To analyze the given statements, we will evaluate each statement one by one and determine their validity based on the principles of physics.
### Step 1: Evaluate Statement 1
**Statement 1:** Kepler's second law is the consequence of the law of conservation of angular momentum.
**Solution:**
Kepler's second law states that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. This implies that the areal velocity (area swept out per unit time) is constant. Mathematically, this can be expressed as:
\[
\frac{dA}{dt} = \text{constant}
\]
The angular momentum \( L \) of a planet moving in an orbit is given by:
\[
L = mvr
\]
where \( m \) is the mass of the planet, \( v \) is its tangential velocity, and \( r \) is the distance from the Sun. Since no external torque acts on the planet, angular momentum is conserved. Thus, the conservation of angular momentum leads to the constancy of areal velocity, confirming that Kepler's second law is indeed a consequence of the conservation of angular momentum.
**Conclusion:** Statement 1 is true.
### Step 2: Evaluate Statement 2
**Statement 2:** In planetary motion, the momentum, angular momentum, and mechanical energy are conserved.
**Solution:**
In the context of planetary motion:
- **Linear momentum** is conserved in the absence of external forces. However, in a gravitational field, the net force is not zero, so linear momentum is not conserved.
- **Angular momentum** is conserved due to the absence of external torque acting on the planet as it orbits the Sun.
- **Mechanical energy** (kinetic + potential energy) is conserved in a closed system where only conservative forces (like gravity) are acting.
Thus, while angular momentum and mechanical energy are conserved, linear momentum is not conserved due to the gravitational force acting on the planet.
**Conclusion:** Statement 2 is partially true but misleading as it implies all three quantities are conserved.
### Step 3: Evaluate Statement 3
**Statement 3:** The gravitational field of a circular ring is maximum at a point upon the axis at a distance \( \frac{R}{\sqrt{2}} \) from the center, where \( R \) is the radius of the ring.
**Solution:**
To find the gravitational field \( E \) at a distance \( x \) along the axis of a circular ring of radius \( R \), we use the formula:
\[
E = \frac{GMx}{(x^2 + R^2)^{3/2}}
\]
To find the maximum gravitational field, we take the derivative of \( E \) with respect to \( x \) and set it to zero:
\[
\frac{dE}{dx} = 0
\]
Solving this gives us \( x = \frac{R}{\sqrt{2}} \), which indicates that the gravitational field is indeed maximum at this point.
**Conclusion:** Statement 3 is true.
### Final Conclusion:
- Statement 1: True
- Statement 2: Partially true (but misleading)
- Statement 3: True
The correct option is that statements 1 and 3 are true, while statement 2 is misleading.
