Class JEE Mathematics Sets, Relations, and Functions Q #1064
KNOWLEDGE BASED
APPLY
4 Marks 2019 JEE Main 2019 (Online) 10th January Morning Slot MCQ SINGLE
In a class of $140$ students numbered $1$ to $140$, all even numbered students opted Mathematics course, those whose number is divisible by $3$ opted Physics course and those whose number is divisible by $5$ opted Chemistry course. Then the number of students who did not opt for any of the three courses is
(A) $42$
(B) $102$
(C) $1$
(D) $38$
Prev Next
Correct Answer: D
Explanation
Let $A$ be the set of students who opted for Mathematics, $B$ be the set of students who opted for Physics, and $C$ be the set of students who opted for Chemistry. We are given:

Total number of students $= 140$

$n(A) = \lfloor\frac{140}{2}\rfloor = 70$

$n(B) = \lfloor\frac{140}{3}\rfloor = 46$

$n(C) = \lfloor\frac{140}{5}\rfloor = 28$

$n(A \cap B) = \lfloor\frac{140}{6}\rfloor = 23$

$n(B \cap C) = \lfloor\frac{140}{15}\rfloor = 9$

$n(C \cap A) = \lfloor\frac{140}{10}\rfloor = 14$

$n(A \cap B \cap C) = \lfloor\frac{140}{30}\rfloor = 4$

Using the principle of inclusion-exclusion:

$n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(C \cap A) + n(A \cap B \cap C)$

$n(A \cup B \cup C) = 70 + 46 + 28 - 23 - 9 - 14 + 4 = 102$

The number of students who did not opt for any of the three courses is:

$140 - n(A \cup B \cup C) = 140 - 102 = 38$

More from this Chapter

MCQ_SINGLE
Let $A = {2, 3, 4}$ and $B = {8, 9, 12}$. Then the number of elements in the relation $R = {((a_1, b_1), (a_2, b_2)) \in (A \times B, A \times B) : a_1$ divides $b_2$ and $a_2$ divides $b_1}$ is :
MCQ_SINGLE
Let $R$ be the set of real numbers. Statement I: $A = \{(x, y) \in R \times R: y - x \text{ is an integer }\}$ is an equivalence relation on $R$. Statement II: $B = \{(x,y) \in R \times R: x = \alpha y \text{ for some rational number } \alpha\}$ is an equivalence relation on $R$.
MCQ_SINGLE
If R is the smallest equivalence relation on the set ${1, 2, 3, 4}$ such that ${((1, 2), (1, 3))} \subset R$, then the number of elements in $R$ is __________.
MCQ_SINGLE
The minimum number of elements that must be added to the relation $R = \{(a, b), (b, c)\}$ on the set $\{a, b, c\}$ so that it becomes symmetric and transitive is :
NUMERICAL
Let $S=\{4,6,9\}$ and $T=\{9,10,11, \ldots, 1000\}$. If $A=\left\{a_{1}+a_{2}+\ldots+a_{k}: k \in \mathbf{N}, a_{1}, a_{2}, a_{3}, \ldots, a_{k}\right.$ $\epsilon S\}$, then the sum of all the elements in the set $T-A$ is equal to __________.
View All Questions