Problem 134 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(\alpha\) and \(\beta\) be the distinct roots of \(a x^{2}+b x+c=0\), then \(\lim _{x \rightarrow \alpha} \frac{1-\cos \left(a x^{2}+b x+c\right)}{(x-\alpha)^{2}}\) is equal to (A) \(\frac{a^{2}}{2}(\alpha-\beta)^{2}\) (B) 0 (C) \(-\frac{a^{2}}{2}(\alpha-\beta)^{2}\) (D) \(\frac{1}{2}(\alpha-\beta)^{2}\)

Step-by-Step Solution

Answer

Option (A) \(\frac{a^2}{2}(\alpha-\beta)^2\).

Step 1: Analyze the Quadratic Equation

Given the quadratic equation is \(a x^2 + bx + c = 0\). The roots of this equation are \(\alpha\) and \(\beta\). This implies that \(a(x - \alpha)(x - \beta) = a x^2 + b x + c\).

Step 2: Understand the Limit Expression

We need to evaluate \(\lim _{x \rightarrow \alpha} \frac{1-\cos \left(a x^{2}+b x+c\right)}{(x-\alpha)^{2}}\). Since \(a\alpha^2 + b\alpha + c = 0\), this term becomes zero at \(x = \alpha\).

Step 3: Apply L'Hôpital's Rule

Since the limit expression results in an indeterminate form \(\frac{0}{0}\) as \(x \to \alpha\), we use L'Hôpital's Rule, which allows us to differentiate the numerator and the denominator until the form is no longer indeterminate. Differentiate the numerator to get \(\sin(a x^2 + b x + c) \, (2ax + b)\) and differentiate the denominator to get \(2(x-\alpha)\).

Step 4: Evaluate the Differentiated Expression

Evaluate \(\lim_{x \to \alpha}\frac{\sin(a x^2 + b x + c) \, (2ax + b)}{2(x-\alpha)}\). As \(x\to\alpha\), \(\sin(a x^2 + b x + c)\to\sin(0) = 0\) and \(2ax + b\to 2a\alpha + b\). We'll apply L'Hôpital's Rule again to this limit.

Step 5: Further Application of L'Hôpital's Rule

Differentiate again: The new numerator becomes \(\cos(a x^2 + b x + c) \, (2ax + b)^2\) and the denominator becomes \(2\). Substitute \(x = \alpha\), that is, evaluate \(\frac{\cos(0) \, (2a\alpha + b)^2}{2}\).

Step 6: Simplify Using Known Values

Since \(\alpha\) and \(\beta\) are roots, and by Vieta's formulas, \(\alpha + \beta = -\frac{b}{a}\) and \(\alpha - \beta = \frac{\sqrt{b^2 - 4ac}}{a}\), apply these values. Substitute into the expression \(\frac{(2a\alpha + b)^2}{2}\), which simplifies further due to zero angle cosine.

Step 7: Final Simplification and Substitute Root Properties

By Vieta’s formulas and symmetry of the problem, substituting \(b = -a(\alpha + \beta)\) yields \((2a\alpha + b) = a(2\alpha - (\alpha + \beta)) = a(\alpha - \beta)\). The expression then reduces to \(\frac{a^2(\alpha - \beta)^2}{2}\).

Step 8: Select the Correct Answer

Compare the simplified expression \(\frac{a^2(\alpha - \beta)^2}{2}\) with the given options. It corresponds to option (A).

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation where the highest degree of the variable is two. It is generally written in the standard form:

• \( ax^2 + bx + c = 0 \)

Here, \( a \), \( b \), and \( c \) are coefficients and \( x \) is the variable. The coefficient \( a \) must be non-zero to make it a quadratic equation.

The main goal when dealing with quadratic equations is to find the values of \( x \) that satisfy the equation, known as the roots. These roots can be real or complex numbers. You can find them by various methods, such as factorization, completing the square, or using the quadratic formula.

Quadratic equations are fundamental in algebra because they appear in various mathematical contexts and have applications in different fields such as physics, engineering, and finance. They serve as a foundation for understanding more complex equations and functions.

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool in calculus used to evaluate limits involving indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). The rule provides a method to simplify these indeterminate forms by differentiating the numerator and the denominator separately.

The rule states that if the limit \( \lim_{x \to c} \frac{f(x)}{g(x)} \) gives an indeterminate form, then:

• \( \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \)

provided \( \lim_{x \to c} \frac{f'(x)}{g'(x)} \) exists.

It's important to remember that differentiation must be applied repeatedly until the limit is no longer indeterminate, and both derivatives \( f'(x) \) and \( g'(x) \) continue to be differentiable at \( x = c \). L'Hôpital's Rule simplifies complex limits into more manageable forms and is widely utilized in integration and finding tangents.

Roots of Quadratic Equations

The roots of a quadratic equation are the solutions for \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). They can be determined using Vieta's formulas or the quadratic formula.

- Vieta's Formulas state:

• \( \alpha + \beta = -\frac{b}{a} \)
• \( \alpha\beta = \frac{c}{a} \)

where \( \alpha \) and \( \beta \) are the roots.

Additionally, the quadratic formula provides another way to determine the roots:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula gives the roots directly by substituting the coefficients from the quadratic equation.

Knowing the roots helps in understanding the behavior of quadratic functions, such as finding the parabola's vertex and intercepts. The nature of the roots, whether real or complex, also offers insights into the graph's appearance and characteristics.