CBSE Sample Paper 2026-27 · Section A · 1 mark

Q20: The general solution as a family of curves

Sample 2026-27Ch 9 · Differential Equations

The question

Select the correct answer from the options (A), (B), (C) and (D) as given below:

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(D) Assertion (A) is false but Reason (R) is true.

Assertion: The general solution of a differential equation represents a family of curves.

Reason: The derivative of an arbitrary constant is zero.

Exam answer

The general solution contains arbitrary constants. For example, integrating $\frac{dy}{dx} = 2x$ gives

y = x^2 + c

Each value of $c$ gives a different curve ( $c = 0$ ⁠: $y = x^2$ ⁠; $c = 1$ ⁠: $y = x^2 + 1$ ⁠), so the general solution represents a family of curves. Assertion (A) is true.

Why?

The general solution of a differential equation of order $n$ has $n$ arbitrary constants. Giving them particular values picks out one curve, a particular solution.

Here the curves $y = x^2 + c$ are copies of the parabola $y = x^2$ ⁠, moved up or down by $c$ ⁠.

The derivative of an arbitrary constant is zero, so Reason (R) is true:

\frac{d}{dx}(x^2 + c) = 2x + 0 = 2x

This is why the constant drops out on differentiation and every curve of the family satisfies the same differential equation, so Reason (R) explains Assertion (A).

Why?

Differentiating loses the constant, so integrating cannot bring back one fixed value of it. Every value of $c$ fits the equation, and that is exactly what makes the solution a whole family.

So both statements are true and the Reason is the cause of the Assertion: the code is (A).

Answer: (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

Where this comes from

Chapter 9, Differential Equations: General solution of a differential equation.

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