Question

If \( u = x^2 + y^2 + 2xy + 2x + 2y \) and \( v = e^{x+y} \) are functionally dependent, then the relation between \( u \) and \( v \) is:

• A. \( u^2 = v^2 + \log uv \)
• B. \( u = (\log v)^2 + 2 \log v \)
• C. \( u = 2 \log v + 2 \log u \)
• D. \( v = (\log u)^2 + 2 \log u \)

Hint:

In functional dependence problems, if you see terms like \( x^2+2xy+y^2 \) alongside \( e^{x+y} \), the common link is almost always the linear term \( x+y \).

Answer 1

The Correct Option is B

Concept: Functional dependence implies that one function can be expressed entirely in terms of the other.

• Look for shared algebraic structures between the two variables.

• Utilize the identity \( (x+y)^2 = x^2 + 2xy + y^2 \).

• Apply logarithmic properties to isolate exponents.

Step 1: Factoring the expression for \( u \).
Group the terms in \( u \) to reveal a quadratic identity. \[ u = (x^2 + 2xy + y^2) + 2(x + y) \] Simplify the quadratic group. \[ u = (x + y)^2 + 2(x + y) \quad \dots (1) \]

Step 2: Finding a substitution from \( v \).
Take the natural logarithm of the second equation. \[ v = e^{x+y} \implies \log v = \log(e^{x+y}) \] By definition of logarithms: \[ \log v = x + y \quad \dots (2) \]

Step 3: Establishing the functional relation.
Substitute equation (2) into equation (1). Replace every instance of \( (x+y) \) with \( \log v \). \[ u = (\log v)^2 + 2(\log v) \] This matches the form given in Option (B).