Statement -1 : Two regular polygons are inscribed in the same circle. The first polygon has 1982 sides and second has 2973 sides. If the polygons have a common vertex, then the number of vertex common to both of them is 991.
Statement -2 : The total number of complex numbers z. satisfying |z-1| = |z +1| =|z| is one
Statement -3 : The locus represented by |2011 z+1| = 2011 |z+1| is a striaght line

Options

A. (a)T T T
B. (b)T F T
C. (c)F T T
D. (d)F F F

Answer

To solve the problem, we need to analyze each statement one by one and determine their validity. Here’s a step-by-step breakdown of the solution:

### Step 1: Analyze Statement 1
**Statement 1**: Two regular polygons are inscribed in the same circle. The first polygon has 1982 sides and the second has 2973 sides. If the polygons have a common vertex, then the number of vertices common to both of them is 991.

1. **Understanding the Polygons**:
- Let \( P_1 \) be the polygon with 1982 sides.
- Let \( P_2 \) be the polygon with 2973 sides.

2. **Finding Common Vertices**:
- The number of common vertices between two polygons inscribed in the same circle can be found using the greatest common divisor (GCD) of the number of sides of the polygons.
- We need to calculate \( \text{GCD}(1982, 2973) \).

3. **Calculating GCD**:
- Factor \( 1982 = 2 \times 991 \).
- Factor \( 2973 = 3 \times 991 \).
- The GCD is \( 991 \) since it is the common factor.

4. **Conclusion for Statement 1**:
- Therefore, the number of common vertices is indeed \( 991 \).
- **Statement 1 is true**.

### Step 2: Analyze Statement 2
**Statement 2**: The total number of complex numbers \( z \) satisfying \( |z-1| = |z + 1| = |z| \) is one.

1. **Understanding the Equation**:
- The equation \( |z - 1| = |z + 1| \) represents the perpendicular bisector of the line segment joining the points \( (1, 0) \) and \( (-1, 0) \).
- The equation \( |z - 1| = |z| \) represents the perpendicular bisector of the segment joining \( (1, 0) \) and the origin \( (0, 0) \).

2. **Finding the Intersection**:
- The intersection of these two lines will give us the point that satisfies both conditions.
- Solving these equations leads us to find that the only solution is the point \( z = 0 \).

3. **Conclusion for Statement 2**:
- Thus, there is exactly one complex number \( z \) that satisfies the conditions.
- **Statement 2 is true**.

### Step 3: Analyze Statement 3
**Statement 3**: The locus represented by \( |2011z + 1| = 2011 |z + 1| \) is a straight line.

1. **Understanding the Locus**:
- Rewrite the equation: \( |2011z + 1| = 2011 |z + 1| \).
- Divide both sides by \( 2011 \) (assuming \( 2011 \neq 0 \)):
\[ |z + \frac{1}{2011}| = |z + 1| \]

2. **Interpreting the Locus**:
- This equation represents the set of points \( z \) that are equidistant from the points \( -\frac{1}{2011} \) and \( -1 \).
- The locus of points equidistant from two fixed points is a straight line, specifically the perpendicular bisector of the segment joining these two points.

3. **Conclusion for Statement 3**:
- Therefore, the locus is indeed a straight line.
- **Statement 3 is true**.

### Final Conclusion
All three statements are true.