Question

a, b and c are three positive integers. What is the value of (a²+b²+c²) ?
Statement 1: a²+b²=17 and c is the arithmetic mean of a and b 
Statement 2: The geometric mean of a and b is 2
Directions: This question has a problem and two statements numbered (1) and (2) giving certain information. You have to decide if the information given in the statements is sufficient for answering the problem. Indicate your answer :

• A. statement (1) alone is sufficient to answer the question
• B. statement (2) alone is sufficient to answer the question
• C. both the statements together are needed to answer the question
• D. either statement (1) alone or statement (2) alone is sufficient to answer the question
• E. neither statement (1) nor statement (2) suffices to answer the question

Answer 1

The Correct Option is C

To determine the value of \(a^2 + b^2 + c^2\), we need to analyze the information given in both statements.

1. Statement 1: \(a^2 + b^2 = 17\) and \(c\) is the arithmetic mean of \(a\) and \(b\).
   • This implies \(c = \frac{a + b}{2}\).
   • We cannot find \(a\) and \(b\) individually from \(a^2 + b^2 = 17\) alone, as there are multiple integer pairs \((a, b)\) that satisfy this.
   • So, Statement 1 alone is insufficient.
2. Statement 2: The geometric mean of \(a\) and \(b\) is 2.
   • This implies \(\sqrt{a \cdot b} = 2\), therefore \(a \cdot b = 4\).
   • This information alone does not specify unique values for \(a\) and \(b\); hence Statement 2 alone is insufficient.

Combining both statements, we have:

• \(a^2 + b^2 = 17\)
• \(c = \frac{a + b}{2}\)
• \(a \cdot b = 4\)

These equations can be used to solve for unique values of \(a\), \(b\), and \(c\).

1. From \(a^2 + b^2 = 17\) and \(a \cdot b = 4\), we know:
   • The identity \((a + b)^2 = a^2 + 2ab + b^2\) can be written as \((a + b)^2 = 17 + 2 \times 4 = 25\).
   • So, \(a + b = 5\).
2. Now, with \(a + b = 5\) and \(ab = 4\), use the quadratic equation:
   • The quadratic equation \(x^2 - 5x + 4 = 0\) has roots \(a\) and \(b\).
   • Solving, \(x^2 - 5x + 4 = (x - 4)(x - 1) = 0\), gives roots \(a = 4\) and \(b = 1\) (or vice versa).
3. Hence, \(c = \frac{a + b}{2} = \frac{5}{2} = 2.5\).
4. Therefore, \(c^2 = (2.5)^2 = 6.25\).
5. Finally, substituting back:
   • \(a^2 = 16\), \(b^2 = 1\), and \(c^2 = 6.25\).
   • Thus, \(a^2 + b^2 + c^2 = 16 + 1 + 6.25 = 23.25\).

Therefore, the statements together are sufficient to find the value of \(a^2 + b^2 + c^2\), confirming the correct answer is:

both the statements together are needed to answer the question.