6 questions from 2021
Let \( S=\{1,\,2,\,3,\,4,\,5,\,6,\,9\} \). Then the number of elements in the set \( T=\{A\subseteq S:\ A\neq \phi \ \text{and the sum of all the elements of }A \text{ is not a multiple of }3\} \) is ______.
Which of the following is not correct for relation \( R \) on the set of real numbers?
If the mean and variance of six observations \( 7, 10, 11, 15, a, b \) are \( 10 \) and \( \dfrac{20}{3} \), respectively, then the value of \( |a-b| \) is equal to:
Out of all the patients in a hospital \( 89\% \) are found to be suffering from heart ailment and \( 98\% \) are suffering from lungs infection. If \( K\% \) of them are suffering from both ailments then \( K \) can not belong to the set:
Let \( A=\{2,3,4,5,\ldots,30\} \) and '\( \simeq \)' be an equivalence relation on \( A\times A \), defined by \( (a,b)\simeq (c,d) \), if and only if \( ad=bc \). Then the number of ordered pairs which satisfy this equivalence relation with ordered pair \( (4,3) \) is equal to:
Let \( A=\{n\in \mathbb{N}\mid n^{2}\leq n+10{,}000\}, B=\{3k+1\mid k\in \mathbb{N}\} \) and \( C=\{2k\mid k\in \mathbb{N}\} \), then the sum of all the elements of the set \( A\cap (B-C) \) is equal to ____________.
