Question:
There are two sets \(A\) and \(B\) with \(|A|=m\) and \(|B|=n\). If \(|P(A)|-|P(B)|=112\), choose the wrong option.
There are two sets \(A\) and \(B\) with \(|A|=m\) and \(|B|=n\). If \(|P(A)|-|P(B)|=112\), choose the wrong option.
Show Hint
Whenever a power set question appears, immediately replace
\[
|P(A)|=2^{|A|}.
\]
Then convert the problem into an equation involving powers of 2.
Updated On: Jun 8, 2026
- \(m+n=11\)
- \(2m-n=3\)
- \(2m-n=1\)
- \(3n-m=5\)
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Concept:
If a set contains \(k\) elements, then its power set contains
\[
2^k
\]
elements.
Therefore,
\[
|P(A)|=2^m
\]
and
\[
|P(B)|=2^n.
\]
Step 1: Use the given condition.
\[ 2^m-2^n=112 \] Factorizing, \[ 2^n(2^{m-n}-1)=112 \] Now \[ 112=2^4\times7 \] Thus \[ 2^n=16 \] and \[ 2^{m-n}-1=7 \] \[ 2^{m-n}=8 \] \[ m-n=3 \] Hence \[ n=4,\qquad m=7 \]
Step 2: Check each option.
Option (A): \[ m+n=7+4=11 \] True. Option (B): \[ 2m-n=14-4=10 \] Not equal to 1. False. Option (D): \[ 3n-m = 12-7 = 5 \] True. Therefore the wrong option is \[ \boxed{(B)} \]
Step 1: Use the given condition.
\[ 2^m-2^n=112 \] Factorizing, \[ 2^n(2^{m-n}-1)=112 \] Now \[ 112=2^4\times7 \] Thus \[ 2^n=16 \] and \[ 2^{m-n}-1=7 \] \[ 2^{m-n}=8 \] \[ m-n=3 \] Hence \[ n=4,\qquad m=7 \]
Step 2: Check each option.
Option (A): \[ m+n=7+4=11 \] True. Option (B): \[ 2m-n=14-4=10 \] Not equal to 1. False. Option (D): \[ 3n-m = 12-7 = 5 \] True. Therefore the wrong option is \[ \boxed{(B)} \]
Was this answer helpful?
0
0
Top NIMCET Mathematics Questions
- Among the given numbers below, the smallest number which will be divided by 9, 10, 15 and 20 and leaves the remainders 4, 5, 10 and 15, respectively, is:
- If one AM (Arithmetic mean) \( a \) and two GM's (Geometric means) \( p \) and \( q \) be inserted between any two positive numbers, the value of \( p^3 + q^3 \) is:
- If one AM (Arithmetic mean) \( a \) and two GM's (Geometric means) \( p \) and \( q \) be inserted between any two positive numbers, the value of \( p^3 + q^3 \) is:
- A coin is thrown 8 times. What is the probability of getting a head in an odd number of throws?
- The value of \( \tan\left(\frac{\pi}{4} + \theta\right) \times \tan\left(\frac{3\pi}{4} + \theta\right) \) is:
View More Questions
Top NIMCET Set Theory Questions
- Out of a group of 50 students taking examinations in Mathematics, Physics, and Chemistry, 37 students passed Mathematics, 24 passed Physics, and 43 passed Chemistry. Additionally, no more than 19 students passed both Mathematics and Physics, no more than 29 passed both Mathematics and Chemistry, and no more than 20 passed both Physics and Chemistry. What is the maximum number of students who could have passed all three examinations?
- Let \( Z \) be the set of all integers, and consider the set \( X = \{(x, y): x^2 + 2y^2 = 3, x, y \in Z \) and \( Y = \{(x, y): x>y, x, y \in Z\} \). Then the number of elements in \( X \cap Y \) is:
Let \( A = (1, 2, 3, \dots, 20) \). Let \( R \subseteq A \times A \) such that \( R = \{(x, y) : y = 2x - 7 \} \). Then the number of elements in \( R \) is equal to:
Let $A = \{5n - 4n - 1 : n \in \mathbb{N}\}$ and $B = \{16(n - 1): n \in \mathbb{N}\}$ be sets. Then:
Let A and B be sets. \[A \cap X = B \cap X = \varnothing \quad \text{and} \quad A \cup X = B \cup X \quad \text{for some set } X,\ \text{find the relation between } A \text{ and } B.\]
View More Questions
Top NIMCET Questions
- How much work is done to slide a crate for a distance of 25 m along a loading dock by pulling on it with a 180 N force where the dock is at an angle of \(45^\circ\) from the horizontal?
- Let \( f : \mathbb{R} \to \mathbb{R} \) be a function such that \( f(0) = \frac{1}{\pi} \) and \( f(x) = \frac{x}{e^x - 1} \) for \( x \ne 0 \). Then:
- The value of the limit \[ \lim_{x \to 0} \left( \frac{1^x + 2^x + 3^x + 4^x}{4} \right)^{\frac{1}{x}} \] is:
- Consider the function \( f(x) = x^{2/3} \cdot (6 - x)^{1/3} \). Which of the following statements is false?
- Lines \( L_1, L_2, \dots, L_{10} \) are distinct, among which the lines \( L_2, L_4, L_6, L_8, L_{10} \) are parallel to each other, and the lines \( L_1, L_3, L_5, L_7, L_9 \) pass through a given point \( C \). The number of points of intersection of pairs of lines from the complete set \( L_1, L_2, L_3, \dots, L_{10} \) is:
View More Questions