If `ax^(2) + bx + c = 0 , a , b, c in` R has no real roots, and if ` c lt ` 0, the which of the following is ture ? (a) a` lt` 0 (b) a + b + c `gt` 0 (c) ` a +b +c lt ` 0

Explanation

To solve the problem, we need to analyze the given quadratic equation \( ax^2 + bx + c = 0 \) under the conditions that it has no real roots and that \( c < 0 \).

### Step-by-Step Solution:

1. **Understanding the Condition of No Real Roots**:
A quadratic equation has no real roots if its discriminant \( D \) is less than 0. The discriminant for the equation \( ax^2 + bx + c = 0 \) is given by:
\[ D = b^2 - 4ac \]
For the equation to have no real roots, we need:
\[ D < 0 \implies b^2 - 4ac < 0 \]

2. **Using the Condition \( c < 0 \)**:
Since we know that \( c < 0 \), we can substitute this into our discriminant condition:
\[ b^2 < 4ac \]
Since \( c \) is negative, \( 4ac \) will also be negative if \( a \) is positive. Therefore, for \( b^2 < 4ac \) to hold true, \( a \) must be negative.

3. **Conclusion about \( a \)**:
From the analysis above, we conclude that:
\[ a < 0 \]
This corresponds to option (a).

4. **Finding \( a + b + c \)**:
Now, we need to check the expression \( a + b + c \). Since \( c < 0 \) and \( a < 0 \), we can analyze \( a + b + c \):
\[ a + b + c = (a + c) + b \]
Since both \( a \) and \( c \) are negative, \( a + c < 0 \). Thus, \( a + b + c \) could be either positive or negative depending on the value of \( b \).

5. **Evaluating \( a + b + c \)**:
However, since \( b^2 < 4ac \) and \( c < 0 \), we can infer that \( b \) must also be negative or small enough to keep \( a + b + c < 0 \). Thus, we conclude:
\[ a + b + c < 0 \]
This corresponds to option (c).

### Final Conclusion:
The true statements based on the conditions given are:
- (a) \( a < 0 \) is true.
- (b) \( a + b + c > 0 \) is not necessarily true.
- (c) \( a + b + c < 0 \) is true.

Thus, the correct answer is option (c).