Question

Let $\alpha, \beta$ are roots of the equation $\left(t-t^2\right) x^2-(1-t) x+t^2=0$ for any real number $t$ belonging to $\left(0, \frac{1}{2}\right)$ than

• A. $\alpha$ and $\beta$ are always imaginary $\forall t \in\left(0, \frac{1}{2}\right)$
• B. for infinitely many values of t, corresponding root $\alpha$ is rational and for infinitely many values of 1 corresponding root $\alpha$ is irrational
• C. $\alpha$ and $\beta$ are real and positive such that one root is greater than 1 and other is smaller than 1
• D. $\displaystyle\lim _{t \rightarrow \frac{1^{-}}{2}}|\alpha-\beta|=0$

Answer 1

Answer: (B), (C), (D)

Correct answer is (b) for infinitely many values of t, corresponding root $\alpha$ is rational and for infinitely many values of 1 corresponding root $\alpha$ is irrationalCorrect answer is (c) $\alpha$ and $\beta$ are real and positive such that one root is greater than 1 and other is smaller than 1Correct answer is (d) $\displaystyle\lim _{t \rightarrow \frac{1^{-}}{2}}|\alpha-\beta|=0$