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 $R = \{(1, 2), (2, 3), (3, 3)\}$ be a relation defined on the set $\{1, 2, 3, 4\}$. Then the minimum number of elements, needed to be added in $R$ so that $R$ becomes an equivalence relation, is:
NUMERICAL
Let X = {n $ \in $ N : 1 $ \le $ n $ \le $ 50}. If A = {n $ \in $ X: n is a multiple of 2} and B = {n $ \in $ X: n is a multiple of 7}, then the number of elements in the smallest subset of X containing both A and B is ________.
MCQ_SINGLE
Consider the following relations $R = \{(x, y) | x, y$ are real numbers and $x = wy$ for some rational number $w\}$; $S = \{(\frac{m}{n}, \frac{p}{q}) | m, n, p$ and $q$ are integers such that $n, q \neq 0$ and $qm = pn\}$. Then
NUMERICAL
Let $A=\{1,2,3, \ldots \ldots \ldots \ldots, 100\}$. Let $R$ be a relation on $\mathrm{A}$ defined by $(x, y) \in R$ if and only if $2 x=3 y$. Let $R_1$ be a symmetric relation on $A$ such that $R \subset R_1$ and the number of elements in $R_1$ is $\mathrm{n}$. Then, the minimum value of $\mathrm{n}$ is _________.
MCQ_SINGLE
Let $N$ denote the set of all natural numbers. Define two binary relations on $N$ as $R_1 = \{(x, y) \in N \times N : 2x + y = 10\}$ and $R_2 = \{(x, y) \in N \times N : x + 2y = 10\}$. Then :
View All Questions