Problem 57 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\alpha \in\left(0, \frac{\pi}{2}\right)\) then \(\sqrt{x^{2}+x}+\frac{\tan ^{2} \alpha}{\sqrt{x^{2}+x}}\) is always greater than or equal to (A) \(2 \tan \alpha\) (B) 1 (C) 2 (D) \(\sec ^{2} \alpha\)

Step-by-Step Solution

Answer

The correct answer is (A) 2\tan \alpha.

Step 1: Simplify the Expression

Given the expression \(\sqrt{x^{2}+x}+\frac{\tan^{2} \alpha}{\sqrt{x^{2}+x}}\), let \(y = \sqrt{x^2 + x}\). Then, the expression becomes \(y + \frac{\tan^2 \alpha}{y}\).

Step 2: Use the AM-GM Inequality

According to the Arithmetic Mean-Geometric Mean Inequality (AM-GM), for any non-negative numbers \(a\) and \(b\), \(\frac{a+b}{2} \geq \sqrt{ab}\). Apply this inequality with \(a = y\) and \(b = \frac{\tan^2 \alpha}{y}\).

Step 3: Calculate the AM-GM Result

The inequality becomes \(\frac{y + \frac{\tan^2 \alpha}{y}}{2} \geq \sqrt{y \cdot \frac{\tan^2 \alpha}{y}} = \tan \alpha\). Simplifying this gives \(y + \frac{\tan^2 \alpha}{y} \geq 2\tan \alpha\).

Step 4: Compare with Options

From Step 3, the expression \(\sqrt{x^2 + x} + \frac{\tan^2 \alpha}{\sqrt{x^2 + x}}\) is always greater than or equal to \(2\tan \alpha\). Among the options, the correct choice is (A) \(2\tan \alpha\).

Key Concepts

Mathematical Inequality

Mathematical inequalities are powerful tools used to compare the sizes of different quantities. They tell us not just about the relative values of expressions but also about bounds and limits that an expression must obey. In our exercise, we use a famous inequality known as the Arithmetic Mean-Geometric Mean (AM-GM) Inequality.

• This inequality states that for any non-negative numbers, the arithmetic mean (average) is always greater than or equal to the geometric mean.
• In simpler terms, \[\frac{a + b}{2} \geq \sqrt{ab}\]where \(a\) and \(b\) are non-negative.
• In our problem, this inequality helps us compare the expression \(\sqrt{x^{2}+x}+\frac{\tan^{2} \alpha}{\sqrt{x^{2}+x}}\) to \(2\tan \alpha\).

This core idea creates a boundary, showing that the expression cannot be less than \(2\tan \alpha\). Inequalities like this are used across various areas of mathematics to establish such critical relationships.

Expression Simplification

Simplification is all about making expressions easier to manage and understand. In mathematical problems, it often involves reducing complex equations or expressions to simpler forms. For example, in our given expression: \[\sqrt{x^{2}+x}+\frac{\tan^{2} \alpha}{\sqrt{x^{2}+x}}\]We introduce a substitution: let \(y = \sqrt{x^{2}+x}\). This allows the expression to be written as:\[y + \frac{\tan^{2} \alpha}{y}\]This substitution helps in applying the AM-GM inequality effectively. Simplification is a critical strategy as it often reveals the underlying structure of a problem, making it easier to solve.

• By isolating simpler components within complex expressions, we can more readily apply mathematical rules or truths.
• It also reduces the potential for errors during further calculations or transformations.

Thus, expression simplification is an essential skill for tackling intricate mathematical challenges.

Trigonometric Identities

Trigonometric identities are equalities involving trigonometric functions that are true for all values of the variable where they are defined. These identities allow us to manipulate and understand expressions involving angles. In our exercise, the trigonometric functions relate specifically to the angle \(\alpha\), with \(\tan \alpha\) representing the tangent of the angle.When dealing with inequalities and expression simplifications, these identities are invaluable because they provide alternative ways to express and work with trigonometric functions. For example:

• \(\tan \alpha = \frac{\sin \alpha}{\cos \alpha}\), which connects tangent to sine and cosine, can help in converting expressions to forms that are simpler to understand or use.
• \(\sec \alpha = \frac{1}{\cos \alpha}\) can also be used to transform or simplify expressions.
• Understanding that \(\tan^2 \alpha + 1 = \sec^2 \alpha\) reveals how trigonometric identities are interlinked and can be used to find equivalent expressions.

Such identities simplify computations and are instrumental in solving trigonometric equations and inequalities. Mastery of these identities equips you to tackle a wide variety of problems in mathematics.