Problem 29 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(p, q, r\) are positive 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 \(r\)

Step-by-Step Solution

Answer

(A) \(\left|\frac{r}{p} - 7\right| \geq 4 \sqrt{3}\) is correct.

Step 1: Identify the relationship between p, q, r

Since \(p, q, r\) are in Arithmetic Progression (A.P), the middle term \(q\) can be expressed as \(q = \frac{p + r}{2}\).

Step 2: Review the condition for real roots

For a quadratic equation \(ax^2 + bx + c = 0\) to have real roots, the discriminant \(b^2 - 4ac\) must be greater than or equal to zero.

Step 3: Apply the real roots condition to the quadratic equation

Substituting \(p, q, r\) into the discriminant condition, we have:\[q^2 - 4pr \geq 0\]Substituting \(q = \frac{p + r}{2}\) gives:\[\left(\frac{p + r}{2}\right)^2 - 4pr \geq 0\]Expanding and simplifying gives:\[\frac{(p + r)^2}{4} - 4pr \geq 0\]\[(p + r)^2 \geq 16pr\]\[p^2 + r^2 + 2pr \geq 16pr\]\[p^2 + r^2 \geq 14pr\].

Step 4: Select the correct option

The inequality \(p^2 + r^2 \geq 14pr\) suggests that, generally, because \(p\) and \(r\) are positive, it leads to constraints when comparing them. This inequality can be transformed using properties of inequalities and modulus operations to show:\[\left|\frac{r}{p} - 7\right| \geq 4\sqrt{3}\]Thus, the statement aligns with option (A).

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental concept in algebra, typically expressed in the form \(ax^2 + bx + c = 0\). In this format, \(a\), \(b\), and \(c\) are constants, and \(a\) is not equal to zero. This is essential because if \(a\) were zero, the equation would not be quadratic but linear. Quadratic equations have a characteristic parabolic graph and can have up to two solutions, known as 'roots'.These equations play a crucial role in many areas of mathematics and science. They often represent relationships where one variable is squared, showing how these two quantities interact. Solving a quadratic equation can be approached through factoring, completing the square, or using the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).Understanding quadratic equations is vital for solving real-life problems, such as calculating projectile motion or optimizing manufacturing processes.

Real Roots Condition

For a quadratic equation to have real roots, its discriminant must be non-negative. This is a crucial condition as it dictates the nature of the roots. The discriminant of a quadratic equation is given by \(b^2 - 4ac\). If this value is greater than or equal to zero, the quadratic equation will have real roots.

• If \(b^2 - 4ac > 0\), the quadratic equation has two distinct real roots.
• If \(b^2 - 4ac = 0\), it has exactly one real root, known as a double root.
• If \(b^2 - 4ac < 0\), the equation has no real roots but two complex roots.

In the given problem, ensuring that the quadratic equation \(px^2 + qx + r = 0\) has real roots involves verifying the condition that the discriminant \(q^2 - 4pr\) satisfies \(q^2 - 4pr \geq 0\). This leads to assessing the relationships between \(p, q,\) and \(r\), especially since they are stated to be in an arithmetic progression.

Discriminant

The discriminant is a key component in understanding the nature of the roots of a quadratic equation. Represented as \(b^2 - 4ac\), the discriminant helps determine whether the equation will have real or complex roots.The value of the discriminant provides specific insights:

• A positive discriminant indicates two distinct real roots. This means the parabola will intersect the x-axis at two points.
• A zero discriminant indicates exactly one real root. Here, the parabola just touches the x-axis, showing that the vertex lies on the x-axis.
• A negative discriminant reveals that there are no real roots, resulting in a parabola that doesn't cross the x-axis at all, implying complex roots.

In the context of the exercise, substituting the arithmetic progression condition \(q = \frac{p+r}{2}\) into the discriminant expression \(q^2 - 4pr\) helps determine whether real roots exist. Simplifying and analyzing this leads to findings about the relationships between \(p\), \(q\), and \(r\), assisting in selecting the correct answer based on mathematical inequalities.