Question

If \[ y(x)=\det\begin{pmatrix} \sin x & \cos x & \sin x+\cos x+1\\ 27 & 28 & 27\\ 1 & 1 & 1 \end{pmatrix} \] for \(x\in\mathbb{R}\), then \(\frac{d^2y}{dx^2}+y\) equals:

• A. \(-1\)
• B. \(0\)
• C. \(1\)
• D. \(2\)

Hint:

Always simplify determinants using row/column operations before differentiation.

Answer 1

The Correct Option is A

Concept: Column operations can simplify determinants before differentiation.

Step 1: Simplify determinant. \[ C_3 \rightarrow C_3 - C_1 - C_2 \] \[ y(x)=\det\begin{pmatrix} \sin x & \cos x & 1\\ 27 & 28 & 27\\ 1 & 1 & 1 \end{pmatrix} \]

Step 2: Expand along third column. \[ y(x)= 1\begin{vmatrix} 27 & 28\\ 1 & 1 \end{vmatrix} - 27\begin{vmatrix} \sin x & \cos x\\ 1 & 1 \end{vmatrix} \] \[ = (27-28) -27(\sin x-\cos x) \] \[ y(x)= -1 +27(\cos x-\sin x) \]

Step 3: Differentiate twice. \[ y'(x)=27(-\sin x-\cos x) \] \[ y''(x)=27(-\cos x+\sin x) \]

Step 4: Compute \(y''+y\). \[ y''+y = -1 \] \[ \boxed{-1} \]