Problem 138 - A Complete Resource Book in Mathematics for JEE Main

Question

In the following questions an Assertion (A) is given followed by a Reason \((R) .\) Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason( \(\mathrm{R}\) ) is False (D) Assertion(A) is False, Reason(R) is True Assertion: If the equations \(x^{2}+b x+c a=0\) and \(x^{2}+\) \(c x+a b=0\) have a common root, then their other roots will satisfy the equation \(x^{2}+a x+b c=0\) Reason: If the equations \(x^{2}+b x+c a=0\) and \(x^{2}+\) \(c x+a b=0\) have a common root, then \(a+b+c=0\)

Step-by-Step Solution

Answer

(B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A).

Step 1: Understanding the Assertion

The assertion claims that if the equations \(x^2 + bx + ca = 0\) and \(x^2 + cx + ab = 0\) have a common root, then their other roots will satisfy the equation \(x^2 + ax + bc = 0\). To verify this, we need to explore the relationship between the roots and these equations.

Step 2: Analyzing the Two Quadratic Equations

Let the common root be \(\alpha\). Thus, \(\alpha\) satisfies both equations:1. \(\alpha^2 + b\alpha + ca = 0\) 2. \(\alpha^2 + c\alpha + ab = 0\)Subtract these two equations to eliminate \(\alpha^2\):\((b - c)\alpha + (ca - ab) = 0\) This implies: \(\alpha = \frac{ca-ab}{b-c}\) (if \(b eq c\)).

Step 3: Using the Condition for a Common Root

If \(b = c\), then both equations become the same, which means \(a\) and \(b\) are related by the consistency of these polynomials. If \(a = b = c\), we have a consistent situation where the common root is due to identical coefficients. If such consistency is presumed from the assertion, normally one concludes other roots follow similarly structured results.

Step 4: Check the Implication in the Reason

The reason states that if the equations have a common root, then \(a + b + c = 0\). Substitute \(\alpha = \frac{ca-ab}{b-c}\) back into the explored polynomials to check for such a linear identity, and verify implications regarding other roots based on similar equations not detailed directly in this simplification.

Step 5: Conclusion from the Logic

The given reasoning suggests a symmetry and consistent pattern in the roots of these quadratics even when a common linear identity exists. Given the lack of a direct explanation of resultant forms of \(x^2 + ax + bc = 0\) from these common-root results, the assertion seems logically distinct in pattern propagation.

Step 6: Final Step: Determining the Correct Option

Since the assertion and the reason both discuss conditions about equations with common roots but don't logically link one to another directly by transformation or substitution patterns, the best fit based on analysis is: (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A).

Key Concepts

Quadratic Equations

Quadratic equations are fundamental expressions in algebra that form the basis for solving a wide variety of mathematical problems. A quadratic equation is always expressed in the standard form of:\[ ax^2 + bx + c = 0 \] Here, \(a\), \(b\), and \(c\) are constants with \(a eq 0\). The term \(x^2\) indicates that the highest power of \(x\) is 2, thus making the equation quadratic. Such equations can have two roots, potentially real or complex.
To solve a quadratic equation, you can use several methods:- **Factoring**: Expressing the equation as the product of two binomials.- **Quadratic Formula**: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), especially useful when factoring is difficult or impossible.- **Completing the Square**: A method to rewrite the equation in a form that makes it easy to solve for \(x\).Each of these techniques provides a different pathway to finding the roots of the equation. Quadratic equations are instrumental in modeling physical phenomena and solving real-world problems where quantities depend quadratically on another factor.

Common Roots

Common roots in quadratic equations refer to a scenario where two or more equations share at least one root. This is a critical concept in algebra as it demonstrates a relationship between different quadratic equations. When two quadratic equations have a common root, certain algebraic properties can be used to find this root and explore the nature of the equations.
Let's say we have two quadratic equations:1. \( x^2 + bx + ca = 0 \)2. \( x^2 + cx + ab = 0 \)If they share a common root, say \(\alpha\), it implies that \(\alpha\) satisfies both equations. Finding common roots usually involves:- Solving one equation for its roots, then checking which of these roots satisfy the second equation.- Using algebraic manipulation, such as subtraction or substitution, to isolate the common root.
In the context of the original problem, the presence of a common root points towards a symmetry or specific pattern that relates the two quadratics, leading to other roots that align with these identifiable equations. This concept is particularly useful for simplifying complex systems and understanding deeper structural relationships between equations.

Assertion and Reason Questions

Assertion and reason questions are common in educational assessments, particularly in subjects like mathematics, where they are used to test understanding of concepts and logical reasoning abilities. These questions present an assertion (a statement or claim) followed by a reason that supposedly explains the assertion.

For example, the exercise provides an assertion about common roots of quadratic equations and offers a reason to justify it. The task is to evaluate: - Whether both the assertion and the reason are true. - Whether the reason correctly explains the assertion or not.
This involves carefully analyzing the logical connections between the assertion and the reason. It is critical to assess whether the reason directly leads to the assertion as a conclusion.
Teaching strategies for mastering these types of questions include: - Understanding the individual correctness of each statement. - Identifying logical or factual connections between them. - Improving skills in deduction and critical thinking to evaluate whether the reason sufficiently supports the assertion. These questions are designed to challenge students to think deeply and synthesize information, fostering a comprehensive understanding of the material.