Q 73. Logical Venn Diagram
Consider the following statements:
1. The minimum number of points of intersection of a square and a circle is 2.
2. The maximum number of points of intersection of a square and a circle is 8.
Which of the above statements is/are correct?
● Statement 1: "The minimum number of points of intersection of a square and a circle is 2."
○ This statement is incorrect. The minimum number of points of intersection between a square and a circle can actually be 0. This occurs when the circle is completely inside the square without touching it, or when the square is completely inside the circle without touching it, or when the circle and square are completely separate from each other.
● Statement 2: "The maximum number of points of intersection of a square and a circle is 8."
○ This statement is correct. A square has four sides, and a circle can intersect each side at most twice. Therefore, the maximum number of intersection points is 4 sides × 2 intersections per side = 8 points.
● Conclusion: Since statement 1 is incorrect and statement 2 is correct, the correct answer is Option B: 2 only.
Q 51. Logical Venn Diagram
The number of persons who read magazine X only is thrice the number of persons who read magazine Y. The number of persons who read magazine Y only is thrice the number of persons who read magazine X. Then, which of the following conclusions can be drawn?
1. The number of persons who read both the magazines is twice the number of persons who read only magazine X.
2. The total number of persons who read either one magazine or both the magazines is twice the number of persons who read both the magazines.
Select the correct answer using the code given below:
Let's define:
○ \( X \) as the number of persons who read magazine X only.
○ \( Y \) as the number of persons who read magazine Y only.
○ \( B \) as the number of persons who read both magazines.
According to the problem:
○ The number of persons who read magazine X only is thrice the number of persons who read magazine Y.
\[
X = 3Y
\]
○ The number of persons who read magazine Y only is thrice the number of persons who read magazine X.
\[
Y = 3X
\]
From these two equations, we have:
\[
X = 3Y \quad \text{and} \quad Y = 3X
\]
Substituting \( Y = 3X \) into \( X = 3Y \):
\[
X = 3(3X) \implies X = 9X
\]
This implies \( X = 0 \). Similarly, substituting \( X = 3Y \) into \( Y = 3X \):
\[
Y = 3(3Y) \implies Y = 9Y
\]
This implies \( Y = 0 \).
Therefore, both \( X \) and \( Y \) must be zero, which means no one reads only magazine X or only magazine Y. This implies that everyone who reads either magazine reads both, i.e., \( B \) is the total number of readers.
Now, let's evaluate the conclusions:
● Conclusion 1: The number of persons who read both the magazines is twice the number of persons who read only magazine X.
○ Since \( X = 0 \), this conclusion implies \( B = 2 \times 0 = 0 \), which is not necessarily true as \( B \) can be any non-zero number.
● Conclusion 2: The total number of persons who read either one magazine or both the magazines is twice the number of persons who read both the magazines.
○ The total number of persons who read either one magazine or both is \( X + Y + B = 0 + 0 + B = B \).
○ This conclusion implies \( B = 2B \), which is not possible unless \( B = 0 \).
Since neither conclusion holds true under the given conditions, the correct answer is Option D: Neither 1 nor 2.