Q.The maximum value of $\Delta = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 1 + \sin\theta & 1 \\ 1 + \cos\theta & 1 & 1 \end{vmatrix}$ is ($\theta$ is real number)
(A) $\dfrac{1}{2}$
(B) $\dfrac{\sqrt{3}}{2}$
(C) $\sqrt{2}$
(D) $\dfrac{2\sqrt{3}}{4}$

Concept: Maximum Value of a Sine-Cosine Expression

We simplify the determinant first.

Step 1 – Row operations

$R_2 \to R_2 - R_1$, $R_3 \to R_3 - R_1$:

Step 2 – Expand along $R_3$

Only the element $\cos\theta$ at position $(3,1)$ contributes:

$\Delta = \cos\theta \cdot (-1)^{3+1} \cdot \begin{vmatrix} 1 & 1 \\ \sin\theta & 0 \end{vmatrix} = \cos\theta \cdot (0 - \sin\theta) = -\sin\theta \cos\theta$ …

Apply the row operations $R_2 \to R_2 - R_1$ and $R_3 \to R_3 - R_1$ (these do not change the value of the determinant):

Expand along the third row, which has two zeros:

$\Delta = \cos\theta\,(+1)\begin{vmatrix} 1 & 1 \\\sin\theta & 0 \end{vmatrix} = \cos\theta\,(0 - \sin\theta) = -\sin\theta\cos\theta.$ …

Method: Simplifying a Determinant to Bound Its Maximum or Minimum Value

To find the extreme value of a determinant whose entries involve $\sin\theta$/$\cos\theta$, first reduce it via row operations to a short trigonometric expression, then apply the standard bound on sine/cosine.

Steps

Step 1: Eliminate the constant entries with row operations

Subtract one row from the others (e.g. $R_2 \to R_2 - R_1$, $R_3 \to R_3 - R_1$) to clear away the entries that don't carry the variable, isolating $\sin\theta$ and $\cos\theta$ in specific positions.

Step 2: Expand along the row/column with the most zeros

This produces a short expression, typically a product of $\sin\theta$ and $\cos\theta$ — write out the cofactor sign $(-1)^{i+j}$ explicitly for whichever entry you expand along.

Step 3: Rewrite the product with a double-angle identity

$\sin\theta\cos\theta = \tfrac12 \sin 2\theta$ …

Common Mistakes

Mistake 1: Forgetting the cofactor sign $(-1)^{i+j}$ when expanding along the third row

Why it's wrong: Expanding along $R_3$ at position $(3,1)$ requires the factor $(-1)^{3+1}=+1$; getting this sign wrong flips the overall sign of the resulting expression for $\Delta$. Correct approach: always write out the cofactor sign explicitly for the exact row/column and position being expanded before multiplying.

Mistake 2: Maximizing $\sin\theta\cos\theta$ instead of the actual signed expression $-\sin\theta\cos\theta$ …