Question

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

• A. Statement I alone is sufficient but statement II aloneq = q + 2 is not sufficient to answer the question.
• B. Statement II alone is sufficient but statement I alone is not sufficient to answer the question asked.
• C. Both statements I and II 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 I and II are not sufficient to answer the question asked and additional data is needed to answer the statements.

Hint:

When an equation involves integers, remember to use number properties like factors and perfect squares to limit the possibilities. For polynomial equations, testing small integer values is an effective strategy to find potential roots.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We are given an equation with two variables, \(p^2\) and \(q\), and we need to find the value of \(p^2\). Let's simplify the main equation first. Let \(P = p^2\). Since \(p^2\) is an integer, P is an integer.
The main equation is: \(\sqrt{P^3 - P^2 - q - 1} = 10\).
Square both sides: \[ P^3 - P^2 - q - 1 = 100 \] \[ P^3 - P^2 - q = 101 \] This can be rewritten as \(P^2(P-1) - q = 101\). Our goal is to find a unique integer value for \(P\).
Step 2: Detailed Explanation:
Analyze Reconstructed Statement I: \(q = -1\)
Substitute \(q = -1\) into our simplified main equation: \[ P^2(P-1) - (-1) = 101 \] \[ P^2(P-1) = 100 \] Since P is an integer, \(P^2\) must be a perfect square that is a divisor of 100. Let's test the possibilities:

If \(P^2=1\), then \(P=1\). Check: \(1^2(1-1) = 1(0) = 0 \neq 100\).
If \(P^2=4\), then \(P=2\). Check: \(2^2(2-1) = 4(1) = 4 \neq 100\).
If \(P^2=25\), then \(P=5\). Check: \(5^2(5-1) = 25(4) = 100\). This is a valid solution.
If \(P^2=100\), then \(P=10\). Check: \(10^2(10-1) = 100(9) = 900 \neq 100\).
The only integer solution is \(P=5\). Therefore, \(p^2=5\), and this statement is sufficient.
Analyze Reconstructed Statement II: \(p^2 + q = 4\)
This can be written as \(P + q = 4\), or \(q = 4 - P\). Substitute this expression for \(q\) into the main equation: \[ P^2(P-1) - (4-P) = 101 \] \[ P^3 - P^2 - 4 + P = 101 \] \[ P^3 - P^2 + P - 105 = 0 \] We need to find integer roots of this cubic equation. Let's test small integer values for P.

P=1: \(1-1+1-105 \neq 0\)
P=2: \(8-4+2-105 \neq 0\)
P=3: \(27-9+3-105 \neq 0\)
P=4: \(64-16+4-105 \neq 0\)
P=5: \(125-25+5-105 = 130 - 130 = 0\). This is a valid solution.
To ensure this is the only integer solution, we can check the function's behavior. Let \(f(P) = P^3 - P^2 + P - 105\). The derivative is \(f'(P) = 3P^2 - 2P + 1\). The discriminant of this derivative is \((-2)^2 - 4(3)(1) = -8\), which is negative. This means the derivative is always positive, so the function \(f(P)\) is strictly increasing. An increasing function can only cross the x-axis once, so \(P=5\) is the unique real solution, and thus the unique integer solution. Therefore, this statement is sufficient.
Step 3: Final Answer:
Since each statement alone is sufficient to find a unique value for \(p^2\), the correct answer is (D).