UFH Mathematics
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Item 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.Item Complex Analysis: MAT 322, Supplementary Examinations November 2024(University of Fort Hare, 2024-11) Prins, A.L.; Nkonkobe, S.This supplementary examination paper for MAT 322, "Complex Analysis," is a 3-hour assessment worth a maximum of 100 marks. Students are required to answer five questions. The paper covers various advanced topics in complex analysis. The first section of the exam consists of multiple-choice questions testing foundational concepts such as the image of a line under a linear transformation, limits of complex functions, differentiability of complex functions, convergence of sequences, and regions of convergence for complex series. It also includes finding values of complex numbers for which an exponential equation holds. The second section focuses on properties of complex functions, requiring students to prove whether a given function is entire, define elementary functions like the complex cosine function and prove its differentiability and derivative, and demonstrate that a specific function is harmonic, subsequently finding its harmonic conjugate. A portion of the exam delves into contour integration and Cauchy's theorems. Students are required to evaluate contour integrals using the Cauchy Integral Formula and the Cauchy-Goursat theorem, and apply the Cauchy Integral Formula for derivatives to compute specific integrals. Further sections cover Taylor and Laurent series expansions and singularity classification. This includes computing the Taylor series for a given function centered at a specific point and finding its interval of convergence. Students must also compute the Laurent series representation for a function in a given domain and classify the singularities of another function. The examination also includes a section on residue theory, requiring students to state the Cauchy Residue Theorem, find residues of functions at given poles, and use the theorem to evaluate contour integrals.Item Introduction to Analytical Methods: MAP 221, Supplementary Examinations November 2024(University of Fort Hare, 2024-11) Katywa, N.; Mahlasela, Z.This supplementary examination paper for MAP 221, "Introduction to Analytical Methods," is a 3-hour assessment worth 100 marks. The paper requires students to answer all questions, covering various topics in analytical mathematics. Question 1 focuses on differential equations, requiring students to define a differential equation, evaluate statements about their properties (like homogeneity, exactness, and linearity), and solve an initial value problem describing a real-world scenario. It also includes solving different types of second-order differential equations. Question 2 is dedicated to Laplace transforms. Students are asked to prove fundamental properties related to Laplace transforms and to find the Laplace transform of specific functions, including those defined piecewise. Question 3 covers Fourier series and special functions. It requires students to analyze a given trigonometric function, including sketching its graph and finding its corresponding Fourier cosine series to evaluate a specific infinite series. Additionally, students must demonstrate properties of the Gamma function and evaluate certain integrals involving Gamma and Beta functions. The examination aims to comprehensively assess students' analytical skills and their ability to apply various mathematical methods.Item Real Analysis: MAT 303, Supplementary Examinations November 2024(University of Fort Hare, 2024-11) Ngcibi, S.; Nkonkobe, S.This abstract summarizes the MAT 303 Supplementary Examination in Real Analysis, held in January 2024 at the University of Fort Hare. The exam is 3 hours long and carries a total of 100 marks. Dr. S. Ngcibi is the Internal Examiner, and Dr. S. Nkonkobe is the External Examiner. Candidates are instructed to answer all questions, and all symbols used retain their usual meaning. The examination covers fundamental concepts in Real Analysis, structured into three main questions: Question 1 focuses on set theory and topology, including definitions of countable and compact sets, the Cantor-Schröder-Bernstein Theorem, proofs related to injective and bijective functions, properties of open sets, and the Monotone Convergence Theorem for sequences. Question 2 delves into limits and continuity of functions. It requires stating and proving the Bolzano-Weierstrass Theorem for sequences, defining the limit of a function, proving limits using the definition, and defining and demonstrating uniform continuity. Question 3 addresses Riemann Integration. Topics include defining upper and lower sums of a function relative to a partition, defining a refinement of a partition, proving relationships between lower sums and refined partitions, defining upper and lower integrals, and proving properties of integrable functions, specifically the additivity of integrals.Item Mathematics MAT 123F, Supplementary Examinations November 2024(University of Fort Hare, 2024-11) Somniso, M.N.; Ndiweni, O.This examination paper, MAT 123 F (SUPP 24) Mathematics (F), is a supplementary exam from the University of Fort Hare, administered in November 2024. The exam is 3 hours long and consists of 3 pages. It is internally examined by Mr. M. N. Somniso and moderated by Dr. O. Ndiweni. The total marks for the paper are 100, and candidates are instructed to answer all questions, with symbols carrying their usual meanings. The paper covers several key areas of mathematics: * **Systems of Linear Equations:** Includes solving systems using the Gauss-Jordan transformation method and Cramer's rule. * **Complex Numbers:** Features operations with complex numbers (addition, subtraction, division using the conjugate method), proving the commutative property of addition, and converting complex numbers to polar form. * **Multivariable Calculus:** Covers finding partial derivatives from first principles and computing various partial derivatives ($f_x, f_{xx}, f_{xy}, f_y, f_{yy}$). * **Differential Equations:** Involves solving differential equations by separating variables, finding particular solutions to initial value problems (IVPs), and identifying and solving homogeneous differential equations.Item Linear Algebra: MAT 223, Supplementary Examinations November 2024(University of Fort Hare, 2024-11) Ngcibi, S.; Mahlasela, Z.Item A Theoretical Approach to Differential Calculus: MAT 121, Supplementary Examinations November 2024(University of Fort Hare, 2024-11) Maqekeni, S.; Ndiweni, O.Item A Practical Approach to Integral Calculus: MAT 123, Supplementary Examinations November 2024(University of Fort Hare, 2024-11) Mahlasela, Z.; Ndiweni, O.Item A Practical Approach to Differential Calculus: MAT 122, Supplementary Examinations November 2024(University of Fort Hare, 2024-11) Funani, S.; Prins, A.L.Item Mathematics: MAT 121F, Supplementary Examinations November 2024(University of Fort Hare, 2024-11) Somniso, M.N.; Ndiweni, O.Item Geometry: MAT 225, Supplementary Examinations November 2024(University of Fort Hare, 2024-11) Appiah, I.K.; Ndiweni, O.Item Real Analysis: MAT 224, Supplementary Examinations November 2024(University of Fort Hare, 2024-11) Ngcibi, S.; Nkonkobe, S.Item Real Analysis, MAT 224, Degree Examinations October/November 2024(University of Fort Hare, 2024-11) Prins, A.L.; Ndiweni, O.Item Mathematics: MAT 123F, Degree Examinations October/November 2024(University of Fort Hare, 2024-11) Somniso, M.N.; Ndiweni, O.Item A Theoretical Approach to Differential Calculus: MAT 121, Degree Examinations October/November 2024(University of Fort Hare, 2024-11) Maqekeni, S.; Ndiweni, O.Item Partial Differential Equations: MAP 321, Degree Examinations October/November 2024(University of Fort Hare, 2024-11) Mahlasela, Z.; Dukuza, K.Item Introduction to Analytical Methods: MAP 221, Degree Examinations October/November 2024(University of Fort Hare, 2024-10) Katywa, N.; Mahlasela, Z.Item A Practical Approach to Differential Calculus: MAT 122, Degree Examinations October/November 2024(University of Fort Hare, 2024-10) Funani, S.; Prins, A.L.Item History and Fundamental Concepts: MAT 304, Degree Examinations October/November 2024(University of Fort Hare, 2024-10) Ngcibi, S.; Nkonkobe, S.Item Complex Analysis: MAT 322, Degree Examinations October/November 2024(University of Fort Hare, 2024-10) Prins, A.L.; Nkonkobe, S.
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