Question

What is the value of \( x^2 + y^2 = ? \)
Statement I: \( xy = 5 \)
Statement II: \( x + y = 10 \)

• A. Choose this option if the question can be answered by using one of the statements alone, but cannot be answered using the other statement.
• B. Choose this option if the question can be answered by using both the statements together, but cannot be answered using the other statement.
• C. Choose this option if the question can be answered by using either statement alone.
• D. Choose this option if the question cannot be answered even by using both the statements together.

Hint:

For expressions involving \( x^2 + y^2 \), use the identity involving \( (x + y)^2 \) and \( xy \) for efficient calculation.

Answer 1

The Correct Option is B

Step 1: Use identity: \[ (x + y)^2 = x^2 + y^2 + 2xy \] Step 2: Plug in known values from both statements. Given: \( x + y = 10 \), \( xy = 5 \) \[ 100 = x^2 + y^2 + 2(5) \Rightarrow 100 = x^2 + y^2 + 10 \Rightarrow x^2 + y^2 = 90 \] Step 3: Can we use either statement alone?
Using only \( x + y = 10 \): Not enough.
Using only \( xy = 5 \): Not enough.
Only with both together can we solve it.