STATEMENT-1 : A compass needle which is allowed to move in horizontal plane is taken to a geomagnetic pole . It will stay in any position .
and
STATEMENT-2 : Angle of dip at poles is `90^(@)`

Options

A. Statement-1 is True , Statement-2 is True , Statement-2 is a correct explanation for Statement-1
B. Statement-1 is True , Statement-2 is True , Statement-2 is NOT a correct explanation for Statement- 1
C. Statement-1 is True , Statement-2 is False
D. Statement-1 is False , Statement-2 is True

Answer

To solve the given question, we need to analyze both statements and their implications regarding the behavior of a compass needle at the geomagnetic pole.

### Step-by-Step Solution:

1. **Understanding Statement 1**:
- The statement claims that a compass needle, when allowed to move in a horizontal plane at a geomagnetic pole, will stay in any position.
- At the geomagnetic pole, the magnetic field lines are vertical, meaning that the compass needle, which aligns itself with the magnetic field, will not have a horizontal component to align with.

2. **Understanding Statement 2**:
- This statement asserts that the angle of dip at the poles is 90 degrees.
- The angle of dip (or magnetic inclination) is defined as the angle that the Earth's magnetic field makes with the horizontal plane. At the geomagnetic poles, the magnetic field is entirely vertical, resulting in an angle of dip of 90 degrees.

3. **Analyzing the Implications**:
- Since the angle of dip is 90 degrees, the horizontal component of the magnetic field (which is responsible for the compass needle's horizontal alignment) becomes zero.
- Mathematically, the horizontal component \( B_H \) can be expressed as \( B_H = B \cos(\theta) \), where \( \theta \) is the angle of dip. At the poles, \( \theta = 90^\circ \), thus \( B_H = B \cos(90^\circ) = 0 \).

4. **Conclusion from the Analysis**:
- With no horizontal magnetic field acting on it, the compass needle experiences no torque to align itself in a particular direction. Therefore, it can remain in any position.
- Both statements are correct:
- Statement 1 is true because the compass needle will not rotate and can stay in any position.
- Statement 2 is true because the angle of dip at the poles is indeed 90 degrees.

5. **Final Answer**:
- Both statements are true, and Statement 2 provides a correct explanation for Statement 1.