L01 Use applications of number theory in practical engineering situations
Assignment Brief and Guidance
Scenario:
You work as an engineer at a multinational manufacturing organisation, they have just taken on some new engineering apprentices. The research and development department has requested your assistance of the design of a manual containing set worked questions and their associated advanced mathematical techniques, which are likely to be consulted by their aspiring engineers in developing solutions to real problems. Therefore, solve the following task describing your thought process.
Task 1
a. Convert the following numbers into decimals. These bases are frequently used in computer engineering:
- 110011002
- 59014388
- 43A62C16
b. Multiply the following numbers:
- A516 * 248
- A12Hex * F8Hex
Task 2
The impedances Z1 and Z2 in an electric circuit are in series. What is the total impedance? State your answer in polar and rectangular forms.
- Z1 = (15 + j 10) Ω
- Z2 = 40 Ω
Task 3
- A system has an open-loop transfer function (G) given by:
G = 7 ej2π * 1.02 ej0.2 / 1.8 ej0.99
Simplify G.
- Express the following current in complex exponential form:
i(t) = 30cos (100t – 60°)
Task 4
Find a formula for cos (3θ) in terms of cos (θ) and sin (θ) using de Moivre’s theorem
| Learning Outcomes and Assessment Criteria | |||
| Learning Outcome | Pass | Merit | Distinction |
| LO1 Use applications of number theory in practical engineering situations. | P1 Apply addition and multiplication methods to numbers that are expressed in different base systems.
P2 Solve engineering problems using complex number theory. P3 Perform arithmetic operations using the polar and exponential form of complex numbers. |
M1 Deduce solutions of problems using de Moivre’s Theorem. | D1 Test the correctness of a trigonometric identity using de Moivre’s Theorem. |
Unit 5006: FM2 Assessment Brief
| Student Name/ID Number | |
| Unit Number and Title | Unit 5006: Further Engineering Mathematics |
| Academic Year | 2025/26 |
| Unit Tutor | Md Akmol Hussain |
| Assignment Title | FM2 – Matrix Methods |
| Review date | 20/10/2025 |
| Submission Date | 03/11/2025 |
| IV Name | Arshad Ahmed Mir |
Submission Format
Instruction to students:
- To maximise your learning outcome, attempt all tasks in this assignment.
- Submit your assignment with your signed front sheet (for authentication) to TurnItIn.
LO2 Solve systems of linear equations relevant to engineering applications using matrix methods
Assignment Brief and Guidance
Scenario:
ou are working as an engineer for a multinational manufacturing organisation who has just taken on some new apprentices. The research and development department has requested your assistance in the design of a booklet containing set worked engineering questions and their associated advanced mathematical techniques. This training booklet will be used aspiring engineers by developing solutions to real world problems. Therefore, solve the following tasks and describing your thought process.
Task 1:
Calculate the determinant of the matrix A.
| 1 | 2 | 7 | |
| det A = | 0 | 4 | 2 |
| 2 | -1 | 3 |
Task 2:
Solve the system of the three linear equations shown below using Gaussian elimination.
| (eq. 1) | 2 x1 | – | 2 x2 | + | 2 x3 | = | 0 |
| (eq. 2) | 8 x1 | + | 10 x2 | + | 2 x3 | = | 6 |
| (eq. 3) | -2 x1 | + | x2 | – | 3 x3 | = | 5 |
Task 3:
Determine the solutions to the set of engineering linear equations below by using the Inverse Matrix Method for a 3×3 matrix.
x+ 2y +3z = 5
2x -3y -z= 3
-3x +4y + 5z = 3
Task 4:
Write a report on the analytical matrix solutions by:
Briefly describing them.
Evaluate your given solutions computed in this assignment by using these methods with appropriate computer software.
| Learning Outcomes and Assessment Criteria | |||
| Learning Outcome | Pass | Merit | Distinction |
| LO2 Solve systems of linear equations relevant to engineering applications using matrix methods | P4 Ascertain the determinant of a given 3×3 matrix.
P5 Solve a system of three linear equations using Gaussian elimination. |
M2 Determine solutions to a set of linear equations using the Inverse Matrix Method. | D2 Evaluate and validate all analytical matrix solutions using appropriate computer software. |
Pearson BTEC Level 5 Higher National Diploma in Electrical Electronic and Engineering Manufacturing Engineering
| COURSE TITLE | Pearson BTEC Level 5 Higher National Diploma in Electrical and Electronic Engineering/Manufacturing Engineering | ||
| STUDENT NAME | |||
| MODULE TITLE | Unit 5006: Further Engineering Mathematics | ||
| TITLE OF ASSIGNMENT | FM3: Graphical and numerical methods
FM4: Differential equations |
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| DEADLINE DATE FOR SUBMISSION BY STUDENTS | 05/01/2026 | ||
| ISSUE DATE | 15/09/2025 | ||
| FORMATIVE FEEDBACK DATE | 29/12/2025 | ||
| SUBMISSION LOCATION | Turnitin | ||
| ASSESSOR(S) | Md Akmol Hussain | ||
| IV NAME | Arshad Ahmed Mir | ||
| LEARNING OUTCOMES ASSESSED
LO3: Approximate solutions of contextualised examples with graphical and numerical methods LO4: Review models of engineering systems using ordinary differential equations |
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NOTES FOR STUDENTS
What is Academic Malpractice?
Academic malpractice relates to academic work that does not meet normal standards of academic practice and encompasses all kinds of academic dishonesty, whether deliberate or unintentional, which infringes the integrity of the College’s assessment procedures. ‘Candidate malpractice’ means malpractice by a candidate in the course of any examination or assessment, including the preparation and authentication of any controlled assessments or coursework, the presentation of any practical work, the compilation of portfolios of assessment evidence and the writing of any examination paper.
| Learner declaration (authentication) |
| I certify that the work submitted for this assignment is my own and research sources are fully acknowledged.
Student signature Date: |
Unit 5006: FM3 Assessment Brief
| Student Name/ID Number | |
| Unit Number and Title | Unit 5006: Further Engineering Mathematics |
| Academic Year | 2025/26 |
| Unit Tutor | Md Akmol Hussain |
| Assignment Title | FM3: Graphical and numerical methods |
| Review date | 22/12/2025 |
| Submission Date | 05/01/2026 |
| IV Name | Arshad Ahmed Mir |
| Submission Format |
|
Instruction to students:
|
LO3: Approximate solutions of contextualised examples with graphical and numerical methods
Assignment Brief & Guidence
Scenario
You are working as an engineer for a multinational manufacturing organisation who has just taken on some new apprentices. The research and development department has requested your assistance in the design of a booklet containing set worked engineering questions and their associated advanced mathematical techniques. This training booklet will be used aspiring engineers by developing solutions to real world problems. Therefore, solve the following tasks and describing your thought process.
Task 1
In operations research, the solutions to problems can frequently be estimated with graphical methods. In this task you have to maximise the profit (Z). The relationship with two other variables (x, y) as well as their constraints is shown below:
Z = 3x + 10y
Constraints: 2x + 5y16, x0, y 0 and x5.
Task 2
A ball is drifting on the surface of the water (see Figure below). The question investigated here is: If the radius and water density are known, how deep does the ball sink in?