National Diploma in Engineering
Further Mathematics for Engineering Technicians
Assignment booklet
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P1 use a graphical technique to solve a pair of simultaneous linear equations
Solve graphically the following pair of simultaneous equations:
4a – 3b = 18
a + 2b = −1
P2 solve a practical engineering problem involving an arithmetical progression
Use an arithmetic progression to solve this problem.
An oil company bores a hole 90m deep. Estimate the cost of boring if the cost is £30 for drilling the first metre with an increase of £2 per metre for each succeeding metre. If the company decides to drill an extra 30 metres, what will be the cost of the extra 30 metres?
P3 solve a practical engineering problem involving geometric progression
Use a geometric progression to solve this problem.
100g of a radioactive material disintegrates at a rate of 3% per annum. How much of the substance is left after (a) 11 years, (b) 20 years?
P4 (see also M2) perform the two basic operations of multiplication and division to a complex number in both rectangular and polar form, to demonstrate the different techniques
Answer the following four questions:-
1)Determine (5 + j6) (3 – j4) working in rectangular form throughout.
2)Determine (3 + j6) ÷ (4 + j3) working in rectangular form throughout.
3)Determine 8∠30º x 4∠40º working in polar form throughout.
4)Determine 6∠20º ÷ 3∠40º working in polar form throughout.
P5 calculate the mean, standard deviation and variance for a set of ungrouped data
The monthly output of a coal pit in thousands of tonnes for twelve consecutive months, are as shown. Determine the monthly mean output, the standard deviation and the variance.
5.3, 5.4, 5.6, 5.5, 5.4, 5.3, 5.2, 5.5, 5.7, 5.4, 5.7 and 5.4.
P6calculate the mean, standard deviation and variance for a set of grouped data
The length in millimetres of a sample of bolts is as shown below. Calculate the mean, the standard deviation and the variance.
165 / 166 / 167 / 168 / 169 / 170 / 171 / 172 / 173 / 1745 / 14 / 21 / 28 / 39 / 29 / 28 / 24 / 19 / 14
P7 sketch the graph of a sinusoidal trigonometrical function and use it to explain the terms and describe amplitude, periodic time and frequency
Sketch the graph of a sinusoidal function and use it to explain the terms and describe amplitude, periodic time and frequency.
P8use two of the compound angle formulae and verify their relationship
Verify (a) that the compound-angle addition formulae are true when A = 25° and B = 40°; and (b) that the compound-angle subtraction formulae are true when A = 110° and B = 75°.
P9 find the differential coefficient for three different functions to demonstrate the use of function of a function and the product and quotient rules
Find the differential coefficient for three different functions to demonstrate the use of
a)The product rule (see question 1)
b)The quotient rule (see question 2)
c)The function of a function rule (see question 3)
Question 1) Use the product rule to differentiate:-
v = 6t sin 3t
Question 2) Use the quotient rule to differentiate:-
Question 3) Use the function of a function rule to differentiate:-
y = 2cos (4x² + 3)
P10 use integral calculus to solve two simple engineering problems involving the
definite and indefinite integral
(i)
A body is fired downwards from a height, and its velocity in metres per second is given by:-
v = 9.8t + 20
It is found that the body hits the ground at t = 5 s. Use the indefinite integral to find the height from which the body has been fired.
(ii)
The force F newtons acting on a body at a distance x metres from a fixed point is given by :-
.
Use a process of definite integration to find the work done when the body moves from the position where x = 2m to that where x = 4m.
M1 use the laws of logarithms to reduce an engineering law of the type y = axn to
straight line form, then using logarithmic graph paper, plot the graph and obtain the
values for the constants a and n
A liquid which is cooling is believed to follow a law of the form θ = θ0ekt where θ0 and k are constants and θ is the temperature of the body at time t. Measurements are made of the temperature and time and the results are:
θ° C / 89.7 / 69.9 / 51.8 / 40.3 / 31.4t minutes / 15 / 20 / 26 / 31 / 36
Plot the data using logarithmic graph paper, and show that these quantities are related by this law and determine the approximate values of θ0 and k.
M2 use complex numbers to solve a parallel arrangement of impedances giving the
answer in both Cartesian and polar form
For the parallel circuit shown below, determine (a) the total admittance, (b) the total impedance, and (c) the supply current and its phase relative to the 240V supply. Use complex numbers to do this, and show your answers in both Cartesian and polar form.
M3 use differential calculus to find the maximum/minimum for an engineering problem.
A shell is projected upwards and the distance vertically,s metres, is given by s = 14t – 4t2 where t is the time in seconds. Find the time at which the missile reaches its maximum height, and find the maximum height reached.
D1using a graphical technique determine the single wave resulting from a combination of two waves of the same frequency and then verify the result using trigonometrical formulae
Using a graphical technique, show how two sine waves:-
y1 = 4 sin (100πt + π/2)
y2 = 5 sin (100πt – π/4)
can be added to give a resultant sine wave. Verify your result using the cosine rule and the sine rule.
D2 use numerical integration and integral calculus to analyse the results of a complex engineering problem.
The velocity of a body is given as v = 2t² −3t – 6.
Draw a graph showing velocity against time for values of t between 0s and 6s.
Using a) integral calculus and b) Simpson`s rule, calculate the distance travelled by the object between t = 0s and t = 5s. Compare the results found from each method.