Statement -1 : A root of the equation `(2^(10)-3)x^(2)-2^(11)x +2^(10)+3 =0 " is " 1`
and
Statement-2 : The sum of the coefficients of a quadratic equation is zero, then 1 is a root of the equation.

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 Flase, Statement -2 is True

Answer

To solve the question, we need to verify the two statements provided and determine if they are true and if one explains the other.

### Step 1: Verify Statement 1
We need to check if \( x = 1 \) is a root of the quadratic equation:

\[ (2^{10} - 3)x^2 - 2^{11}x + (2^{10} + 3) = 0 \]

Substituting \( x = 1 \):

\[ (2^{10} - 3)(1^2) - 2^{11}(1) + (2^{10} + 3) = 0 \]

This simplifies to:

\[ (2^{10} - 3) - 2^{11} + (2^{10} + 3) = 0 \]

Combining like terms:

\[ 2^{10} - 3 - 2^{11} + 2^{10} + 3 = 0 \]

The \(-3\) and \(+3\) cancel out:

\[ 2^{10} + 2^{10} - 2^{11} = 0 \]

This can be rewritten as:

\[ 2 \cdot 2^{10} - 2^{11} = 0 \]

Since \(2^{11} = 2 \cdot 2^{10}\), we have:

\[ 2^{11} - 2^{11} = 0 \]

Thus, Statement 1 is confirmed to be true.

### Step 2: Verify Statement 2
Statement 2 claims that if the sum of the coefficients of a quadratic equation is zero, then \( x = 1 \) is a root.

Let the quadratic equation be represented as:

\[ ax^2 + bx + c = 0 \]

The sum of the coefficients is:

\[ a + b + c \]

If \( a + b + c = 0 \), we substitute \( x = 1 \):

\[ a(1^2) + b(1) + c = a + b + c = 0 \]

Thus, if the sum of the coefficients is zero, \( x = 1 \) is indeed a root. Therefore, Statement 2 is also confirmed to be true.

### Step 3: Explanation of Statements
Since Statement 1 is true and Statement 2 correctly explains why \( x = 1 \) is a root of the quadratic equation given in Statement 1, we conclude that both statements are true, and Statement 2 is the correct explanation of Statement 1.

### Conclusion
Both statements are true, and Statement 2 explains Statement 1. Therefore, the answer is that both statements are correct, and Statement 2 is the correct explanation of Statement 1.

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