JEE Mains · Maths · STD 12 - 3 and 4 . metrices and determinant

Consider the system of linear equation \(x+y+z=\) \(4 \mu, x+2 y+2 \lambda z=10 \mu, x+3 y+4 \lambda^2 z=\mu^2+15\), where \(\lambda, \mu \in R\). Which one of the following statements is \(NOT\) correct?

A The system has unique solution if \(\lambda \neq \frac{1}{2}\) and \(\mu \neq 1,15\)
B The system is inconsistent if \(\lambda=\frac{1}{2}\) and \(\mu \neq 1\)
C The system has infinite number of solutions if \(\lambda=\frac{1}{2}\) and \(\mu=15\)
D The system is consistent if \(\lambda \neq \frac{1}{2}\)

Verified Solution

Answer & Solution

Correct Answer

(B) The system is inconsistent if \(\lambda=\frac{1}{2}\) and \(\mu \neq 1\)

Step-by-step Solution

Detailed explanation

\( x+y+z=4 \mu, x+2 y+2 \lambda z=10 \mu, x+3 y+4 \lambda \) \( { }^2 z=\mu^2+15\) \(\Delta=\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & 2 & 2 \lambda \\ 1 & 3 & 4 \lambda^2\end{array}\right|=(2 \lambda-1)^2\) For unique solution…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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