Problem 40 - A Complete Resource Book in Mathematics for JEE Main

Question

The number of real roots of the equation \(2^{\sin ^{4} x}-2^{\cos ^{2} x}=\) 1 is (A) 2 (B) 1 (C) infinite (D) None of these

Step-by-Step Solution

Answer

The equation has infinite real roots, option (C).

Step 1: Understand the Given Equation

The equation given is \(2^{\sin^4 x} - 2^{\cos^2 x} = 1\). We are tasked to find the number of real roots. The equation involves exponential functions of trigonometric expressions, specifically \(\sin^4 x\) and \(\cos^2 x\).

Step 2: Analyze the Range of Trigonometric Functions

Recall that for any real number \(x\), \(\sin^2 x + \cos^2 x = 1\). Therefore, both \(\sin^2 x\) and \(\cos^2 x\) take values between 0 and 1. Consequently, \(\sin^4 x\) is between 0 and 1, and \(\cos^2 x\) is also between 0 and 1.

Step 3: Simplify the Problem

The equation can be rewritten by setting \(a = \sin^4 x\) and \(b = \cos^2 x\), thus the equation becomes \(2^a - 2^b = 1\) where \(0 \leq a, b \leq 1\). This implies we are searching for integer solutions \(a, b\) such that the difference in their exponential forms is 1.

Step 4: Evaluate Potential Solutions

Consider possible values for \(a\) and \(b\) within the given range.- If \(a = 0\), we get \(2^0 = 1\), and thus \(2^b = 1 + 1 = 2\), resulting in \(b = 1\).- If \(a = 0.5\), \(2^{0.5} \approx 1.414\), and \(2^b = 1 + 1.414\) is not a power of 2.- If \(a = 1\), we get \(2^1 = 2\), and thus \(2^b = 2 + 1 = 3\), which is not a power of 2.Hence, the pair (0, 1) represents a valid solution.

Step 5: Check Solution Consistency

For \(\sin^4 x = 0\), we have \(\sin x = 0\) leading to \(x = n\pi\) for integer \(n\). For \(\cos^2 x = 1\), we have \(\cos x = \pm 1\) leading to similar solutions. Both conditions can simultaneously hold for even integer multiples of \(\pi\). Therefore, there are multiple \(x\) values corresponding to \(x = 2n\pi\).

Step 6: Conclude the Solution

Since there are infinitely many even integer multiples of \(\pi\) that satisfy the condition, the equation has infinite real roots.

Key Concepts

Exponential Equations

Exponential equations are equations in which variables appear in the exponent. In the given exercise, the equation is of the form \(2^{\sin^4 x} - 2^{\cos^2 x} = 1\). Here, the terms with exponential functions are driven by trigonometric inputs. Understanding exponential equations involves knowing how to manipulate and rationalize expressions involving powers. Such equations are versatile in modeling growth and decay processes and require special techniques for solving, such as matching bases, logarithms, or bounds by specific numeric constraints.

In the practice of solving these, ensure that the exponent expressions meet the base criteria precisely to get the right solution, as exponential functions are highly sensitive to changes in the exponent. This sensitivity underscores the importance of correctly identifying the base values and any potential integer solutions.

Real Roots

The real roots of an equation are the values that satisfy the equation when substituted in place of the unknown variable. Real roots are part of solving equations and play a crucial role in determining where functions cross the x-axis on a graph. For the given equation \(2^{\sin^4 x} - 2^{\cos^2 x} = 1\), determining the real roots involves finding the values of \(x\) that make the equation true.

This requires consideration of the possible pairs of \(a\) and \(b\) values (\(a = \sin^4 x\) and \(b = \cos^2 x\)) within their valid ranges. By employing this strategy, we identified solutions where the exponents equal 1 unit apart, allowing us to conclude the number of solutions. Knowing the set of real roots informs us about the behavior and solutions possible for the equation, such as whether it has a finite number of solutions or infinitely many, as this exercise concludes.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring angles. One of the most fundamental identities, used often in solving trigonometric equations, is \( \sin^2 x + \cos^2 x = 1\). This fundamental identity helps in simplifying expressions and solving equations involving sine and cosine functions.

In the provided exercise, leveraging the identity \( \sin^2 x + \cos^2 x = 1\) was critical in understanding the range these functions can take, and thus in unearthing values for \(a\) and \(b\). By transforming trigonometric functions into equivalent forms, and then using their potential range from 0 to 1, it was possible to identify the condition that would lead to the correct real roots. Mastery of these trigonometric identities amplifies the understanding of how trigonometric functions interact within equations, especially those involving exponential forms as seen here.