Problem 27 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The means of five observations is 4 and their variance is \(5.2\). If three of these observations are 1,2 and 6 , then the other two are (A) 2 and 9 (B) 3 and 8 (C) 4 and 7 (D) 5 and 6

Step-by-Step Solution

Answer

(B) 3 and 8

Step 1: Understanding the Mean

The mean of the observations is given by the formula: \( \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \). We are told that the mean is 4 and there are 5 observations. Thus, the total sum of the observations must be \( 4 \times 5 = 20 \).

Step 2: Utilize Known Information

We know three of the observations: 1, 2, and 6. Let's calculate their sum: \( 1 + 2 + 6 = 9 \). The sum of the remaining two observations must be \( 20 - 9 = 11 \).

Step 3: Understanding the Variance

The variance is given by the formula: \( \text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{\text{Number of observations}} \). We are told the variance is 5.2, so \( \sum (x_i - 4)^2 = 5.2 \times 5 = 26 \).

Step 4: Utilizing Known Partial Variances

For the known observations of 1, 2, and 6, calculate their squared deviations from the mean: \( (1-4)^2 = 9, (2-4)^2 = 4, (6-4)^2 = 4 \). The sum of these is 9 + 4 + 4 = 17. Thus, the remaining two observations must have a sum of squared deviations equal to \( 26 - 17 = 9 \).

Step 5: Solve System of Equations

Let the two unknown observations be \( x \) and \( y \). We have two equations: \( x + y = 11 \) and \( (x-4)^2 + (y-4)^2 = 9 \). Simplifying the second equation: \( (x - 4)^2 + (y - 4)^2 = x^2 - 8x + 16 + y^2 - 8y + 16 = 9 \), which simplifies to \( x^2 + y^2 - 8x - 8y = -23. \)

Step 6: Substitute and Solve

Using \( x + y = 11 \), express \( y \) as \( y = 11 - x \) and substitute into the squared deviation equation. This leads to \( x^2 + (11-x)^2 - 8x - 8(11-x) = -23 \). Simplify and solve for \( x \), leading to potential solutions for \( x \) and \( y \).

Step 7: Verify and Choose Solution

Upon solving, one pair of possible values is \( x = 3 \) and \( y = 8 \). Verify that they satisfy both \( x+y=11 \) and the variance equation: \((3-4)^2 + (8-4)^2 = 1 + 16 = 9 \). This confirms that the observations are indeed 3 and 8.

Key Concepts

Mean

The concept of mean, commonly referred to as the average, is a fundamental aspect of mathematics. It represents the central value of a data set. Imagine you have five different numbers. To find their mean, you simply add all these numbers together, and then divide the sum by the quantity of numbers you had, which, in this case, is five.

• Formula: \( \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \)
• Example: If you have observations 4, 5, 6, 7, 8, the mean is \( \frac{4+5+6+7+8}{5} = 6 \).

Its simplicity makes it a great measure of central tendency, giving you a sense of where the middle ground of the number set lies. This makes it extremely useful not just in academics but in everyday life scenarios where you might want to quickly find a rough estimate or summary of data.

Variance

Variance helps us understand how spread out the numbers in a data set are. In simpler terms, it tells us the degree to which numbers differ from the mean. If the variance is small, it means the numbers are very close to the mean. If it's large, the numbers are spread out.

• Formula: \( \text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{\text{Number of observations}} \)
• Steps to calculate: Subtract the mean from each number, square the result, sum all the squared results, and then divide by the number of observations.

Variance is important because it provides insight into the stability and reliability of the mean. For instance, in the context of the exercise, a variance of 5.2 indicates how aligned the five observations are around the average of 4. This informs about the consistency of the set.

System of Equations

A system of equations consists of multiple equations considered simultaneously. Each equation within a system holds certain variables, and your goal is to find the values of these variables that satisfy all the equations at once. This concept is crucial in solving complex mathematical problems where relationships between multiple variables must be defined and solved.
For example, in the exercise, we had two equations based on the unknown observations. The equations were \( x + y = 11 \) and \( (x-4)^2 + (y-4)^2 = 9 \). By solving these, we determine the values of \( x \) and \( y \) that satisfy both conditions.

• Understanding involved: Using substitution or elimination methods for solving these systems.
• Substitution: Replace one variable with another expression to simplify and solve the equations.

Grasping systems of equations allows you to tackle a variety of mathematical and real-world problems, facilitating complex decision-making and critical thinking.

JEE Main

JEE Main is a highly competitive engineering entrance exam conducted in India. It serves as a gateway for students aspiring to gain admissions into prestigious engineering institutions such as the NITs (National Institutes of Technology) and other engineering colleges.
The Mathematics section of JEE Main often includes problems on topics like mean, variance, and systems of equations. This implies that understanding these concepts is crucial not just academically, but also for achieving excellence in such exams.

• Preparation: Regular practice of mathematical problems and familiarization with their applications is necessary.
• Skill refinement: Mastering time management and problem-solving techniques is essential, given the timed nature of the exam.

Excelling in JEE Main requires a solid understanding of fundamental and advanced mathematical concepts, with an emphasis on applying them strategically during the exam.