For what values of `m` the equation `(1+m)x^(2)-2(1+3m)x+(1+8m)=0` has `(m in R)`
(i) both roots are imaginary?
(ii) both roots are equal?
(iii) both roots are real and distinct?
(iv) both roots are positive?
(v) both roots are negative?
(vi) roots are opposite in sign?
(vii)roots are equal in magnitude but opposite in sign?
(viii) atleast one root is positive?
(iv) atleast one root is negative?
(x) roots are in the ratio `2:3`?

Explanation

To solve the given quadratic equation \( (1+m)x^2 - 2(1+3m)x + (1+8m) = 0 \) for the various conditions related to the roots, we will analyze the discriminant and the nature of the roots based on the coefficients of the equation.

### Step 1: Identify Coefficients
The coefficients of the quadratic equation are:
- \( A = 1 + m \)
- \( B = -2(1 + 3m) \)
- \( C = 1 + 8m \)

### Step 2: Calculate the Discriminant
The discriminant \( D \) of a quadratic equation \( Ax^2 + Bx + C = 0 \) is given by:
\[ D = B^2 - 4AC \]
Substituting the values of \( A \), \( B \), and \( C \):
\[ D = [-2(1 + 3m)]^2 - 4(1 + m)(1 + 8m) \]
Calculating \( D \):
\[ D = 4(1 + 3m)^2 - 4(1 + m)(1 + 8m) \]
\[ = 4[(1 + 9m + 9m^2) - (1 + 9m + 8m^2)] \]
\[ = 4(1 + 9m + 9m^2 - 1 - 9m - 8m^2) \]
\[ = 4(m^2 + m) \]
\[ = 4m(m + 1) \]

### Step 3: Analyze Conditions for Roots

#### (i) Both roots are imaginary
For the roots to be imaginary, the discriminant must be less than zero:
\[ 4m(m + 1) < 0 \]
This inequality holds when:
- \( m < -1 \) or \( 0 < m < 1 \)

#### (ii) Both roots are equal
For the roots to be equal, the discriminant must be equal to zero:
\[ 4m(m + 1) = 0 \]
This gives:
- \( m = 0 \) or \( m = -1 \)

#### (iii) Both roots are real and distinct
For the roots to be real and distinct, the discriminant must be greater than zero:
\[ 4m(m + 1) > 0 \]
This inequality holds when:
- \( m < -1 \) or \( m > 0 \)

#### (iv) Both roots are positive
The sum of the roots \( \frac{-B}{A} > 0 \) and the product of the roots \( \frac{C}{A} > 0 \):
1. \( \frac{2(1 + 3m)}{1 + m} > 0 \) implies \( m > -\frac{1}{3} \)
2. \( \frac{1 + 8m}{1 + m} > 0 \) implies \( m > -\frac{1}{8} \)

Combining these conditions with \( D \geq 0 \):
- \( m \in (-\frac{1}{8}, \infty) \)

#### (v) Both roots are negative
The sum of the roots \( \frac{-B}{A} < 0 \) and the product of the roots \( \frac{C}{A} > 0 \):
1. \( \frac{2(1 + 3m)}{1 + m} < 0 \) implies \( m < -\frac{1}{3} \)
2. \( \frac{1 + 8m}{1 + m} > 0 \) implies \( m > -\frac{1}{8} \)

No common values exist, so there are no solutions.

#### (vi) Roots are opposite in sign
The product of the roots \( \frac{C}{A} < 0 \):
\[ 1 + 8m < 0 \implies m < -\frac{1}{8} \]
And \( D > 0 \) gives:
\[ m < -1 \text{ or } m > 0 \]
Combining gives \( m < -1 \).

#### (vii) Roots are equal in magnitude but opposite in sign
The sum of the roots \( \frac{-B}{A} = 0 \):
\[ 2(1 + 3m) = 0 \implies m = -\frac{1}{3} \]
And \( D > 0 \) gives \( m < 0 \) or \( m > 3 \). Thus, \( m = -\frac{1}{3} \).

#### (viii) At least one root is positive
This condition can be satisfied if either both roots are positive or one is positive:
- From (iv), \( m \in (-\frac{1}{8}, \infty) \).

#### (ix) At least one root is negative
This condition can be satisfied if either both roots are negative or one is negative:
- From (v), \( m < -\frac{1}{3} \) or \( m > 0 \).

#### (x) Roots are in the ratio \( 2:3 \)
Let the roots be \( 2k \) and \( 3k \):
1. Sum: \( 5k = \frac{2(1 + 3m)}{1 + m} \)
2. Product: \( 6k^2 = \frac{1 + 8m}{1 + m} \)

Solving these equations gives the values of \( m \).

### Summary of Results
- (i) \( m \in (-1, 0) \)
- (ii) \( m = 0 \) or \( m = -1 \)
- (iii) \( m < -1 \) or \( m > 0 \)
- (iv) \( m \in (-\frac{1}{8}, \infty) \)
- (v) No solutions
- (vi) \( m < -1 \)
- (vii) \( m = -\frac{1}{3} \)
- (viii) \( m \in (-\frac{1}{8}, \infty) \)
- (ix) \( m < -\frac{1}{3} \) or \( m > 0 \)
- (x) Solve the equations derived from the ratio condition.