Let a, b, c be distinct real number and D be the determinant given by
`D = |(a,1,1),(1,b,1),(1,1,c)|`
Statement1 : If `D gt 0 " then " abc gt - 8`
Statement - 2: `A.M. gt G.M.`.

Options

A. Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 5
B. Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 5
C. Statement 1 is true, Statement 2 is False
D. Statement 1 is False, Statement 2 is true

Explanation

To solve the problem step by step, we will calculate the determinant \( D \) and analyze the statements provided.

### Step 1: Calculate the Determinant \( D \)

Given the determinant:

\[ D = \begin{vmatrix} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \end{vmatrix} \]

We can calculate this determinant using the formula for a \( 3 \times 3 \) determinant:

\[ D = a \begin{vmatrix} b & 1 \\ 1 & c \end{vmatrix} - 1 \begin{vmatrix} 1 & 1 \\ 1 & c \end{vmatrix} + 1 \begin{vmatrix} 1 & b \\ 1 & 1 \end{vmatrix} \]

Calculating the \( 2 \times 2 \) determinants:

1. \( \begin{vmatrix} b & 1 \\ 1 & c \end{vmatrix} = bc - 1 \)

2. \( \begin{vmatrix} 1 & 1 \\ 1 & c \end{vmatrix} = c - 1 \)

3. \( \begin{vmatrix} 1 & b \\ 1 & 1 \end{vmatrix} = 1 - b \)

Substituting these back into the determinant \( D \):

\[ D = a(bc - 1) - (c - 1) + (1 - b) \]

Expanding this gives:

\[ D = abc - a - c + 1 + 1 - b \]

Combining like terms:

\[ D = abc - (a + b + c) + 2 \]

### Step 2: Analyze Statement 1

Statement 1 states: If \( D > 0 \), then \( abc > -8 \).

From our expression for \( D \):

\[ D > 0 \implies abc - (a + b + c) + 2 > 0 \]

Rearranging gives:

\[ abc + 2 > a + b + c \]

### Step 3: Analyze Statement 2

Statement 2 states: \( A.M. > G.M. \).

The Arithmetic Mean (A.M.) of \( a, b, c \) is:

\[ A.M. = \frac{a + b + c}{3} \]

The Geometric Mean (G.M.) of \( a, b, c \) is:

\[ G.M. = (abc)^{1/3} \]

According to the A.M.-G.M. inequality:

\[ \frac{a + b + c}{3} \geq (abc)^{1/3} \]

This implies:

\[ a + b + c \geq 3(abc)^{1/3} \]

### Step 4: Combine Results

From \( D > 0 \):

\[ abc + 2 > a + b + c \]

Using the A.M.-G.M. inequality:

\[ a + b + c \geq 3(abc)^{1/3} \]

Substituting this into our previous inequality:

\[ abc + 2 > 3(abc)^{1/3} \]

Letting \( x = (abc)^{1/3} \), we rewrite:

\[ x^3 + 2 > 3x \]

Rearranging gives:

\[ x^3 - 3x + 2 > 0 \]

### Step 5: Find Roots and Analyze

To find the roots of \( x^3 - 3x + 2 = 0 \), we can use the Rational Root Theorem or synthetic division. Testing \( x = 1 \):

\[ 1^3 - 3(1) + 2 = 0 \]

Thus, \( x - 1 \) is a factor. We can factor the cubic polynomial:

\[ x^3 - 3x + 2 = (x - 1)(x^2 + x - 2) \]

Factoring \( x^2 + x - 2 \):

\[ x^2 + x - 2 = (x - 1)(x + 2) \]

Thus:

\[ x^3 - 3x + 2 = (x - 1)^2(x + 2) \]

### Step 6: Determine Inequality

The inequality \( (x - 1)^2(x + 2) > 0 \) holds when:

1. \( x - 1 \neq 0 \) (i.e., \( x \neq 1 \))
2. \( x + 2 > 0 \) (i.e., \( x > -2 \))

Since \( x = (abc)^{1/3} \), we conclude:

\[ (abc)^{1/3} > -2 \implies abc > -8 \]

### Conclusion

Thus, we have shown that if \( D > 0 \), then \( abc > -8 \). Therefore, both statements are true, and Statement 2 provides a valid explanation for Statement 1.