GATE EC 2026 Question Paper (15-Feb-2026) (Shift 1)

Consider the differential equation $\dot{\vec{w}}=A\vec{w}$, with $\vec{w}(t=0)=\left[\begin{array}{c}1\\ 1\end{array}\right]$.
If $\vec{w}(t)=e^t{\vec{u}}_x+e^{-2t}{\vec{u}}_y$ be the solution to the equation where ${\vec{u}}_x$ and ${\vec{u}}_y$ are unit vectors along the positive x and y axes respectively, then which of the following options is the correct matrix representing $A$?

A surface is given by $z^2=2x^2-y^2$ and $\vec{n}$ and $-\vec{n}$ are unit normal vectors to the surface at the point $\vec{P}=\hat{i}+\sqrt{2}\hat{k}$.
Which of the following vectors can be $\vec{n}$, where $\hat{i},\hat{j}$ and $\hat{k}$ are the unit vectors along x, y and z axes, respectively?

The Laplace Transform of the signal $x(t)=u(t-2)\ast (tu(t))$ is given by which of the following expressions?
["*"] represents convolution operator]

Consider carrier transport in a Zener diode in the breakdown region.
Which is the dominant transport mechanism for current flow in this case?

Two analog signals $x_1(t)$ and $x_2(t)$ ($t$ in second), are sampled at a rate $F_s=40\text{ Hz}$, where

$$x_1(t)=\cos (20\pi t),t\ge 0,\text{ and }{x}_2(t)=\cos (100\pi t),t\ge 0.$$
The first ten samples (starting from $t=0$) are considered for the analysis.
Which of the following statements is TRUE?