Let a,b and c be three distinct real roots of the cubic `x ^(3) +2x ^(2)-4x-4=0.` If the equation `x ^(3) +qx ^(2)+rx+le =0` has roots `1/a, 1/b and 1/c,` then the vlaue of `(q+r+s)` is equal to :

Options

A. `3/4`
B. `1/2`
C. `1/4`
D. `1/6`

Answer

To solve the problem, we need to find the coefficients \( q \), \( r \), and \( s \) of the cubic equation whose roots are \( \frac{1}{a} \), \( \frac{1}{b} \), and \( \frac{1}{c} \), where \( a, b, c \) are the roots of the cubic equation \( x^3 + 2x^2 - 4x - 4 = 0 \).

### Step 1: Identify the roots and their properties
The roots \( a, b, c \) satisfy the following relationships derived from Vieta's formulas:

1. \( a + b + c = -\frac{\text{coefficient of } x^2}{\text{coefficient of } x^3} = -\frac{-2}{1} = -2 \)
2. \( ab + ac + bc = \frac{\text{coefficient of } x}{\text{coefficient of } x^3} = -4 \)
3. \( abc = -\frac{\text{constant term}}{\text{coefficient of } x^3} = -\frac{-4}{1} = 4 \)

### Step 2: Find \( q \)
The sum of the roots of the new cubic equation \( x^3 + qx^2 + rx + s = 0 \) is given by:

\[ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{bc + ac + ab}{abc} \]

Substituting the values we found:

\[ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{-4}{4} = -1 \]

Thus, \( q = -\left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) = -(-1) = 1 \).

### Step 3: Find \( r \)
The sum of the products of the roots taken two at a time is given by:

\[ \frac{1}{ab} + \frac{1}{ac} + \frac{1}{bc} = \frac{c + b + a}{abc} \]

Substituting the values:

\[ \frac{1}{ab} + \frac{1}{ac} + \frac{1}{bc} = \frac{-2}{4} = -\frac{1}{2} \]

Thus, \( r = \frac{1}{ab} + \frac{1}{ac} + \frac{1}{bc} = -\frac{1}{2} \).

### Step 4: Find \( s \)
The product of the roots is given by:

\[ \frac{1}{abc} \]

Substituting the value we found:

\[ s = \frac{1}{4} \]

### Step 5: Calculate \( q + r + s \)
Now we can find \( q + r + s \):

\[ q + r + s = 1 - \frac{1}{2} + \frac{1}{4} \]

To combine these, we can convert them to a common denominator (which is 4):

\[ q + r + s = \frac{4}{4} - \frac{2}{4} + \frac{1}{4} = \frac{4 - 2 + 1}{4} = \frac{3}{4} \]

### Final Answer
Thus, the value of \( q + r + s \) is:

\[ \frac{3}{4} \]