Problem 91 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(a_{n}=\int_{0}^{\pi / 2} \frac{1-\cos 2 n x}{1-\cos 2 x} d x\), then (A) \(a_{n+1}\) is A.M. between \(a_{n}\) and \(a_{n+2}\) (B) \(a_{n+1}\) is G.M between \(a_{n}\) and \(a_{n+2}\) (C) \(a_{n+1}\) is H.M. between \(a_{n}\) and \(a_{n+2}\) (D) \(\left|\begin{array}{ccc}a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\\ a_{7} & a_{8} & a_{9}\end{array}\right|=0\)

Step-by-Step Solution

Answer

(D) The determinant is zero.

Step 1: Analyze the given integral function

We are given the function \(a_{n}=\int_{0}^{\pi / 2} \frac{1-\cos 2 n x}{1-\cos 2 x} dx.\) This integral can be simplified or evaluated by using trigonometric identities or properties of definite integrals.

Step 2: Use trigonometric identities

The function \(1-\cos 2x\) can be written as \(2\sin^2 x\). Thus, the integral simplifies to \(a_{n}=\int_{0}^{\pi / 2} \frac{1-\cos 2 n x}{2\sin^2 x} dx.\)

Step 3: Simplify the expression inside the integral

The expression \((1-\cos 2nx)\) can be rewritten using trigonometric identities: \(1-\cos 2nx = 2\sin^2(nx)\). Therefore, \(a_{n}=\int_{0}^{\pi / 2} \frac{2\sin^2(nx)}{2\sin^2 x} dx = \int_{0}^{\pi / 2} \frac{\sin^2(nx)}{\sin^2 x} dx.\)

Step 4: Simplify or evaluate the integral

The function inside the integral simplifies via periodic properties of sine. So, the expression modifies, but we focus on relationships between consecutive terms \(a_n, a_{n+1}, a_{n+2}\).

Step 5: Establish recurrence relation

The major hint is the application of properties of trigonometric integrals leading us to express \(a_n\) terms dependent on each other possibly indicating that these sequences follow a known relationship like arithmetic, geometric, or harmonic means.

Step 6: Analyze options

By analyzing the setup for examining arithmetic, geometric or harmonic mean properties, consider equalities and properties in sequence like arithmetic: \(a_{n+1} = \frac{a_n + a_{n+2}}{2}\). The symmetry and potential linear progression can instead align more with an arithmetic mean.

Step 7: Check the condition for determinants

If the sequence follows linear relations or behaves symmetrically, this may equalize conditions leading to zero determinants as structures or coefficients balance out.

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in simplifying expressions and solving integrals involving trigonometric functions. One such identity is used to transform the expression \(1 - \cos 2x\) into \(2 \sin^2 x\). This makes it much easier to work with these integrals. When dealing with products and powers of sine and cosine, substituting using identities helps simplify complex expressions.
These identities can include:

• \( \cos 2x = 2\cos^2 x - 1 = 1 - 2\sin^2 x \)
• \( \sin 2x = 2\sin x \cos x \)

Such transformations consolidate the integrals into more manageable forms and can be used to solve problems involving periodic functions. Understanding and applying trigonometric identities accurately lead to streamlined calculations when dealing with integrals.

Definite Integrals

Definite integrals are used to find the area under a curve within specified limits. They provide a way to accumulate quantities and can be represented as \( \int_{a}^{b} f(x) \, dx \).

Definite integrals can handle various functions, and their properties become instrumental in solving complex integrals. These properties are:

• The limits of integration determine the bounds of calculation.
• The value of a definite integral is impacted by the function's nature over the given interval.
• Applications of trigonometric identities within definite integrals aid in simplifying these calculations.

Understanding these fundamentals can help you solve for values like \(a_n\) in integral calculus, where symmetry and periodic properties considerably ease the calculations and lead to meaningful results in sequences and series relations, such as relating consecutive \(a_n\) terms.

Arithmetic Mean

Arithmetic Mean (A.M.) is a central concept in algebra and statistics. It is the ordinary average of a set of values and is calculated by summing up all the values, then dividing by their number. The formula used is: \[ \text{A.M.} = \frac{x_1 + x_2 + ... + x_n}{n} \]The mean provides a single value that represents a balance point of the dataset.
This concept is handy in analyzing sequences, such as when verifying if \(a_{n+1}\) is an arithmetic mean between two consecutive terms. You would see if:

• \(a_{n+1} = \frac{a_n + a_{n+2}}{2}\)

The arithmetic mean concept thus explains how data distributes around the mean in a linear approach, providing insights into sequence behavior and mathematical symmetry as seen in the original exercise.

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Problem 94

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