Comprehension
Let α, β and γ be real numbers such that the system of linear equations
\(x + 2y + 3z = α\)
\(4x + 5y + 6z = β\)
\(7x + 8y + 9z = γ – 1 \)
is consistent. Let\( |M|\) represent the determinant of the matrix.
\(M = \begin{bmatrix} \alpha & 2 & \gamma &\\ \beta & 1 & 0 & \\-1& 0&1 \end{bmatrix}\)
Let P be the plane containing all those \((α, β, γ)\) for which the above system of linear equations is consistent, and D be the square of the distance of the point (0, 1, 0) from the plane P.
\(x + 2y + 3z = α\)
\(4x + 5y + 6z = β\)
\(7x + 8y + 9z = γ – 1 \)
is consistent. Let\( |M|\) represent the determinant of the matrix.
\(M = \begin{bmatrix} \alpha & 2 & \gamma &\\ \beta & 1 & 0 & \\-1& 0&1 \end{bmatrix}\)
Let P be the plane containing all those \((α, β, γ)\) for which the above system of linear equations is consistent, and D be the square of the distance of the point (0, 1, 0) from the plane P.
Question: 1
The value of \(|M|\) is _____ ?
The value of \(|M|\) is _____ ?
Updated On: May 22, 2024
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Correct Answer: 1
Solution and Explanation
The provided system of equations remains consistent when \(α=β=γ−1=0\), indicating that the equations form a homogeneous system.
Therefore, we conclude that \(α=0,β=0\), and \(γ=1.\)
\(M = \begin{vmatrix} 0 & 2 & 1 \\ 0 & 1 & 0 \\ -1 & 0 & 1 \end{vmatrix}\)
\(= -1×(-1)\) \(= 1\)
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Question: 2
The value of 𝐷 is ___ .
The value of 𝐷 is ___ .
Updated On: May 22, 2024
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Correct Answer: 1.5
Solution and Explanation
Answer is 1.50
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