Problem 55 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The contrapositive of the following statement, "If the side of a square doubles, then its area increases four times", is : \(\quad\) (a) If the area of a square increases four times, then its side is not doubled. (b) If the area of a square increases four times, then its side is doubled. (c) If the area of a square does not increases four times, then its side is not doubled. (d) If the side of a square is not doubled, then its area does not increase four times.

Step-by-Step Solution

Answer

The contrapositive is statement (c).

Step 1: Understand the Original Statement

The original statement is: "If the side of a square doubles, then its area increases four times." This can be expressed as: \( p \rightarrow q \), where \( p \) is "the side of a square doubles" and \( q \) is "its area increases four times."

Step 2: Identify the Contrapositive

The contrapositive of a statement \( p \rightarrow q \) is \( eg q \rightarrow eg p \), where \( eg q \) is the negation of "its area increases four times", and \( eg p \) is the negation of "the side of a square doubles."

Step 3: Translate the Contrapositive

Negating \( q \): "its area does not increases four times". Negating \( p \): "the side of a square is not doubled." So, the contrapositive is: "If the area of a square does not increase four times, then its side is not doubled."

Step 4: Match with Given Options

Check the given options to see which matches our contrapositive: (c) "If the area of a square does not increases four times, then its side is not doubled." matches our derived contrapositive statement exactly.

Key Concepts

Contrapositive Statement

A contrapositive statement is a powerful logical tool in mathematics. When you have an implication that reads "if \( p \) then \( q \)", its contrapositive is inversely formed by negating both the conclusion and the hypothesis. Thus, the contrapositive of \( p \rightarrow q \) becomes \( eg q \rightarrow eg p \). These two statements are logically equivalent, meaning they always share the same truth value.
Understanding how to find the contrapositive involves:

• Identifying the hypothesis (\( p \)) and the conclusion (\( q \)) from the original statement.
• Negating both \( p \) and \( q \) in the original statement.
• Reversing the order to form the contrapositive: from \( p \rightarrow q \) to \( eg q \rightarrow eg p \).

By practicing these steps, you'll gain confidence in constructing and understanding the significance of contrapositive statements.

Square Geometry

Square geometry is a fundamental concept where a square is defined as a four-sided polygon, or quadrilateral, with equal sides and angles. Each angle in a square measures 90 degrees, resulting in a total of 360 degrees for all corners combined. Square geometry becomes especially important when evaluating relationships within the shape, such as side lengths and area.
For a square:

• Side Length: Each side has the same length, denoted as \( s \).
• Area: The area is calculated using the formula \( s^2 \), which reflects the square's side length.
• Perimeter: The perimeter is the sum of all four sides, so \( 4s \).

If the side of a square changes, geometric properties like area respond proportionally. For example, if a side doubles, area increases by \( 4 \) since \( (2s)^2 = 4s^2 \). Understanding these relationships is pivotal in solving problems involving squares.

Negation in Logic

Negation in logic involves transforming a statement to its opposite, typically using the word "not." If a statement is true, the negation of that statement is false, and vice versa. The symbol \( eg \) represents negation, and it can alter both individual and compound statements in logical forms.
Steps to negate a statement:

• Identify the part of the statement that requires negation.
• Introduce the term "not" to change its truth value.
• If the statement involves a compound, carefully negate each part and adjust any logical connectors like "and" to "or," and vice versa.

For instance, if "the side of a square doubles" is considered true, its negation is "the side of a square is not doubled." Similarly, from" its area increases four times," the negation is "its area does not increase four times." Understanding negation is essential for forming contrapositives and other logical statements in mathematics.