Class CBSE Class 12 Mathematics Vector Algebra Q #1728
COMPETENCY BASED
APPLY
1 Marks 2026 AISSCE(Board Exam) MCQ SINGLE
If $(3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0}$ then the values of p and q are :
(A) $p=-\frac{2}{3}, q=\frac{5}{3}$
(B) $p=-\frac{8}{3}, q=\frac{20}{3}$
(C) $p=\frac{20}{3}, q=-\frac{8}{3}$
(D) $p=0, q=0$
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Correct Answer: B

AI Tutor Explanation

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Detailed Solution

Step 1: Understand the Condition

Two vectors are collinear if their cross product is the zero vector. Given that the cross product of (3, -2, 5) and (4, p, q) is 0, the vectors must be proportional.

Step 2: Set up Proportionality

Since the vectors are parallel, their components must be in the same ratio: $$ \frac{4}{3} = \frac{p}{-2} = \frac{q}{5} $$

Step 3: Solve for p

Using the ratio: $$ \frac{p}{-2} = \frac{4}{3} $$ $$ p = -2 \times \frac{4}{3} = -\frac{8}{3} $$

Step 4: Solve for q

Using the ratio: $$ \frac{q}{5} = \frac{4}{3} $$ $$ q = 5 \times \frac{4}{3} = \frac{20}{3} $$

Final Answer: p = -8/3, q = 20/3

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must utilize the geometric property of the cross product (collinearity) to solve for unknown variables.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the application of a specific algorithm (ratio of components) derived from the definition of vector cross products.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. It tests the conceptual understanding of vector algebra beyond mere calculation, requiring the student to recognize the relationship between parallel vectors.

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