1429 Past Paper Autumn 2024 Solution

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This comprehensive solution guide provides step-by-step answers to all eight questions of the AIOU Business Mathematics 1429 past paper for Autumn 2024. Each problem is solved in detail, covering inequalities, systems of equations, matrix operations, probability distributions, limits, concavity, matrix inversion, optimization, and profit maximization. Designed for BS, B.Ed., and Associate Degree students, these solutions will help you understand key concepts and prepare effectively for exams.

Course Content

1429 Past Paper Autumn 2024 Question 1: Solving Absolute Value Inequalities
In this session, we tackle two absolute value inequalities. The first, |x² – 2| ≥ 2, is solved by considering the two cases that arise from the definition of absolute value: the expression inside is either ≥ 2 or ≤ –2. We find the solution set in interval notation. The second inequality, |6t – 15| ≤ –6, appears impossible because an absolute value can never be less than or equal to a negative number; we conclude that there is no solution. Step‑by‑step explanations help you understand the logic behind each case. This lesson is part of the Business Mathematics (1429) course, designed for BS, B.Ed., and Associate Degree students preparing for their Autumn 2024 exams.

  • Question 1: Solving Absolute Value Inequalities
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1429 Past Paper Autumn 2024 Question 2: Solving System of Linear Equations
In this lesson, we solve a system of three linear equations with three variables using the elimination method. By strategically adding equations to eliminate variables step by step, we find the unique solution (x₁, x₂, x₃). This problem demonstrates how to handle systems where some equations already have missing variables, making the elimination process more efficient. The solution is verified by substituting back into the original equations. This topic is essential for mastering linear algebra concepts in Business Mathematics.

1429 Code Past Paper Autumn 2024 Question 3: Finding Unknown Matrix Elements
In this lesson, we solve a matrix equation to find unknown elements a, b, c, and d. Given two matrices multiplied together equal a resulting matrix, we perform matrix multiplication and set up a system of linear equations. By solving these equations systematically, we determine the values of all unknown entries. This problem reinforces the fundamentals of matrix multiplication and solving linear systems, which are essential skills in business mathematics.

1429 Code Past Paper Autumn 2024 Question 4: Probability Distribution and Limits
This lesson covers two important mathematical concepts. First, we construct a discrete probability distribution for the sum of dots when rolling a pair of fair dice. We identify all possible outcomes, count the number of ways each sum can occur, and calculate the corresponding probabilities. Second, we evaluate a limit by direct substitution. Both topics are fundamental in business mathematics for understanding probability theory and calculus concepts.

1429 Code Past Paper Autumn 2024 Question 5: Concavity and Points of Inflection
This question focuses on analyzing the concavity of a polynomial function and identifying its points of inflection. In part (a), we find the open intervals where the function is concave upward and concave downward by examining the second derivative. In part (b), we determine all points of inflection where the concavity changes. These concepts are fundamental in calculus for understanding the shape and behavior of graphs, which has applications in optimization and economic analysis.

1429 Code Past Paper Autumn 2024 Question 6: Finding Inverse of Matrix Using Cofactors
In this lesson, we learn how to find the inverse of a 3×3 matrix using the cofactor method. The process involves calculating the determinant, finding all cofactors, forming the cofactor matrix, taking its transpose to obtain the adjugate, and finally multiplying by the reciprocal of the determinant. This systematic approach is essential for solving systems of linear equations and has numerous applications in business and economics.

1429 Code Past Paper Autumn 2024 Question 7: Optimization – Maximizing Volume of an Open Box
In this lesson, we solve a classic optimization problem from calculus. We have a 12-inch by 12-inch tin sheet, and we cut congruent squares from each corner and bend up the sides to form an open-top box. We need to find the size of the squares to cut that will maximize the volume of the box. This problem demonstrates how to apply derivative techniques to real-world optimization scenarios in business and manufacturing.

1429 Code Past Paper Autumn 2024 Question 8: Verifying Maximum Profit at Given Point
In this lesson, we verify that maximum profit occurs at a given point for a multivariable profit function. Using partial derivatives and the second derivative test, we confirm that the point (20, 10) is indeed a critical point and that it yields a maximum profit. This problem demonstrates the application of calculus to business optimization, specifically in profit maximization with two decision variables.

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