Problem 57 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the standard deviation of the numbers \(2,3, a\) and 11 is \(3.5\), then which of the following is true? \([2016]\) (A) \(3 a^{2}-23 a+44=0\) (B) \(3 a^{2}-26 a+55=0\) (C) \(3 a^{2}-32 a+84=0\) (D) \(3 a^{2}-34 a+91=0\)

Step-by-Step Solution

Answer

Option (C) is true: \( 3a^2 - 32a + 84 = 0 \).

Step 1: Understand Standard Deviation Formula

The standard deviation of a set of numbers is calculated using the formula \( \sigma = \sqrt{ \frac{ \sum{(x_i - \mu)^2} }{n} } \), where \( \mu \) is the mean of the numbers, \( x_i \) are the individual numbers, and \( n \) is the number of data points.

Step 2: Calculate Mean

Find the mean \( \mu \) of the numbers \(2, 3, a, 11\). The mean is calculated as \( \mu = \frac{2 + 3 + a + 11}{4} = \frac{16 + a}{4} \).

Step 3: Set Up the Variance Equation

Using the standard deviation \( \sigma = 3.5 \), we know the variance is \( \sigma^2 = 12.25 \). The variance is \( \frac{(2 - \mu)^2 + (3 - \mu)^2 + (a - \mu)^2 + (11 - \mu)^2}{4} = 12.25 \).

Step 4: Simplify the Variance Expression

Substitute \( \mu = \frac{16 + a}{4} \) into the variance expression: \[ (2 - \frac{16 + a}{4})^2 + (3 - \frac{16 + a}{4})^2 + (a - \frac{16 + a}{4})^2 + (11 - \frac{16 + a}{4})^2 = 49. \]

Step 5: Solve the Equation

Expand each term and combine like terms to form a quadratic equation in terms of \( a \). Simplify completely to find: \( 3a^2 - 32a + 84 = 0 \).

Step 6: Compare Provided Options

Compare the derived equation \( 3a^2 - 32a + 84 = 0 \) with the given options (A), (B), (C), and (D) to identify the correct one.

Key Concepts

Variance

Variance is a measure of how far each number in a set is from the mean, and it helps us understand the spread or dispersion of the dataset. In essence, it's the average of the squared differences from the mean. This can give you a sense of how much variation exists within your data.

To calculate variance, follow these steps:

• Find the mean (average) of the dataset.
• Subtract the mean from each data point to find the deviation for each point.
• Square each deviation to eliminate negative numbers and emphasize larger deviations.
• Calculate the average of these squared deviations.

In the context of the exercise, after determining the mean of the numbers, we set up an equation for variance based on the given standard deviation. Knowing that the variance is the square of the standard deviation (i.e., variance = 3.5² = 12.25), we can express how each number deviates from the mean squared in an equation. This leads us to derive a quadratic equation to solve for the missing variable in the dataset.

Quadratic Equation

A quadratic equation is a type of polynomial equation where the highest degree of the variable is 2. They are generally written in the form of \( ax^2 + bx + c = 0 \) where \( a \), \( b \), and \( c \) are constant coefficients.

Quadratic equations are solved primarily by:

• Factoring the quadratic when possible.
• Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Completing the square to transform the equation into a solvable one.

In the exercise, we derived a quadratic equation of the form \( 3a^2 - 32a + 84 = 0 \) by expanding and simplifying the variance expression as a polynomial in terms of \( a \). Comparing this with the given options allowed us to identify the correct equation as one of the options. Quadratic equations are vital in algebra because they appear frequently in various mathematical and real-world scenarios.

Mean

The mean, or average, is a central value of a set of numbers, providing a numerical summary of the dataset. It's calculated by adding up all the numbers and dividing by the count of numbers.

To calculate the mean:

• Add all the values together.
• Divide the total by the number of values.

In this exercise, we calculated the mean of four numbers \( 2, 3, a, \) and \( 11 \) as follows:
The mean is \( \mu = \frac{2 + 3 + a + 11}{4} = \frac{16 + a}{4} \).

This mean is subsequently used in forming the variance equation, which is necessary for solving the problem. Understanding the mean is crucial since it serves as the baseline for measuring deviations, thus facilitating further calculations such as finding variances and standard deviations.