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Qualification - Pearson BTEC Level 5 HND in Mechanical Engineering

Unit Name - Further Maths

Unit Level - Level 5

Unit Number - Unit 39

Unit Credit - 15

Assignment Title - Further Maths

Learning Outcome 1: Use applications of number theory in practical engineering situations

Learning Outcome 2: Solve systems of linear equations relevant to engineering applications using matrix methods

Assignment Brief:

Question 1: As the new Engineering intern for a programming firm, you are required to prove your knowledge of base conversions to be placed on a specialized team. Perform the following base conversions and calculations to provide as samples to the Engineering Manager:
A. Convert 101001012 into its hexadecimal equivalent.
B. Convert 9F16 into its binary and decimal equivalent.
C. Convert 9816 into its binary and decimal equivalent.
D. Convert 23810 into its hexadecimal equivalent.

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Question 2: In an electrical circuit, three resistors are in series with each other. Their resistances are represented by the complex numbers shown below:

R1 = 2+4i ohms R2 = 1+3i ohms R3 = 3-5i ohms

Determine the effective resistance in the circuit.

Question 3: In an electrical system the two current values are represented by the following complex numbers: 3+4i Amps and 2-3i Amps find the following for each:
A. Modulus
B. Argument
C. Polar forms of the current
D. Exponential forms of the current
E. Conjugate
F. Solve 2-3i/3+4i
G. Add 3+4i to 2-3i in the polar form
H. Add 3+4i to 2-3i in the exponential form

Question 4: The complex numbers (1 + 2i)6 and (6 + 5i)3 represent the points at which eccentric masses must be placed on a system to provide dynamic balance. Use De Moivre's Theorem to convert the complex numbers into their polar and rectangular forms as requested to determine the exact distance from the origin required.
A. (1 + 2i)6 in the polar form (r,θ)
B. (6 + 5i)3 in rectangular form (a + bi)

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Question 5: While performing design calculations as the lead Mechanical Engineer for ABC Engineering Solutions Ltd, you are required to use the known Trigonometric identity: (cosθ + isinθ)2 = cos 2θ + isin 2θ. An Engineering trainee enquires about the correctness of this identity to which you indicate there is a procedure to prove that it is correct. Using De Moivre's Theorem test the correctness of the following trigonometric identity:
(cosθ + isinθ)2 = cos 2θ + isin 2θ

Question 6: Your analysis of a free body diagram indicates that there are three variables. Via a series of test, you are confident that the following 3 x 3 matrix can be used to assist in relating the three variables. To prove that the three variables are related, Find the determinant of the 3x3 matrix:

4 -7 6
[-2 4 0 ]
5 7 -4

Question 7: The following system of linear equations represent the relationship between quality (x) duration (y) and cost (z) variables for an engineering project. Using the Gaussian Elimination Method solve the system of equations below and determine the required quality, duration, and cost variables.
2 x + 4 y + 6 z = 4
x + 5 y + 9 z = 2
2 x + y + 3 z = 7

Question 8: The variables mass (x) and velocity (y) of a system in motion are related by the following pair of linear equations. Using the inverse Matrix method, solve the following set of linear equations:
3x +4y = 0
2x +5y +7 =0

Question 9: Using the software on matrixcalc validate the solutions to the system of linear equations from question (7):

2 x + 4 y + 6 z = 4
x + 5 y + 9 z = 2
2 x + y + 3 z = 7

 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. (1) P2 Solve engineering problems using complex number theory. (2) P3 Perform arithmetic operations using the polar and exponential form of complex numbers. (3) M1 Deduce solutions of problems using de Moivre's Theorem. (4) - D1 Test the correctness of a trigonometric identity using de Moivre's Theorem. (5) LO2 Solve systems of linear equations relevant to engineering applications using matrix methods P4 Ascertain the determinant of a given 3x3 matrix. (6) P5 Solve a system of three linear equations using Gaussian elimination. (7) M2 Determine solutions to a set of linear equations using the Inverse Matrix Method. (8) D2 Evaluate and validate all analytical matrix solutions using appropriate computer software. (9)

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