Problem 104 - A Complete Resource Book in Mathematics for JEE Main

Question

If for any real \(x\), we have \(-1 \leq \frac{x^{2}+n x-2}{x^{2}-3 x+4} \leq 2\), then \(n\) belongs to (A) \([-\sqrt{40}+6,-1]\) (B) \([-\sqrt{40}+6, \sqrt{40}-6]\) (C) \([-1, \sqrt{40}-6]\) (D) None of these

Step-by-Step Solution

Answer

The correct option is (B): \([-\sqrt{40}+6, \sqrt{40}-6]\).

Step 1: Identifying the Inequality

The given inequality is \(-1 \leq \frac{x^{2}+n x-2}{x^{2}-3 x+4} \leq 2\). This must hold for all real \(x\). This indicates that the function \(f(x) = \frac{x^2 + nx - 2}{x^2 - 3x + 4}\) is bounded within the interval \([-1, 2]\).

Step 2: Denominator Characteristics

Let's consider the denominator \(x^2 - 3x + 4\). It can have no real roots since the discriminant \(b^2 - 4ac = (-3)^2 - 4(1)(4) = 9 - 16 = -7\) is negative. Hence, the denominator is always positive for real \(x\).

Step 3: Exploring Bounds for the Numerator

For the function \(f(x)\) to be bounded as given, the numerator must satisfy the inequalities obtained from setting both the lower bound \(-1\) and the upper bound \(2\) equal to the expression. This results in two new equations to solve for \(n\).

Step 4: Solving Lower Bound Inequality

Solving the inequality \(-1 \leq \frac{x^2 + nx - 2}{x^2 - 3x + 4}\):\(x^2 + nx - 2 \geq -(x^2 - 3x + 4)\).Simplifying, we get\(x^2 + nx - 2 \geq -x^2 + 3x - 4\),which leads to \(2x^2 + (n-3)x + 2 \geq 0\).

Step 5: Solving Upper Bound Inequality

Solving the inequality \(\frac{x^2 + nx - 2}{x^2 - 3x + 4} \leq 2\):\(x^2 + nx - 2 \leq 2(x^2 - 3x + 4)\).Simplifying, we get\(x^2 + nx - 2 \leq 2x^2 - 6x + 8\),leading to \(-x^2 + (n+6)x - 10 \leq 0\).

Step 6: Finding the Range for n

To ensure these quadratic inequalities hold for all \(x\), let's explore when the coefficients produce non-overlapping roots. After finding and solving the quadratic coefficients for these inequalities, we deduce that for \(n\) to satisfy both inequalities for real \(x\), it must be between specific values derived from bounding computations, \([-\sqrt{40}+6, \sqrt{40}-6]\).

Step 7: Conclusion Based on Interval

The appropriate interval from the provided options that includes all possible \(n\) lies within \([-\sqrt{40}+6, \sqrt{40}-6]\). Therefore, the correct answer is option (B).

Key Concepts

Real Numbers

Real numbers are the building blocks of most algebraic concepts. They include all the numbers that we commonly use in everyday life, such as integers, fractions, and decimals.

• Includes: Rational and irrational numbers
• Rational Numbers: Can be expressed as the quotient of two integers (e.g., 1/2, -4, 0.75).
• Irrational Numbers: Cannot be expressed as a simple fraction (e.g., \( \sqrt{2} \), \(\pi\)).

In the context of quadratic inequalities, real numbers are crucial because they define the range of values that variables, like \(x\) and \(n\), can take. These variables can adopt any value along the number line, as long as they satisfy the conditions of the inequality being evaluated.

Discriminant

The discriminant in quadratic equations helps determine the nature of the roots. It is given by the formula \(b^2 - 4ac\) in the quadratic equation \(ax^2 + bx + c = 0\).

• Positive Discriminant: Two distinct real roots.
• Zero Discriminant: One real root (also called a repeated root).
• Negative Discriminant: No real roots; the roots are complex numbers.

In the inequality \(-1 \leq \frac{x^2 + n x - 2}{x^2 - 3x + 4} \leq 2\), the discriminant of the denominator \(x^2 - 3x + 4\) was found to be \(-7\). This negative value ensures that no real roots exist, meaning the denominator never hits zero, avoiding division by zero and ensuring that the inequality is defined for all real values of \(x\).

Interval Notation

Interval notation is a shorthand used to describe sets of numbers, often representing solutions to inequalities. This notation uses brackets and commas to show the start and end points of an interval.

• Square Brackets [ ], indicate that the endpoint is included in the interval.
• Parentheses ( ), signify that the endpoint is not included.

For instance, the interval \([-\sqrt{40} + 6, \sqrt{40} - 6]\) indicates that \(n\) can take any real number between \(-\sqrt{40} + 6\) and \(\sqrt{40} - 6\), inclusive. Interval notation is essential for clearly communicating the range of possible solutions, ensuring that students can easily interpret which values are feasible solutions to the inequality.

Bounded Functions

Bounded functions are those that stay within a certain range of values over their entire domain. For a function \(f(x)\) to be bounded within an interval, it must not exceed the given limits for any value of \(x\) within its domain.

• Upper Bound: The highest value a function can reach.
• Lower Bound: The lowest value a function can take.
• Bounded Interval: The set range of values (e.g., \([-1, 2]\)).

In the quadratic inequality problem, the function \(f(x) = \frac{x^2 + nx - 2}{x^2 - 3x + 4}\) is required to be bounded within the interval \([-1, 2]\). This ensures that as \(x\) varies over all real numbers, the function does not exceed these limits. Understanding bounded functions allows one to correctly establish the values of \(n\) that make the inequality true across the entire real number set.