Complex Numbers And Quadratic Equations question 173
Question: The values of $ ‘a’ $ for which $ (a^{2}-1)x^{2}+2(a-1)x+2 $ is positive for any $ x $ are [UPSEAT 2001]
Options:
A) $ a\ge 1 $
B) $ a\le 1 $
C) $ a>-3 $
D) $ a<-3 $ or $ a>1 $
Show Answer
Answer:
Correct Answer: D
Solution:
We know that the expression $ ax^{2}+bx+c>0 $ for all x, if $ a>0 $ and $ b^{2}<4ac $ \ $ (a^{2}-1)x^{2}+2(a-1)x+2 $ is positive for all x if $ a^{2}-1>0 $ and $ 4{{(a-1)}^{2}}-8(a^{2}-1)<0 $
Þ $ a^{2}-1>0 $ and $ -4(a-1)(a+3)<0 $
Þ $ a^{2}-1>0 $ and $ (a-1)(a+3)>0 $
Þ $ a^{2}>1 $ and $ a<-3ora>1 $
Þ $ a<-3 $ or $ a>1 $
BETA
