GATE EE 2024 Question Paper (11-Feb-2024) (Shift-2)

Consider a vector $\bar{u}=2\hat{x}+\hat{y}+2\hat{z}$, where $\hat{x},\hat{y},\hat{z}$ represent unit vectors along the coordinate axes $x,y,z$ respectively. The directional derivative of the function $f(x,y,z)=2\ln (xy)+\ln (yz)+3\ln (xz)$ at the point $(x,y,z)=(1,1,1)$ in the direction of $\bar{u}$ is

The input $x(t)$ and the output $y(t)$ of a system are related as
$y(t)=e^{-t}{\int }_{-\infty }^t{e}^{\tau }x(\tau )d\tau$, $-\infty <t<\infty$.
The system is

Consider the discrete-time systems $T_1$ and $T_2$ defined as follows:
$\{T_{1}x\}[n]=x[0]+x[1]+\cdots +x[n]$
$\{T_{2}x\}[n]=x[0]+\frac{1}{2}x[1]+\cdots +\frac{1}{2^n}x[n]$
Which one of the following statements is true?

If the Z-transform of a finite-duration discrete-time signal $x[n]$ is $X(z)$, then the Z-transform of the signal $y[n]=x[2n]$ is

A 3-phase, 11 kV, 10 MVA synchronous generator is connected to an inductive load of power factor ($\sqrt{3}/2$) via a lossless line with a per-phase inductive reactance of 5 $\Omega$. The per-phase synchronous reactance of the generator is 30 $\Omega$ with negligible armature resistance. If the generator is producing the rated current at the rated voltage, then the power factor at the terminal of the generator is