A uniform wire of resistance `R` is shaped into a regular `n-`sided polygon (`n` is even), The equivalent resistance between any two corners can have.
(i) the maximum value `(R )/(4)`
(ii) the minimum value `( R)/(n)`
(iii) the minimum value `R((n -1)/(n^2))`
(iv) the minimum value `(R )/(n)`.

Options

A. (i),(ii)
B. (i),(iii)
C. (i) only
D. (i),(iv)

Explanation

To solve the problem of finding the equivalent resistance between any two corners of a uniform wire shaped into a regular n-sided polygon (where n is even), we will analyze the configuration step by step.

### Step 1: Understanding the Configuration
We have a uniform wire of resistance \( R \) shaped into a regular n-sided polygon. Since \( n \) is even, we can visualize the polygon as having equal sides, with each side having a resistance of \( \frac{R}{n} \).

**Hint:** Remember that the total resistance of the wire is distributed evenly across the n sides.

### Step 2: Maximum Resistance Calculation
To find the maximum equivalent resistance between any two corners, we consider the case when \( n = 2 \) (the simplest polygon, a straight line). In this case, the resistance between the two corners is simply the sum of the two halves of the wire:

\[ R_{max} = \frac{R}{2} + \frac{R}{2} = R \]

However, since we are looking for the equivalent resistance between two adjacent corners in a polygon with more sides, we can find that the maximum resistance occurs when the two corners are directly connected through the wire, leading to:

\[ R_{max} = \frac{R}{4} \]

**Hint:** The maximum occurs when the two corners are adjacent, and the total resistance is divided among the paths available.

### Step 3: Minimum Resistance Calculation
To find the minimum equivalent resistance, we consider the case where the two corners are opposite each other in the polygon. For a regular n-sided polygon, the equivalent resistance between two opposite corners can be calculated as follows:

1. Each side has a resistance of \( \frac{R}{n} \).
2. The two corners are connected through \( n-1 \) sides in parallel.

The equivalent resistance \( R_{min} \) can be calculated using the formula for resistors in parallel:

\[ R_{min} = \frac{R/n}{n-1} = \frac{R(n-1)}{n^2} \]

**Hint:** When calculating minimum resistance, consider the paths available and how they combine in parallel.

### Step 4: Summary of Results
From our calculations, we can summarize:

- The maximum equivalent resistance between any two corners is \( \frac{R}{4} \).
- The minimum equivalent resistance between any two corners is \( \frac{R(n-1)}{n^2} \).

### Conclusion
Based on our analysis, the correct statements regarding the equivalent resistance between any two corners of the n-sided polygon are:
- (i) The maximum value is \( \frac{R}{4} \) - **Correct**
- (ii) The minimum value is \( \frac{R}{n} \) - **Incorrect**
- (iii) The minimum value is \( \frac{R(n-1)}{n^2} \) - **Correct**
- (iv) The minimum value is \( \frac{R}{n} \) - **Incorrect**

Thus, the correct options are (i) and (iii).