If p, q, r are + ve and are in A . P., the roots of...

If $p, q, r$ are $+$ ve and are in $A . P$., the roots of quadratic equation $p x^{2}+$ $q x+r=0$ are all real for
(a) $\left|\frac{r}{p}-7\right| \geq 4 \sqrt{3}$
(b) $\left|\frac{p}{r}-7\right| \geq 4 \sqrt{3}$
(c) all $p$ and $r$
(d) no $p$ and $r$

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Key Concepts

Parameter Inequalities

Deriving inequalities involving parameters often follows from substituting relationships (like the arithmetic progression condition) into the discriminant condition. This process yields inequalities that the ratios of the coefficients must satisfy to ensure real roots, forming a key step in bridging abstract relationships to specific solution criteria.

Quadratic Equation

A quadratic equation is a second?degree polynomial equation in the form ax² + bx + c = 0. Its properties, such as the nature of its roots (real or complex), are determined primarily by its coefficients and the discriminant.

Discriminant

The discriminant of a quadratic equation, given by b² - 4ac, is key in determining the nature of the roots. For the equation to have real roots, the discriminant must be non-negative. This concept is crucial when analyzing quadratic equations with parameters affecting the coefficients.

Arithmetic Progression

When coefficients of a quadratic equation are in arithmetic progression, they satisfy a linear relation, typically written as b = (a + c)/2. This relation reduces the independent parameters in the equation and has significant implications when substituting into the discriminant.

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