1 questions with worked solutions.
Q1. Suppose a, b, c are in arithmetic progression and alpha, beta, gamma are positive real numbers in geometric progression. Consider the determinant D formed as follows: Row 1: (x+a), (x² + log(alpha)), k Row 2: (x+b), (x² + log(beta)), k Row 3: (x+c), (x² + log(gamma)), k Which of the following is true?
- D = 0 is an identity (true for all x)
- D = 0 has a root x = 1
- D = 0 has a root x = 0
- D = 0 has real and identical roots
Answer: D = 0 is an identity (true for all x)
Since a, b, c are in AP: b = (a+c)/2, so Row 2 = (Row 1 + Row 3)/2, making the rows linearly dependent and D = 0 for all x. This is an identity.