Directions: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as
Assertion (A): The order and degree of the differential equation `sqrt((d^2y)/(dx^2))=sqrt((dy)/(dx)+5)` are 2 and 1 respectively
Reason (R): The differential equation
`((dx)/(dy))^3+2y^(1//2)=x`
is of order 1 and degree 3.

Options

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

Answer

To solve the problem, we need to analyze both the assertion (A) and the reason (R) provided in the question.

### Step 1: Analyze the Assertion (A)
The assertion states:
\[ \sqrt{\frac{d^2y}{dx^2}} = \sqrt{\frac{dy}{dx} + 5} \]
We need to determine the order and degree of this differential equation.

**Step 1.1: Remove the square roots**
To eliminate the square roots, we square both sides:
\[ \frac{d^2y}{dx^2} = \frac{dy}{dx} + 5 \]

**Step 1.2: Identify the order and degree**
- The highest derivative present is \(\frac{d^2y}{dx^2}\), which indicates that the order of the differential equation is **2**.
- The equation is now in the form where the highest power of the derivative \(\frac{dy}{dx}\) is 1. Therefore, the degree of the differential equation is **1**.

### Conclusion for Assertion (A)
The order is 2 and the degree is 1, which confirms that the assertion is **true**.

### Step 2: Analyze the Reason (R)
The reason states:
\[ \left(\frac{dx}{dy}\right)^3 + 2\sqrt{y} = x \]
We need to determine the order and degree of this differential equation.

**Step 2.1: Identify the order and degree**
- The term \(\frac{dx}{dy}\) indicates that this is a first-order differential equation since it involves only the first derivative.
- The highest power of the derivative \(\frac{dx}{dy}\) is 3, which means the degree of the differential equation is **3**.

### Conclusion for Reason (R)
The order is 1 and the degree is 3, which confirms that the reason is also **true**.

### Final Evaluation
Both the assertion (A) and reason (R) are true, but the reason does not serve as a correct explanation for the assertion. Therefore, the correct choice is:
**Option 2: A and R both are true but R is not the correct explanation for A.**