Question

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : If \(u=x^n f\left(\dfrac{y}{x}\right)\), then \[ x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=nu \] Reason (R) : Given function \(u\) is homogeneous of degree \(n\) in \(x\) and \(y\). In the light of the above statements, choose the most appropriate answer from the options given below :

• A. Both (A) and (R) are correct and (R) is the correct explanation of (A)
• B. Both (A) and (R) are correct but (R) is not the correct explanation of (A)
• C. (A) is correct but (R) is not correct
• D. (A) is not correct but (R) is correct

Hint:

Whenever a function is of the form: \[ u=x^n f\left(\frac{y}{x}\right) \] it is automatically homogeneous of degree \(n\). Then Euler’s theorem can be directly applied: \[ x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=nu \]

Answer 1

The Correct Option is A