if the equation `sin (pi x^2) - sin(pi x^2 + 2 pi x) = 0` is solved for positive roots, then in the increasing sequence of positive root (A) first term is `(-1+sqrt(7))/(2)` (B) first term is `(-1+sqrt(3))/(2)` (C) third term is `1` (D) third term is `(-1+sqrt(11))/(2)`

Options

A. first term is `(-1+sqrt(7))/(2)`
B. first term is `(-1+sqrt(3))/(2)`
C. third term is `1`
D. third term is `(-1+sqrt(11))/(2)`

Answer

To solve the equation \( \sin(\pi x^2) - \sin(\pi x^2 + 2\pi x) = 0 \) for positive roots, we can follow these steps:

### Step 1: Use the sine subtraction formula
We start with the equation:
\[ \sin(\pi x^2) - \sin(\pi x^2 + 2\pi x) = 0 \]
Using the sine subtraction formula, we can rewrite this as:
\[ \sin A - \sin B = 0 \implies 2 \cos\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) = 0 \]
where \( A = \pi x^2 \) and \( B = \pi x^2 + 2\pi x \).

### Step 2: Set up the equations
From the sine subtraction formula, we have:
\[ 2 \cos\left(\frac{\pi x^2 + (\pi x^2 + 2\pi x)}{2}\right) \sin\left(\frac{\pi x^2 - (\pi x^2 + 2\pi x)}{2}\right) = 0 \]
Simplifying gives:
\[ 2 \cos\left(\pi x^2 + \pi x\right) \sin\left(-\pi x\right) = 0 \]
This leads to two cases:
1. \( \cos(\pi x^2 + \pi x) = 0 \)
2. \( \sin(\pi x) = 0 \)

### Step 3: Solve \( \sin(\pi x) = 0 \)
The equation \( \sin(\pi x) = 0 \) implies:
\[ \pi x = n\pi \quad \Rightarrow \quad x = n \quad (n \in \mathbb{Z}) \]
For positive roots, we have \( x = 1, 2, 3, \ldots \)

### Step 4: Solve \( \cos(\pi x^2 + \pi x) = 0 \)
The equation \( \cos(\pi x^2 + \pi x) = 0 \) implies:
\[ \pi x^2 + \pi x = \left(2n + 1\right)\frac{\pi}{2} \quad \Rightarrow \quad x^2 + x = \frac{2n + 1}{2} \]
This simplifies to:
\[ x^2 + x - \frac{2n + 1}{2} = 0 \]

### Step 5: Use the quadratic formula
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{-1 \pm \sqrt{1 + 2(2n + 1)}}{2} = \frac{-1 \pm \sqrt{4n + 3}}{2} \]
Thus, the roots are:
\[ x = \frac{-1 + \sqrt{4n + 3}}{2} \quad \text{and} \quad x = \frac{-1 - \sqrt{4n + 3}}{2} \]
We only consider the positive root:
\[ x = \frac{-1 + \sqrt{4n + 3}}{2} \]

### Step 6: Calculate the roots for different values of \( n \)
1. For \( n = 0 \):
\[ x = \frac{-1 + \sqrt{3}}{2} \]
2. For \( n = 1 \):
\[ x = \frac{-1 + \sqrt{7}}{2} \]
3. For \( n = 2 \):
\[ x = \frac{-1 + \sqrt{11}}{2} \]
4. For \( n = 3 \):
\[ x = \frac{-1 + \sqrt{15}}{2} \]

### Step 7: Arrange the roots in increasing order
The roots in increasing order are:
1. \( \frac{-1 + \sqrt{3}}{2} \)
2. \( \frac{-1 + \sqrt{7}}{2} \)
3. \( 1 \)
4. \( \frac{-1 + \sqrt{11}}{2} \)

### Conclusion
From the options given:
- The first term is \( \frac{-1 + \sqrt{3}}{2} \) (Option B).
- The third term is \( 1 \) (Option C).

Thus, the correct answers are B and C.