Sequence And Series Question 5
Question: If $ p,\ q,\ r $ are in A.P. and are positive, the roots of the quadratic equation $ px^{2}+qx+r=0 $ are all real for
[IIT 1995]
Options:
A) $ | ,\frac{r}{p}-7\ |\ \ge 4\sqrt{3} $
B) $ | \ \frac{p}{r}-7\ |\ <4\sqrt{3} $
C) All $ p $ and $ r $
D) No $ p $ and $ r $
Show Answer
Answer:
Correct Answer: A
Solution:
$ p,\ q,\ r $ are positive and are in A.P.
$ \therefore \ q=\frac{p+r}{2} $ ……(i) $ \because $ The roots of $ px^{2}+qx+r=0 $ are real
$ \Rightarrow $ $ q^{2}\ge 4pr $
$ \Rightarrow $ $ {{[ \frac{p+r}{2} ]}^{2}}\ge 4pr $ [using (i)]
$ \Rightarrow $ $ p^{2}+r^{2}-14pr\ge 0 $
$ \Rightarrow $ $ {{( \frac{r}{p} )}^{2}}-14( \frac{r}{p} )+1\ge 0 $ $ (\because \ p>0\ and\ p\ne 0) $
$ \Rightarrow $ $ {{( \frac{r}{p}-7 )}^{2}}-48\ge 0 $
$ \Rightarrow $ $ {{( \frac{r}{p}-7 )}^{2}}-{{(4\sqrt{3})}^{2}}\ge 0 $
$ \Rightarrow $ $ | \ \frac{r}{p}-7\ |\ \ge 4\sqrt{3} $ .
BETA
