Problem 8 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let S denotes the set of all values of \(x\) for which the equation \(2 x^{2}-2(2 a+1) x+a(a+1)=0\) has one root less than \(a\) and other root greater than \(a\), then \(S\) equals (A) \((0,1)\) (B) \((-1,0)\) (C) \((0,1 / 2)\) (D) None of these

Step-by-Step Solution

Answer

Option (D) None of these.

Step 1: Identify conditions for roots

We know that for a quadratic equation of the form \(ax^2 + bx + c = 0\), the given condition is for one root \(\alpha\) to be less than \(a\) and the other root \(\beta\) to be greater than \(a\). This means the roots are on either side of \(a\), which implies the condition \(\alpha < a < \beta\).

Step 2: Apply the roots condition to quadratic equation

Given the quadratic equation \(2x^2 - 2(2a+1)x + a(a+1) = 0\), let's assume roots are \(\alpha\) and \(\beta\). We use the Vieta's formulas: \(\alpha + \beta = \frac{2(2a+1)}{2} = 2a+1\) and \(\alpha \beta = \frac{a(a+1)}{2}\). The condition \(\alpha < a < \beta\) combined with \(\alpha + \beta = 2a + 1\) implies symmetry around \(a\).

Step 3: Verify symmetry condition

Since the sum \(\alpha + \beta = 2a + 1\) holds, and roots are symmetrically placed around \(a\), we split it as \(\beta = a + c\) and \(\alpha = a - c\). This implies \(2a = \alpha + \beta - 1 = (a - c) + (a + c) - 1\). Simplifying, this confirms \(\alpha < a < \beta\) is naturally fulfilled by any valid roots.

Step 4: Check completeness of conditions

To ensure there are two distinct roots, we need to check the discriminant condition: \( b^2 - 4ac > 0 \). For our quadratic: \((2(2a+1))^2 - 4 \cdot 2 \cdot a(a+1) > 0\). Simplifying, we calculate: \((4a + 2)^2 - 8a(a+1) > 0\).

Step 5: Solve discriminant inequality

Simplifying, \((4a + 2)^2 - 8a^2 - 8a > 0\) becomes \(16a^2 + 16a + 4 - 8a^2 - 8a > 0\). Simplifying gives \(8a^2 + 8a + 4 > 0\). Simplifying further yields \(2a + 1 > 0\). Thus, \(a > -\frac{1}{2}\).

Step 6: Combine results for final solution

We combine conditions: \(\alpha < a < \beta\) ensures symmetry, while \(a > -\frac{1}{2}\) gives the restriction on \(a\). The combination of these regions pertaining to missing options implies choice (D) None of these, is correct.

Key Concepts

Roots of a Quadratic Equation

Quadratic equations are essential in mathematics, and understanding their roots is key. These equations take the form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. The solutions of this equation, or the roots, are the values of \( x \) that make the equation true. Typically, a quadratic equation has two solutions.

• When the roots are real numbers, they can be either equal or distinct.
• In this context, the roots are also referred to as \( \alpha \) and \( \beta \).

The relationship between the roots is crucial in our solution. For instance, if one root is less than a certain value \( a \) and the other root is greater than \( a \), this creates certain conditions that must be satisfied by the coefficients \( a \), \( b \), and \( c \). This exercise requires us to understand how these roots behave relative to a parameter \( a \).

Vieta's Formulas

Vieta’s Formulas are powerful tools for analyzing quadratic equations without explicitly solving them. For a quadratic equation of the form \( ax^2 + bx + c = 0 \), Vieta’s Formulas are expressed as:

• The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
• The product of the roots \( \alpha \beta = \frac{c}{a} \)

Using these formulas, one can derive profound insights about the equation's roots. For example, in our specific quadratic equation expressed as \( 2x^2 - 2(2a+1)x + a(a+1) = 0 \), Vieta’s tells us:

• \( \alpha + \beta = 2a + 1 \)
• \( \alpha \beta = \frac{a(a+1)}{2} \)

These relationships allow us to understand the symmetry around \( a \), as the sum \( \alpha + \beta = 2a + 1 \) implies that the roots are positioned evenly around the value \( a \). This is a key aspect of the problem's solution.

Discriminant of a Quadratic Equation

The discriminant \( \Delta \) is another vital part of understanding quadratic equations. The discriminant is found using the formula \( \Delta = b^2 - 4ac \). It indicates the nature of the roots:

• If \( \Delta > 0 \), the equation has two distinct real roots.
• If \( \Delta = 0 \), the equation has one real root (or a double root).
• If \( \Delta < 0 \), the equation has two complex roots.

In our exercise, we require the discriminant of the quadratic to be greater than zero to ensure that the roots are distinct and real.The inequality \((2(2a+1))^2 - 8a(a+1) > 0\) simplifies to \(8a^2 + 8a + 4 > 0\). Through further simplification, this tells us \(2a + 1 > 0\), which translates into \(a > -\frac{1}{2}\).This discriminant-based condition, combined with the symmetry condition derived from Vieta's, helps us narrow down the set of possible values for \( a \), ultimately leading to the conclusion of the original problem.