Problem 30 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\alpha, \beta\) be the roots of \(x^{2}+p x-q=0\) and \(\gamma, \delta\) be the roots of \(x^{2}+p x+r=0, q+r \neq 0\), then \(\frac{(\alpha-\gamma)(\alpha-\delta)}{(\beta-\gamma)(\beta-\delta)}=\) (A) 1 (B) \(q\) (C) \(r\) (D) \(q+r\)

Step-by-Step Solution

Answer

The answer is (A) 1.

Step 1: Identify the Roots

The roots of the first equation are \( \alpha \) and \( \beta \), satisfying \( \alpha + \beta = -p \) and \( \alpha \beta = -q \). The roots of the second equation are \( \gamma \) and \( \delta \), satisfying \( \gamma + \delta = -p \) and \( \gamma \delta = r \).

Step 2: Write Expressions for the Products

Consider the expression \((\alpha - \gamma)(\alpha - \delta)\) which can be rewritten using roots properties as \( \alpha^2 - \alpha(\gamma + \delta) + \gamma\delta = \alpha^2 + p\alpha + r \).

Step 3: Transform and Simplify

Similarly, consider \((\beta - \gamma)(\beta - \delta)\) which simplifies to \( \beta^2 - \beta(\gamma + \delta) + \gamma\delta = \beta^2 + p\beta + r \).

Step 4: Substitute and Simplify

Substitute \( \alpha \) and \( \beta \) as roots of the first equation: \( \alpha^2 = -p\alpha + q \) and \( \beta^2 = -p\beta + q \). This leads to the expressions \( \alpha^2 + p\alpha = q \) and \( \beta^2 + p\beta = q \).

Step 5: Plug into Main Expression

Substitute these results into the fraction: \( \frac{(q + r)}{(q + r)} = 1 \).

Step 6: Conclusion with Answer

The calculations confirm that the expression simplifies to 1. Thus, the answer is option (A) 1.

Key Concepts

Roots of Equations

In mathematics, particularly in algebra, the roots of equations are the values that satisfy the given equation. For a quadratic equation like \( ax^2 + bx + c = 0 \), the roots are represented as \( x = \alpha \) and \( x = \beta \). These roots reflect where the quadratic polynomial intersects the x-axis on a graph.

Understanding the properties of roots is essential:

• The sum of the roots, \( \alpha + \beta \), is given by \( -\frac{b}{a} \).
• The product of the roots, \( \alpha \beta \), is given by \( \frac{c}{a} \).

In the problem, the equations provided show us these relationships. For the first quadratic equation, the roots \( \alpha \) and \( \beta \) follow \( \alpha + \beta = -p \) and \( \alpha \beta = -q \). This is based on our familiarity with these fundamental roots properties.

Similarly, for the second equation with roots \( \gamma \) and \( \delta \), they satisfy \( \gamma + \delta = -p \) and \( \gamma \delta = r \). Careful analysis of these properties allows us to solve complex quadratic equations efficiently and can sometimes lead to elegant solutions, as seen in our exercise above.

Polynomial Functions

Polynomial functions are expressions made up of variables and coefficients, involving terms in the form of \( ax^n \), where \( n \) is a non-negative integer. The quadratic function, which is of degree two, is a simple form of polynomial that often arises in algebraic problems.

In this exercise:

• The first polynomial function is \( x^2 + px - q \).
• The second polynomial function is \( x^2 + px + r \).

Both are quadratic polynomials characterized by their coefficients: a common \( x^2 \) term, a linear \( px \) term, and a constant term \(-q\) or \(r\), respectively. Solving these involves finding their roots, which are useful when analyzing polynomial properties.

Polynomial functions are essential because they provide a fundamental means to model and solve higher-degree equations. They are utilized in various mathematical modeling scenarios, from physics to economics, where understanding their roots and behavior can reveal valuable insights into the problem at hand.

Mathematical Identity

A mathematical identity is an equation that holds true for all variables within its domain. These identities are often used to simplify and transform expressions, which can be crucial when solving equations.

In the context of quadratic equations, identities enable us to express complex relationships through simpler forms:

• Consider \( (\alpha - \gamma)(\alpha - \delta) \) transforming into \( \alpha^2 + p\alpha + r \).
• This transformation uses the identity derived from the sum and product of roots.

Mathematical identities help in reducing the intricacies of algebraic manipulations, turning a daunting problem into a manageable one. By understanding these identities, like those exhibited in quadratic equations, students can greatly enhance their problem-solving toolkit. They are akin to shortcuts, enabling quicker solutions and clearer insights about the characteristics of equations being dealt with.

Algebraic Expressions

Algebraic expressions, consisting of numbers, variables, and operations, form the basis of many algebra problems. These expressions can represent real-world quantities and relationships, which is why understanding them is vital.

For instance, in our exercise, expressions like \( \alpha^2 + p\alpha + r \) or \( \beta^2 + p\beta + r \) represent specific rearrangements of original equation terms:

• The initial expressions \( \alpha^2 = -p\alpha + q \) and \( \beta^2 = -p\beta + q \) play integral roles.
• These expressions help express relations in simpler terms, making the solution process methodical.

When working with algebraic expressions, always be conscious of how terms are manipulated through basic algebra rules like distribution, factoring, and combining like terms. Mastery of these expressions empowers students to tackle not only theoretical problems but also practical applications, which often involve evaluating or simplifying such expressions in real-life scenarios.