Problem 22 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The number of real solutions of \(\tan ^{-1} \sqrt{x(x+1)}+\sin ^{-1} \sqrt{x^{2}+x+1}=\frac{\pi}{2}\) is (A) 0 (B) 1 (C) 2 (D) infinite

Step-by-Step Solution

Answer

The number of real solutions is 0 (Option A).

Step 1: Understand the Expression

Recognize that the given equation is \( \tan^{-1}(\sqrt{x(x+1)}) + \sin^{-1}(\sqrt{x^2+x+1}) = \frac{\pi}{2} \). We need to find the values of \( x \) which satisfy this equality.

Step 2: Analyze the Inverse Functions

Recall that for \( \tan^{-1} y + \sin^{-1} z = \frac{\pi}{2} \), we can have the case where \( \tan^{-1} y = \tan^{-1} \frac{1}{z} \). Thus, for \( \sqrt{x(x+1)} = \frac{1}{\sqrt{x^2+x+1}} \), squaring both sides will simplify the analysis.

Step 3: Simplify and Solve the Relationship

Square both sides of \( \sqrt{x(x+1)} = \frac{1}{\sqrt{x^2+x+1}} \) to get \( x(x+1) = \frac{1}{x^2+x+1} \). Cross-multiply to form the equation \( x(x+1)(x^2+x+1) = 1 \).

Step 4: Solve the Polynomial Equation

Expand and simplify the equation to \( x^4 + 2x^3 + 2x^2 + x - 1 = 0 \). We aim to find the roots of this quartic equation to determine the possible \( x \) values that satisfy this.

Step 5: Check for Real Roots

Use the Rational Root Theorem or online calculators to check the real roots of \( x^4 + 2x^3 + 2x^2 + x - 1 = 0 \). Determine which roots are valid solutions to the original trigonometric equation.

Step 6: Verify the Solutions

Substitute any found real roots back into the original inverse trigonometric equation to ensure they satisfy the condition \( \tan^{-1} y + \sin^{-1} z = \frac{\pi}{2} \). Confirm any solutions found in Step 5.

Step 7: Conclude the Number of Real Solutions

After verification, it turns out there is no real solution that satisfies the condition fully due to contradictions in solving trigonometric identities or real domain issues.

Key Concepts

Understanding Inverse Trigonometric Functions

Inverse trigonometric functions help us find angles when given specific trigonometric values. For example, \(\tan^{-1}(x)\) provides the angle whose tangent is \(x\). Similarly, \(\sin^{-1}(x)\) gives the angle with sine \(x\). These functions are essential since they reverse trigonometric operations, which is especially useful in geometry and calculus.
In problems, these functions help determine angles that fulfill particular conditions. Importantly, each inverse trigonometric function has a specific range of output, known as its principal value. For instance:

• \(\tan^{-1}(x)\) ranges from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). This means the result we get will always lie in this interval.
• \(\sin^{-1}(x)\) has a range from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).

Understanding these ranges allows us to correctly solve equations involving the sum of inverse trigonometric functions, such as \(\tan^{-1}(y) + \sin^{-1}(z) = \frac{\pi}{2}\). Adding both functions can help relate different trigonometric expressions, facilitating the equation's simplification and solution.

Solving Quartic Equations

Quartic equations are polynomials of degree four. They take the general form \(ax^4 + bx^3 + cx^2 + dx + e = 0\), where \(a \, eq \, 0\). Solving quartic equations can be complex compared to quadratic or cubic equations, but several methods exist.
An effective approach is factorization, where possible, or using numerical methods for precision. In other instances, it's helpful to apply advanced mathematical techniques like Descartes' Rule of Signs to evaluate the number and types of roots. If manual solving becomes overwhelming, utilizing technology or specialized calculators aids in finding real solutions.
For the quartic equation \(x^4 + 2x^3 + 2x^2 + x - 1 = 0\) from our example, determining solutions involves looking for real \(x\) values that fulfill the equation. A solution might include:

• Checking for rational roots using the Rational Root Theorem, which proposes potential fractions as solutions to the polynomial.
• Exploiting symmetric properties if applicable.
• Using numerical tools for approximating roots when analytical solutions are challenging.

Exploring these gives insight and possible roots, which are then tested to confirm their validity.

Understanding Polynomial Roots

Finding polynomial roots solves for \(x\) values where the polynomial equals zero. These roots can be real or complex. Real solutions directly intersect the x-axis on a graph of the polynomial, while complex roots do not.
Analyzing the polynomial \(x^4 + 2x^3 + 2x^2 + x - 1 = 0\), each potential root is tried to see if it satisfies the equation \(f(x) = 0\). Techniques for analyzing roots include:

• The Rational Root Theorem, used to suggest rational solutions derived from the equation's coefficients.
• Graphical methods to visually assess where the graph intercepts the x-axis, highlighting real root locations.
• Using synthetic division to simplify the equation once possible roots are identified.

Understanding these methods helps in tackling more complex polynomials, aiming for solutions that potentially satisfy the original condition set by the trigonometric equation.