given \_____\/
1
as 3 /7 or 3.142.
e.g. For a circle or radius 7 cm, the perimeter is 44 cm
and the area is 154 sq.cm.
_VOLUME_AND_SURFACE_AREA_OF_SOLID_SHAPES_
The _volume_ of a solid shape is measured in cubic units.
The _surface_area_ of a solid shape is the total area of its
surface and is measured in square units.
As with plane shapes, it is very important when calculating
volumes and surface areas to make sure that all the dimensions
used are measured in the same units.
For example if the linear units are in cm, the surface area is in
sq.cm (or cm²) and the volume is in cu.cm (or cm²).
You will be required to know the following basic solid shapes
but problems may combine two or more of these shapes.
______
(a) _CUBE_ |\ \
a| \a \
All edges of a cube are the | \__a__\
same length. Let this be a. \ | |
volume = a³ \| |
surface area = 6a² \_____|
e.g. For a square of side 3 cm,
the volume = 27 cm³ and the surface area = 54 cm²
(b) _CUBOID_
______
A cuboid is a rectangular block. / /|
Let the lengths of the sides be / / /
a,b and c. / c/ /
Then / / /
volume = abc /_____/ /
surface area = 2(ab+bc+ac) b |______|/
a
e.g. For a cuboid with sides 2,3 and 4 cm,
volume = 24 cm³
surface area = 52 cm²
(c) _PRISM_ ___
_∕ |
A prism is a solid with a uniform _∕ /
cross section. _∕__ |
/ | _∕
The prism shown has an end section P |_ A / _∕l
of area A and perimeter P. \__| ∕
The length is l.
volume = Al
surface area = 2A + pl
e.g. For a prism whose end face has perimeter 4 cm and
area 2 cm² and whose length is 3 cm, then the surface
area = 16 cm²
(d) _TRIANGULAR-FACED_PRISM_ _∕\
_∕ \
This is just a special case of a _∕ \
prism. _∕ _∕
/\ _∕l
If the sides of the end face are of a/ A\ b_∕
length 3, 4 and 5 cm and its area /____\∕
is 6 cm² and the length of the prism c
is 8 cm then the volume = 6x8 = 48 cm³
and the surface area = 2x6 + 12x8 = 108 cm²
(e) _RIGHT_CIRCULAR_CYLINDER_
This is again a special case of a prism. ___
_∕ \
If r is the radius of the _∕ |
end section (or base) and _∕ _/
h is the height of the _∕ _∕
cylinder then / \ _∕
volume = πr²h | r |_∕ h
surface area = 2πr² + 2πrh = 2πr(r+h) \/_/∕
e.g. For a circular cylinder of height 10 cm and radius 7 cm
22
volume = —— x 7² x 10 = 1540 cm³
7
22 22
surface area = 2 x —— x 7² + 2 x —— x 7 x 10 = 308 + 440
7 7 = 748 cm²
(f) _SPHERE_
For a sphere of radius, r _____
_/ \_
4 / \
volume = —πr³ | __r__|
3 | |
\_ _/
surface area = 4πr² \_____/
e.g. If a sphere has radius 6 cm,
4
the volume = —π6³ = 904.9 cm³
3
surface area = 4π6² = 452.4 cm²
(results given to 1dp using π = 3.142)
(g) _PYRAMID_
For a pyramid, let the area
of the base be A and the ^
perpendicular height be h then /│\
/_│_\
1 /∕ │h \
volume = —Ah \ A│ /
3 \___/
surface area = A + (area of triangular faces)
Two special cases of the pyramid which you should be aware
of are the _square-based_ pyramid and the _triangular-based_
pyramid (usually termed a tetrahedron).
(h) _RIGHT-CIRCULAR_CONE_
A right-circular cone is a cone with ^
a circular base whose apex is /│\
directly above the centre of the / │h\
circle. / │ \l
/___│___\
If r is the radius of the circular base / │ \
h is the perpendicular height | r_∕ |
l is the slant height, \_∕_______/
1
Then the volume of the cone = —πr²h
3
and the surface area of the cone = πr² + πrl
e.g If a right circular cone has radius 3 cm, height
4 cm and slant height 5 cm then
1 22
the volume = — x —— x 9 x 4 = 17.7 cm³
3 7
22 22
and the surface area = —— x 9 + —— x 3 x 5 = 75.4 cm²
7 7
*21