Question

Let \(\mathrm{f}(\mathrm{x})=\tan \sqrt{\mathrm{m}} \cdot \mathrm{x}\) where \(\mathrm{m}=[\mathrm{P}]=\) greatest integer less then or equal to P and Principal period of \(\mathrm{f}(\mathrm{x})\) is \(\pi\). then (a) \(2 \leq \mathrm{P} \leq 3\) (b) \(1 \leq \mathrm{P} \leq 2\) (c) \(1 \leq \mathrm{P}<2\) (d) \(3 \leq \mathrm{P}<4\)

Short answer

The correct answer is (c) \(1 \leq \mathrm{P}<2\).

Step-by-step solution

Understand the function

The given function is \( \mathrm{f}(\mathrm{x})=\tan \sqrt{\mathrm{m}} \cdot \mathrm{x} \), where \( \mathrm{m}=\left[\mathrm{P}\right] \), the greatest integer less than or equal to P.

Find the period for the given function

The period of this function will be \( \pi / \sqrt{m} \).

Apply the Principal Period condition

Given that the principal period of the function is \( \pi \), equate this to the period of the function. So, \( \pi / \sqrt{m} = \pi \). This implies \( m=1 \).

Find the range of P based on \( m=1 \)

Since \( m=[P] \) is the greatest integer less than or equal to P, the range of P that gives \( m=1 \) is \( 1 \leq \mathrm{P}<2 \). Hence the correct answer is (c) \(1 \leq \mathrm{P}<2\).

Key concepts

principal period

The principal period is a key concept when exploring functions, especially periodic ones. It is the smallest positive interval over which a function repeats its pattern. When dealing with trigonometric functions like \( \tan x \), the principal period helps us understand how often the graph replicates on the x-axis.
For the function \( \tan(\sqrt{m} \cdot x) \), the principal period depends on \((\pi/\sqrt{m})\). In our exercise, we are tasked with ensuring this is equal to \( \pi \). This implies \((\sqrt{m}) = 1\), resulting in \((m = 1\)). Understanding principal periods is crucial in analyzing and comparing functions over fixed intervals.

periodic functions

Periodic functions are functions that repeat their values at regular intervals along the x-axis. An easy example is the sine or cosine functions, widely known for their repetitive wave patterns. Periodic functions are characterized by a period, which indicates the length of the interval over which they complete one full cycle.
In our exercise, the function \( \tan(\sqrt{m} \cdot x)\) is periodic. Its period is calculated by dividing \( \pi\) by \( \sqrt{m}\). This calculation is foundational as it helps determine the frequency of repetition of the function's values.

• Periodic functions provide essential insights into patterns and occurrences in different mathematical and physical contexts.
• They are used extensively in analyzing cycles, such as sound waves or oscillations.

trigonometric functions

Trigonometric functions, such as sine, cosine, and tangent, are essential in mathematics. They relate the angles of triangles to their side lengths and are crucial in various applications like physics, engineering, and navigation.
In our exercise, we examine the function \( \tan(\sqrt{m} \cdot x)\). The tangent function has a basic period of \( \pi\), meaning every \( \pi\) units along the x-axis, the function's pattern repeats. We took advantage of this property to solve our exercise by equating this period to the modified function's desired period.
Key properties of trigonometric functions include:

• Their periodic nature, which makes them predictable and easy to model in diverse scenarios.
• They can describe waves, circular motion, and oscillations.

In many cases, understanding these functions allows us to decipher complex periodic patterns within a defined scope.