Problem 38 - A Complete Resource Book in Mathematics for JEE Main

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

The possible values of \(a\) and \(b\) are \(a=3, b=4\) (Option D).

Step 1: Calculate the Mean Equation

The mean of the numbers is given as 6. The numbers are \(a, b, 8, 5, 10\), which makes up a set of five numbers. The mean can be calculated by the equation:\[ \frac{a+b+8+5+10}{5} = 6 \] Simplifying, we find:\[ a + b + 23 = 30 \] Thus,\[ a + b = 7 \]

Step 2: Understand Variance Formula

Variance is given by the formula:\[ \text{Variance} = \frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + (x_3 - \mu)^2 + (x_4 - \mu)^2 + (x_5 - \mu)^2}{5} \]Where \(\mu\) is the mean, which is 6 in this case. We have a variance of 6.80, so we can set up the equation:\[ 6.80 = \frac{(a - 6)^2 + (b - 6)^2 + (8 - 6)^2 + (5 - 6)^2 + (10 - 6)^2}{5} \]

Step 3: Substitute Known Values in Variance Equation

We know that \((8-6)^2 = 4, (5-6)^2 = 1,\) and \((10-6)^2 = 16\). Substitute these into the variance equation:\[ 6.80 = \frac{(a-6)^2 + (b-6)^2 + 4 + 1 + 16}{5} \] Further simplify:\[ 6.80 = \frac{(a-6)^2 + (b-6)^2 + 21}{5} \] Multiply through by 5:\[ 34 = (a-6)^2 + (b-6)^2 + 21 \] So,\[ (a-6)^2 + (b-6)^2 = 13 \]

Step 4: Solve for Possible Values of a and b

We have two equations now:1. \( a + b = 7 \)2. \( (a-6)^2 + (b-6)^2 = 13 \)Check the given options:- **(A)** \( a=0, b=7 \): \( (0-6)^2 + (7-6)^2 = 36 + 1 = 37 eq 13 \)- **(B)** \( a=5, b=2 \): \( (5-6)^2 + (2-6)^2 = 1 + 16 = 17 eq 13 \)- **(C)** \( a=1, b=6 \): \( (1-6)^2 + (6-6)^2 = 25 + 0 = 25 eq 13 \)- **(D)** \( a=3, b=4 \): \( (3-6)^2 + (4-6)^2 = 9 + 4 = 13 \)Thus, option **(D)** satisfies both equations.

Key Concepts

Variance calculation

Variance measures the spread of a set of numbers from their mean. It tells us how much the numbers deviate from the average. To calculate variance, first determine the mean (or average), which is the center point of the data set.
For example, the variance formula for a data set including numbers like \(a, b, 8, 5, 10\) is:

• Find the mean \(\mu\).
• Subtract \(\mu\) from each number to obtain differences.
• Square these differences; this emphasizes larger deviations.
• Average the squared differences.

For our exercise, with a mean of 6, the variance is calculated as:\[\text{Variance} = \frac{(a - 6)^2 + (b - 6)^2 + (8 - 6)^2 + (5 - 6)^2 + (10 - 6)^2}{5}\]Once calculated, this variance provides insight into how much the numbers \(a, b, 8, 5,\) and \(10\) differ from their mean value.

Mean calculation

The mean or average of a set of numbers is a basic statistical measure that provides a central value for the data. To compute it, add all the numbers together and divide the sum by the total count of numbers.
In our problem, we have the set \(a, b, 8, 5, 10\). Given that the mean of this set is 6, we write the equation as: \[\frac{a+b+8+5+10}{5} = 6\] This simplifies to: \[a + b + 23 = 30\] Solving it gives us the equation \(a + b = 7\).
Knowing how to calculate means is crucial because it helps you summarize large data sets into a single measure that represents the whole data set, simplifying further calculations like variance.

Simultaneous equations in Algebra

Simultaneous equations involve solving for multiple variables at the same time. These equations often represent shared constraints between variables, and solving them reveals values satisfying all equations simultaneously. In this exercise, we work with two equations:

• \(a + b = 7\)
• \((a-6)^2 + (b-6)^2 = 13\)

To solve these equations, we test different possible values for \(a\) and \(b\) and check which satisfy both equations.
For example, checking if \(a = 3\) and \(b = 4\):- Substitute into the first equation: \(3 + 4 = 7\). Checks out.- Substitute into the second equation: \((3-6)^2 + (4-6)^2 = 9 + 4 = 13\). Checks out too.
Thus, learning simultaneous equations is essential for solving complex problems in mathematics, where multiple conditions must be met at once.