The mean of x and `(1)/(x)` is N then the mean of ` x^(2) and (1)/( x^(2))` is

Options

A. `N^(2) `
B. ` 2N^(2) - 1 `
C. ` N^(2) - 2 `
D. ` 4 N^(2) - 2 `

Answer

To solve the problem step by step, we start with the information given and derive the required mean.

### Step 1: Understand the Mean of x and (1/x)
The mean of \( x \) and \( \frac{1}{x} \) is given as \( N \). Therefore, we can express this mathematically as:
\[ \text{Mean} = \frac{x + \frac{1}{x}}{2} = N \]
From this, we can derive:
\[ x + \frac{1}{x} = 2N \]

**Hint:** Remember that the mean is calculated by summing the values and dividing by the number of values.

### Step 2: Square Both Sides
Next, we want to find the mean of \( x^2 \) and \( \frac{1}{x^2} \). To do this, we first square the equation we derived:
\[ \left( x + \frac{1}{x} \right)^2 = (2N)^2 \]
Expanding the left side gives:
\[ x^2 + 2 + \frac{1}{x^2} = 4N^2 \]

**Hint:** When squaring a binomial, remember to apply the formula \( (a + b)^2 = a^2 + 2ab + b^2 \).

### Step 3: Rearranging the Equation
From the equation \( x^2 + 2 + \frac{1}{x^2} = 4N^2 \), we can isolate \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} = 4N^2 - 2 \]

**Hint:** To isolate a term, you can move other terms to the opposite side of the equation.

### Step 4: Calculate the Mean of \( x^2 \) and \( \frac{1}{x^2} \)
Now that we have \( x^2 + \frac{1}{x^2} \), we can find the mean:
\[ \text{Mean} = \frac{x^2 + \frac{1}{x^2}}{2} = \frac{4N^2 - 2}{2} \]
This simplifies to:
\[ \text{Mean} = 2N^2 - 1 \]

**Hint:** The mean is calculated by dividing the sum of the observations by the number of observations.

### Final Answer
Thus, the mean of \( x^2 \) and \( \frac{1}{x^2} \) is:
\[ \boxed{2N^2 - 1} \]