QCF157 Advanced Mathematics 1 Assignment

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QCF157 Advanced Mathematics 1

Assignment

Directions:              - Show all working and setting out.

(Total Marks: 60)

QUESTION 1

This semester you have learnt how to solve simultaneous equations algebraically. Another method of solving these equations is using matrices and elementary row operations on an augmented  matrix to  reduce  it  to  row-echelon  form.  This  method  is  called  Gaussian Elimination.

(i)      Explain in detail how this process works providing definitions of any new terms used such as augmented matrix, elementary row operations, row-echelon form etc. for both two-variable and three-variable equations.

(ii)     Give an example of solving a two-variable and a three-variable set of simultaneous equations using an augmented matrix with a detailed explanation.

(iii)    Are there any special cases? Give an explanation and an example of any special cases. (20 marks)

QUESTION 2

Solve the following:

(i)      By reducing the augmented matrix to row-echelon form and also by (ii)    Using Algebra

(a)

3x − 4y = −2

−3x = −3 + y

(b)

u + 3v − w  = 8

3u + v − w  = 2

u + 2v − 2w = 9

(20 marks)

QUESTION 3

Mr Boss will be twice as old as his son in 8 years time. If Mr Boss was three times as old as his son 10 years ago, find their ages today using matrices and elementary row operations on an augmented matrix to reduce it to row-echelon form. (10 marks)

QUESTION 4

Salespeople at a car dealer are paid 15% commission on their sales. For a certain model of car, the car dealer buys the base model at a cost price of $25 000, the sports model at $30 000 and luxury model at $37 500.

During a week, Kate sells 2 base models, 1 sports model and 1 luxury model. Ben sells 1 base model, 3 sports models and 1 luxury model. Simon sells 2 base models and 2 luxury models.

For the week, Kate’s commission was $22 912.50, Ben’s was $29737.50 and Simon’s was $24 375.00.

Find the sale price of each model of car using elementary row operations on an augmented matrix to reduce it to row-echelon form. Also, what is the percentage mark-up added to the cost price of each model of car. (10 marks)


2023-09-22