Class CBSE Class 12 Mathematics Vector Algebra Q #564
COMPETENCY BASED
APPLY
1 Marks 2025 AISSCE(Board Exam) MCQ SINGLE
The respective values of \(|\vec{a}|\) and \(|\vec{b}|\), if given \((\vec{a}-\vec{b})\cdot(\vec{a}+\vec{b})=512\) and \(|\vec{a}|=3|\vec{b}|\), are:
(A) 48 and 16
(B) 3 and 1
(C) 24 and 8
(D) 6 and 2
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AI Tutor Explanation

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Step-by-Step Solution

  1. Given \((\vec{a}-\vec{b})\cdot(\vec{a}+\vec{b})=512\).
  2. Expanding the dot product, we get \(|\vec{a}|^2 - |\vec{b}|^2 = 512\).
  3. Also, given \(|\vec{a}|=3|\vec{b}|\).
  4. Substituting \(|\vec{a}|=3|\vec{b}|\) into the first equation, we get \((3|\vec{b}|)^2 - |\vec{b}|^2 = 512\).
  5. This simplifies to \(9|\vec{b}|^2 - |\vec{b}|^2 = 512\), which means \(8|\vec{b}|^2 = 512\).
  6. Dividing both sides by 8, we get \(|\vec{b}|^2 = 64\).
  7. Taking the square root, we find \(|\vec{b}| = 8\).
  8. Since \(|\vec{a}|=3|\vec{b}|\), we have \(|\vec{a}| = 3 \times 8 = 24\).
  9. Therefore, the respective values of \(|\vec{a}|\) and \(|\vec{b}|\) are 24 and 8.

Correct Answer: 24 and 8

AI Suggestion: Option C

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because it requires students to apply their knowledge of dot products and vector magnitudes to solve for the unknown magnitudes.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to follow a specific procedure to solve the problem, involving algebraic manipulation and substitution.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. It assesses the student's ability to apply vector algebra concepts to solve a problem, rather than just recalling definitions or theorems.

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