Let `C_(1):y=x^(2)sin3x,C_(2):y=x^(2)and C_(3):y=-y^(2),` then

Options

A. `C_(1)` touches `C_(2)` at infinite points
B. `C_(1)` touches `C_(3)` at infinite points
C. `C_(1) and C_(2) and C_(1)andC_(3)` meet at alternate points
D. none of these

Answer

To solve the problem, we need to analyze the three curves given:

1. \( C_1: y = x^2 \sin(3x) \)
2. \( C_2: y = x^2 \)
3. \( C_3: y = -y^2 \)

### Step 1: Analyze \( C_1 \) and \( C_2 \)
We start by examining the curves \( C_1 \) and \( C_2 \).

- The function \( C_2: y = x^2 \) is a standard upward-opening parabola.
- The function \( C_1: y = x^2 \sin(3x) \) oscillates between \( -x^2 \) and \( x^2 \) because the sine function oscillates between -1 and 1.

**Finding points of intersection:**
To find the points where \( C_1 \) and \( C_2 \) intersect, we set:
\[ x^2 \sin(3x) = x^2 \]
Dividing both sides by \( x^2 \) (assuming \( x \neq 0 \)):
\[ \sin(3x) = 1 \]
The sine function equals 1 at:
\[ 3x = \frac{\pi}{2} + 2n\pi \quad \text{for } n \in \mathbb{Z} \]
Thus,
\[ x = \frac{\pi}{6} + \frac{2n\pi}{3} \]
This shows that \( C_1 \) and \( C_2 \) intersect at infinite points.

### Step 2: Analyze \( C_1 \) and \( C_3 \)
Next, we analyze the curves \( C_1 \) and \( C_3 \).

- The equation for \( C_3 \) can be rearranged as:
\[ y + y^2 = 0 \implies y(y + 1) = 0 \]
This gives us the lines \( y = 0 \) and \( y = -1 \).

**Finding points of intersection:**
To find the points where \( C_1 \) intersects \( C_3 \), we need to set:
\[ x^2 \sin(3x) = 0 \]
This occurs when:
\[ x^2 = 0 \quad \text{or} \quad \sin(3x) = 0 \]
From \( x^2 = 0 \), we have \( x = 0 \).
From \( \sin(3x) = 0 \), we have:
\[ 3x = n\pi \quad \text{for } n \in \mathbb{Z} \implies x = \frac{n\pi}{3} \]
Thus, \( C_1 \) intersects \( C_3 \) at infinite points as well.

### Step 3: Analyze \( C_2 \) and \( C_3 \)
Lastly, we analyze \( C_2 \) and \( C_3 \).

- Since \( C_3 \) consists of the lines \( y = 0 \) and \( y = -1 \), we can check the intersections with \( C_2: y = x^2 \).

**Finding points of intersection:**
1. For \( y = 0 \):
\[ x^2 = 0 \implies x = 0 \]
2. For \( y = -1 \):
\[ x^2 = -1 \quad \text{(no real solutions)} \]

Thus, \( C_2 \) and \( C_3 \) intersect only at the point \( (0, 0) \).

### Conclusion
- \( C_1 \) and \( C_2 \) touch at infinite points.
- \( C_1 \) and \( C_3 \) touch at infinite points.
- \( C_2 \) and \( C_3 \) touch at only one point.

### Final Answer
- \( C_1 \) touches \( C_2 \) at infinite points.
- \( C_1 \) touches \( C_3 \) at infinite points.
- \( C_2 \) and \( C_3 \) meet only at one point.