If the equation \(x^2-2\theta x+3\theta=0\) has two real and distinct roots, which condition on \(\theta\) is correct?

Correct answer: \(\theta<0\) या \(\theta>3\)

A quadratic equation has two real and distinct roots only when its discriminant is positive. Here, \(a=1\), \(b=-2\theta\), and \(c=3\theta\), so \(D=b^2-4ac=4\theta^2-12\theta=4\theta(\theta-3)\). Therefore, \(4\theta(\theta-3)>0\), which gives \(\theta<0\) or \(\theta>3\). At \(\theta=0\) and \(\theta=3\), the roots are equal, while for \(0<\theta<3\), the roots are not real. Exam tip: For two real and distinct roots, always impose \(D>0\).