Class CBSE Class 12 Mathematics Matrices and Determinants Q #740
KNOWLEDGE BASED
APPLY
1 Marks 2024 MCQ SINGLE
If \(A=[a_{ij}]\) be a \(3\times3\) matrix, where \(a_{ij}=i-3j\), then which of the following is false ?
(A) \(a_{11}\lt0\)
(B) \(a_{12}+a_{21}=-6\)
(C) \(a_{13}\gt a_{31}\)
(D) \(a_{31}=0\)
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AI Tutor Explanation

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Step-by-Step Solution

  1. Calculate \(a_{11}\):

    \(a_{11} = 1 - 3(1) = 1 - 3 = -2\). Since \(-2 \lt 0\), option (A) is true.

  2. Calculate \(a_{12}\) and \(a_{21}\):

    \(a_{12} = 1 - 3(2) = 1 - 6 = -5\)

    \(a_{21} = 2 - 3(1) = 2 - 3 = -1\)

    Then, \(a_{12} + a_{21} = -5 + (-1) = -6\). So, option (B) is true.

  3. Calculate \(a_{13}\) and \(a_{31}\):

    \(a_{13} = 1 - 3(3) = 1 - 9 = -8\)

    \(a_{31} = 3 - 3(1) = 3 - 3 = 0\)

    Since \(-8 \gt 0\) is false, \(a_{13} \gt a_{31}\) is false. So, option (C) is false.

  4. Calculate \(a_{31}\):

    \(a_{31} = 3 - 3(1) = 3 - 3 = 0\). So, option (D) is true.

Correct Answer: C

APPLY|||KNOWLEDGE|||PROCEDURAL|||MEDIUM|||
Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply the given formula \(a_{ij} = i - 3j\) to calculate the values of the matrix elements and then evaluate the given conditions.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to follow a specific procedure: first, calculate the matrix elements using the given formula, and then, verify each of the given conditions.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the understanding and application of matrix element calculation, a core concept covered in the textbook.

Step-by-Step Solution

  1. Calculate \(a_{11}\):

    \(a_{11} = 1 - 3(1) = 1 - 3 = -2\). Since \(-2 \lt 0\), option (A) is true.

  2. Calculate \(a_{12}\) and \(a_{21}\):

    \(a_{12} = 1 - 3(2) = 1 - 6 = -5\)

    \(a_{21} = 2 - 3(1) = 2 - 3 = -1\)

    Then, \(a_{12} + a_{21} = -5 + (-1) = -6\). So, option (B) is true.

  3. Calculate \(a_{13}\) and \(a_{31}\):

    \(a_{13} = 1 - 3(3) = 1 - 9 = -8\)

    \(a_{31} = 3 - 3(1) = 3 - 3 = 0\)

    Since \(-8 \gt 0\) is false, \(a_{13} \gt a_{31}\) is false. So, option (C) is false.

  4. Calculate \(a_{31}\):

    \(a_{31} = 3 - 3(1) = 3 - 3 = 0\). So, option (D) is true.

Correct Answer: C

AI Suggestion: Option C

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply the given formula \(a_{ij} = i - 3j\) to calculate the values of the matrix elements and then evaluate the given conditions.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to follow a specific procedure: first, calculate the matrix elements using the given formula, and then, verify each of the given conditions.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the understanding and application of matrix element calculation, a core concept covered in the textbook.

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