Question

If the matrix \[ \begin{bmatrix} 1 & 3 & 1 \\ 2 & 1 & \alpha \\ 0 & 1 & -1 \end{bmatrix} \] is singular. Given a function \( f(x) = \int_{0}^{x} (t^2 + 2t + 3)\, dt \), \( \forall x \in [1, \alpha] \). If \( m \) and \( n \) are the maximum and minimum values of the function \( f(x) \), then the value of \( 3(m - n) \) is:

• A. 550
• B. 510
• C. 490
• D. 540

Hint:

If the derivative of an integral function \(f'(x) = g(x)\) is always positive, you don't need to evaluate the integral twice. The difference \(m-n\) is simply the definite integral over the given interval.

Answer 1

The Correct Option is D