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Homogeneous System of Linear Equations - Practice Questions & MCQ
Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main
Quick Facts
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17 Questions around this concept.
Solve by difficulty
Let $\lambda$ and $\alpha$ be real. Find the set of all values $\lambda$ for which the system of linear equations
$
\begin{aligned}
& \lambda x+\sin \alpha \cdot \mathrm{y}+\cos \alpha \cdot \mathrm{z}=0 \\
& \mathrm{x}+\cos \alpha \cdot \mathrm{y}+\sin \alpha \cdot \mathrm{z}=0 \\
& -\mathrm{x}+\sin \alpha \cdot \mathrm{y}-\cos \alpha \cdot \mathrm{z}=0
\end{aligned}
$
Has a non-trivial solution. For $\lambda=1$, The possible values of $\alpha$ are
An ordered pair $(\alpha, \beta)$ for which the system of linear equations
$
\begin{aligned}
& (1+\alpha) x+\beta y+z=2 \\
& \alpha x+(1+\beta) y+z=3 \\
& \alpha x+\beta y+2 z=2
\end{aligned}
$
has a unique solution, is :
Concepts Covered - 1
System of Homogeneous linear equations
Homogeneous Equation
A linear equation with constant value as zero is called a homogeneous equation.
Note that x = y = z = 0 will always satisfy this system of equations. So system of homogeneous equations will always have at least one solution.
Also the solution x = 0, y = 0, z = 0 is called trivial solution and other solutions are called non-trivial solutions.
- If ? ≠ 0, then x= 0, y = 0, z = 0 is the only solution of the above system. This solution is also known as a trivial solution.
- If ? = 0, at least one of x, y and z are non-zero. In this case we will have non-trivial solutions as well. Also there would be infinite solutions of such a system of equations.
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