Question

Set of values of ' $K$ ' for which roots of the quadratic $x^2-(2 K-1) x+K(K-1)=0$ are -

• A. both less than 2 is $K \in(2, \infty)$
• B. of opposite sign is $K \in(-\infty, 0) \cup(1, \infty)$
• C. of same sign is $K \in(-\infty, 0) \bigcup(1, \infty)$
• D. both greater than 2 is $K \in(2, \infty)$

Answer 1

Answer: (C)

(A)
Expression is $\left( x ^2+( K + K -1) x + K ( K -1)=0\right.$
$\Rightarrow (x-K)(x-K+1)=0$
$\Rightarrow x=K \text { or } x=K-1 $
$\Rightarrow \text { greater part }< 2 \Rightarrow K< 2$
(B) Opposite sign $\left(\frac{ c }{ a }<0\right) \Rightarrow K ( K -1)<0$
$\Rightarrow K \in(0,1)$
(C) $K ( K -1) >0 \Rightarrow K \in(-\infty, 0) \cup(1, \infty)$
(D) $K -1 >2 \Rightarrow K >3 \Rightarrow K \in(3, \infty)$