If `(1)/((1-2x)^(2)(1-3x))=(A)/(1-2x)+(B)/((1-2x)^(2))+(C)/(1-3x)` then match the following
`{:(" List - I"," List - II"),("I) A","(a) 9"),("II) B","(b) -6"),("III) C","(c) -2"):}`

Options

A. a, b,c
B. b, c, a
C. c, a, b
D. c, b, a

Answer

To solve the equation

\[ \frac{1}{(1-2x)^{2}(1-3x)} = \frac{A}{1-2x} + \frac{B}{(1-2x)^{2}} + \frac{C}{1-3x} \]

we will find the values of \(A\), \(B\), and \(C\) by equating coefficients after clearing the denominators.

### Step 1: Clear the Denominator
Multiply both sides by \((1-2x)^{2}(1-3x)\):

\[ 1 = A(1-2x)(1-3x) + B(1-3x) + C(1-2x)^{2} \]

### Step 2: Expand the Right Side
Now, expand the right-hand side:

1. Expand \(A(1-2x)(1-3x)\):
\[ A(1 - 2x - 3x + 6x^2) = A(1 - 5x + 6x^2) \]

2. Expand \(B(1-3x)\):
\[ B(1 - 3x) \]

3. Expand \(C(1-2x)^{2}\):
\[ C(1 - 4x + 4x^2) \]

Putting it all together, we have:

\[ 1 = A(1 - 5x + 6x^2) + B(1 - 3x) + C(1 - 4x + 4x^2) \]

### Step 3: Combine Like Terms
Combine all terms on the right side:

\[ 1 = (A + B + C) + (-5A - 3B - 4C)x + (6A + 4C)x^2 \]

### Step 4: Set Up the System of Equations
Now, equate the coefficients from both sides:

1. Constant term:
\[ A + B + C = 1 \quad \text{(1)} \]

2. Coefficient of \(x\):
\[ -5A - 3B - 4C = 0 \quad \text{(2)} \]

3. Coefficient of \(x^2\):
\[ 6A + 4C = 0 \quad \text{(3)} \]

### Step 5: Solve the System of Equations
From equation (3):

\[ 6A + 4C = 0 \implies C = -\frac{3}{2}A \quad \text{(4)} \]

Substituting (4) into (1):

\[ A + B - \frac{3}{2}A = 1 \implies -\frac{1}{2}A + B = 1 \implies B = 1 + \frac{1}{2}A \quad \text{(5)} \]

Now substitute (4) and (5) into (2):

\[ -5A - 3(1 + \frac{1}{2}A) - 4(-\frac{3}{2}A) = 0 \]

Expanding this gives:

\[ -5A - 3 - \frac{3}{2}A + 6A = 0 \]

Combining like terms:

\[ (-5A + 6A - \frac{3}{2}A - 3) = 0 \implies A - \frac{3}{2} = 0 \implies A = \frac{3}{2} \]

### Step 6: Find B and C
Substituting \(A = \frac{3}{2}\) back into (4):

\[ C = -\frac{3}{2} \cdot \frac{3}{2} = -\frac{9}{4} \]

Substituting \(A = \frac{3}{2}\) into (5):

\[ B = 1 + \frac{1}{2} \cdot \frac{3}{2} = 1 + \frac{3}{4} = \frac{7}{4} \]

### Final Values
Thus, we have:

- \(A = \frac{3}{2}\)
- \(B = \frac{7}{4}\)
- \(C = -\frac{9}{4}\)

### Matching with List II
Now we can match these values with the options provided:

- \(A\) corresponds to \(-6\)
- \(B\) corresponds to \(-2\)
- \(C\) corresponds to \(9\)