Which of the following compound statements are false?
p : 17 is odd and prime number
q : 23 is even and prime number
r : Volume of cube is `a^3`and total surface area is `4a^2` where a is the length of side of a cube

Options

A. p and q
B. q and r
C. p and r
D. None of these

Answer

To determine which of the given compound statements are false, we will analyze each statement one by one.

### Step 1: Analyze Statement P
**Statement P:** "17 is an odd and prime number."

- **Sub-statement A:** "17 is odd."
- This is true because 17 is not divisible by 2.

- **Sub-statement B:** "17 is a prime number."
- This is also true because 17 has no divisors other than 1 and itself.

Since both sub-statements A and B are true, the compound statement P is true.

### Step 2: Analyze Statement Q
**Statement Q:** "23 is an even and prime number."

- **Sub-statement A:** "23 is even."
- This is false because 23 is not divisible by 2 (it is odd).

- **Sub-statement B:** "23 is a prime number."
- This is true because 23 has no divisors other than 1 and itself.

Since sub-statement A is false and sub-statement B is true, the compound statement Q is false (because both need to be true for the statement to be true).

### Step 3: Analyze Statement R
**Statement R:** "Volume of a cube is \(a^3\) and total surface area is \(4a^2\)."

- **Sub-statement A:** "Volume of a cube is \(a^3\)."
- This is true because the volume of a cube with side length \(a\) is indeed \(a^3\).

- **Sub-statement B:** "Total surface area is \(4a^2\)."
- This is false because the total surface area of a cube is \(6a^2\), not \(4a^2\).

Since sub-statement A is true and sub-statement B is false, the compound statement R is false.

### Conclusion
The false statements among the given options are:
- Q: "23 is an even and prime number." (False)
- R: "Total surface area is \(4a^2\)." (False)

Thus, the compound statements that are false are Q and R.