Class JEE Mathematics Sets, Relations, and Functions Q #1063
KNOWLEDGE BASED
APPLY
4 Marks 2019 JEE Main 2019 (Online) 12th January Morning Slot MCQ SINGLE
Let $S = {1, 2, 3, … , 100}$. The number of non-empty subsets A of S such that the product of elements in A is even is :
(A) $2^{50} – 1$
(B) $2^{50} (2^{50} – 1)$
(C) $2^{100} – 1$
(D) $2^{50} + 1$
Prev Next
Correct Answer: B
Explanation
Let $S = {1, 2, 3, ..., 100}$. The total number of non-empty subsets is $2^{100} - 1$. The number of subsets with an odd product consists only of odd numbers. There are 50 odd numbers in the set $S$. So, the number of such subsets is $2^{50} - 1$. Therefore, the number of subsets with an even product is $(2^{100} - 1) - (2^{50} - 1) = 2^{100} - 2^{50} = 2^{50}(2^{50} - 1)$.

More from this Chapter

NUMERICAL
For $n \geq 2$, let $S_n$ denote the set of all subsets of $\{1,2, \ldots, n\}$ with no two consecutive numbers. For example $\{1,3,5\} \in S_6$, but $\{1,2,4\} \notin S_6$. Then $n\left(S_5\right)$ is equal to ________
MCQ_SINGLE
Which of the following is not correct for relation $R$ on the set of real numbers?
MCQ_SINGLE
Let $R = \{(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)\}$ be a relation on the set $A = \{3, 6, 9, 12\}$. The relation is :
NUMERICAL
Let $\mathrm{A}=\{-4,-3,-2,0,1,3,4\}$ and $\mathrm{R}=\left\{(a, b) \in \mathrm{A} \times \mathrm{A}: b=|a|\right.$ or $\left.b^{2}=a+1\right\}$ be a relation on $\mathrm{A}$. Then the minimum number of elements, that must be added to the relation $\mathrm{R}$ so that it becomes reflexive and symmetric, is __________
MCQ_SINGLE
Let $\bigcup_{i=1}^{50} X_i = \bigcup_{i=1}^{n} Y_i = T$ where each $X_i$ contains $10$ elements and each $Y_i$ contains $5$ elements. If each element of the set $T$ is an element of exactly $20$ of sets $X_i$'s and exactly $6$ of sets $Y_i$'s, then $n$ is equal to:
View All Questions