Assertion (A) : If `(1)/(x^(3)(x+2))=A/x + B/x^(2) +C/x^(3) + D/(x+2)` then `A = 1/8, B= -1/4, C = 1/2, D= -1/8`
Reason (R) : `(1)/(x^(3)(x+a))=(1)/(a^(3)x)-(1)/(a^(2)x^(2))+(1)/(ax^(3))-(1)/(a^(3)(x+a))`

Options

A. Both A & R are true and R is correct explanation of A
B. Both A & R are true and R is not correct explanation of A
C. A is true but R is false
D. A is false but R is true

Answer

To solve the assertion and verify the values of A, B, C, and D in the equation

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

we will follow these steps:

### Step 1: Set Up the Equation

Start with the equation:

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

Multiply both sides by the common denominator \(x^3(x+2)\) to eliminate the fractions:

\[ 1 = A \cdot x^2(x+2) + B \cdot x(x+2) + C \cdot (x+2) + D \cdot x^3 \]

### Step 2: Expand the Right Side

Now, expand the right-hand side:

\[ 1 = A(x^3 + 2x^2) + B(x^2 + 2x) + C(x + 2) + Dx^3 \]

Combine like terms:

\[ 1 = (A + D)x^3 + (2A + B)x^2 + (2B + C)x + 2C \]

### Step 3: Set Up the System of Equations

Now, we can compare coefficients from both sides of the equation. Since the left side is just 1 (which can be written as \(0x^3 + 0x^2 + 0x + 1\)), we can set up the following equations:

1. \(A + D = 0\) (coefficient of \(x^3\))
2. \(2A + B = 0\) (coefficient of \(x^2\))
3. \(2B + C = 0\) (coefficient of \(x\))
4. \(2C = 1\) (constant term)

### Step 4: Solve the System of Equations

From equation (4):

\[ C = \frac{1}{2} \]

Substituting \(C\) into equation (3):

\[ 2B + \frac{1}{2} = 0 \implies 2B = -\frac{1}{2} \implies B = -\frac{1}{4} \]

Substituting \(B\) into equation (2):

\[ 2A - \frac{1}{4} = 0 \implies 2A = \frac{1}{4} \implies A = \frac{1}{8} \]

Substituting \(A\) into equation (1):

\[ \frac{1}{8} + D = 0 \implies D = -\frac{1}{8} \]

### Step 5: Summary of Values

Thus, we have found:

- \(A = \frac{1}{8}\)
- \(B = -\frac{1}{4}\)
- \(C = \frac{1}{2}\)
- \(D = -\frac{1}{8}\)

### Conclusion

The assertion is true, as we have verified the values of A, B, C, and D.

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