Problem 5 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The mean of a set of 30 observations is 75 . If each other observation is multiplied by a non-zero number \(\lambda\) and then each of them is decreased by 25 , their mean remains the same. The \(\lambda\) is equal to [Online April 15, 2018] (a) \(\frac{10}{3}\) (b) \(\frac{4}{3}\) (c) \(\frac{1}{3}\) (d) \(\frac{2}{3}\)

Step-by-Step Solution

Answer

\( \lambda = \frac{4}{3} \)

Step 1: Understand the Original Mean

The original mean of the set of 30 observations is given as 75. This means that the sum of all the observations is:\[ \text{Sum} = 30 \times 75 = 2250 \]

Step 2: Apply the Transformation

Each observation is multiplied by \( \lambda \) and then decreased by 25. Thus, an original observation \( x_i \) becomes \( \lambda x_i - 25 \).

Step 3: Calculate the New Sum

If each \( x_i \) is transformed to \( \lambda x_i - 25 \), then the sum of the transformed observations is:\[ \sum (\lambda x_i - 25) = \lambda \sum x_i - 25 \times 30 \]Substituting the original sum:\[ \lambda \times 2250 - 750 \]

Step 4: Equating Means

According to the problem, the new mean remains the same as the original mean, which is 75. Therefore, we set up the equation:\[ \frac{\lambda \times 2250 - 750}{30} = 75 \]

Step 5: Solve for \( \lambda \)

Simplify and solve the equation:\[ \lambda \times 2250 - 750 = 2250 \]\[ \lambda \times 2250 = 2250 + 750 \]\[ \lambda \times 2250 = 3000 \]\[ \lambda = \frac{3000}{2250} = \frac{4}{3} \]

Step 6: Conclusion

The value of \( \lambda \) that satisfies the condition of the transformed observations having the same mean as the original is \( \frac{4}{3} \).

Key Concepts

Mathematical Transformations

In mathematics, transformations refer to operations that alter the form of a mathematical object, such as numbers, functions, or equations. It is a key step in problem solving and can make complex problems simpler. In this exercise, each observation in a dataset is first multiplied by a constant \( \lambda \) and then decreased by a set amount, making it a classic example of a linear transformation.

Mathematical transformations can include:

• Scaling, where each data point is multiplied by a constant.
• Translation, where a constant is added or subtracted from each data point.

Combining a scaling and a translation creates the transformation that was employed in this exercise. By understanding the effect of these transformations, you can predict how they will impact the mean or other properties of a dataset.
The problem illustrates how transformations can help maintain certain characteristics of datasets, like the mean in this case, even as the individual data points themselves are altered.

Algebraic Equations

Algebraic equations are mathematical statements that use algebraic expressions. They show the relationship between different quantities and often involve variables that need to be solved. In the problem, we derived an equation to find the value of \( \lambda \), the constant affecting each observation.

The key steps involved in solving equations such as this include:

• Setting up the equation based on problem constraints, like maintaining the mean.
• Simplifying the equation using algebraic rules, such as distributing and combining like terms.
• Isolating the variable in question to find its value.

In this particular algebraic scenario, we started with the equation \( \frac{\lambda \times 2250 - 750}{30} = 75 \) and solved for \( \lambda \), eventually finding \( \lambda = \frac{4}{3} \). Algebra allows us to systematically handle the transformation of observations and keep the mean unchanged, using these structured mathematical tools.

Problem Solving in Mathematics

Problem solving in mathematics involves identifying, investigating, and resolving mathematical problems using systematic and logical approaches. It often requires critical thinking alongside mathematical techniques like transformations and equation solving.

Here are some important steps in mathematical problem solving, demonstrated by this exercise:

• Understanding the problem: Identify what is given and what needs to be found. In this case, we know the original mean and the transformation and need to find \( \lambda \).
• Translation into mathematical terms: Converting the verbal problem description into mathematical equations or expressions, as seen when writing the expression for the transformed mean.
• Solving the mathematical model: Using mathematical techniques and calculations to find the answer, as with solving the algebraic equation for \( \lambda \).
• Interpreting the solution: Verify that the solution meets the problem's conditions, like confirming the mean stays at 75 after the transformation.

Using these structured methods, students can tackle a wide range of problems effectively, allowing them to apply these strategies to similar mathematical challenges.