Problem 21 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(x\) is real, then the expression \(\frac{x^{2}+34 x-71}{x^{2}+2 x-7}\) (A) cannot lie between 5 and 9 (B) always lies between 5 and 9 (C) is not real (D) None of these

Step-by-Step Solution

Answer

Option (A) cannot lie between 5 and 9.

Step 1: Understanding the Problem

We need to find when the expression \( \frac{x^{2}+34x-71}{x^{2}+2x-7} \) will lie between, is always between, or can never be between 5 and 9. We will solve inequalities to determine this.

Step 2: Setting Up the Inequality

To find when the expression lies between 5 and 9, we set up the inequality: \[ 5 < \frac{x^{2}+34x-71}{x^{2}+2x-7} < 9. \] This breaks into two separate inequalities to solve: \( \frac{x^{2}+34x-71}{x^{2}+2x-7} > 5 \) and \( \frac{x^{2}+34x-71}{x^{2}+2x-7} < 9 \).

Step 3: Solving the First Inequality

Solve \( \frac{x^{2}+34x-71}{x^{2}+2x-7} > 5 \). Subtract 5 from both sides to get: \[ \frac{x^{2}+34x-71 - 5(x^{2}+2x-7)}{x^{2}+2x-7} > 0 \]. This simplifies to: \[ \frac{-4x^{2}+24x-36}{x^{2}+2x-7} > 0. \] Factor the numerator: \[ \frac{-4(x^2 - 6x + 9)}{x^{2}+2x-7} > 0. \] The quadratic \( x^2 - 6x + 9 \) factors further to \( (x-3)^2 \), hence: \[ \frac{-4(x-3)^2}{x^{2}+2x-7} > 0. \]

Step 4: Analyzing the First Inequality

Notice that \( (x-3)^2 \) is always non-negative. Hence, \( -4(x-3)^2 \) is always non-positive. This implies the inequality \( \frac{-4(x-3)^2}{x^{2}+2x-7} > 0 \) cannot be true for any real \( x \).

Step 5: Conclusion from the First Inequality

Since the inequality \( \frac{x^{2}+34x-71}{x^{2}+2x-7} > 5 \) is never true, the expression \( \frac{x^{2}+34x-71}{x^{2}+2x-7} \) can never be greater than 5.

Step 6: Solving the Second Inequality

Not necessary since \( \frac{x^{2}+34x-71}{x^{2}+2x-7} > 5 \) cannot happen, making it clear that the expression does not lie between 5 and 9.

Step 7: Final Answer

The value of \( \frac{x^{2}+34x-71}{x^{2}+2x-7} \) cannot lie between 5 and 9 for any real \( x \). Thus, the correct answer is (A) cannot lie between 5 and 9.

Key Concepts

Inequalities

Inequalities are mathematical expressions involving comparisons. Here, we are dealing with an inequality of the form \( \frac{x^2 + 34x - 71}{x^2 + 2x - 7} > 5 \). To solve inequalities that involve a rational expression, we set the inequality and manipulate the terms as if solving an equation, while being careful with the inequality direction. However, inequalities involving rational expressions often require us to consider the signs of the numerator and the denominator separately.

When solving inequalities, it's essential to identify the critical points where the expression equals zero or is undefined. These critical points help determine the intervals where the inequality holds. For rational expressions, critical points can arise from:

• Values that make the numerator zero.
• Values that make the denominator zero, which are points of discontinuity.

Analyzing these intervals involves checking the signs of the expression in each interval divided by the critical points. This way, you can accurately determine where the inequality is satisfied or not.

Quadratic Functions

A quadratic function is an essential form in algebra represented by \( ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants and \( a eq 0 \). In our exercise, the expression we dealt with in the numerator simplifies to a quadratic function \( -4(x - 3)^2 \) after factoring.

Understanding the behavior of quadratic functions is crucial here. Specifically,

• The quadratic \( (x - 3)^2 \) is always non-negative because square terms are always zero or positive.
• This made it impossible for the inequality \( -4(x - 3)^2 > 0 \) to hold, as multiplying a non-negative expression by a negative constant results in a non-positive product.

Such behaviors help us conclude that the quadratic in this context makes the inequality not hold for any real \( x \). Understanding these characteristics can help effectively analyze and solve complex inequalities involving quadratics.

Rational Expressions

Rational expressions represent the quotient of two polynomials. In our exercise, the expression \( \frac{x^2 + 34x - 71}{x^2 + 2x - 7} \) is a rational expression. Handling these carefully is essential due to their properties, especially concerning division by zero.

For rational expressions:

• Identify any restrictions caused by the denominator, where the expression is undefined. These values make the denominator zero.
• When simplifying, the expression inside may transform into a simpler form, revealing more about its graph or solutions.
• Consider how the numerator and denominator behave separately to understand the complete movement and behavior of the rational expression as a whole.

The complexity of the given problem is primarily due to the rational expression's behavior concerning its denominator and the required simplifications. Mastery over manipulation and simplification of rational expressions aids greatly in resolving such complex situations in analysis.