Question

If \( p^2 \) is an integer and \( \sqrt{p^6 - p^4 - q - 1} = 10 \), what is the value of \( p^2 \)?
(1) \( p^2 = \sqrt{p^2 + 20} \)
(2) \( q = \sqrt{q + 2} \)

• A. Statement 1 alone is sufficient but statement 2 alone is not sufficient to answer the question asked.
• B. Statement 2 alone is sufficient but statement 1 alone is not sufficient to answer the question asked.
• C. Both statements 1 and 2 together are sufficient to answer the question but neither statement is sufficient alone.
• D. Each statement alone is sufficient to answer the question.
• E. Statements 1 and 2 are not sufficient to answer the question asked and additional data is needed to answer the statements.

Hint:

When solving systems of equations, ensure to combine all available information from the statements to solve for the unknowns.

Answer 1

The Correct Option is C

Step 1: Analyze Statement 1.
From Statement 1, we have: \[ p^2 = \sqrt{p^2 + 20} \] Squaring both sides: \[ p^4 = p^2 + 20 \] This equation alone is not sufficient to solve for \( p^2 \), as it does not provide enough information about \( q \).
Step 2: Analyze Statement 2.
From Statement 2, we have: \[ q = \sqrt{q + 2} \] Squaring both sides: \[ q^2 = q + 2 \] This does not provide enough information to solve for \( p^2 \) either.
Step 3: Combine Both Statements.
When we combine both statements, we have a system of two equations involving \( p^2 \) and \( q \), which we can solve to find the value of \( p^2 \). Hence, both statements together are sufficient to answer the question.
Step 4: Conclusion.
The correct answer is (C).