Browsing by Author "Mtakazi, S."

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    Computational Methods: MNU 122, Supplementary Examinations November 2024
    (University of Fort Hare, 2024-11) Mtakazi, S.; Katywa, N.
    This supplementary examination paper for MNU 122, "Computational Methods," is a 3-hour assessment worth 100 marks. Students are required to answer all questions, which cover various topics in linear algebra, numerical methods, and mathematical modeling. Question 1 (18 marks) focuses on linear equations and matrix properties. It requires solving a system of linear equations using the Gaussian Elimination Method and proving a property related to matrix multiplication and addition for given matrices. Question 2 (35 marks) delves into linearizing non-linear equations and applying the least squares method. It asks students to express various non-linear equations in linear form. Subsequently, it presents a practical problem involving torque and diameter, where students must determine the best values for constants in a power law relationship using the least squares method from given data. Another part of this question involves finding the best values for constants in a reciprocal relationship from a given set of data. Question 3 (35 marks) covers differential equations and their applications. It defines a differential equation and asks students to determine the truthfulness of statements regarding the homogeneity, exactness, and linearity of given differential equations, correcting false statements. It then presents a real-world scenario involving the growth of supermarkets using a computerized checkout system, described by an initial value problem. Students must find the number of supermarkets at any time t and compute the number of supermarkets at a specific time (t=10). The question concludes with solving various types of differential equations (homogeneous, non-homogeneous, and Cauchy-Euler equations). Question 4 (12 marks) focuses on Laplace transforms. It requires showing properties of Laplace transforms, specifically for scaling and exponential functions, and finding the Laplace transform of a trigonometric function. It also involves working with a piecewise function and finding its Laplace transform. The examination aims to comprehensively assess students' theoretical understanding and practical application of computational methods in solving mathematical and real-world problems.