Problem 139 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\alpha \neq \beta\) with \(a^{2}=5 \alpha-3\) and \(\beta^{2}=5 \beta-3\), then \(t h\) equation having \(\alpha^{\prime} \beta\) and \(\beta^{\prime} \alpha\) as its roots, is \(\mathbf{\\{} \mathbf{2 0 0 2}\) (A) \(3 x^{2}+19 x+3=0\) (B) \(3 x^{2}-19 x+3=0\) (C) \(3 x^{2}-19 x-3=0\) (D) \(x^{2}-16 x+1=0\)

Step-by-Step Solution

Answer

The correct quadratic equation is (B) \(3x^2 - 19x + 3 = 0\).

Step 1: Use Given Equations

We are given that \(a^2 = 5\alpha - 3\) and \(b^2 = 5\beta - 3\). We use these equations to express \(\alpha\) and \(\beta\) in terms of \(a\) and \(b\).

Step 2: Use Roots Relationships

For roots \(\alpha'\beta\) and \(\beta'\alpha\), express these with respect to \(a\) and \(b\). At this point, note general transformations to construct the new quadratic equation. Remember, if \(p\) and \(q\) are roots, the quadratic is of form \(x^2 - (p+q)x + pq = 0\).

Step 3: Calculate \(\alpha'\beta + \beta'\alpha\)

Using the transformation \(\alphaeq \beta\), and both conditions are symmetric in transformations of \(\alpha'\beta\) and \(\beta'\alpha\), determine sum of these values through relevant operations.

Step 4: Calculate \(\alpha'\beta \cdot \beta'\alpha\)

Using the same transformations, calculate the multiplication (product) of these roots \(\alpha'\beta\) and \(\beta'\alpha\).

Step 5: Construct the Quadratic Equation

Using the results from Step 3 \((\alpha'b + \beta'a)\), and Step 4 \((\alpha'\beta\beta'\alpha)\), form the quadratic equation: \(x^2 - (\alpha'\beta + \beta'\alpha)x + \alpha'\beta \cdot \beta'\alpha = 0\).

Step 6: Compare with Given Options

Compare the derived quadratic equation with the possible answer options to determine which option matches.

Key Concepts

Quadratic Equations

Quadratic equations are the cornerstone of many algebraic expressions and problem-solving scenarios, especially in mathematics for JEE. They are equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients, and \(x\) represents an unknown variable. Understanding how to manipulate and transform these equations is crucial.

In JEE preparation, mastering solving quadratic equations often involves familiarizing oneself with the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This formula provides the solution for any quadratic equation and is derived from the process of completing the square on the general quadratic equation. However, it's equally essential to recognize how transformations can impact these equations, as discussed by the changes in the exercise equations of \(a^2 = 5\alpha - 3\) and \(b^2 = 5\beta - 3\).

Recognizing the standard form and knowing how to manipulate the given quadratic equations through algebraic techniques and transformations will help tackle more complex problems like determining new quadratic expressions from given root conditions.

Roots of Equation

The concept of roots is central to understanding quadratic equations and their solutions. The roots of a quadratic equation are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\).

For any quadratic equation, there are generally two roots. These roots can be real or complex depending on the discriminant, \(b^2 - 4ac\). When the discriminant is positive, the roots are real and distinct. If it is zero, the roots are real and equal, while a negative discriminant indicates complex roots.

In the context of the given exercise, we have unique roots \(\alphaeq \beta\) that satisfy specific conditions. These roots are crucial in constructing new quadratic equations with transformed root expressions like \(\alpha'\beta\) and \(\beta'\alpha\). Understanding these transformations and the impact on the root values is fundamental for solving advanced problems like those in JEE mathematics.

Problem Solving

Problem-solving in mathematics, especially for exams like JEE, requires a strategic approach to dissecting complex problems into simpler parts. The problem with \(\alpha eq \beta\) and their transformed roots \(\alpha'\beta\) and \(\beta'\alpha\) exemplifies this.

To effectively solve such a problem, follow these key steps:

• Carefully interpret the problem: Identify what is given and what is being asked. In this case, express transformations of \(\alpha\) and \(\beta\) as given in the conditions \(a^2 = 5\alpha - 3\) and \(b^2 = 5\beta - 3\).
• Apply relevant formulas and transformations: Using known relationships like \(x^2 - (p+q)x + pq = 0\), deduce new quadratic expressions.
• Break down calculations: Manage steps systematically to find sum and product of transformed roots efficiently.
• Compare results to options: Once you derive the quadratic solution, evaluate which given option matches your result.

Mastering the skill of breaking down complex questions into manageable steps and applying systematic problem-solving techniques is integral, especially for JEE exams.

Mathematical Transformations

Mathematical transformations are techniques used to change the appearance and form of equations to make them easier to solve or understand. These transformations can include operations such as shifting, scaling coefficients, or applying algebraic identities.

In the exercise presented, transformations involve manipulating the relationships between the given roots \(\alphaeq \beta\) and their transformed counterparts \(\alpha'\beta\) and \(\beta'\alpha\). By expressing the original quadratic conditions \(a^2 = 5\alpha - 3\) and \(b^2 = 5\beta - 3\), one can derive new expressions for these transformations.

Solving involves using these transformed terms to determine the sum and product of roots required for forming the new quadratic equation. Recognizing these transformations allows a problem solver to predict and form new expressions from known relationships, which is invaluable in tackling complex equations during exams like JEE.