Problem 36 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(a\) and \(b\) are odd integers then \([x]^{2}+a[x]+b=0\) (where [. ] denotes greatest integer function) has (A) finite number of roots (B) infinite number of roots (C) no roots (D) None of these

Step-by-Step Solution

Answer

(C) no roots

Step 1: Understanding the Problem

We need to determine the nature of the roots of the quadratic expression \([x]^{2}+a[x]+b=0\), where \(a\) and \(b\) are odd integers. Here, \([x]\) represents the greatest integer function, which means the largest integer less than or equal to \(x\).

Step 2: Analyzing the Properties of the Equation

Since \([x]\) is an integer, let \(n = [x]\). The equation becomes \(n^2 + an + b = 0\), which is a standard quadratic in terms of \(n\).

Step 3: Checking for Integer Solutions

For the quadratic equation \(n^2 + an + b = 0\) to have integer solutions, we need the discriminant \(D = a^2 - 4b\) to be a perfect square, say \(k^2\). Thus, \(a^2 - 4b = k^2\) for some integer \(k\).

Step 4: Properties of Odd Integers

Both \(a\) and \(b\) are odd. Substituting this into the discriminant condition: \(a^2\) is odd since the square of an odd number is odd, and \(4b\) is even (since 4 is even and \(b\) is an odd integer). Thus, \(a^2 - 4b\) results in an odd number, which cannot be a perfect square because a perfect square that results from an integer \(k\) is always even.

Step 5: Conclusion

Since \(a^2 - 4b\) cannot be a perfect square, the quadratic equation \(n^2 + an + b = 0\) does not have integer solutions for \(n\). Consequently, the original equation \([x]^{2}+a[x]+b=0\) has no solutions. Thus, the answer is (C) no roots.

Key Concepts

Odd Integers

Odd integers are whole numbers that cannot be evenly divided by 2. These numbers occur in sequence as every other whole number, such as 1, 3, 5, 7, etc. Understanding odd integers is crucial because they exhibit unique properties that influence mathematical concepts and solutions, as seen in the exercise involving the quadratic equation.

Odd integers have special characteristics:

• When two odd integers are added or subtracted, the result is always even. For instance, 3 + 5 = 8.
• The product of two odd integers is always odd, such as 3 × 5 = 15.
• Squaring an odd integer results in another odd integer, for example, 3² = 9.

In the context of the original exercise, both coefficients \(a\) and \(b\) were odd. This peculiarity played a crucial role in determining the nature of the roots of the given quadratic equation.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in a single variable, typically written in the form \(ax^2 + bx + c = 0\). In the exercise, we deal with a specific quadratic equation wherein variables are treated within the context of the greatest integer function.

The general properties of quadratic equations include:

• They graph as parabolas, opening upwards if \(a > 0\) and downwards if \(a < 0\).
• They can have two distinct roots, one repeated root, or no real roots depending on their discriminant.
• The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(b^2 - 4ac\). It determines the nature of the solutions:
• If \(D > 0\), there are two distinct real roots.
• If \(D = 0\), there is one repeated real root.
• If \(D < 0\), there are no real roots.

In this exercise, the equation turned into a quadratic form \(n^2 + an + b = 0\) in terms of \(n\). Evaluating the discriminant's nature directly influenced the determination that no integer solutions exist for specific integer values.

Integer Solutions

Integer solutions refer to the values that satisfy an equation where the entire expression evaluates to an integer. Finding integer solutions in mathematical problems often involves checking conditions or properties that integers must meet.

In quadratic equations, determining if integer solutions exist may require inspecting the discriminant and ensuring it results in a perfect square. The exercise showed that if the discriminant of a quadratic equation represented by \(n^2 + an + b = 0\) is not a perfect square, integer solutions are not possible.

Here are key points to understand integer solutions:

• An integer solution to an equation means substituting the integer into the equation yields a true statement.
• Quadratic equations with integer coefficients need the discriminant to be a perfect square for integer solutions to exist.
• When both coefficients \(a\) and \(b\) are odd, their mathematical interaction, as explained in the exercise, can prevent the discriminant from being a perfect square, eliminating the possibility of integer solutions.

Thus, evaluating whether integer solutions exist requires both conditions for consistency in the equations and careful calculus of involved parameters.