2023 · Q50
Consider the following statements:
- 1One of the roots of the equation (a + b + c)x² − (2a + 2b)x + (a + b − c) = 0 is always less than 1 if a, b and c are all positive.
- 2One of the roots of the equation is always negative if a, b and c are all negative.
Which of the statements given above is/are correct?
Elementary MathematicsAlgebra
- A1 only
- B2 only
- CBoth 1 and 2
- DNeither 1 nor 2
Explanation
Substituting x = 1 gives (a+b+c) − (2a+2b) + (a+b−c) = 0, so x = 1 is always a root. The other root equals the product of roots, (a+b−c)/(a+b+c); for positive a, b, c this is less than 1 since a+b−c < a+b+c. Hence statement 1 is correct. For statement 2, with a, b, c all negative (write a = −a′ etc.), the other root is (a′+b′−c′)/(a′+b′+c′), which is negative only when c′ > a′+b′ — not always (e.g. a′=b′=c′=1 gives +1/3). So statement 2 is false. The answer is (A) 1 only.