Problem 24 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The set of possible values of \(\lambda\) for which \(\lambda^{2}-\left(\lambda^{2}-\right.\) \(5 \lambda+5) x+\left(2 \lambda^{2}-3 \lambda-4\right)=0\) has roots whose sum and product are both less than 1 is (A) \(\left(-1, \frac{5}{2}\right)\) (B) \((1,4)\) (C) \(\left[1, \frac{5}{2}\right]\) (D) \(\left(1, \frac{5}{2}\right)\)

Step-by-Step Solution

Answer

The correct set of values for \( \lambda \) is (D) \( \left(1, \frac{5}{2}\right) \).

Step 1: Identify the quadratic equation

The equation given is \( \lambda^{2}-(\lambda^{2}-5\lambda+5)x+(2\lambda^{2}-3\lambda-4)=0 \). It is a quadratic equation in terms of \( x \). We can write it in standard form as \( ax^{2} + bx + c = 0 \) with \( a = 1 \), \( b = -(\lambda^{2} - 5\lambda + 5) \), and \( c = 2\lambda^{2} - 3\lambda - 4 \).

Step 2: Find the sum of the roots

According to Vieta's formulas, the sum of the roots \( S \) of a quadratic equation \( ax^{2} + bx + c = 0 \) is given by \( S = -\frac{b}{a} \). For this problem, \( S = (\lambda^{2} - 5\lambda + 5) \). We need this sum to be less than 1, which implies solving the inequality \( \lambda^{2} - 5\lambda + 5 < 1 \). Simplifying, we have \( \lambda^{2} - 5\lambda + 4 < 0 \).

Step 3: Find the product of the roots

The product of the roots \( P \) is given by \( P = \frac{c}{a} \), so \( P = 2\lambda^{2} - 3\lambda - 4 \). We require this product to be less than 1, which results in the inequality \( 2\lambda^{2} - 3\lambda - 4 < 1 \). Simplifying gives \( 2\lambda^{2} - 3\lambda - 5 < 0 \).

Step 4: Solve the inequalities

First solve \( \lambda^{2} - 5\lambda + 4 < 0 \). This factors to \( (\lambda - 4)(\lambda - 1) < 0 \). The solution for this inequality is \( 1 < \lambda < 4 \). Next, solve \( 2\lambda^{2} - 3\lambda - 5 < 0 \). Factoring gives \( (\lambda - 1)(2\lambda + 5) < 0 \). The solution to this inequality is also \( \lambda < 1 \) or \( \lambda > \frac{5}{2} \).

Step 5: Find the common interval

The interval that satisfies both inequalities is where the intervals intersect. The common solution from \(( \lambda - 4)(\lambda - 1) < 0\), which is \( (1, 4) \), and \((\lambda < 1 \text{ or } \lambda > \frac{5}{2})\) is \( \lambda \in (1, \frac{5}{2}) \).

Step 6: Choose the correct option

Comparing our solution \( (1, \frac{5}{2}) \) with the given options, we find that option (D) \( \left(1, \frac{5}{2}\right) \) matches our solution.

Key Concepts

Vieta's formulas

Vieta's formulas are an essential tool in solving quadratic equations. They provide a straightforward way to relate the coefficients of the equation to the sum and product of its roots. For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the sum of the roots \( S \) is given by \( S = -\frac{b}{a} \), and the product of the roots \( P \) is \( P = \frac{c}{a} \).In the given problem, we treat the equation from the assignment as a quadratic in terms of \( x \). The sum of the roots for this equation, using Vieta’s formulas, is \( \lambda^{2} - 5\lambda + 5 \). The product of the roots is \( 2\lambda^{2} - 3\lambda - 4 \). These expressions are crucial in setting up the inequalities that determine valid \( \lambda \) values.These connections not only simplify complex calculations but also offer insights into the behavior of the roots based on the equation's structure. Understanding Vieta's formulas empowers you to tackle further polynomial problems with confidence.

Inequalities

Inequalities describe a range of possible values rather than a single solution. They are crucial in identifying valid intervals for variables within a given problem.In this exercise, we established two inequalities based on the conditions that the sum and product of the roots must both be less than one. For the sum of the roots to be less than one, we form the inequality \( \lambda^{2} - 5\lambda + 4 < 0 \), which simplifies to \( (\lambda - 4)(\lambda - 1) < 0 \). This inequality tells us that \( \lambda \) must fall between 1 and 4.For the product of the roots to be less than one, we derive \( 2\lambda^{2} - 3\lambda - 5 < 0 \), which factors to \( (\lambda - 1)(2\lambda + 5) < 0 \). This inequality suggests two possibilities: \( \lambda < 1 \) or \( \lambda > \frac{5}{2} \).Combining these inequalities helps us identify where both conditions are satisfied, highlighting the importance of inequalities in finding a feasible solution range.

Roots of polynomials

Roots of polynomials are solutions where the polynomial evaluates to zero. In a quadratic polynomial such as \( ax^2 + bx + c = 0 \), the roots can be found using the quadratic formula, factoring, or through Vieta's relations, which offer a shortcut as seen in this problem.For higher-degree polynomials, understanding the roots allows you to sketch graphs and predict behavior over different intervals, providing insight into complex functions. Deciphering the conditions to ensure certain properties of roots (such as those being less than or within a particular value) is often challenging without management through inequalities.Solving through roots requires practice in pattern recognition and algebraic manipulation. Whether checking conditions in exams or just enhancing mathematical problem-solving skills, getting comfortable with roots strengthens every student’s foundational knowledge in algebra.