Q566: If the sides AB and AC of $\triangle ABC$ are represented ...

COMPETENCY BASED

APPLY

1 Marks 2025 AISSCE(Board Exam) MCQ SINGLE

If the sides AB and AC of $\triangle ABC$ are represented by vectors $\hat{j}+\hat{k}$ and $3\hat{i}-\hat{j}+4\hat{k}$ respectively, then the length of the median through A on BC is:

(A) $2\sqrt{2}$ units

(B) $\sqrt{18}$ units

(D) $\frac{\sqrt{48}}{2}$ units

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Correct Answer: C

Explanation

The position vector of the midpoint $D$ is the average of the position vectors of $B$ and $C$ relative to $A$:\[\vec{AD} = \frac{\vec{AB} + \vec{AC}}{2}\]
\[\vec{AD} = \frac{3}{2}\hat{i} + \frac{5}{2}\hat{k}\]
The length of the median is the magnitude of the vector $\vec{AD}$, denoted as $|\vec{AD}|$:\[|\vec{AD}| = \sqrt{\left(\frac{3}{2}\right)^2 + \left(0\right)^2 + \left(\frac{5}{2}\right)^2}\]
\[|\vec{AD}| = \frac{\sqrt{34}}{2}\]

AI Tutor Explanation

Step-by-Step Solution

Let the position vectors of A, B, and C be $\vec{a}$, $\vec{b}$, and $\vec{c}$ respectively.

Given: $\vec{AB} = \hat{j} + \hat{k}$ and $\vec{AC} = 3\hat{i} - \hat{j} + 4\hat{k}$
Let M be the midpoint of BC. Then AM is the median.

$\vec{AM} = \vec{m} - \vec{a} = \frac{\vec{b} + \vec{c}}{2} - \vec{a} = \frac{1}{2}(\vec{b} - \vec{a} + \vec{c} - \vec{a}) = \frac{1}{2}(\vec{AB} + \vec{AC})$
Substitute the given vectors:

$\vec{AM} = \frac{1}{2}[(\hat{j} + \hat{k}) + (3\hat{i} - \hat{j} + 4\hat{k})] = \frac{1}{2}(3\hat{i} + 5\hat{k})$
Find the magnitude of $\vec{AM}$:

$|\vec{AM}| = \frac{1}{2}\sqrt{3^2 + 0^2 + 5^2} = \frac{1}{2}\sqrt{9 + 25} = \frac{1}{2}\sqrt{34}$
Therefore, the length of the median through A on BC is $\frac{\sqrt{34}}{2}$ units.

Correct Answer: $\frac{\sqrt{34}}{2}$ units

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 understanding of vectors and medians to solve a specific problem. They need to use vector addition/subtraction and the midpoint formula to find the length of the median.

Knowledge Dimension: PROCEDURAL

Justification: The question requires a series of steps to solve, including vector addition/subtraction, finding the midpoint, and calculating the magnitude of a vector. These are all procedural skills.

Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. The question assesses the student's ability to apply vector concepts in a geometric context, which aligns with competency-based assessment principles.

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