Class CBSE Class 12 Mathematics Matrices and Determinants Q #1842
COMPETENCY BASED
APPLY
1 Marks 2026 AISSCE(Board Exam) MCQ SINGLE
One of the values of $x$ for which $\begin{vmatrix} \cos x & \sin x \\ -\cos x & \sin x \end{vmatrix} = 1$ is
(A) $0$
(B) $\frac{\pi}{4}$
(C) $\frac{\pi}{3}$
(D) $\frac{\pi}{2}$
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Correct Answer: B

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Detailed Solution

Step 1: Evaluate the determinant

The determinant of a $2 \times 2$ matrix $\begin{vmatrix} a & b \\ c & d \end{vmatrix}$ is calculated as $ad - bc$. Applying this to the given matrix:

$$(\cos x)(\sin x) - (-\cos x)(\sin x) = 1$$

Step 2: Simplify the expression

Simplify the equation by combining the terms:

$$\cos x \sin x + \cos x \sin x = 1$$ $$2 \sin x \cos x = 1$$

Step 3: Use trigonometric identity

Recall the double angle identity $\sin(2x) = 2 \sin x \cos x$. Substitute this into the equation:

$$\sin(2x) = 1$$

Step 4: Solve for x

We know that $\sin(\theta) = 1$ when $\theta = \frac{\pi}{2}$. Therefore:

$$2x = \frac{\pi}{2}$$ $$x = \frac{\pi}{4}$$

Final Answer: $\frac{\pi}{4}$

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must apply the definition of a determinant and trigonometric identities to solve for an unknown variable.
Knowledge Dimension: PROCEDURAL
Justification: The question requires a step-by-step algorithmic process involving matrix algebra and trigonometric simplification.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. It tests the integration of two distinct chapters (Determinants and Trigonometry) which is a hallmark of board-level competency-based assessments.

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