Class CBSE Class 12 Mathematics Matrices and Determinants Q #1841
COMPETENCY BASED
APPLY
1 Marks 2026 AISSCE(Board Exam) MCQ SINGLE
If $\Delta_{1}=\begin{vmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{vmatrix}$ and $\Delta_{2}=\begin{vmatrix} 0 & 2 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 6 \end{vmatrix}$, then
(A) $\Delta_{1}=2 \Delta_{2}$
(B) $\Delta_{2}=-2 \Delta_{1}$
(C) $\Delta_{1}=\Delta_{2}$
(D) $\Delta_{2}=-\Delta_{1}$
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Correct Answer: B

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Detailed Solution

Step 1: Calculate the value of determinant $\Delta_{1}$

Since $\Delta_{1}$ is a diagonal matrix, its determinant is the product of its diagonal elements: $$ \Delta_{1} = 1 \times 2 \times 3 = 6 $$

Step 2: Calculate the value of determinant $\Delta_{2}$

Expand $\Delta_{2}$ along the first row: $$ \Delta_{2} = 0(0 - 0) - 2(6 - 0) + 0(0 - 0) $$ $$ \Delta_{2} = -2(6) = -12 $$

Step 3: Compare the values

We have $\Delta_{1} = 6$ and $\Delta_{2} = -12$. Observing the relationship: $$ \Delta_{2} = -2 \times 6 = -2 \Delta_{1} $$

Final Answer: Option (B)

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must apply the properties of determinants and matrix expansion techniques to solve for specific values.
Knowledge Dimension: PROCEDURAL
Justification: The question requires a step-by-step algorithmic approach to evaluate determinants and establish a mathematical relationship between them.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. It tests the student's ability to perform matrix operations accurately, which is a core competency in the Matrices and Determinants unit.
||KEY:B

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