Question

Let S be the set of values of λ, for which the system of equations  
6λx – 3y + 3z = 4λ2 ,  
2x + 6ly + 4z = 1, 
3x + 2y + 3λz = l has no solution. Then  12\( ∑ _{λ ∈ S}\) | λ | 12 ∑ λ ∈ | λ |  is equal to ________ .

Answer 1

Correct Answer: 24

Step 1: Form the augmented matrix of the system 
The given system of equations can be written as the augmented matrix: \[ \Delta = \begin{vmatrix} 6\lambda & -3 & 3 2 & 6\lambda & 4 3 & 2 & 3\lambda \end{vmatrix} \] For the system to have no solution, the determinant of the coefficient matrix must be zero. 
Step 2: Calculate the determinant We calculate the determinant \( \Delta \): \[ \Delta = 6\lambda \begin{vmatrix} 6\lambda & 4 2 & 3\lambda \end{vmatrix} - (-3) \begin{vmatrix} 2 & 4 3 & 3\lambda \end{vmatrix} + 3 \begin{vmatrix} 2 & 6\lambda 3 & 2 \end{vmatrix} \] Simplifying the 2x2 determinants: \[ \Delta = 6\lambda \left( (6\lambda)(3\lambda) - (4)(2) \right) + 3 \left( (2)(3\lambda) - (4)(3) \right) + 3 \left( (2)(2) - (6\lambda)(3) \right) \] \[ \Delta = 6\lambda \left( 18\lambda^2 - 8 \right) + 3 \left( 6\lambda - 12 \right) + 3 \left( 4 - 18\lambda \right) \] Expanding the terms: \[ \Delta = 6\lambda (18\lambda^2 - 8) + 3(6\lambda - 12) + 3(4 - 18\lambda) \] \[ \Delta = 108\lambda^3 - 48\lambda + 18\lambda - 36 + 12 - 54\lambda \] \[ \Delta = 108\lambda^3 - 84\lambda - 24 \] 
Step 3: Solve for \( \lambda \) 
For the system to have no solution, the determinant must be zero: \[ 108\lambda^3 - 84\lambda - 24 = 0 \] Divide the entire equation by 12: \[ 9\lambda^3 - 7\lambda - 2 = 0 \] 
Step 4: Find the roots of the cubic equation Solving the cubic equation \( 9\lambda^3 - 7\lambda - 2 = 0 \) by using trial and error or factoring, we find the roots: \[ \lambda = 1, -\frac{1}{3}, \frac{2}{3} \] 
Step 5: Calculate \( 12 \sum_{\lambda \in S} |\lambda| \) 
Now, the values of \( \lambda \) are \( 1, -\frac{1}{3}, \frac{2}{3} \). The sum of their absolute values is: \[ |1| + \left| -\frac{1}{3} \right| + \left| \frac{2}{3} \right| = 1 + \frac{1}{3} + \frac{2}{3} = 2 \] Thus, \[ 12 \sum_{\lambda \in S} |\lambda| = 12 \times 2 = 24 \] Thus, the required value is \( \boxed{24} \).