Section A (20 Marks)
Write short notes on any four of the following
1. Quantitative decisions
2. Linear equations
3. Use of Matrices for Production Planning
4. Objective and subjective probability
5. Binomial distribution
Section B (30 marks)
(Attempt any three)
1. Discuss the various types of functions.
2. Explain the supply and demand functions.
3. Elaborate solution of linear equations using determinants.
4. Write short notes on the following:
a. Random Variables
b. Poisson distribution.
Section C (50 marks)
(Attempt all questions. Every question carries 10 marks)
Read the case “Solving Systems of Linear Equations.” and answer the following questions:
Case Study: Solving Systems of Linear Equations
We will motivate our study of linear algebra by considering the problem of solving several linear equations simultaneously. The word “solve” tends to get abused somewhat, as in “solve this problem.” When talking about equations we understand a more precise meaning: find all of the values of some variable quantities that make an equation, or several equations, simultaneously true.
Our first example is of a type we will not pursue further. While it has two equations, the first is not linear. So this is a good example to come back to later, especially after you have seen.
Solving two (nonlinear) equations
In order to discuss systems of linear equations carefully, we need a precise definition. And before we do that, we will introduce our periodic discussions about “Proof Techniques.” Linear algebra is an excellent setting for learning how to read, understand and formulate proofs. But this is a difficult step in your development as a mathematician, so we have included a series of short essays containing advice and explanations to help you along. These will be referenced in the text as needed, and are also collected as a list you can consult when you want to return to re-read them.
With a definition next, now is the time for the first of our proof techniques. So study Proof Technique D. We'll be right here when you get back. See you in a bit.
Definition SLE: System of Linear Equations
A system of linear equations is a collection of m equations in the variable quantities x1, x2, x3, …, xn of the form,
Where, the values of aij, bi and xj, 1≤i≤m, 1≤j≤n, are from the set of complex numbers, C.
Don't let the mention of the complex numbers, C, rattle you. We will stick with real numbers exclusively for many more sections, and it will sometimes seem like we only work with integers! However, we want to leave the possibility of complex numbers open, and there will be occasions in subsequent sections where they are necessary. You can review the basic properties of complex numbers in Section CNO, but these facts will not be critical until we reach Section O.
Now we make the notion of a solution to a linear system precise.
Solution of a System of Linear Equations
A solution of a system of linear equations in n variables, x1, x2, x3, …, xn, is an ordered list of n complex numbers, s1, s2, s3, …, sn such that if we substitute s1 for x1, s2 for x2, s3 for x3, …, sn for xn, then for every equation of the system the left side will equal the right side, i.e. each equation is true simultaneously.
More typically, we will write a solution in a form like x1=12, x2=−7, x3=2 to mean that s1=12, s2=−7, s3=2 in the notation of Definition SSLE. To discuss all of the possible solutions to a system of linear equations, we now define the set of all solutions. (So Section SET is now applicable, and you may want to go and familiarize yourself with what is there.)
Definition SSSLE: Solution Set of a System of Linear Equations
The solution set of a linear system of equations is the set which contains every solution to the system, and nothing more.
Be aware that a solution set can be infinite, or there can be no solutions, in which case we write the solution set as the empty set, ∅={}.
Questions:
1. Explain how system of equations are solved.
2. Discuss how two non-linear equations are solved.
3. Throw light on how solution is derived of linear system of equations.
4. Give example of non-linear system of equations.
5. Give an example of linear system of equations.