Problem 27 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(X=\\{x \in \mathbf{N}: 1 \leq x \leq 17\\}\) and \(Y=\\{a x+b: x \in X\) and \(a, b \in \mathbf{R}, a>0\\} .\) If mean and variance of elements of \(Y\) are 17 and 216 respectively then \(a+b\) is equal to : [Sep. \(\mathbf{0 2}, \mathbf{2 0 2 0}(\mathbf{I})]\) (a) 7 (b) \(-7\) (c) \(-27\) (d) 9

Step-by-Step Solution

Answer

\(a + b = -7\)

Step 1: Understand the Transformation of X to Y

Elements of set X are transformed to create set Y using the formula \( Y = \{ax + b: x \in X\} \). Here, \(a\) and \(b\) are constants with \(a>0\). This means each element \(x\) from set X is multiplied by \(a\) and then \(b\) is added to it.

Step 2: Identify the Mean of X

To find the mean of set X, which includes all natural numbers from 1 to 17, we calculate \[\text{mean}(X) = \frac{1 + 2 + \cdots + 17}{17} = \frac{\frac{17 \cdot (17 + 1)}{2}}{17} = 9\]

Step 3: Relate Mean of Y to Mean of X

The mean of set Y is given as 17. Since Y is derived by \(ax + b)\), it implies:\[\text{mean}(Y) = a \cdot \text{mean}(X) + b = 17\]Substitute the mean of X from Step 2:\[a \cdot 9 + b = 17\]

Step 4: Identify the Variance of X

Variance for natural numbers 1 to 17 is calculated using the formula for variance of first n natural numbers:\[\text{Var}(X) = \frac{(17)(17+1)(2 \cdot 17 + 1)}{6} - 9^2 = 24\]

Step 5: Relate Variance of Y to Variance of X

According to the transformation, \(\text{Var}(Y) = a^2 \cdot \text{Var}(X)\). Given \(\text{Var}(Y) = 216\), substitute the variance from Step 4:\[a^2 \cdot 24 = 216\]

Step 6: Solve for a and b

From Step 5:\[a^2 = \frac{216}{24} = 9 \]\(a = 3\) (since \(a > 0\))From Step 3, substitute \(a = 3\):\[3 \cdot 9 + b = 17\]\[27 + b = 17\]\[b = -10\]

Step 7: Calculate a + b

Now that we know \(a = 3\) and \(b = -10\), we find:\[a + b = 3 - 10 = -7\]

Key Concepts

Natural Numbers

Natural numbers are a fundamental concept in mathematics, representing the sequence of positive integers starting from 1 and increasing indefinitely. They are denoted by the symbol \( \mathbf{N} \). In this exercise, we consider the set \( X \) which includes natural numbers from 1 to 17. This means all elements are whole numbers without decimals that form a complete list from the smallest natural number up to 17.

Natural numbers have a few key properties:

• They are countable and infinite.
• They follow a predictable sequence: each number is exactly one unit larger than the previous.
• They are used in set theory to define sets with non-negative integer elements.

For our purposes, the natural numbers in set \( X \) allow us to calculate collective properties such as the mean or variance, which inform us about the central tendency and dispersion of these numbers respectively.

Set Theory

Set theory is an important branch of mathematical logic that deals with collections of objects known as sets. A set is essentially a group of distinct objects, considered as a whole. In the context of the exercise, set theory allows us to define the set \( X \) and apply transformations to create a new set \( Y \).

Here are a few principles of set theory that are relevant:

• Sets can be represented using curly brackets \( \{ \} \).
• Elements within a set are unordered and unique.
• Operations can be performed within sets, such as transformations or determining means and variances.

When we transform a set, as in transforming \( X \) into \( Y \), we apply some operation (in this case, multiplication by \( a \) and addition of \( b \)) to each element of the original set. Set theory provides the conceptual framework to perform these operations systematically.

Transformation of Variables

Transformation of variables refers to the process of changing the scale or distribution of a set of data by applying some mathematical operation to its elements. In the given problem, the transformation is executed by converting set \( X \) into set \( Y \) using the equation \( Y = \{ ax + b : x \in X \} \).

This transformation includes the following steps:

• Scaling: Each element \( x \) in set \( X \) is multiplied by a constant factor \( a \).
• Translation: The products \( ax \) are then incremented by another constant \( b \).

The transformation alters the statistical properties of the set. For example, the mean and variance of \( Y \) are directly affected by the values of \( a \) and \( b \).

Understanding how transformations influence these properties aids in solving the given exercise, by establishing relationships like \( \text{mean}(Y) = a \cdot \text{mean}(X) + b \) and \( \text{Var}(Y) = a^2 \cdot \text{Var}(X) \), which were critical to determining the solution to the problem.