19. Five mark exercise problems - VIII

1. If \(a\), \(b\) are real then show that the roots of the equation \((a - b)x^2 - 6(a + b)x - 9(a - b) = 0\) are real and unequal.

 

2. If the roots of the equation \((c^2 - ab)x^2 - 2(a^2 - bc)x + b^2 - ac = 0\) are real and equal, prove that either \(a = 0\) or \(a^3 + b^3 + c^3 = 3abc\).

 

3. If the roots of \((a - b)x^2 + (b - c)x + (c - a) = 0\) are real and equal, then prove that \(b\), \(a\), \(c\) are in arithmetic progression.