49. If `[x^2-2x + a] = 0 `has no solution then

Options

A. `-oo lt a lt 1`
B. `2 le a lt oo`
C. `1 lt a lt 2`
D. ` a in R`

Explanation

To solve the problem, we need to analyze the expression \( [x^2 - 2x + a] = 0 \) and determine the conditions under which it has no solutions. Here, the brackets represent the greatest integer function.

### Step-by-Step Solution:

1. **Understanding the Greatest Integer Function**:
The expression \( [x^2 - 2x + a] = 0 \) implies that \( x^2 - 2x + a \) must lie in the interval \([0, 1)\) for it to equal 0. This means:
\[ 0 \leq x^2 - 2x + a < 1 \]

2. **Setting Up the Inequalities**:
From the above, we can derive two inequalities:
- \( x^2 - 2x + a \geq 0 \) (1)
- \( x^2 - 2x + a < 1 \) (2)

3. **Analyzing the First Inequality**:
The first inequality \( x^2 - 2x + a \geq 0 \) is a quadratic equation in \( x \). For this quadratic to have no real solutions, its discriminant must be less than or equal to zero:
\[ D = b^2 - 4ac = (-2)^2 - 4(1)(a) = 4 - 4a \]
Setting the discriminant \( D \leq 0 \):
\[ 4 - 4a \leq 0 \]
This simplifies to:
\[ a \geq 1 \]

4. **Analyzing the Second Inequality**:
The second inequality \( x^2 - 2x + a < 1 \) can be rearranged to:
\[ x^2 - 2x + (a - 1) < 0 \]
For this quadratic to have no real solutions, its discriminant must also be less than or equal to zero:
\[ D = (-2)^2 - 4(1)(a - 1) = 4 - 4(a - 1) = 4 - 4a + 4 = 8 - 4a \]
Setting this discriminant \( D < 0 \):
\[ 8 - 4a < 0 \]
This simplifies to:
\[ a > 2 \]

5. **Combining the Results**:
From the two inequalities derived:
- From the first inequality, we have \( a \geq 1 \).
- From the second inequality, we have \( a > 2 \).

The condition that satisfies both inequalities is:
\[ a > 2 \]

### Final Answer:
Thus, the values of \( a \) for which the equation \( [x^2 - 2x + a] = 0 \) has no solutions are:
\[ a \in (2, \infty) \]