Class CBSE Class 12 Mathematics Vector Algebra Q #860
COMPETENCY BASED
APPLY
1 Marks 2023 MCQ SINGLE
In $\Delta ABC$, $\vec{AB}=\hat{i}+\hat{j}+2\hat{k}$ and $\vec{AC}=3\hat{i}-\hat{j}+4\hat{k}$. If D is mid-point of BC, then vector $\vec{AD}$ is equal to :
(A) $4\hat{i}+6\hat{k}$
(B) $2\hat{i}-2\hat{j}+2\hat{k}$
(C) $\hat{i}-\hat{j}+\hat{k}$
(D) $2\hat{i}+3\hat{k}$
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AI Tutor Explanation

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Step-by-Step Solution

  1. First, we need to find the vector $\vec{BC}$. We know that $\vec{AC} = \vec{AB} + \vec{BC}$, so $\vec{BC} = \vec{AC} - \vec{AB}$.

    $\vec{BC} = (3\hat{i} - \hat{j} + 4\hat{k}) - (\hat{i} + \hat{j} + 2\hat{k}) = 2\hat{i} - 2\hat{j} + 2\hat{k}$

  2. Since D is the midpoint of BC, $\vec{BD} = \frac{1}{2}\vec{BC}$.

    $\vec{BD} = \frac{1}{2}(2\hat{i} - 2\hat{j} + 2\hat{k}) = \hat{i} - \hat{j} + \hat{k}$

  3. Now, we need to find $\vec{AD}$. We know that $\vec{AD} = \vec{AB} + \vec{BD}$.

    $\vec{AD} = (\hat{i} + \hat{j} + 2\hat{k}) + (\hat{i} - \hat{j} + \hat{k}) = 2\hat{i} + 0\hat{j} + 3\hat{k} = 2\hat{i} + 3\hat{k}$

Correct Answer: $2\hat{i}+3\hat{k}$

AI Suggestion: Option D

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because it requires students to apply their understanding of vector algebra and midpoint formula to find the vector AD.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to follow a specific procedure to calculate the position vector of point D and then use it to find the vector AD. This involves applying formulas and algebraic manipulation.
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 geometric problem, which aligns with competency-based assessment principles.

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