Problem 66 - A Complete Resource Book in Mathematics for JEE Main

Question

The function \(f(x)=\frac{\sin ^{101} x}{\left[\frac{x}{\pi}\right]+\frac{1}{2}}\), where \([x]\) denotes the integral part of \(x\) is (A) an odd function (B) an even function (C) neither odd nor even function (D) both odd and even function

Step-by-Step Solution

Answer

(C) neither odd nor even function.

Step 1: Understand Even and Odd Functions

A function is even if, for all values of \(x\), \(f(x) = f(-x)\). A function is odd if, for all values of \(x\), \(f(-x) = -f(x)\). Our goal is to check these properties for the given function \(f(x) = \frac{\sin^{101} x}{\left[\frac{x}{\pi}\right] + \frac{1}{2}}\).

Step 2: Examine Oddness

To verify if \(f(x)\) is odd, we calculate \(f(-x)\). We have \(f(-x) = \frac{\sin^{101}(-x)}{\left[\frac{-x}{\pi}\right] + \frac{1}{2}}\). Because sine is an odd function, we obtain \(\sin(-x) = -\sin(x)\), so \(\sin^{101}(-x) = -\sin^{101}(x)\). Thus, \(f(-x) = \frac{-\sin^{101}(x)}{\left[\frac{-x}{\pi}\right] + \frac{1}{2}}\).

Step 3: Simplify Oddness Condition

Compare \(f(-x)\) with \(-f(x)\). For the function to be odd, \(\left[\frac{-x}{\pi}\right] + \frac{1}{2}\) should equal \(-\left(\left[\frac{x}{\pi}\right] + \frac{1}{2}\right)\). Check whether \([\frac{-x}{\pi}] = -[\frac{x}{\pi}] - 1\) holds for all \(x\). This relation generally does not hold true for all values of \(x\), thus \(f(x)\) is not odd.

Step 4: Examine Evenness

To check if \(f(x)\) is even, calculate \(f(-x) = \frac{-\sin^{101}(x)}{[\frac{-x}{\pi}] + 0.5}\). We need to verify \(f(-x) = f(x) = \frac{\sin^{101}(x)}{[\frac{x}{\pi}] + 0.5}\). This equation simplifies to \(\sin^{101}(x) = -\sin^{101}(x)\), which is impossible unless \(\sin^{101}(x) = 0\). Thus, \(f(x)\) is not even.

Step 5: Conclusion

Since \(f(x)\) is neither equal to \(-f(x)\) nor equal to \(f(-x)\), the function is neither odd nor even.

Key Concepts

Odd and Even Functions

In mathematics, functions are classified as either odd, even, or neither based on their symmetry properties. The symmetrical behavior of a function is crucial for understanding various mathematical concepts and solving problems efficiently.
If a function is even, it has symmetry about the y-axis, meaning that for every input value, the output is the same whether we plug in a positive or negative value. This can be formally written as:

• For an even function: \(f(x) = f(-x)\) for all \(x\).

Odd functions, on the other hand, have rotational symmetry about the origin, so the function exhibits this property:

• For an odd function: \(f(-x) = -f(x)\) for all \(x\).

In our exercise, we determine that neither of these conditions is satisfied for the given function, hence it is neither even nor odd.

Integer Part Function

The integer part function, often denoted as \([x]\), is a fundamental concept in mathematics that represents the largest integer less than or equal to a given number \(x\). This function is sometimes referred to as the floor function.
For instance:

• For \(x = 3.7\), \([x] = 3\).
• For \(x = -1.2\), \([x] = -2\).

The integer part function plays a critical role in our problem. It appears in the denominator of the function, disturbing the symmetry required for the function to be classified as odd or even.
Because it causes differing integer results for positive and negative values of \(x\), \(f(x)\) cannot maintain necessary symmetry.

Sine Function

The sine function is one of the fundamental trigonometric functions, representing the y-coordinate of a point on the unit circle. The key identity for sine is that it is an odd function, meaning:

• \(\sin(-x) = -\sin(x)\)

This property impacts the overall function in the problem by flipping the sign of the numerator when \(x\) becomes \(-x\). In our problem, we raise the sine function to the 101st power, leading to:

• \(\sin^{101}(x)\), retaining oddness as long as the exponent is odd.

In scenarios like this, checking oddness and evenness requires careful attention, as exponentiation could alter function properties.

JEE Main Mathematics

The Joint Entrance Examination (JEE) Main is a crucial exam for students in India seeking admission to engineering colleges. Successfully solving problems related to trigonometric functions is vital, as they are a core part of the JEE Main mathematics curriculum.
Trigonometry involves analyzing angles and the relationships between the angles and sides of triangles. Mastery of concepts like odd and even functions, the integer part function, and understanding trigonometric identities, significantly aids in tackling this subject effectively.
Students preparing for JEE Main should focus on solving varied problems to strengthen their understanding and fluency with trigonometric functions' concepts and other algebraic properties.