Two dice are thrown simultaneously. The events A, B, C, D are described as follows:
A = Getting an even number on the first die
B = Getting an odd number on the first die
C = Getting atmost 5 as sum of the numbers on the two dice
D = Getting the sum of the numbers on the dice greater than 5 but less than 10
Which of the following statements is true?

Options

A. A and D are mutually exclusive
B. A and B are mutually exclusive and exhaustive events
C. A and C are mutually exclusive events
D. C and D are mutually exclusive and exhaustive events

Answer

To solve the problem, we need to analyze the events A, B, C, and D defined in the context of rolling two dice. We will determine whether these events are mutually exclusive and/or exhaustive.

### Step 1: Define the Sample Space
When two dice are thrown, the total number of outcomes is given by:
\[ \text{Total outcomes} = 6 \times 6 = 36 \]
This is because each die has 6 faces.

### Step 2: Analyze Event A
Event A is defined as getting an even number on the first die. The even numbers on a die are 2, 4, and 6.

- Possible outcomes for Event A:
- For 2: (2,1), (2,2), (2,3), (2,4), (2,5), (2,6)
- For 4: (4,1), (4,2), (4,3), (4,4), (4,5), (4,6)
- For 6: (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)

Total outcomes for Event A:
\[ \text{Total outcomes for A} = 3 \times 6 = 18 \]

### Step 3: Analyze Event B
Event B is defined as getting an odd number on the first die. The odd numbers on a die are 1, 3, and 5.

- Possible outcomes for Event B:
- For 1: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6)
- For 3: (3,1), (3,2), (3,3), (3,4), (3,5), (3,6)
- For 5: (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)

Total outcomes for Event B:
\[ \text{Total outcomes for B} = 3 \times 6 = 18 \]

### Step 4: Analyze Event C
Event C is defined as getting at most 5 as the sum of the numbers on the two dice. We will count the combinations that yield a sum of 2, 3, 4, or 5.

- Possible outcomes for Event C:
- Sum = 2: (1,1)
- Sum = 3: (1,2), (2,1)
- Sum = 4: (1,3), (2,2), (3,1)
- Sum = 5: (1,4), (2,3), (3,2), (4,1)

Total outcomes for Event C:
\[ \text{Total outcomes for C} = 1 + 2 + 3 + 4 = 10 \]

### Step 5: Analyze Event D
Event D is defined as getting a sum of the numbers on the dice greater than 5 but less than 10. We will count the combinations that yield a sum of 6, 7, 8, or 9.

- Possible outcomes for Event D:
- Sum = 6: (1,5), (2,4), (3,3), (4,2), (5,1)
- Sum = 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
- Sum = 8: (2,6), (3,5), (4,4), (5,3), (6,2)
- Sum = 9: (3,6), (4,5), (5,4), (6,3)

Total outcomes for Event D:
\[ \text{Total outcomes for D} = 5 + 6 + 5 + 4 = 20 \]

### Step 6: Check for Mutual Exclusivity and Exhaustiveness
1. **Mutually Exclusive**: Events A and B cannot occur simultaneously since the first die cannot be both even and odd at the same time. Thus, A and B are mutually exclusive.
2. **Exhaustive**: Events A and B cover all possible outcomes for the first die (either even or odd). Therefore, they are exhaustive.

### Conclusion
Based on the analysis:
- A and B are mutually exclusive and exhaustive.
- C and D are not mutually exclusive with A and B since they can occur simultaneously.

### Final Answer
The correct statement is that **A and B are mutually exclusive and exhaustive events**.