Problem 67 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

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 are \(a = 3\) and \(b = 4\) (option d).

Step 1: Calculate the Mean Equation

Given the mean of the numbers \(a, b, 8, 5, 10\) is 6, we start by writing the equation for the mean:\[\frac{a+b+8+5+10}{5} = 6\]Simplify this: \[a + b + 8 + 5 + 10 = 30\]\[a + b + 23 = 30\]Thus, \(a + b = 7\).

Step 2: Calculate the Variance Equation

Given the variance of the numbers is 6.80, the formula for variance is:\[\text{Variance} = \frac{1}{n}\sum_{i=1}^{n}(x_i - \text{mean})^2\]Here, \(n = 5\) and the mean is 6.

Step 3: Setup the Variance Equation

We substitute the known mean into the variance formula:\[\frac{1}{5}[(a-6)^2 + (b-6)^2 + (8-6)^2 + (5-6)^2 + (10-6)^2] = 6.80\]\[(a-6)^2 + (b-6)^2 + (8-6)^2 + (5-6)^2 + (10-6)^2 = 34\]

Step 4: Simplify Known Values

Calculate the squared differences for known numbers:\((8-6)^2 = 4, (5-6)^2 = 1, (10-6)^2 = 16\)Sum these up:\[4 + 1 + 16 = 21\]So the equation becomes:\[(a-6)^2 + (b-6)^2 + 21 = 34\]\[(a-6)^2 + (b-6)^2 = 13\]

Step 5: Evaluate Options

We must check which pair \( (a, b) \) satisfies both equations: \(a + b = 7\) and \((a-6)^2 + (b-6)^2 = 13\).- Option (a): \( a = 0, b = 7\) - Check: \(0 + 7 = 7\) and \((0-6)^2 + (7-6)^2 = 36 + 1 = 37\)- Option (b): \( a = 5, b = 2\) - Check: \(5 + 2 = 7\) and \((5-6)^2 + (2-6)^2 = 1 + 16 = 17\)- Option (c): \( a = 1, b = 6\) - Check: \(1 + 6 = 7\) and \((1-6)^2 + (6-6)^2 = 25 + 0 = 25\)- Option (d): \( a = 3, b = 4\) - Check: \(3 + 4 = 7\) and \((3-6)^2 + (4-6)^2 = 9 + 4 = 13\)Only option (d) satisfies both conditions.

Key Concepts

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules to manipulate those symbols. These symbols can represent numbers or other mathematical entities, allowing for the creation of equations and expressions. In this problem, algebra is used to solve for unknown variables, \(a\) and \(b\), through the equations derived from given conditions. The use of algebra here simplifies the process of finding the solution by breaking down the problem into manageable parts and forming equations such as \(a + b = 7\). By substituting in different values from the options provided, one can use algebraic manipulation to determine which values satisfy both the mean and variance conditions.

Problem-solving

Problem-solving strategies involve techniques and logical steps that help in approaching complex problems in a systematic way. For this exercise, the problem-solving approach starts by identifying the options given for \(a\) and \(b\), and then checks each option to see if it fulfills the conditions derived using algebra. One key problem-solving tactic is to take the equations, \(a + b = 7\) and \((a-6)^2 + (b-6)^2 = 13\), and test through substitution for each answer choice. This logical approach ensures that the solution is correct by verifying compliance with all constraints of the problem.

Mean Calculation

Calculating the mean involves finding the average of a set of numbers. The mean is determined by summing up the numbers and then dividing by the quantity of the numbers. In this problem, the mean of the numbers \(a, b, 8, 5, 10\) is provided as \(6\). To find the mean, the formula is used:

• Sum the numbers: \(a + b + 8 + 5 + 10\)
• Divide by the number of terms, which in this case is \(5\)

This creates the equation \(\frac{a + b + 23}{5} = 6\), which simplifies to \(a + b = 7\), forming the first constraint in solving the problem.

Variance Calculation

Variance is a measure of how much the numbers in a set differ from the mean of the set. It is an important concept in statistics because it gives an idea of the data's spread and variability. To calculate variance, one follows these steps:

• Determine the mean (which is already provided as \(6\) in this problem)
• Find the squared difference between each number and the mean
• Averaging these squared differences gives the variance

For this problem, substituting the known mean into the variance formula, \[\frac{1}{5}[(a-6)^2 + (b-6)^2 + (8-6)^2 + (5-6)^2 + (10-6)^2] = 6.80\]This simplifies to \((a-6)^2 + (b-6)^2 + 21 = 34\), further reducing to \((a-6)^2 + (b-6)^2 = 13\), which forms the basis for determining \(a\) and \(b\). This calculation of variance ensures that the spread of data is correctly articulated and used in the solution process.