Problem 127 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The values of ' \(a\) ' for which the roots of the equation \((a+1) x^{2}-3 a x+4 a=0(a \neq-1)\) to be greater than unity are (A) \(\frac{-16}{7} \leq a<-1\) (B) \(-2

Step-by-Step Solution

Answer

None of these (Option D).

Step 1: Understand the Root Condition

The problem states that the roots should be greater than unity (greater than 1). For a quadratic equation, if both roots are greater than 1, then their sum is greater than 2 and their product is greater than 1.

Step 2: Write the Quadratic Equation

The quadratic equation given is \( (a+1)x^2 - 3ax + 4a = 0 \). We need to analyze this equation under the condition where \( a eq -1 \).

Step 3: Use Vieta's Formulas

Vieta's formulas give the sum of the roots \( r_1 + r_2 = \frac{3a}{a+1} \) and the product of the roots \( r_1r_2 = \frac{4a}{a+1} \). We set the sum greater than 2 and the product greater than 1: \((r_1 + r_2) > 2\) and \((r_1r_2) > 1\).

Step 4: Solve the Sum Inequality

Set up the inequality for the sum of the roots: \[ \frac{3a}{a+1} > 2 \] Solve this inequality: \((3a > 2(a + 1))\) simplifies to \((3a > 2a + 2)\), and further to \(a > 2\).

Step 5: Solve the Product Inequality

Set up the inequality for the product of the roots: \[ \frac{4a}{a+1} > 1 \] Solve this inequality: \((4a > a+1)\) simplifies to \((3a > 1)\), and further to \(a > \frac{1}{3}\).

Step 6: Combine Inequalities

The solution must satisfy both inequalities. Combining \(a > 2\) and \(a > \frac{1}{3}\) gives a stronger restriction \(a > 2\).

Step 7: Check Result

From the combined inequalities, \(a > 2\). Compare with the given options: None of the options directly represent \(a > 2\), so the correct answer is option (D) None of these.

Key Concepts

Vieta's Formulas

Vieta's Formulas are a set of relationships that link the coefficients of a polynomial to sums and products of its roots. These formulas are particularly useful when dealing with quadratic equations, such as the one in your problem.
For a quadratic equation in the form of \( ax^2 + bx + c = 0 \), Vieta's Formulas tell us:

• The sum of the roots \( r_1 + r_2 = -\frac{b}{a} \)
• The product of the roots \( r_1r_2 = \frac{c}{a} \)

These relationships allow us to express properties of the roots directly by using the coefficients of the quadratic equation. In the given problem, the quadratic equation is \( (a+1)x^2 - 3ax + 4a = 0 \). Thus:

• The sum of the roots is \( r_1 + r_2 = \frac{3a}{a+1} \)
• The product of the roots is \( r_1r_2 = \frac{4a}{a+1} \)

This setup helped in analyzing the conditions where both roots are greater than unity (greater than 1). By using these formulas, we can derive inequalities that must be solved to find valid values for \( a \).

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, which means they have a term with a variable raised to the power of two, like \( ax^2 + bx + c = 0 \). These equations often appear in various mathematical contexts, and understanding how to solve them is crucial.
The solutions to a quadratic equation are known as its "roots". These roots can be found using several methods, such as:

• Factoring, when the quadratic can be expressed as a product of two binomials.
• The quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Completing the square, a method that rearranges the equation to reveal perfect squares.

In the given problem, understanding the structure of the equation \\((a+1)x^2 - 3ax + 4a = 0\) and using Vieta's Formulas reveals conditions on the values of a. These conditions come from ensuring the sum of the roots is more than 2 and the product greater than 1. This leads directly to solving inequalities to find possible values for \( a \).

Inequalities

Inequalities describe the relative size or order of two values and are a fundamental concept in mathematics. They are often used to find a range of solutions rather than a precise value.
When solving inequalities, we express the solution as a range or set of values that satisfy the condition. For example:

• \( x > 2 \) signifies that x is any number greater than 2.
• \( a \leq -1 \) denotes that a is any number less than or equal to -1.

In quadratic contexts, inequalities might involve the roots of quadratic equations. In the original exercise, two inequalities needed to be solved to ensure both roots are greater than 1:

• Sum Inequality: \( \frac{3a}{a+1} > 2 \)
• Product Inequality: \( \frac{4a}{a+1} > 1 \)

These provide conditions on \( a \) that need to be met concurrently. The solution combines these conditions, resulting in \( a > 2 \). Such problems provide insight into how inequalities can define regions over which mathematical relationships hold.