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soundofheaven.info - Download as PDF File .pdf), Text File .txt) or read online. EASA Part Module-1 (Mathematics) - Download as Word Doc .doc), PDF File .pdf), Text File .txt) or read online. EASA Part Module-1 (Mathematics). View Notes - MODULE 1 from EASA PART MODULE 01 at KDU University College. EASA PART 66 CATEGORY B MODULE 1 - MATHEMATICS TRAINING.

It should be noted that AP is always at right-angles to the radius OP. In this case the number is the same but the system or numeration is different. Choose scales so that completed graph fits the sheet of graph paper. Study the graph paper to find how many large squares there are from left to right. This is a function which is commonly found in physical situations. Nedu Japsi. To determine this.

Brackets 2. Equations 2. Linear equations 2. Equations containing simple fractions 2. Simultaneous equations 2. Indices and powers 2. Binary system 2. Second degree equations with one unknown 2. Simple geometrical constructions 3.

Graphical representation 3. Nature and use of graphs 3. Graphs and mathematical formulae 3. Simple trigonometry 3. Trigonometrically relationships 3. Co-ordinate geometry 3. Polar coordinates. This is a review course so it is important that you spend time studying the material in preparation for your examination — see also www. To register for this training, please email office sassofia. The Early Bird Discount Program is applicable for bookings taken and payment received more than 4 weeks before commencement of the training!

Check all discounts. Site Search Keywords. In this case we must eliminate the value of —6 from the LHS. As in all cases of solving equations. Example 1. Example 2. If we swap them around and change the signs i. In this one. Here is another a formula involving several algebraic symbols.

To help. What is it? By manipulating transposing is the word the equation. It is important to get a 'feel' for the form of the equation. Look again at the equation. Find N. One important point. Two unknowns require two different equations. Three unknowns required three different equations. He uses formulas and equations. This is what transposition is all about. Design is the creation of a component or mechanism on paper.

One unknown quantity can be deduced from one equation. This is known as 'solving an equation'. Maths serves as a "tool" for Engineers at the design stage. The rule is. Find r the radius. The design engineer hopefully makes it strong enough. We are re-arranging formulas expressed as equations.

To calculate the area we multiply one side by the other. Example 2 If one side of a rectangular field is twice as long as the other. Doubling this number gives 2A.

What is the answer? Let the number you think of be A. He studies the situation and then makes the statement. This formula can be used to calculate the answer no matter what number you think of.

The long side is therefore 2 x L or 2L. If 6 is then added. Let the short side of the field be L. Calculate the area of the field.

Let the cost of a motor cycle be M another unknown. Let the cost of a car be C at present C is an unknown. How do we construct equations from the facts contained within a scenario? Example 1 Think of a number. Here 2 equations were constructed from the facts. There are 2 unknowns x and y in one equation. Find the cost of each.

For example: In the next example. There are an infinite number of values of x for which there are corresponding values of y. The process is simple and involves modifying the equations. Using the two equations above as an example: We do not need to manipulate either of the equations because the co-efficient of y is the same in both equations. The other having become zero. The result is: If you had used equation 2 to find x. An equation of this type will produce a curve called a parabola.

I have selected 1. Do not use equation 1 because it will not highlight an error. The actual value for coefficients a. It can be shown that the Roots are found to be equal to: This concept is not considered in these notes.

When b2. P and S are known as the roots of the equation. It has been considered impossible to find the square root of a negative value. The equation concerned is then said to have no real roots. In the above examples. Where the power is expressed as a decimal. To indicate this. To allow the use of numbers involving powers and indices.

Rule 1. The method of notation used is that: Power 2 and power 3 are generally referred to as the square and the cube. It is independent of the nature of the individual items in the set. The first part is a number between 1 and 10 but does not equal A decimal point enables numbers less than one to be represented.

To express a number in standard form. A number in standard form has two parts. Number is the property associated with a set or collection of things. L and C are also well known and understood. The value of which equals the number of places by which the decimal point has been moved. The number fourteen may be written as 15 or XIV. In this case the number is the same but the system or numeration is different.

In general a system of numeration consists of a set of symbols together with a rule by which the symbols can be combined together. The first part is called the Mantissa. If the point was moved Left. This calculation can be written as 8. When is written as 8. The base. In practice only the digits of the system are written. The index indicates the power to which the base is raised.

The two digits 0 and 1 are referred to as bits. Example 1 is really: All digits to the right of the binary point refer to negative powers. There is no ambiguity. A switch is either off or on corresponding. Again the system of numeration is the same as that used for decimal and binary.

All digits to the right of the octal point refer to negative powers. The remainder gives the number in the new base and should be read from bottom to top. Example — convert to binary.

Example 1 — convert 1 0 1 1 1 0 0 1 to octal Binary No Weighting Octal No sum 4 1 0 1 1 1 0 0 1 2 1 4 2 1 4 2 1 2 7 1 Answer 1 0 1 1 1 0 0 12 is equal to The reverse process should be used to convert octal to binary. The easiest way to convert from binary to decimal is to remember the weightings.

Example 1 — convert 1 0 1 1 0 1 to decimal. Each octal digit can be represented by 3 binary digits. Convert each digit into a 3 digit binary number keeping the order of digits the same.

Work from the bottom to the top of the table shown above to convert 8 to binary. Convert each hexadecimal digit into its binary equivalent keeping the order the same. Remembering that when. Each hexadecimal digit can be represented by 4 binary digits.

Example 1 — convert A to binary. The weightings this time are 8 — 4 — 2 — 1. Example Calculate 6. The general definition is. An example of the function of logarithms is shown below. The use of logarithms is no longer so widespread as the electronic calculator has become so readily available.

Logarithms enable multiplication and division to be performed using addition and subtraction. The process for converting a binary number to a hexadecimal one. If a division is to be performed. It is the concept of a logarithm that is important at this stage. If the calculator is used to solve 6. It is important to realise that this example shows how logarithms can be used.

The calculator shows this to be But how are 'angles' expressed or measured. Consider a single line, and rotate it through a complete revolution.

A degree is of a revolution. Note that 1 degree can be sub-divided into 60 minutes and 1 minute can be subdivided into 60 seconds very small. A few definitions are included here: An Acute angle. The more common ones are described in the following text. A triangle obviously has 3 sides and 3 internal angles. The sides are often represented by the 3 small letters a, b and c; the angles by the large letters A, B and C.

The construction of a dotted line parallel to AB and an extension of BC proves this. There are many different types of triangle. The main types and features are summarised as follows: Scalene triangle has three sides of different lengths. Isosceles triangle has two sides and two angles equal. The equal angles lie opposite to the equal sides. Similar triangles do not have to be the same size. A triangle has the least number of sides.

We need to be more precise. One triangle may have sides twice or ten times as large as another triangle and still be classified as similar. If they have the same shape.

Other multi-sided figures have names indicating the number of sides. It is sometimes necessary to determine whether triangles are Congruent.

A simple criteria exists to assist us. Octagon — 8 sided Issue 0 Page 4. Two triangles are congruent if: Their corresponding sides are of equal length. The term poly means multi. Hexagon — 6 sided. Pentagon — 5 sided. If they are exactly the same shape and size. Some are not so common. Each pair of opposite angles are equal The diagonals bisect each other The diagonals bisect the parallelogram and form two congruent triangles 4.

The following properties apply to parallelograms: Each pair of opposite sides is equal in length. It has the same properties as a parallelogram with the addition that the diagonals are equal in length.

There are various types. Since a quadrilateral has four sides. Issue 0 Page 6. It also has all of the properties of a parallelogram and the following additional properties: The diagonals bisect at right angles 4. It has all the properties of a parallelogram.

The distance. We already know that if OP is rotated through 1 complete revolution. It will certainly be found on a scientific calculator. Put another way. The length OP is the Radius of the circle. If the line OP is fixed at O and rotated around O. Consider a circle of radius R and consider an arc AB. How far does it move from its start point? The distance moved in 1 rev. But there is another important unit of angular measurement.

The angle at the centre of the circle. A wheel. AOB is then equal to I Radian. It should be noted that AP is always at right-angles to the radius OP.

If an arc of a circle. Note also that if a point P is moving with speed N. If we want to fit a picture-rail along a wall. Obviously we also need to know the width. This twodimensional concept of size is termed Area.

Each square has an area of 1 square meter. Clearly it can be divided up into 12 equal squares. But if we wish to fit a carpet to the room floor. A theorem exists stating that triangles with the same base and drawn between the same parallels will have the same area. Note again that base in meters x height in meters gives m 2. Inspection reveals the 2 triangles are congruent. If we consider this diagram.

This is true for any triangle. What is the area of a semi-circle where the diameter is 30cm? So the area of a circle must include a 'squared' term.

Don't forget to include units in the answer e. The rectangle has dimensions mm x mm The semi-circle has a diameter of mm Total area is the sum of the two individual areas. What is the area of the surround? With a problem like this. An office 8. Sometimes it is made up from a combination of shapes. We may need to find the area of an object that is a combination of shapes: In this case the shape comprises a rectangle and a semi-circle.

Volumes are calculated in cubic units such as cubic centimetres. If we have a box. For a box. There are 2 layers. For irregular or particular shapes. Having the ability to calculate volume enables you to determine the capacity of a fuel tank or reservoir. Instead of squares. This is a 3-dimensional concept and the typical units of volume are cubic metres m3.

The volume would normally be given in cm3. When calculating areas or volumes. What is the cubic capacity of a 2 cylinder engine. The data can be presented in many different ways as shown below.

The data will data will generally comprise 2 variables. If we are plotting temperature with respect to time then a continuous line graph is better. First of all. That is. Study the graph paper to find how many large squares there are from left to right. As the temperature of the gas increased. The horizontal x-axis will represent the independent variable and the vertical yaxis the dependent variable.

Issue 0 Page 2. The most useful scales are 1. The next stage is to plan the use of the graph-paper so as to present the graph in the clearest manner possible. So the graph must be drawn so that each value appears or fits on the paper. Choose scales so that completed graph fits the sheet of graph paper. This should give an idea of a suitable scale.

Look at the largest right-hand. Now divide the value found by the subtraction. The scales cross at the origin O. The graph constructed by plotting a series of points. The first quantity. There is no merit in drawing small graphs.

Subtract the smallest lower value from the largest upper value to give the range. When all the points have been plotted. The transfer is very simple. The procedure is repeated for each pair of values in turn. Having done this. The intersection represents one plotted point of the graph. The graph paper has now been prepared for the object of the exercise. But try to draw a continuous smooth line. This topic looks at the shape and characteristics of these functions when expressed graphically.

An example is that the drag force D varies according to the square of the airspeed V. In this case. There should be roughly the same number of points on both sides of the smooth curve.

It is probably true to say that most mathematical or scientific data change gradually or progressively. This value of C measured along the y axis is known as the intercept. Obviously for a straight line. Here several different functions are considered graphically.

If m is -ve. C is a constant. As k increases the value of kx 2 also increases. This is the characteristic shape. Note that the slope is no longer constant. This is a function which is commonly found in physical situations. Functions within this family are less likely to be encountered during this course. Note that the graph has Turning points. Some variations on the basic function are also shown.

Both of these functions are repetitive but the word used to describe such behaviour is periodic in this case. Engineers are often interested in slope.

Reference has already been made to the slope of a graph. It is also often found in Engineering applications. Curves have variable slopes. These graphs are often found.

Straight lines have a constant slope. In fact. In this example. Most nomographs contain a great deal of information and require the use of scales on three sides of the chart. Nomographs are a special type of graph that enable you to solve complex problems involving more than one variable.

Illustrated is a fairly typical graph of three variables. If any two of the three variables is known. The area can be calculated by: Considering simple shapes and approximating Counting squares. The resulting distance travelled can be extracted from the graph at the point where these two dashed lines meet. Using calculus 5. How can these ratios be used in practice? Consider a triangle with side lengths 3. From our definition of sine. Some students find the mnemonic "Sohcahtoa" to be helpful in this respect.

These ratios are used very extensively in Maths and Science and very many modifications to the basic ratio have been evolved. When the top is viewed through a theodolite. Example A church spine is known to be 60 metres high. How far is the theodolite from the base of the spine? The distance D is the unknown quantity. They can now be applied to practical situations. The basic trigonometry ratios were explained with reference to a right-angled triangle. But their use can be extended for use with any triangle.

Sketch a diagram if necessary. Example ABC is any triangle. AD is now the height of the triangle. Page 4.