BTEC NC - Further Mathematics for Technicians - Applied Algebraic and Graphical Techniques

Brendan Burr BTEC National Certificate in ElectronicsApplied Algebraic and Graphical Techniques

Task 11.1 Solve the following simultaneous equations using a graphical technique

e.g. Graphmatica.

4x – 3y = 18x + 2y = -1

This Graph shows that x = 3.0 and y=-2.0.

(4 x 3) – (3 x -2) = 18(3) + (2 x -2) = -1

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1.2 A body is in free fall. In the first second of motion it falls 5m, in the second second of motion it falls 15m, and in the third second of motion it falls 25m etc.

(a) How far does it fall in 12 seconds?

a = 5 metres r = 10 metresN = 12 seconds

(b) How far does it fall in 14 seconds?

(c) How long will it take to fall 3000m?

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1.3 Car production in the first week of a new model was 150.If production continues with a fall per week of 2%:

(a) Find the total car production output after 52 weeks, from the commencement of production in the new model.

to 0 decimal place

(b) How many cars will be produced in the second 52 weeks, assuming the same fall per week?

to 0 decimal place

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1.4 Given that Z1 = 1 + j2 and Z2 = 4 – j3 evaluate in Cartesian Form:

(a)

(b)

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1.5 Given that Z1 = 20 / 40° and Z2 = 4 / 20°, evaluate in Polar Form:

(a)

(b)

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1.6 The values of capacitance in microfarads of 10 capacitors selected at random are:

34.3, 25.0, 30.4, 34.6, 29.6, 28.7, 33.4, 32.7, 29.0, 31.3 µF

Determine correct to 2 decimal places:

(a) The mean value

(b) The variance

x x lx-xI Ix-xI2

25.0 30.9 5.9 34.8128.7 30.9 2.2 4.8429.0 30.9 1.9 3.6129.6 30.9 1.3 1.6930.4 30.9 0.5 0.2531.3 30.9 0.4 0.1632.7 30.9 1.8 3.2433.4 30.9 2.5 6.2534.3 30.9 3.4 11.5634.6 30.9 3.7 13.69

∑= 80.1

Variance =

Variance =

(c) The standard deviation

Standard Deviation =

Standard Deviation = 2 Decimal Places

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1.7 The frequency distribution for these values of resistance in ohms of 48 resistors is as follows:

20.5-20.9 321.0-21.4 1021.5-21.9 1122.0-22.4 1322.5-22.9 923.0-23.4 2

Determine correct to 3 significant figures:

(a) The mean value

3 Significant Figures

(b) The variance

Ω (x) f Ix-xI Ix-xI2 f . Ix-xI2

20.7 3 1.21875 1.485351563 4.45605468921.2 10 0.71875 0.5166015625 5.16601562521.7 11 0.21875 0.0478515625 0.526367187522.2 13 0.28125 0.0791015625 1.02832031322.7 9 0.78125 0.6103515625 5.54316406323.2 2 1.28125 1.641601563 3.283203126

48 20.0031250035

to 3 Decimal Places

(c) The standard deviation

to 3 Decimal Places

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Task 22.1 The intensity of radiation R from certain radioactive materials at a

particular time t is determined by the following engineering law:

Where k and t are constants.

In an experiment to test this law the following values were obtained:

R 58 43.5 26.5 14.5 10t 1.5 2 3 5 7

(a) Using the laws of logarithms reduce the above law to straight line form.

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(b) Plot the graph of the straight line on log log graph paper.

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(c) From the graph determine the values of the constants k and n.

Two sets of coordinates from the line of best fit:

(3.453 , 22.564)(10.316 , 6.304)

Subtract Equation from Equation :

to 3d.p.

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2

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Sub “n” into Equation :

The LAW is given by:

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2.2 A simple circuit has the following two impedances connected in parallel:

(a) Z1 = 8 + j2 Ω(b) Z2 = 2 – j8 Ω

Calculate the total impedance Z giving the answer in both Polar and Cartesian forms:

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Top Line:

Bottom Line:

Addition and Subtraction is not possible in Polar Form, therefore the Cartesian Form must be used.

To convert to Cartesian Form the following equation can be used:

Therefore:

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Polar to Cartesian:

Therefore the answer to 3 Decimal Places is:

Cartesian FormPolar Form

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BTEC NC - Further Mathematics for Technicians - Applied Algebraic and Graphical Techniques