If x = u^2 - v^2, y = 2uv and z = u^2 + v^2, and if x = 11,...

Understanding the Question

We're given three related expressions involving variables u and v:

• \(\mathrm{x = u^2 - v^2 = 11}\) (given)
• \(\mathrm{y = 2uv}\)
• \(\mathrm{z = u^2 + v^2}\)

What we need to determine: The exact value of z.

Key Insight

Notice the elegant relationship between x and z:

• \(\mathrm{x + z = (u^2 - v^2) + (u^2 + v^2) = 2u^2}\)
• \(\mathrm{z - x = (u^2 + v^2) - (u^2 - v^2) = 2v^2}\)

Since x = 11, we can write \(\mathrm{z = 11 + 2v^2}\). This tells us that to find z, we need to determine \(\mathrm{v^2}\) (which we can get from knowing either u or v).

Analyzing Statement 1

Statement 1 gives us: y = 60

Since \(\mathrm{y = 2uv}\), we now have: \(\mathrm{uv = 30}\)

Can We Find \(\mathrm{v^2}\) From This?

We have two equations:

1. \(\mathrm{u^2 - v^2 = 11}\)
2. \(\mathrm{uv = 30}\)

From equation 2: \(\mathrm{u = 30/v}\)

Substituting into equation 1:
\(\mathrm{(30/v)^2 - v^2 = 11}\)

Expanding: \(\mathrm{900/v^2 - v^2 = 11}\)

Multiplying through by \(\mathrm{v^2}\): \(\mathrm{900 - v^4 = 11v^2}\)

Rearranging: \(\mathrm{v^4 + 11v^2 - 900 = 0}\)

This is a quadratic equation in \(\mathrm{v^2}\). Let \(\mathrm{w = v^2}\), then:
\(\mathrm{w^2 + 11w - 900 = 0}\)

Using the quadratic formula:
\(\mathrm{w = \frac{-11 \pm \sqrt{121 + 3600}}{2} = \frac{-11 \pm \sqrt{3721}}{2} = \frac{-11 \pm 61}{2}}\)

This gives us: \(\mathrm{w = 25}\) or \(\mathrm{w = -36}\)

Since \(\mathrm{v^2}\) must be positive, we have \(\mathrm{v^2 = 25}\).

Therefore: \(\mathrm{z = 11 + 2(25) = 61}\)

[STOP - Statement 1 is Sufficient!]

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2 tells us: u = 6

Finding z Directly

From our original equation \(\mathrm{x = u^2 - v^2 = 11}\), with u = 6:
\(\mathrm{36 - v^2 = 11}\)

Therefore: \(\mathrm{v^2 = 25}\)

Now we can calculate z directly:
\(\mathrm{z = u^2 + v^2 = 36 + 25 = 61}\)

[STOP - Statement 2 is Sufficient!]

The Answer: D

Each statement alone is sufficient to determine that z = 61.

• Statement 1 alone → z = 61 ✓
• Statement 2 alone → z = 61 ✓

Answer Choice D: Each statement alone is sufficient.