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 $X = {1, 2, 3, 4, 5}$. The number of different ordered pairs $(Y, Z)$ that can be formed such that $Y \subseteq X$, $Z \subseteq X$ and $Y \cap Z$ is empty, is:
MCQ_SINGLE
Let $A = {2, 3, 6, 8, 9, 11}$ and $B = {1, 4, 5, 10, 15}$. Let $R$ be a relation on $A \times B$ defined by $(a, b)R(c, d)$ if and only if $3ad - 7bc$ is an even integer. Then the relation $R$ is
NUMERICAL
Let $A=\{1,2,3,4\}$ and $R=\{(1,2),(2,3),(1,4)\}$ be a relation on $\mathrm{A}$. Let $\mathrm{S}$ be the equivalence relation on $\mathrm{A}$ such that $R \subset S$ and the number of elements in $\mathrm{S}$ is $\mathrm{n}$. Then, the minimum value of $n$ is __________.
MCQ_SINGLE
The number of non-empty equivalence relations on the set ${1, 2, 3}$ is :
NUMERICAL
Let $A = \sum\limits_{i = 1}^{10} {\sum\limits_{j = 1}^{10} {\min \,\{ i,j\} } } $ and $B = \sum\limits_{i = 1}^{10} {\sum\limits_{j = 1}^{10} {\max \,\{ i,j\} } } $. Then A + B is equal to _____________.
View All Questions