Problem 34 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\sum_{i=1}^{2 n} \cos ^{-1} x_{i}=0\), then \(\sum_{i=1}^{2 n} x_{i}\) is (A) \(n\) (B) \(2 n\) (C) \(\frac{n(n+1)}{2}\) (D) none of these

Step-by-Step Solution

Answer

The sum is \(2n\), so the correct answer is (B) \(2n\).

Step 1: Understanding the Problem

We are given that the sum of inverse cosines, \(\cos^{-1}(x_i)\), for \(2n\) terms equals zero: \[ \sum_{i=1}^{2n} \cos^{-1}(x_i) = 0 \] Our task is to find out the value of \(\sum_{i=1}^{2n} x_i\). The equation implies that each term \(\cos^{-1}(x_i)\) must also be zero.

Step 2: Analyzing the Condition Given

For \(\cos^{-1}(x_i) = 0\), it is necessary that \(x_i = 1\). This is because \(\cos^{-1}(1) = 0\). Since all \(2n\) terms satisfy this condition, it implies that each \(x_i = 1\) for all \(i\).

Step 3: Summing Up the Values

Since all \(x_i = 1\), the sum \(\sum_{i=1}^{2n} x_i\) adds each \(x_i\) being 1, exactly \(2n\) times. Thus, \[ \sum_{i=1}^{2n} x_i = 1 + 1 + \ldots + 1 \quad (2n\text{ times}) = 2n \]

Step 4: Selecting the Correct Option

From the previous step, we find that \(\sum_{i=1}^{2n} x_i = 2n\). Therefore, the correct choice is (B) \(2n\).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential in solving equations that involve angles and sides of triangles. They include functions such as \(\sin^{-1}(x)\), \(\cos^{-1}(x)\), and \(\tan^{-1}(x)\). These functions are the inverses of the standard trigonometric functions, commonly used to find angles when the ratio of sides is known.
For instance, \(\cos^{-1}(x)\) gives us the angle whose cosine is \(x\). It helps in cases where the cosine value is provided, and we need to determine the angle. The range of \(\cos^{-1}(x)\) is from 0 to \(\pi\) radians.
In the given problem, we utilized the identity that when \(\cos^{-1}(x_i) = 0\), then \(x_i\) must equal 1. Thus, understanding inverse trigonometric functions greatly assists in problem-solving by simplifying the analysis of angles and their relationships to specific values.

Summing Series

Summing a series involves adding up all terms in a sequence of numbers. It can include a finite series (a series with a definite number of terms) or an infinite series (where terms continue indefinitely).
In mathematical problems, especially those involving patterns or regular sequences, it's crucial to recognize these patterns and apply appropriate formulas for their sum.

• A simple arithmetic series has terms that differ by a constant value, and its sum can be calculated using the formula for the sum of an arithmetic series: \(S_n = \frac{n}{2} (a + l)\), where \(a\) is the first term, \(l\) is the last term, and \(n\) is the number of terms.
• In our problem, the series consists entirely of repeated ones (since \(x_i = 1\)). Hence, summing the series is equivalent to multiplying the value 1 by the number of terms, \(2n\), resulting in the sum \(2n\).

This type of problem highlights how recognizing the simplicity in a series can lead quickly to the correct sum.

Problem Solving in Mathematics

Mathematics problem-solving is about applying logical steps to find a solution. It involves several techniques and strategies that help in breaking down complex problems into manageable parts.

• Understand the Problem: Start by clearly understanding the conditions and what is being asked. This might involve translating a word problem into equations.
• Analyze and Simplify: Identify patterns or use known identities and properties to simplify the problem. In our example, recognizing that \(\cos^{-1}(x_i) = 0\) implies \(x_i = 1\) was crucial.
• Execute the Plan: Carry out the calculations needed to achieve the solution. This often involves algebraic manipulation or applying mathematical formulas.
• Review the Solution: Once a solution is found, review each step to ensure there are no errors and the solution makes sense.

Effective problem-solving not only helps in finding the right answer but also enhances learning by applying theoretical concepts in practical scenarios.