Question

Let \(a, b, c, p, q\) be real numbers. Suppose \(\alpha, \beta\) are the roots of the equation \(x^{2}+2 p x+q=0\) and \(\alpha, \frac{1}{\beta}\) are the roots of the equation \(a x^{2}+2 b x+c=0\), where \(\beta^{2} \notin\\{-1,0,1\\}\) STATEMENT-1: \(\quad\left(p^{2}-q\right)\left(b^{2}-a c\right) \geq 0\) and STATEMENT-2: \(b \neq p a\) or \(c \neq q a\) (A) STATEMENT- 1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1 (B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1 (C) STATEMENT-1 is True, STATEMENT- 2 is False (D) STATEMENT-1 is False, STATEMENT-2 is True

Short answer

STATEMENT-1 is True, STATEMENT-2 is also True; however, STATEMENT-2 is NOT a correct explanation for STATEMENT-1. The correct answer is Option (B).

Step-by-step solution

Identify the Relationships Between Roots and Coefficients for the First Equation

According to Vieta's formulas, for the equation \(x^{2}+2px+q=0\), the sum of the roots \(\alpha + \beta\) is equal to \(-\frac{2p}{1}\), and the product of the roots \(\alpha\beta\) is equal to \(\frac{q}{1}\). Thus, \(\alpha + \beta = -2p\) and \(\alpha\beta = q\).

Identify the Relationships Between Roots and Coefficients for the Second Equation

Similarly, for the equation \(ax^{2}+2bx+c=0\) with roots \(\alpha\) and \(\frac{1}{\beta}\), the sum of the roots is \(-\frac{2b}{a}\), and their product is \(\frac{c}{a}\). Therefore, \(\alpha + \frac{1}{\beta} = -\frac{2b}{a}\), and \(\alpha \cdot \frac{1}{\beta} = \frac{c}{a}\).

Use the Relationship Between the Roots to Form Equations

Since \(\alpha\) is a common root in both equations, equate their sum and products. For the sums, we have \(-2p = -\frac{2b}{a}\) leading to the equation \(ap = b\). For the products, we have \(q = \alpha\beta\) and \(\frac{c}{a} = \alpha \cdot \frac{1}{\beta}\), resulting in \(q = \alpha\beta\) and \(\frac{c}{a} = \frac{\alpha}{\beta}\). Multiplying the latter by \(\beta\), we get \(q\beta = c\).

Connect the Products of Roots

The relationship between the products of the roots gives us \(\alpha\beta^{2} = c\), since we know \(q\beta = c\). Therefore, we have \(q = \alpha\beta\) and \(q\beta^{2} = c\). This implies \(\beta^{2} = \frac{c}{q}\).

Analyze the Given Statements

Now we evaluate STATEMENT-1 \((p^{2}-q)(b^{2}-ac) \geq 0\). Since \(p = \frac{b}{a}\) and \(q \beta^{2} = c\), we substitute \(p\) and \(c\) in terms of \(b\), \(a\), \(q\), and \(\beta\): \(\left(\frac{b^2}{a^2}-q\right)\left(b^2-a\cdot q\beta^2\right) = \left(\frac{b^2}{a^2}-q\right)\left(b^2-a\cdot c\right)\). Since \(\beta^2\) cannot be -1, 0, or 1, \(\frac{b^2}{a^2} - q\) will never be 0 and therefore, \((p^2 - q)(b^2 - ac)\) coincides with the sign of \(b^2 - ac\). As a result, STATEMENT-1 is True when \(b^2 - ac \geq 0\).

Examine the Second Statement

To test STATEMENT-2, \(b eq pa\) or \(c eq qa\), we have already found that \(b = ap\) from equating the sums of the roots, which contradicts the first part of STATEMENT-2. However, since \(c = q\beta^2\) and \(\beta^2 eq 1\), \(c eq qa\) and thus meets only the second condition of STATEMENT-2. Therefore, STATEMENT-2 is True.

Link the Statements and Choose the Correct Option

Since we found that STATEMENT-1 is True when \(b^2 - ac \geq 0\) and STATEMENT-2 is True, we need to determine if STATEMENT-2 is a correct explanation for STATEMENT-1. Given that \(b = ap\), the inequality in STATEMENT-1 simply reflects the sign of \(b^2 - ac\), and since \(c = q\beta^2\), and \(\beta^2\) is not equal to 1, the inequality in STATEMENT-1 does not depend on the conditions stated in STATEMENT-2. Therefore, STATEMENT-2 is not a correct explanation for STATEMENT-1, and the answer is Option (B).

Key concepts

Vieta's Formulas

Vieta's formulas serve as a critical bridge connecting the coefficients of a polynomial to the sums and products of its roots. Particularly in quadratic equations, they offer a handy set of equations that can help us understand the relationships between the roots without actually solving the equation.

For any quadratic equation of the form, \( ax^2 + bx + c = 0 \), with roots \( \rho_1 \) and \( \rho_2 \), Vieta's formulas state that:

• \( \rho_1 + \rho_2 = -\frac{b}{a} \)
• \( \rho_1 \rho_2 = \frac{c}{a} \)

These formulas are named after the French mathematician François Viète, and their simplicity aids in the analysis of polynomials for unknown variables or coefficients when certain conditions are applied, such as in the exercise provided.

In the given problem, Vieta's formulas were applied to two different quadratic equations, identifying the relationships between their roots and coefficients. This established an equality that could be used to deduce further properties about the unknowns involved. Utilizing Vieta's formulas is an essential concept in JEE Advanced mathematics, particularly when handling algebraic equations and their properties.

Quadratic Equations

Quadratic equations, which are polynomials of degree two, often appear in various mathematics problems, especially in JEE Advanced problems. They are generally expressed in the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are real numbers and \( a eq 0 \).

The solutions to these equations, known as roots, can be found through several methods including factoring, completing the square, using the quadratic formula, or by graphing. The quadratic formula, \( x = \frac{{-b \text{pm} \text{sqrt}{b^2 - 4ac}}}{2a} \), is a direct application of completing the square and provides the roots in terms of the coefficients of the equation.

In JEE Advanced exams, quadratic equations are commonly explored for their intricate properties and relationships between their roots and coefficients, not just their solutions. For example, by manipulating these equations or using Vieta's formulas, students can unveil symmetry properties, the behavior of graphs, or solve higher-degree polynomial systems by breaking them into quadratic factors.

Roots of Equations

The roots of an equation are the solutions that satisfy the equation; in other words, they 'zero out' the polynomial when substituted into it. In the context of quadratic equations, these are the values of \( x \) that make the expression \( ax^2 + bx + c \) equal to zero.

Understanding the nature of roots is fundamental in higher mathematics and JEE Advanced algebra. There can be two roots in a quadratic equation, which may be real and distinct, real and equal (repeated), or complex conjugates depending on the discriminant \( (b^2 - 4ac) \).

When students master the concept of roots, they are not just solving for numerical values; they are delving into the structure of equations. They learn how roots can be manipulated to form new equations, or how they relate to the coefficients of the original equation, as highlighted by the exercise. This exercise also emphasizes the non-trivial aspect that roots can have relationships among themselves, leading to more profound algebraic identities and theorems, such as the one used in the step-by-step solution.