Question

If the system of equations \(11x + y + \lambda z = -5,\) \(2x + 3y + 5z = 3,\) \(8x - 19y - 39z = \mu\) has infinitely many solutions, then \( \lambda^4 - \mu \) is equal to:

• A. 49
• B. 45
• C. 47
• D. 51

Answer 1

The Correct Option is C

For the system to have infinitely many solutions, the determinant of the coefficient matrix must be zero. Set up the determinant \( D \) as follows:

\[ D = \begin{vmatrix} 11 & 1 & \lambda \\ 2 & 3 & 5 \\ 8 & -19 & -39 \end{vmatrix} = 0 \]

Expanding the determinant:

\[ = 11(3 \cdot (-39) - 5 \cdot (-19)) - 1(2 \cdot (-39) - 5 \cdot 8) + \lambda(2 \cdot (-19) - 3 \cdot 8) \]

\[ = 11(-117 + 95) - 1(-78 - 40) + \lambda(-38 - 24) \]

\[ = 11(-22) + 118 + \lambda(-62) = 0 \]

Solving for \( \lambda \):

\[ -242 + 118 - 62\lambda = 0 \]

\[ 62\lambda = -124 \implies \lambda = -2 \]

Now substitute \( \lambda = -2 \) and calculate \( \mu \) by setting up the augmented determinant \( D_1 \):

\[ D_1 = \begin{vmatrix} -5 & 1 & -2 \\ 3 & 3 & 5 \\ \mu & -19 & -39 \end{vmatrix} = 0 \]

Expanding \( D_1 \):

\[ = -5(3 \cdot (-39) - 5 \cdot (-19)) - 1(3 \cdot (-39) - 5\mu) - 2(3 \cdot (-19) + 3\mu) \]

\[ = -5(-117 + 95) + (-117 + 5\mu) + 6\mu = 0 \]

\[ -341 + 6\mu = 0 \implies \mu = -31 \]

Finally, calculate \( \lambda^4 - \mu \): \[ \lambda^4 - \mu = (-2)^4 - (-31) = 16 + 31 = 47 \]