Class CBSE Class 12 Mathematics Vector Algebra Q #1721
COMPETENCY BASED
APPLY
1 Marks 2026 AISSCE(Board Exam) MCQ SINGLE
If vectors $\vec{a}=3\hat{i}+2\hat{j}+\lambda\hat{k}$ and $\vec{b}=2\hat{i}-4\hat{j}+5\hat{k}$, represent the two strips of the Red Cross sign placed outside a doctor's clinic, then the value of $\lambda$ is :
(A) 1
(B) $\frac{5}{2}$
(C) $\frac{2}{5}$
(D) 0
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Correct Answer: C

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Detailed Solution

Step 1: Identify the condition for perpendicularity

The Red Cross sign implies that the two strips are perpendicular to each other. For two vectors $\vec{a}$ and $\vec{b}$ to be perpendicular, their dot product must be equal to zero.

Step 2: Set up the dot product equation

The dot product is given by $\vec{a} \cdot \vec{b} = 0$. Substituting the given vectors:

$$(3\hat{i} + 2\hat{j} + \lambda\hat{k}) \cdot (2\hat{i} - 4\hat{j} + 5\hat{k}) = 0$$

Step 3: Calculate the dot product

Multiply the corresponding components:

$$(3 \times 2) + (2 \times -4) + (\lambda \times 5) = 0$$ $$6 - 8 + 5\lambda = 0$$

Step 4: Solve for $\lambda$

Simplify the equation:

$$-2 + 5\lambda = 0$$ $$5\lambda = 2$$ $$\lambda = \frac{2}{5}$$

Final Answer: \frac{2}{5}

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must translate a real-world scenario (Red Cross sign) into a mathematical condition (perpendicular vectors) and apply the dot product formula.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the execution of a specific mathematical algorithm (dot product calculation) to solve for an unknown variable.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. The question embeds a standard vector algebra problem within a contextual application, aligning with NEP 2020's focus on competency-based assessment.

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