Previous Year Questions on System of Linear Equations

System of Linear Equations - Previous Year Questions

Previous Year Questions on System of Linear Equations

Previous year questions on System of Linear Equations provide a pathway to understanding their evolution and applications in mathematics and engineering. Originating from the works of ancient mathematicians, the study of systems of linear equations was formalized in the field of Linear Algebra Notes. This topic is widely used in optimization, network theory, and computational solutions to real-world problems.

Vidyasagar University

2023-24 (CBCS)

For what values of \(k\), the system of equations has a non trivial solution? Also, solve the system for any one value of \(k\): \begin{align*} & x+2y+3z=kx \\ & 2x+y+3z=ky \\ & 2x+3y+z=kz \end{align*}. [6]

2021-22 (CBCS)

For what values of \(\lambda\), for which the system of equations: \begin{align*} & 2x_{1}-x_{2}+x_{3}+x_{4}=1 \\ & x_{1}+2x_{2}-x_{3}+4x_{4}=2 \\ & x_{1}+7x_{2}-4x_{3}+11x_{4}=\lambda \end{align*} is solvable. [4]
Determine the conditions for wich the system of equations \begin{align*} & x+2y+z=1 \\ & 2x+3y+3z=b \\ & x+ay+3z=b+1 \end{align*} has
1. a unique solution
2. many solutiona
3. no solution
. [4]

2020-21 (CBCS)

For what values of \(a\) and \(b\) the following system of equations has
1. a unique solution
2. no solution
3. infinite number of solutions
over the field of rational numbers \begin{align*} & x_{1}+4x_{2}+2x_{3}=1 \\ & 2x_{1}+7x_{2}+5x_{3}=2b \\ & 4x_{1}+ax_{2}+10x_{3}=2b+1 \end{align*} . [8]

2019-20 (CBCS)

For what values of \(k\), the planes : \begin{align*} & x-4y+5z=k \\ & x-y+2z=3 \\ & 2x+y+z=0 \end{align*} intersect in a line? [2]
Solve the system of equations : \begin{align*} & x_{2}+x_{3}=a \\ & x_{1}+x_{3}=b \\ & x_{1}+x_{2}=c \end{align*} and use the solution to find the inverse of the matrix \(A=\begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix} \) . [5]
Determine the conditions for which the system : \begin{align*} & x+y+z=b \\ & 2x+2y+3z=b+1 \\ & 5x+7y+az=b^{2} \end{align*} has
1. only one solution
2. no solution
3. many solutions
. [5]

2018-19 (CBCS)

Show that the planes : \begin{align*} & 2x-y+z=5 \\ & x+2y+4z=7 \\ & 5x+3y-z=0 \end{align*} are concucrrent. [2]
Investigate for what values of \(\alpha\) and \(\mu\), the following rquations : \begin{align*} & x+y+z=6 \\ & x+2y+3z=10 \\ & x+2y+\lambda z=\mu \end{align*} have
1. no solution
2. a unique solution
3. an infinite number of solutions.
. [5]
For what values of \(k\), such that the following system of linear equation is consistent : \begin{align*} & 2x+y-z=12 \\ & x-y+2z=-3 \\ & 3y+3z=k \end{align*}. [2]

2017-18 (CBCS)

Solve the system of equations
\begin{align*} & x+2y-z-3w=1 \\ & 2x+4y+3z+w=3 \\ & 3x+6y+4z-2w=5 \end{align*} If possible. [2]
For what values of \(k\), the system of equations : \begin{align*} & x+2y+3z=kx \\ & 2x+y+3z=ky \\ & 2x+3y+z=kz \end{align*} has a non trivial solution. [2]
Determine the conditions for which the system : \begin{align*} & x+y+z=1 \\ & x+2y-z=b \\ & 5x+7y+az=b^{2} \end{align*} admits of
1. only one solution
2. no solution
3. many solutions
. [5]
For what values of \(k\), the planes : \begin{align*} & x-4y+5z=k \\ & x-y+2z=3 \\ & 2x+y+z=0 \end{align*} intersect in a line. [3]

FAQs

What is a system of linear equations?
It is a collection of two or more linear equations involving the same set of variables.
How are systems of linear equations solved?
They can be solved using substitution, elimination, or matrix methods like Gaussian elimination.
What is Gaussian elimination?
It is a systematic method of solving systems of equations by transforming the matrix into row-echelon form.
What is the role of determinants in solving linear equations?
Determinants help determine the uniqueness of solutions and are used in Cramer’s rule.
What is the difference between consistent and inconsistent systems?
A consistent system has at least one solution, while an inconsistent system has none.
How are matrix inverses used to solve linear systems?
If the coefficient matrix is invertible, the system can be solved using the formula Ax = b, where x = A^-1b.
What is a homogeneous system of equations?
It is a system where all constant terms are zero, often having either a trivial or infinite number of solutions.
How does rank relate to solving systems of linear equations?
The rank of the coefficient matrix determines the number of independent equations and the nature of the solutions.
Can nonlinear systems be solved using linear methods?
No, linear methods are specifically for linear systems, but some nonlinear systems can be approximated as linear near specific points.
What are real-world applications of systems of linear equations?
They are used in engineering, economics, computer science, and operations research to model and solve practical problems.