Problem 108 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the equation \(x^{2}+\left[a^{2}-5 a+b+4\right] x+b=0\) has roots \(-5\) and 1, where \([a]\) denotes the greatest integer less than or equal to \(a\), then the set of values of \(a\) is (A) \(\left(\frac{5-3 \sqrt{5}}{2}, \frac{5+3 \sqrt{5}}{2}\right)\) (B) \(\left(0, \frac{5+3 \sqrt{5}}{2}\right)\) (C) \(\left(-1, \frac{5-3 \sqrt{5}}{2}\right] \cup\left[\frac{5+3 \sqrt{5}}{2}, 6\right)\) (D) None of these

Step-by-Step Solution

Answer

Option D: None of these.

Step 1: Identify the Coefficients

The quadratic equation given is \(x^2 + [a^2 - 5a + b + 4]x + b = 0\). We know the roots are \(-5\) and \(1\). By Vieta's formulas, for a quadratic \(ax^2 + bx + c = 0\), the sum of the roots \((-\frac{b}{a})\) is \(-5 + 1 = -4\) and the product of the roots \((\frac{c}{a})\) is \(-5 \cdot 1 = -5\). Here, the coefficients can be related to \([a^2 - 5a + b + 4]\) and \(b\).

Step 2: Set Up Equations from Known Information

Given the sum of roots is \(-4\), we equate \(-[a^2 - 5a + b + 4] = -4\), resulting in the equation \([a^2 - 5a + b + 4] = 4\). The product of the roots implies \(b = -5\).

Step 3: Solve for \(b\)

From the product of the roots, we identify \(b = -5\). Substituting \(b = -5\) into the sum equation: \([a^2 - 5a - 5 + 4] = 4\), simplifying to \([a^2 - 5a - 1] = 4\).

Step 4: Find Values of \(a^2 - 5a\)

The equation \([a^2 - 5a - 1] = 4\) simplifies to \(4 \leq a^2 - 5a < 5\). Solving \(4 = a^2 - 5a\) and \(a^2 - 5a = 5\) gives the boundaries for \(a\): solve \(a^2 - 5a - 4 = 0\) and \(a^2 - 5a - 5 = 0\).

Step 5: Solve Quadratic Inequalities

First, solve \(a^2 - 5a - 4 = 0\) for \(a\): The solutions found are \(a = \frac{5 \pm \sqrt{41}}{2}\). Second, solving \(a^2 - 5a - 5 = 0\) gives roots \(a = \frac{5 \pm \sqrt{45}}{2}\).

Step 6: Identify Range of \(a\) from Inequalities

Determine where \([a] = 4\). This occurs between the fractional root boundary values of the two inequalities. Integrate fractions \(\frac{5 - \sqrt{41}}{2} < a < \frac{5 + \sqrt{41}}{2}\), using intersections identified by solving the inequalities. Result cv|| integrate the ranges into continuous inequality.

Step 7: Select the Correct Set of Values for \(a\)

Comparing the ranges calculated and choosing the corresponding set option from the provided choices. The correct set option is based on intersections: Apply elimination to find that the calculated range overlaps with none.

Key Concepts

Greatest Integer Function

The Greatest Integer Function, often denoted as \([a]\), refers to the largest integer that is less than or equal to a given number \(a\). This function is also called the floor function. It simplifies expressions or equations into integer values by essentially rounding down to the nearest whole number.
For example,

• For the number 3.7, the greatest integer is 3.
• For -2.3, the greatest integer is -3, since it's still smaller than -2.

It's crucial in problems involving discrete variables or steps and often appears in quadratic equations where interpretation of integer values is necessary. In the context of our exercise, greatest integer functions like \([a^2 - 5a + b + 4]\) are essential in determining constraints on the variables when working with known roots of the quadratic equation. This allows translation of complicated expressions into manageable whole number ranges, aligning better with the problem's requirements.

Vieta's Formulas

Vieta's Formulas are key principles in algebra that relate the coefficients of a polynomial to sums and products of its roots. For a quadratic equation of the form \(ax^2 + bx + c = 0\), Vieta's formulas are stated as:

• The sum of the roots, \(r_1 + r_2 = -\frac{b}{a}\), gives a direct link to the coefficient \(b\).
• The product of the roots, \(r_1 \cdot r_2 = \frac{c}{a}\), connects to the constant term \(c\).

These formulas simplify the process of solving quadratic equations by enabling quick derivation of root relations without explicitly calculating the roots. In our problem, Vieta's formulas allow us to set up an equation for the sum of the roots which aids in identifying parts of the quadratic's coefficients. Knowing the roots in advance, we use Vieta's formulas to find and verify \([a^2 - 5a + b + 4]\) and determine \(b\) as part of resolving the quadratic's relationships.

Quadratic Inequalities

Quadratic inequalities involve expressions where a quadratic polynomial is compared to a number or variable using inequality symbols such as \(<\), \(>\), \(\leq\), or \(\geq\). They require finding the range of values that both satisfy the equation and adhere to the inequality's constraints.
Solving a quadratic inequality generally involves:

• Finding the roots of the corresponding quadratic equation by setting the inequality to equality.
• Using these roots to test intervals between and outside the roots to determine which parts satisfy the inequality condition.

In our exercise, the step of solving \([a^2 - 5a - 1] = 4\) as an inequality determines the potential values of \(a\). It is necessary to consider that \(4 \leq a^2 - 5a < 5\), solving for these bounds to find intervals where \([a]\) is consistent with the equation's constraints on \(a\). Carefully addressing these inequalities helps identify the accurate range of \(a\) that fulfills the entire set of conditions presented.