Statement I Differential equation corresponding to all lines, ax+by+c=0 has the order 3.
Statement II Gereral solution of a differential equation of nth order contains n independent arbitaray constanis.

Options

A. Statement I is true ,and Statement II is the correct explanation for Statement I.
B. Statement I is true, Statement II is true and Statement II is the correct explanation for Statment I
C. Statement I is true, Statement II is false.
D. Statement I is false, Statement II is true.

Answer

To solve the problem, we need to analyze both statements provided and determine their validity.

### Step 1: Analyze Statement I
**Statement I**: The differential equation corresponding to all lines of the form \( ax + by + c = 0 \) has order 3.

1. The general equation of a line in two dimensions is given by \( ax + by + c = 0 \), where \( a, b, c \) are constants.
2. To find the corresponding differential equation, we can express the line in terms of derivatives. The equation can be differentiated to eliminate the constants.
3. The first derivative \( \frac{dy}{dx} \) can be expressed in terms of \( a \) and \( b \).
4. However, since we have three constants \( a, b, c \), we need to differentiate the equation enough times to eliminate all three constants.
5. Differentiating the equation twice will yield a second-order differential equation, but we need to differentiate it three times to confirm that the order is indeed 3.

Thus, we conclude that the statement is **true**.

### Step 2: Analyze Statement II
**Statement II**: The general solution of a differential equation of \( n \)th order contains \( n \) independent arbitrary constants.

1. A differential equation of order \( n \) means that the highest derivative present is the \( n \)th derivative.
2. The general solution of such a differential equation typically includes \( n \) arbitrary constants. This is a fundamental property of differential equations.
3. For example, the general solution of a second-order differential equation will contain two arbitrary constants, while a third-order differential equation will contain three arbitrary constants.

Thus, we conclude that this statement is also **true**.

### Conclusion
Both statements are true, and Statement II serves as a correct explanation for Statement I.

### Final Answer
- **Statement I**: True
- **Statement II**: True
- **Explanation**: Statement II correctly explains why the differential equation corresponding to all lines has an order of 3 due to the presence of three arbitrary constants.