Problem 12 - A Complete Resource Book in Mathematics for JEE Main

Question

If one of the roots of the equation \(x^{2}-(p+1) x+p^{2}+\) \(p-8=0\) is greater than 2 and the other root is smaller than 2 , then \(p\) is such that (A) \(-\frac{11}{3}

Step-by-Step Solution

Answer

The correct option is (A) \(-\frac{11}{3} < p < 3\).

Step 1: Understand the Problem

We need to find the values of \(p\) such that the quadratic equation \(x^2 - (p+1)x + p^2 + p - 8 = 0\) has roots where one is greater than 2 and the other is less than 2.

Step 2: Set Up Conditions for Roots

For a quadratic equation \(ax^2 + bx + c = 0\), if \(\alpha\) and \(\beta\) are the roots, then the sum is \(\alpha + \beta = -\frac{b}{a}\) and the product is \(\alpha \beta = \frac{c}{a}\). Here, \(a = 1\), \(b = -(p+1)\), \(c = p^2 + p - 8\).

Step 3: Derive Equations for the Roots

Since one root is greater than 2: \(\alpha > 2\). Another root is smaller than 2: \(\beta < 2\). From sum of roots \(\alpha + \beta = p + 1\), write: \((\alpha - 2) + (\beta - 2) = p + 1 - 4 = p - 3\).

Step 4: Analyze the Conditions

Given \(\alpha > 2\) and \(\beta < 2 \), we have \((\alpha - 2) > 0\) and \((\beta - 2) < 0\), leading to this conclusion: \((\alpha - 2) + (\beta - 2) < 0\). Hence, \(p - 3 < 0\), or \(p < 3\).

Step 5: Find the Range of p Considering Discriminant Condition

The discriminant of the quadratic \((p+1)^2 - 4(p^2 + p - 8)\) must be positive for real roots. Solve this inequality: \(-(3p^2 + 3p - 33) > 0\), leading to \(3p^2 + 3p - 33 < 0\). Solve this inequality to find \(p\) values resulting in \(-\frac{11}{3} < p < 3\).

Step 6: Conclusion Based on Conditions

Combine the two conditions: \(p < 3\) (from root conditions) and \(-\frac{11}{3} < p < 3\) (from discriminant condition). The intersection of both gives \(-\frac{11}{3} < p < 3\).

Key Concepts

Roots of Equations

When you have a quadratic equation, such as \(x^2 - (p+1)x + p^2 + p - 8 = 0\), finding the roots means determining the values for \(x\) that satisfy the equation. The roots are where the equation equals zero. In most simple terms, they are the solutions to the equation. This particular quadratic equation can be written in standard form as \(ax^2 + bx + c = 0\), where \(a = 1\), \(b = -(p+1)\), and \(c = p^2 + p - 8\). The properties of the roots \(\alpha\) and \(\beta\) are important. They relate to the equation through:

• The sum of the roots: \(\alpha + \beta = -\frac{b}{a} = p + 1\)
• The product of the roots: \(\alpha \beta = \frac{c}{a} = p^2 + p - 8\)

For this exercise, one root is greater than 2, and the other is less than 2. These constraints guide the values of \(p\) that ensure the equation's roots fit this scenario.

Discriminant Condition

The discriminant in a quadratic equation, given by the expression \(b^2 - 4ac\), tells us about the nature of the roots. For real and distinct roots, the discriminant must be positive. Applying this to our equation, the discriminant becomes \((p+1)^2 - 4(p^2 + p - 8)\). Simplifying this, you get the inequality: \[-(3p^2 + 3p - 33) > 0\]This simplifies further to \(3p^2 + 3p - 33 < 0\). Solving this quadratic inequality will give the range of values for \(p\) where the roots are real and distinct, meaning they exist and are not equal. Solving this inequality, you find that \(p\) must satisfy \[-\frac{11}{3} < p < 3\].This condition ensures the equation has two separate and real solutions, which confirms the variable \(p\) falls within an acceptable range to fit the given constraints of the root values.

Range of Values for Roots

Finding the range of possible values for \(p\) is crucial to ensuring the roots fall into the specified conditions. The problem states that one root is greater than 2, and the other is less than 2. This imposes the condition \((\alpha - 2) + (\beta - 2) < 0\) when considering the shifted roots relation from their position around 2. Solving, we deduct that \(p - 3 < 0\), meaning \(p < 3\).When combined with the condition from solving the discriminant inequality, as mentioned earlier < \(-\frac{11}{3} < p < 3\), we find that both sets of conditions intersect only in the range \[-\frac{11}{3} < p < 3\]. This range represents the values \(p\) can take for the quadratic equation's roots to meet the problem's criteria, ensuring one is larger than 2 and the other smaller. Thus, the conditions line up perfectly to satisfy the requirements of the exercise.