Problem 64 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(a<\frac{1}{32}\), then the number of solutions of \(\left(\sin ^{-1} x\right)^{3}+\) \(\left(\cos ^{-1} x\right)^{3}=a \pi^{3}\) is (A) 0 (B) 1 (C) 2 (D) infinite

Step-by-Step Solution

Answer

0 solutions.

Step 1: Define the Function and Its Range

Consider the equation \( (\sin^{-1} x)^3 + (\cos^{-1} x)^3 = a \pi^3 \). Recall that \( \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \) for any \(-1 \leq x \leq 1\). This implies that both \( \sin^{-1} x \) and \( \cos^{-1} x \) range between \( 0 \) and \( \frac{\pi}{2} \). Thus, \( (\sin^{-1} x)^3 \) and \( (\cos^{-1} x)^3 \) range from \( 0 \) to \( \left(\frac{\pi}{2}\right)^3 \).

Step 2: Derive the Maximum Value of the Function

The expression \( (\sin^{-1} x)^3 + (\cos^{-1} x)^3 \) reaches its maximum value when one term is \( 0 \) and the other is at its maximum. However, since both terms cannot be zero at the same time, evaluate the combination as both reach their maximum at \( x=1 \) and \( x=-1 \). This gives \( \left(\frac{\pi}{2}\right)^3 \approx 3.875 \).

Step 3: Compare with Given \( a \)

We are given that \( a < \frac{1}{32} \). Translating this condition to the equation, we have \( a \pi^3 < \frac{1}{32} \pi^3 \). Since the maximum value of our function is \( \left(\frac{\pi}{2}\right)^3 \approx 3.875 \pi^3 \), and even the smallest plausible combination \( 0 + 0^3 = 0 \), both are greater than \( a \pi^3 \) when \( a < \frac{1}{32} \).

Step 4: Conclusion on Number of Solutions

Given the ranges and maximum values, \( (\sin^{-1} x)^3 + (\cos^{-1} x)^3 \) cannot be equal to \( a \pi^3 \) for any \( x \) in \([-1, 1]\), when \( a < \frac{1}{32} \). Therefore, there are no real solutions to the equation in this scenario.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find the angles for given trigonometric values. They are the inverse operations of sine, cosine, and other basic trigonometric functions.
For instance, \(\sin^{-1} x\) asks for the angle \(\theta\) such that \(\sin \theta = x\). These functions have specific ranges to ensure they return unique values:

• \(\sin^{-1} x\) returns values between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \).
• \(\cos^{-1} x\) returns values between \( 0 \) to \( \pi \).

When considering the sum \(\sin^{-1} x + \cos^{-1} x\), it conveniently simplifies to a constant \( \frac{\pi}{2} \), thanks to the complementary nature of sine and cosine. This balance ensures that if one angle increases, the other decreases proportionally to maintain this sum.
This is essential for our work with inverse trigonometric functions and serves as a baseline for calculations and transformations.

Solution of Equations

Solving equations involves finding values for variables that make the equation true. In our case, we analyze the equation \(\left(\sin^{-1} x\right)^{3} + \left(\cos^{-1} x\right)^{3} = a \pi^{3}\). The approach involves understanding the behavior and limits of each component.
We know that \(\sin^{-1} x\) and \(\cos^{-1} x\) both move within \(0\) to \(\frac{\pi}{2}\). After cubing these, the new range becomes between \(0\) and \(\left(\frac{\pi}{2}\right)^{3}\).

• Find the maximum achievable sum by evaluating boundary conditions, typically when one value is maximal and the other minimal.
• Verification involves matching this sum against the target \(a \pi^{3}\), ensuring we consider all the values that \(a\) can take.

Using these insights, if the maximum summed value is lower or higher than any \(a\pi^3\), it guides us to the existence or absence of solutions. Understanding these principles is critical to mastering intra-variable behavior within equations.

Function Range Analysis

Function range analysis helps determine where (in terms of input value) a function exists and can be calculated. The range is the collection of possible outputs of a function. \(\sin^{-1} x\) and \(\cos^{-1} x\) have specific ranges, both impacting the outcome when they are manipulated through other operations like squaring or cubing.
For example, in \(\left(\sin^{-1} x\right)^{3}\), the range is \(0\) to \(\left(\frac{\pi}{2}\right)^{3}\) because cubing the values expands the potential results.

• By analyzing the function's operation on the specified range, you determine maximum and minimum possible values.
• In our equation setup, the essence of comparing this range with the stipulated \(a \pi^{3}\) points to zero or infinite solutions.

Carefully mapping how a function transforms values gives the necessary insight to predict behavior and guarantees you are considering all potential solutions or the absence thereof.