JEE Mathematics 2023 Question with Solution

Given: Two dice are thrown independently. Events $A, B, C$ are defined as in the question.

Find: Which statement is correct.

From the solution:

$$n(A) = 5 + 4 + 3 + 2 + 1 = 15$$

For $B$, the $1^{\text{st}}$ die is even and the $2^{\text{nd}}$ die is odd, so the working states

$$n(B) = 3 \times 3 = 9$$

For $C$, the $1^{\text{st}}$ die is odd and the $2^{\text{nd}}$ die is even, so the working states

$$n(C) = 3 \times 3 = 9$$

Now,

$$(A \cup B) \cap C = (A \cap C) \cup (B \cap C)$$

The favourable pairs for $A \cap C$ are listed as

$$(1,2), (1,4), (1,6), (3,4), (3,6), (5,6)$$

Hence,

$$n(A \cap C) = 6$$

Also, $B$ and $C$ are disjoint, so

$$n(B \cap C) = 0$$

Therefore,

$$n((A \cup B) \cap C) = n(A \cap C) + n(B \cap C) = 6 + 0 = 6$$

So statement A is true.

The solution also marks the correct option as D, but that contradicts the displayed working. In addition, option C is false because the working gives $n(B) = 9$ and $n(C) = 9$, not $6$ and $6$. Therefore, based on the solution working, the correct option is A.

Given: Outcomes are ordered pairs $(i,j)$ with $i, j \in \{1,2,3,4,5,6\}$.

Find: Which option matches the event counts.

For event $A$, count pairs with $i < j$:

$$(1,2),(1,3),(1,4),(1,5),(1,6),$$ $$(2,3),(2,4),(2,5),(2,6),$$ $$(3,4),(3,5),(3,6),$$ $$(4,5),(4,6),$$ $$(5,6)$$

So,

$$n(A) = 15$$

For $B$, first die even and second die odd:

$$i \in \{2,4,6\}, \quad j \in \{1,3,5\}$$

Thus,

$$n(B) = 3 \times 3 = 9$$

For $C$, first die odd and second die even:

$$i \in \{1,3,5\}, \quad j \in \{2,4,6\}$$

Thus,

$$n(C) = 3 \times 3 = 9$$

Now check option $A$:

$$(A \cup B) \cap C = (A \cap C) \cup (B \cap C)$$

Since a pair cannot have first die both even and odd, and second die both odd and even, we get

$$B \cap C = \emptyset$$

Hence only $A \cap C$ remains. Under $C$, the possible pairs are

$$(1,2),(1,4),(1,6),(3,2),(3,4),(3,6),(5,2),(5,4),(5,6)$$

Among these, keeping only $i < j$ gives

$$(1,2),(1,4),(1,6),(3,4),(3,6),(5,6)$$

So,

$$n((A \cup B) \cap C) = 6$$

Therefore, the correct statement is A.