If p,;q,;r are in A.P. and are positive, the | vedclass

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Question

(a) $p,\;q,\;r$ are positive and are in $A.P.$

$	herefore \;q = \frac{{p + r}}{2}$ ......(i)

The roots of $p{x^2} + qx + r = 0$ are real

$ \

Vedclass · Accepted Answer

$\left| {\,\frac{r}{p} - 7\;} \right|\; \ge 4\sqrt 3 $

Vedclass · Answer

(a) $p,\;q,\;r$ are positive and are in $A.P.$

$	herefore \;q = \frac{{p + r}}{2}$ ......(i)

The roots of $p{x^2} + qx + r = 0$ are real

$ \Rightarrow $ ${q^2} \ge 4pr$

$ \Rightarrow $${\left[ {\frac{{p + r}}{2}} ight]^2} \ge 4pr$ [using (i)]

$ \Rightarrow $ ${p^2} + {r^2} - 14pr \ge 0$

$ \Rightarrow $ ${\left( {\frac{r}{p}} ight)^2} - 14\left( {\frac{r}{p}} ight) + 1 \ge 0$

$(\because \;p > 0\;{	ext{and}}\;p 
e 0)$

$ \Rightarrow $ ${\left( {\frac{r}{p} - 7} ight)^2} - 48 \ge 0$ 

$ \Rightarrow $${\left( {\frac{r}{p} - 7} ight)^2} - {(4\sqrt 3 )^2} \ge 0$

$ \Rightarrow $ $\left| {\;\frac{r}{p} - 7\;} ight|\; \ge 4\sqrt 3 $.

If $p,\;q,\;r$ are in $A.P.$ and are positive, the roots of the quadratic equation $p{x^2} + qx + r = 0$ are all real for

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