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

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(A) Given $p, q, r$ are in $A.P.$ and are positive.Therefore,$q = \frac{p + r}{2}$ ......$(i)$For the roots of $px^2 + qx + r = 0$ to be real,the disc

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

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(A) Given $p, q, r$ are in $A.P.$ and are positive.Therefore,$q = \frac{p + r}{2}$ ......$(i)$For the roots of $px^2 + qx + r = 0$ to be real,the discriminant $D \ge 0$.$D = q^2 - 4pr \ge 0$Substituting $(i)$ into the inequality:$\left( \frac{p + r}{2} \right)^2 - 4pr \ge 0$$p^2 + r^2 + 2pr - 16pr \ge 0$$p^2 + r^2 - 14pr \ge 0$Dividing by $p^2$ (since $p > 0$):$1 + \left( \frac{r}{p} \right)^2 - 14\left( \frac{r}{p} \right) \ge 0$$\left( \frac{r}{p} \right)^2 - 14\left( \frac{r}{p} \right) + 1 \ge 0$Completing the square:$\left( \frac{r}{p} - 7 \right)^2 - 49 + 1 \ge 0$$\left( \frac{r}{p} - 7 \right)^2 \ge 48$$\left( \frac{r}{p} - 7 \right)^2 \ge (4\sqrt{3})^2$Taking the square root on both sides:$\left| \frac{r}{p} - 7 \right| \ge 4\sqrt{3}$.

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

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