If p, q, r are +ve and are in A.P., the roots of quadratic equation px2 + qx + r = 0 are all real for

For real roots q2−4pr≥0⇒p+r22−4pr≥0 (∵p,q,r are in A.P. )⇒p2+r2−14pr≥0⇒p2r2−14pr+1≥0⇒pr−7−48≥0 ⇒pr−7≥43

If p, q, r are +ve and are in A.P., the roots of quadratic equation px2 + qx + r = 0 are all real for

For real roots q2−4pr≥0⇒p+r22−4pr≥0 (∵p,q,r are in A.P. )⇒p2+r2−14pr≥0⇒p2r2−14pr+1≥0⇒pr−7−48≥0 ⇒pr−7≥43

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If p, q, r are +ve and are in A.P., the roots of quadratic equation px² + qx + r = 0 are all real for

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$|\frac{p}{r} - 7| \geq 4 \sqrt{3}$

no p and r

all p and r

$|\frac{r}{p} - 7| \geq 4 \sqrt{3}$

answer is B.

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Detailed Solution

For real roots $q^{2} - 4 pr \geq 0$

$\begin{array}{l} \Rightarrow {(\frac{p + r}{2})}^{2} - 4 pr \geq 0 (∵ p, q, r are in A.P.) \\ \Rightarrow p^{2} + r^{2} - 14 pr \geq 0 \Rightarrow \frac{p^{2}}{r^{2}} - 14 \frac{p}{r} + 1 \geq 0 \\ \Rightarrow (\frac{p}{r} - 7) - 48 \geq 0 \Rightarrow (\frac{p}{r} - 7) \geq 4 \sqrt{3} \end{array}$