CBSE Class 12 Mathematics Differential Equations Q #2092
COMPETENCY BASED
APPLY
1 Marks 2026-27 CBSE Sample QP ASSERTION REASON
Assertion: The general solution of a differential equation represents a family of curves.
Reason: The derivative of an arbitrary constant is zero.
(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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Detailed Solution

Step 1: Analyze the Assertion

The general solution of a differential equation of order n contains n arbitrary constants. Geometrically, these constants represent parameters that define a family of curves. For example, the solution y = mx + c represents a family of straight lines.

Step 2: Analyze the Reason

The derivative of any constant C with respect to x is d/dx(C) = 0. This property is fundamental in the process of forming a differential equation from a general solution, as it allows the elimination of arbitrary constants.

Step 3: Evaluate the Relationship

While both statements are mathematically true, the reason provided (derivative of a constant is zero) is the mechanism used to derive the differential equation from the family of curves, but it does not explain why the general solution represents a family of curves. Therefore, the reason is not the correct explanation for the assertion.

Final Answer: Assertion is true, Reason is true, but Reason is not the correct explanation for Assertion.

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because it requires the student to connect the geometric interpretation of differential equations with the algebraic rules of differentiation.
Knowledge Dimension: CONCEPTUAL
Justification: The student must understand the conceptual link between arbitrary constants in solutions and the formation of differential equations.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. This question tests the conceptual clarity of the 'Differential Equations' chapter, specifically the transition from general solutions to differential forms.

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