Statement -1 If the equation `(4p-3)x^(2)+(4q-3)x+r=0` is satisfied by `x=a,x=b` nad `x=c` (where a,b,c are distinct) then `p=q=3/4` and `r=0`
Statement -2 If the quadratic equation `ax^(2)+bx+c=0` has three distinct roots, then a, b and c are must be zero.

Options

A. Statement -1 is true, Statement -2 is true, Statement -2 is a correct explanation for Statement-1
B. Statement -1 is true, Statement -2 is true, Statement -2 is not a correct explanation for Statement -1
C. Statement -1 is true, Statement -2 is false
D. Statement -1 is false, Statement -2 is true

Explanation

To solve the given problem, we need to analyze the statements provided and prove their validity step by step.

### Step 1: Analyze Statement 1
We start with the equation given in Statement 1:
\[ (4p - 3)x^2 + (4q - 3)x + r = 0 \]
This equation is satisfied by three distinct roots \(x = a\), \(x = b\), and \(x = c\).

### Step 2: Use the Condition for Roots
Since a quadratic equation can have at most two distinct roots, if we assume that \(x = a\), \(x = b\), and \(x = c\) are distinct roots, it implies that the polynomial must be identically zero. This means that all coefficients of the polynomial must equal zero.

### Step 3: Set Coefficients to Zero
From the equation, we can set the coefficients equal to zero:
1. \(4p - 3 = 0\)
2. \(4q - 3 = 0\)
3. \(r = 0\)

### Step 4: Solve for \(p\), \(q\), and \(r\)
Now we solve each of these equations:
1. From \(4p - 3 = 0\):
\[ 4p = 3 \implies p = \frac{3}{4} \]
2. From \(4q - 3 = 0\):
\[ 4q = 3 \implies q = \frac{3}{4} \]
3. From \(r = 0\):
\[ r = 0 \]

### Conclusion for Statement 1
Thus, we have shown that if the equation is satisfied by three distinct roots, then:
\[ p = \frac{3}{4}, \quad q = \frac{3}{4}, \quad r = 0 \]
This proves Statement 1 to be true.

### Step 5: Analyze Statement 2
Now, we examine Statement 2:
"If the quadratic equation \(ax^2 + bx + c = 0\) has three distinct roots, then \(a\), \(b\), and \(c\) must be zero."

### Step 6: Understand the Implication
A quadratic equation can only have a maximum of two distinct roots. If we assume it has three distinct roots, it implies that the polynomial must be the zero polynomial. Therefore, all coefficients \(a\), \(b\), and \(c\) must indeed be zero.

### Conclusion for Statement 2
Thus, Statement 2 is also true as it logically follows from the conclusion of Statement 1.

### Final Conclusion
Both statements are true:
- Statement 1 is true with \(p = \frac{3}{4}\), \(q = \frac{3}{4}\), and \(r = 0\).
- Statement 2 is true as a consequence of Statement 1.

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