Class CBSE Class 12 Mathematics Vector Algebra Q #1837
COMPETENCY BASED
APPLY
1 Marks 2026 AISSCE(Board Exam) ASSERTION REASON
Assertion: Lines given by $x = py + q, z = ry + s$ and $x = p'y + q', z = r'y + s'$ are perpendicular to each other when $pp' + rr' = 1$.
Reason: Two lines $\vec{r} = \vec{a_1} + \lambda \vec{b_1}$ and $\vec{r} = \vec{a_2} + \mu \vec{b_2}$ are perpendicular to each other if $\vec{b_1} \cdot \vec{b_2} = 0$.
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true but R is NOT the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.
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Correct Answer: D

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Detailed Solution

Step 1: Express the lines in symmetric form

The given lines are $x = py + q$ and $z = ry + s$. Rearranging these, we get: $$ \frac{x - q}{p} = y = \frac{z - s}{r} $$ The direction vector of the first line is $\vec{b_1} = (p, 1, r)$.

Step 2: Express the second line in symmetric form

Similarly, for the second line $x = p'y + q'$ and $z = r'y + s'$, we get: $$ \frac{x - q'}{p'} = y = \frac{z - s'}{r'} $$ The direction vector of the second line is $\vec{b_2} = (p', 1, r')$.

Step 3: Apply the condition for perpendicularity

Two lines are perpendicular if the dot product of their direction vectors is zero: $$ \vec{b_1} \cdot \vec{b_2} = 0 $$ $$ (p)(p') + (1)(1) + (r)(r') = 0 $$ $$ pp' + 1 + rr' = 0 $$ $$ pp' + rr' = -1 $$

Step 4: Conclusion

The statement claims the condition is $pp' + rr' = 1$, but the derivation shows the condition is $pp' + rr' = -1$. Therefore, the assertion is false.

Final Answer: The assertion is False.

AI Suggestion: Option D

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must transform the given linear equations into standard symmetric form and apply the vector dot product condition.
Knowledge Dimension: PROCEDURAL
Justification: The student must follow a specific sequence of algebraic manipulations to extract direction ratios and verify the condition.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. It tests the ability to interpret 3D geometry equations beyond standard textbook formats.

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