NIMCET 2022 — Mathematics PYQ

1. Express the cardinality of power sets: The cardinality of the power set of a set with $k$ elements is given by $2^k$. Therefore, $|P(A)|=2^m$ and $|P(B)|=2^n$. 

2. Formulate the equation: The given condition is $|P(A)|-|P(B)|=112$. Substituting the expressions from step $1$, the equation becomes $2^m-2^n=112$. 

3. Factor out the common term: The equation can be rewritten as $2^n(2^{m-n}-1)=112$. 

4. Prime factorize the constant: The number $112$ can be prime factorized as $112=2^4\times 7$. 

5. Equate the factors: By comparing the factored equation with the prime factorization of $112$, it is deduced that $2^n=2^4$ and $2^{m-n}-1=7$. 

6. Solve for n: From $2^n=2^4$, it is concluded that $n=4$. 

7. Solve for m: From $2^{m-n}-1=7$, it follows that $2^{m-n}=8$. Since $8=2^3$, it is deduced that $m-n=3$. Substituting $n=4$ into this equation, $m-4=3$, which yields $m=7$. 

8. Evaluate the given options: 

• (a) $m+n=7+4=11$. This option is correct. 

• (b) $2n-m=2(4)-7=8-7=1$. This option is correct. 

• (c) $2m-n=2(7)-4=14-4=10$. This option is incorrect. 

• (d) $3n-m=3(4)-7=12-7=5$. This option is correct. 

Final Answer 

The wrong option is (c).