NCERT Class 12 · Ch 4 Miscellaneous Exercise · Q1

NCERT Class 12 Maths · Chapter 4 Determinants · Miscellaneous Exercise Question 1

Prove that the determinant \(\left[\begin{matrix}x & \sin{\left(\theta \right)} & \cos{\left(\theta \right)}\\- \sin{\left(\theta \right)} & - x & 1\\\cos{\left(\theta \right)} & 1 & x\end{matrix}\right]\) is independent of θ.

💡 Concept Hints

Identify whether this needs determinant expansion, an inverse, or the absolute determinant formula for area.
For a determinant use signed cofactors. For a system first check invertibility; for an area take half the absolute determinant.

✅ Step-by-Step Solution

Final Answer: \(- x^{3}\)

Expand along the first row: \(- x^{3} + x \sin^{2}{\left(\theta \right)} + x \cos^{2}{\left(\theta \right)} - x\).
The mixed sin θ cos θ terms cancel; combine sin²θ+cos²θ=1.
The result is \(- x^{3}\), which contains no θ.

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NCERT Class 12 Maths · Chapter 4 Determinants · Miscellaneous Exercise Question 1

💡 Concept Hints

✅ Step-by-Step Solution

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