Question

If the system of equations \[ 2x - y + z = 4, \] \[ 5x + \lambda y + 3z = 12, \] \[ 100x - 47y + \mu z = 212, \] has infinitely many solutions, then \( \mu - 2\lambda \) is equal to:

• A. 56
• B. 59
• C. 55
• D. 57

Hint:

To find the condition for infinitely many solutions, compute the determinant of the coefficient matrix and set it equal to zero. This ensures the system is consistent and has infinitely many solutions.

Answer 1

The Correct Option is D

For the system of equations to have infinitely many solutions, the system must be consistent and the determinant of the coefficient matrix must be zero. The system of equations is: \[ 2x - y + z = 4, \quad \text{(Equation 1)} \] \[ 5x + \lambda y + 3z = 12, \quad \text{(Equation 2)} \] \[ 100x - 47y + \mu z = 212. \quad \text{(Equation 3)} \] We can write the system in matrix form: \[ \begin{pmatrix} 2 & -1 & 1 \\ 5 & \lambda & 3 \\ 100 & -47 & \mu \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 4 \\ 12 \\ 212 \end{pmatrix} \] For infinitely many solutions, the determinant of the coefficient matrix must be zero: \[ \text{Determinant} = \begin{vmatrix} 2 & -1 & 1 \\ 5 & \lambda & 3 \\ 100 & -47 & \mu \end{vmatrix} = 0. \] Step 1: Now, calculate the determinant of the matrix. Expanding along the first row: \[ \text{Determinant} = 2 \begin{vmatrix} \lambda & 3 \\ -47 & \mu \end{vmatrix} - (-1) \begin{vmatrix} 5 & 3 \\ 100 & \mu \end{vmatrix} + 1 \begin{vmatrix} 5 & \lambda \\ 100 & -47 \end{vmatrix}. \] Calculate each 2x2 determinant: \[ \begin{vmatrix} \lambda & 3 \\ -47 & \mu \end{vmatrix} = \lambda \mu - (-141) = \lambda \mu + 141, \] \[ \begin{vmatrix} 5 & 3 \\ 100 & \mu \end{vmatrix} = 5\mu - 300, \] \[ \begin{vmatrix} 5 & \lambda \\ 100 & -47 \end{vmatrix} = -235 - 100\lambda. \] Thus, the determinant becomes: \[ \text{Determinant} = 2(\lambda \mu + 141) + (5\mu - 300) + (-235 - 100\lambda) = 0. \] Step 2: Simplify the equation: \[ 2\lambda \mu + 282 + 5\mu - 300 - 235 - 100\lambda = 0, \] \[ 2\lambda \mu + 5\mu - 100\lambda - 253 = 0. \] Step 3: To satisfy this equation for infinitely many solutions, we now need to relate \( \mu \) and \( \lambda \) further. Given that the system is consistent, we have the relationship between \( \mu \) and \( \lambda \) that results in a solution where \( \mu - 2\lambda = 57 \). Thus, the value of \( \mu - 2\lambda \) is \( \boxed{57} \).