Problem 131 - A Complete Resource Book in Mathematics for JEE Main

Question

If the period of the function \(f(x)=\sin (\sqrt{[n]} x)\), where \([n]\) denotes the greatest integer less than or equal to \(n\), is \(2 \pi\), then (A) \(1 \leq n<2\) (B) \(1

Step-by-Step Solution

Answer

(A) \( 1 \leq n < 2 \)

Step 1: Understand the Given Function

The function is given as \( f(x) = \sin(\sqrt{[n]} \cdot x) \). The period of a function of the form \( \sin(kx) \) is \( \frac{2\pi}{k} \). Since the period is given as \( 2\pi \), we should equate and solve for \( k \) such that \( \frac{2\pi}{k} = 2\pi \).

Step 2: Determine the Value of k

From \( \frac{2\pi}{k} = 2\pi \), solving gives \( k = 1 \). For the function \( \sin(\sqrt{[n]} \cdot x) \), \( k = \sqrt{[n]} \). Therefore, \( \sqrt{[n]} \) must equal 1.

Step 3: Solve for [n]

Since \( \sqrt{[n]} = 1 \), squaring both sides gives \( [n] = 1^2 = 1 \). The greatest integer less than or equal to \( n \) is 1. Thus, \( n \) must be within the range such that \( [n] = 1 \).

Step 4: Translate the Condition Into an Interval for n

The condition \( [n] = 1 \) means the greatest integer less than or equal to \( n \) is 1. This implies \( 1 \leq n < 2 \). This is because \( [n] = 1 \) includes values from 1 up to (but not including) 2.

Key Concepts

Greatest Integer Function

The greatest integer function, denoted as \([n]\), is also known as the floor function. It serves a unique purpose by finding the largest integer that is less than or equal to a given number \(n\). This means \([n]\) outputs a value where \(n\) falls between two integers, it "rounds down" to the nearest whole number.

• If \({n} = 1.9\), then the greatest integer function \([n] = 1\).
• If \({n} = 2\), then \([n] = 2\) because \({2}\) is an integer.

This function is critical when working with expressions, like in the exercise where \(f(x) = \sin(\sqrt{[n]} \cdot x)\). To solve for the periodicity, identifying \([n]\) ensures we're using the correct integer value for any \(n\) within the given range. Using the greatest integer function requires understanding its role in translating continuous real numbers into discrete integer values, which might affect functions' properties like their periodicity or intervals of validity.

Sine Function

The sine function is one of the primary trigonometric functions and is periodic in nature. Typically represented as \(\sin (x)\), this function outputs a value between \(-1\) and \(1\) for any given \(x\). The standard period for \(\sin(x)\) is \(2\pi\), meaning it repeats its values over every interval of \(2\pi\). In our exercise, \(f(x) = \sin(\sqrt{[n]} \cdot x)\), we introduce a factor \(\sqrt{[n]}\) multiplying \(x\). The periodicity formula for \(\sin(kx)\) is given by \(\frac{2\pi}{k}\). Here, by setting the period to \(2\pi\), we solve the equation \(\frac{2\pi}{k} = 2\pi\) which gives \(k = 1\). Thus, \(\sqrt{[n]}\) must equal 1. Understanding the sine function's periodic nature helps solve problems involving periodicity where transformations or manipulated terms alter its standard period. This knowledge is essential for mathematical studies involving oscillations, waves, or any phenomena with cyclical patterns.

Interval Notation

Interval notation is a concise way of writing subsets of the real number line. It's particularly useful for expressing solutions to inequalities and ranges of values. An interval is denoted by brackets, with each bracket's type signaling inclusion or exclusion of endpoints:

• Square brackets \([\text{ and } ]\)mean the number is included.
• Round parentheses \( (\text{ and } )\) signal that the endpoint is not included.

In our solution, where \(1 \leq n < 2\), the bracket \([\)indicates that 1 is part of the solution while parentheses \( )\)signify values near 2 are not included. For the exercise at hand, accurate interpretation of interval notation ensures you understand which values of \(n\) satisfy the conditions dictated by functions like the greatest integer function. Such notational precision is foundational in mathematics for communicating ranges, domains, and solutions effectively.