University Engineering Mathematics Assignment - Semester 1

This document presents a comprehensive solution to a mathematics assignment, encompassing a range of topics essential for engineering students. The solution begins with problems related to arithmetic and geometric progressions, calculating salaries and determining speeds in a geometric sequence. It then delves into complex numbers, requiring calculations in both rectangular and polar forms. Statistical techniques are applied to analyze ungrouped and grouped data, calculating measures of central tendency and dispersion. Further, the assignment addresses simultaneous equations through graphical methods and reduction to straight-line form using logarithmic axes. Trigonometric graphs are explored, including plotting waveforms and explaining key terms. Compound angle formulae are utilized for expansion and simplification. Finally, the solution covers differentiation and integration, providing solutions to various functions and calculating areas under curves. This solved assignment is designed to help students understand and master core mathematical concepts. The assignment is contributed by a student to be published on Desklib, a platform providing AI-based study tools for students.

P2 - Arithmetic Progressions
1. A nurse is paid a salary of £21500 per annum and receives annual
increments of £1075. Determine her salary in the 6th year and calculate the
total she will have received in the first 15 years.
P3 - Geometric Progressions
2. A drilling machine is to have 7 speeds ranging from 75 rev/min to 850
rev/min. If the speeds form a geometric progression determine their values,
each correct to the nearest whole number.
P4 - Complex Numbers
Enter your surname to find your personal values for Z1, Z2, Z3 and
frequency. ~(See sheet for values)
3. For your personal numbers (Z1, Z2, Z3), calculate the following, using the
rectangular (Cartesian) form. Show your working and give your answers in
rectangular form.
a) Z1Z2 b) Z1Z2 / Z3
4. Using the polar form of Z1, Z2, Z3, determine in polar form, with angles
stated in radians:
a) Z1Z2 b) Z1Z2 / Z3
P5 - Ungrouped Data
For this part of assignment (Statistical Techniques) open the personal
data: ~(See sheet for values)
5. The values of capacitances, in microfarads, of ten capacitors selected at
random from a large batch of similar capacitors are given in the personal

data generator file. Determine the mean, the standard deviation from the
mean and the variance for these results correct to 3 decimal places.
P6 - Grouped Data
6. The values of resistance for a number of resistors are shown in the
frequency distribution table in the personal data generator file. Determine
the estimated mean, the standard deviation from the estimated mean and
the variance for these resistors, correct to 4 significant figures.
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Simultaneous equations – both of these questions should be solved by
plotting 2 straight lines on a graph. ~(See sheet for values)
P1
1. The law connecting friction F and load L for an experiment is of the form
F = aL + b, where a and b are constants.
When F = 12.0N, L = 7.0N and when F = 8.0N, L = 1.5N. Find the values
of ‘a’ and ‘b’ by plotting a graph of ‘b’ against ‘a’.
Reduction to straight line form – this question should be solved by
plotting the data from the personal data generator on a graph with
logarithmic axes
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Enter your surname to find your individual data sets for P7, P8 See
Sheet
P7 Trigonometrical Graphs
2. An A.C. voltage is given by the expression V = V0 sin(2ft + Φ0)
where V is the voltage at time t in Volts, f is frequency in Hz (or cycles per
second) and Φ0 is phase angle in radians.
Plot instantaneous voltage V against time over 1 complete cycle. Use

values for amplitude V0, frequency f and phase angle Φ0 from the personal
data generator for assignment 3. (Hint: you may plot the waveform using a
scatter chart in Excel.)
Using your graph, describe and explain the terms amplitude, periodic
time, phase angle, and frequency of the waveform.
P8 Compound angle formulae
3. Use the compound angle formulae to expand and simplify the following
expression: sin(A + B) − sin(A - B)
Verify that your answer is correct by substituting any 2 values for A and B.
Differentiation
1. Differentiate the following functions with respect to x or t using the chain
rule, the product rule or the quotient rule. Simplify your answers as far as
possible.
(a) y=−4 e2 t sin 2 t
(b) y=2cos (3 x4+5 )
(c)
y=(5 x3+2 )
( x2+3 )
Integration
2. An electric current is given by i = 24 sin θ amps. Find its mean value
over half a cycle.
Hint: The mean value of the current is given by the area under the curve
divided by the length of the base, where the base is measured in
radians.
3. Determine the area underneath the curve y = 3x2 + 4, the axis and
ordinates are x=4 and x = -1 by integration.