National Diploma in Engineering
Further Mathematics for
Engineering Technicians
Assignment booklet
Don’t forget that when submitting work you must declare which outcome you are
claiming. (P1, M3, D2,for example)
Don’t forget to put your name on all submitted work.
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Make sure that you understand the work you have submitted. You may be asked
questions upon submission.
Work which is not reasonably presented might not be accepted.
Deadline
Outcome
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
Grading Criteria
use a graphical technique to
solve a pair of simultaneous
linear equations
solve a practical engineering
problem involving an
arithmetical progression
solve a practical engineering
problem involving geometric
progression
perform the two basic
operations of multiplication and
division to a complex number in
both rectangular and polar
form, to demonstrate the
different techniques
Evidence Type
2/11/2015
11/12/2015
11/12/2015
15/1/2016
calculate the mean, standard
deviation and variance for a set
of ungrouped data
calculate the mean, standard
deviation and variance for a set
of grouped data
sketch the graph of a sinusoidal
trigonometric function and use
it to explain the terms and
describe amplitude, periodic
time and frequency
11/2/2016
use two of the compound angle
formulae and verify their
relationship
find the differential coefficient
for three different functions to
demonstrate the use of function
of a function and the product
and quotient rules
use integral calculus to solve
two simple engineering
problems involving the definite
and indefinite integral
24/3/2016
26/2/2016
11/3/2016
29/4/2016
20/5/2016
Acheivement
Date
M1
M2
M3
D1
D2
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
exam
use complex numbers to solve a
parallel arrangement of
impedances giving the answer in
both Cartesian and polar form
exam
use differential calculus to find
the maximum/minimum for an
engineering problem.
exam
using a graphical technique
determine the single wave
resulting from a combination of
two waves of the same
frequency and then verify the
result using trigonometric
formulae
use numerical integration and
integral calculus to analyse the
results of a complex engineering
problem.
exam
exam
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 + 6j) (3 – 4j) working in rectangular form throughout.
2) Determine (3 + 6j) ÷ (4 + 3j) 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.
P6 calculate 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.
length
165
number 5
166
14
167
21
168
28
169
39
170
29
171
28
172
24
173
19
174
14
P7 sketch the graph of a sinusoidal trigonometric 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.
P8 use 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:3𝑐𝑜𝑠2𝑥
𝑥²
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 the formula 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 the formula 𝐹 = 4𝑥+6.
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
t minutes
89.7
15
69.9
20
51.8
26
40.3
31
31.4
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.
D1 using a graphical technique determine the single wave resulting from a
combination of two waves of the same frequency and then verify the result
using trigonometric 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.