GATE 2025 EE Q.53: solution and answer

Official answer: 0 to 1Source: Official answer key, GATE 2025 EE (IIT Roorkee)

1. The rate of change of a scalar field in a given direction is the directional derivative $D_{\hat{u}}V=\nabla V\cdot \hat{u}$, where $\hat{u}$ is a unit vector along that direction.

2. Gradient: $\nabla V=\left(\frac{\partial V}{\partial x},\frac{\partial V}{\partial y}\right)=(2x+1,2y)$.

3. At the origin: $\nabla V(0,0)=(1,0)$.

4. The direction from the origin $(0,0)$ towards the point $(1,2)$ is $\vec{d}=(1,2)-(0,0)=(1,2)$, with $|\vec{d}|=\sqrt{1^2+2^2}=\sqrt{5}$, so $\hat{u}=\left(\frac{1}{\sqrt{5}},\frac{2}{\sqrt{5}}\right)$.

5. Directional derivative: $$D_{\hat{u}}V(0,0)=(1)\left(\frac{1}{\sqrt{5}}\right)+(0)\left(\frac{2}{\sqrt{5}}\right)=\frac{1}{\sqrt{5}}\approx 0.447$$

6. Rounded to the nearest integer, as the question asks: $0.447\to 0$.