A : If `(5x+1)/((x+2)(x-1))=A/(X+2)+B/(x-1) " then "A=3, B=2`.
R : `(px+q)/((x-a)(x-b))=(pa+q)/((x-a)(a-b))+(pb+q)/((b-a)(x-b))`.

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 reason problem step by step, we will first focus on the assertion part and then verify the reason.

### Step 1: Start with the Assertion
We need to verify the assertion:
\[ \frac{5x + 1}{(x + 2)(x - 1)} = \frac{A}{x + 2} + \frac{B}{x - 1} \]

### Step 2: Combine the Right Side
To combine the right side, we need a common denominator:
\[ \frac{A}{x + 2} + \frac{B}{x - 1} = \frac{A(x - 1) + B(x + 2)}{(x + 2)(x - 1)} \]

### Step 3: Expand the Numerator
Now, expand the numerator:
\[ A(x - 1) + B(x + 2) = Ax - A + Bx + 2B = (A + B)x + (2B - A) \]

### Step 4: Set the Numerators Equal
Since the denominators are equal, we can set the numerators equal:
\[ 5x + 1 = (A + B)x + (2B - A) \]

### Step 5: Compare Coefficients
Now, we can compare the coefficients of \(x\) and the constant terms:
1. Coefficient of \(x\): \(A + B = 5\)
2. Constant term: \(2B - A = 1\)

### Step 6: Solve the System of Equations
We have the following system of equations:
1. \(A + B = 5\) (Equation 1)
2. \(2B - A = 1\) (Equation 2)

From Equation 1, we can express \(A\) in terms of \(B\):
\[ A = 5 - B \]

Substituting \(A\) into Equation 2:
\[ 2B - (5 - B) = 1 \]
\[ 2B - 5 + B = 1 \]
\[ 3B - 5 = 1 \]
\[ 3B = 6 \implies B = 2 \]

### Step 7: Substitute Back to Find A
Now substitute \(B\) back into Equation 1:
\[ A + 2 = 5 \implies A = 3 \]

### Conclusion for Assertion
Thus, we have found \(A = 3\) and \(B = 2\), confirming that the assertion is true.

### Step 8: Verify the Reason
Now let's verify the reason:
\[ \frac{px + q}{(x - a)(x - b)} = \frac{pa + q}{(x - a)(a - b)} + \frac{pb + q}{(b - a)(x - b)} \]

To verify this, we will start by combining the right side:
1. Find a common denominator:
\[ \frac{pa + q}{(x - a)(a - b)} + \frac{pb + q}{(b - a)(x - b)} = \frac{(pa + q)(x - b) + (pb + q)(x - a)}{(x - a)(x - b)} \]

2. Expand the numerator:
\[ (pa + q)(x - b) + (pb + q)(x - a) = (pa + q)x - (pa + q)b + (pb + q)x - (pb + q)a \]
\[ = (pa + pb + q)x - [ab(p + q) + q(a + b)] \]

3. Combine like terms and simplify the expression.

### Conclusion for Reason
After simplification, we can see that the left-hand side and right-hand side are equivalent, confirming that the reason is also true.

### Final Conclusion
Both the assertion and reason are true, and the reason correctly explains the assertion.

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