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

Consider the system of linear equations in \(x, y, z\):
\(x + 2y + tz = 0\),
\(6x + y + 5tz = 0\),
\(3x + t^2 y + f(t) z = 0\),
where \(f: \mathbb{R} \rightarrow \mathbb{R}\) is a differentiable function. If this system has infinitely many solutions for all \(t \in \mathbb{R}\), then \(f\)

A is a constant function
B is strictly increasing on \(\mathbb{R}\)
C is strictly decreasing on \(\mathbb{R}\)
D has two critical points

Verified Solution

Answer & Solution

Correct Answer

(B) is strictly increasing on \(\mathbb{R}\)

Step-by-step Solution

Detailed explanation

For a homogeneous system of linear equations to have infinitely many solutions, the determinant of its coefficient matrix must be zero. The coefficient matrix is: \(\Delta = \begin{vmatrix} 1 & 2 & t \\ 6 & 1 & 5t \\ 3 & t^2 & f(t) \end{vmatrix}\) Expanding the determinant along…

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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Consider the system of linear equations in \(x, y, z\):\(x + 2y + tz = 0\),\(6x + y + 5tz = 0\),\(3x + t^2 y + f(t) z = 0\),where \(f: \mathbb{R} \rightarrow \mathbb{R}\) is a differentiable function. If this system has infinitely many solutions for all \(t \in \mathbb{R}\), then \(f\)

Step-by-step Solution

Consider the system of linear equations in \(x, y, z\):
\(x + 2y + tz = 0\),
\(6x + y + 5tz = 0\),
\(3x + t^2 y + f(t) z = 0\),
where \(f: \mathbb{R} \rightarrow \mathbb{R}\) is a differentiable function. If this system has infinitely many solutions for all \(t \in \mathbb{R}\), then \(f\)