Problem 49 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The mean of the numbers \(a, b, 8,5,10\) is 6 and the variance is \(6.80 .\) Then which one of the following gives possible values of \(a\) and \(b ?\) (A) \(a=0, b=7\) (B) \(a=5, b=2\) (C) \(a=1, b=6\) (D) \(a=3, b=4\)

Step-by-Step Solution

Answer

Option (D) \(a = 3, b = 4\) is correct.

Step 1: Understand the Mean Formula

The mean of a set of numbers is the sum of the numbers divided by the number of values. For the numbers \( a, b, 8, 5, 10 \), the mean is given as 6. Therefore, the equation is \( \frac{a + b + 8 + 5 + 10}{5} = 6 \). Solve this equation to find the relationship between \( a \) and \( b \).

Step 2: Simplify the Mean Equation

Multiply both sides of the mean equation by 5 to eliminate the fraction: \( a + b + 8 + 5 + 10 = 30 \). Simplify the left side: \( a + b + 23 = 30 \). Then isolate \( a + b \) to get \( a + b = 7 \).

Step 3: Understand the Variance Formula

Variance is calculated as the mean of the squares of the deviations from the mean. With variance given as 6.80, set up the formula for variance using \( a, b, 8, 5, 10 \). The deviations from the mean (6) are \( (a-6), (b-6), 2, -1, 4 \). Thus, the variance equation is: \( \frac{(a-6)^2 + (b-6)^2 + 2^2 + (-1)^2 + 4^2}{5} = 6.80 \).

Step 4: Solve for Variance

Substitute \( a + b = 7 \) into the variance equation: \( \frac{(a-6)^2 + (b-6)^2 + 4 + 1 + 16}{5} = 6.80 \). This simplifies to \( \frac{(a-6)^2 + (b-6)^2 + 21}{5} = 6.80 \). Multiply through by 5 to clear the fraction: \( (a-6)^2 + (b-6)^2 + 21 = 34 \), resulting in \( (a-6)^2 + (b-6)^2 = 13 \).

Step 5: Check Each Option

Using \( a + b = 7 \) and \( (a-6)^2 + (b-6)^2 = 13 \), check each option: - For (A) \( a=0, b=7\), \( a+b=7 \) is true and \( (0-6)^2 + (7-6)^2 = 36 + 1 = 37 \), false.- For (B) \( a=5, b=2\), \( a+b=7 \) and \( (5-6)^2 + (2-6)^2 = 1 + 16 = 17 \), false.- For (C) \( a=1, b=6\), \( a+b=7 \) and \( (1-6)^2 + (6-6)^2 = 25 + 0 = 25 \), false.- For (D) \( a=3, b=4\), \( a+b=7 \) and \( (3-6)^2 + (4-6)^2 = 9 + 4 = 13 \), true.

Key Concepts

Mean Calculation

When you're working with a set of numbers, the mean, or average, is an essential concept often required for further calculations. To find the mean, add all the numbers together and divide by the total count of those numbers. For example, if we have the numbers \( a, b, 8, 5, 10 \), the mean is calculated as follows:

• First, sum up all the numbers: \( a + b + 8 + 5 + 10 \).
• Since there are 5 numbers, divide the total sum by 5.

The problem states this mean is 6, so we have the equation: \[ \frac{a + b + 8 + 5 + 10}{5} = 6 \] By solving this, you determine the relationship \( a + b = 7 \). This equation is fundamental for understanding the relationship between any potential values of \( a \) and \( b \), given any constraints provided by the problem. Understanding the mean is crucial in statistics and data analysis because it gives a central value that helps compare various data points.

Deviation from Mean

Deviation from the mean is a term used to express how far a particular number in your set is from the mean value. Understanding this concept is vital because it helps one make sense of the variance and standard deviation, which measure the spread of a dataset. In the exercise, finding deviations involves calculating how far off each number is from the mean, which is 6 in this case. For instance:

• The deviation of 8 from the mean is \(8 - 6 = 2\).
• The deviation of 5 is \(5 - 6 = -1\).
• Similarly, calculate the deviations for each number.

This set of deviations is then squared, and these squared values are used in the variance calculation. It's important to consider every number in the dataset, including unknowns like \(a\) or \(b\). Learning how to calculate deviations from the mean can be incredibly beneficial because it forms the foundation for understanding how data varies within a dataset. In practical terms, large deviations indicate numbers that are spread out over wide ranges, while small deviations suggest that the numbers are clustered closely near the mean.

Mathematical Problem Solving

Mathematical problem-solving is a systematic process used to tackle mathematical problems, particularly when they require multiple steps, like calculating variance or mean. Here's a simple framework that can be applied:

• Identify: Clearly identify what the problem requires. In our case, the key was finding possible values of \( a \) and \( b \) given the conditions.
• Formulate: Develop equations from information given, like using the mean to form \( a + b = 7 \).
• Solve: Use arithmetic and algebra to simplify and solve these equations, verifying each potential solution against all conditions.

Working through the variance involves understanding and applying the formula correctly: \[ \frac{(a-6)^2 + (b-6)^2 + 2^2 + (-1)^2 + 4^2}{5} = 6.80 \] Substituting from the mean equation helps narrow options by revealing which choices are mathematically viable. In global problem-solving, whether you're dealing with numbers or symbols, it's essential to logically follow steps to reach the correct solution. Thus, choices require testing until one fits all criteria, as demonstrated where option D was verified as true.