GATE EC 2022 Question Paper (06-Feb-2022) (Shift 1)

A symbol stream contains alternate QPSK and 16-QAM symbols. If symbols from this stream are transmitted at the rate of 1 mega-symbols per second, the raw (uncoded) data rate is _______ mega-bits per second (rounded off to one decimal place).

The function f(x) = 8 loge x - x² + 3 attains its global minimum over the interval [1, e] at x = ________.

(Here loge x is the natural logarithm of x and e${}^{2=7.39)}$

Let α, β be two non-zero real numbers and v₁, v₂ be two non-zero real vectors of size 3 × 1. Suppose that v₁ and v₂ satisfy $v_1^T{v}_2=0,v_1^T{v}_1=1$, and $v_2^T{v}_2=1$. Let A be the 3 × 3 matrix given by:

$\mathrm{A}=\alpha {\mathrm{v}}_1{\mathrm{v}}_1^{\mathrm{T}}+\beta {\mathrm{v}}_2{\mathrm{v}}_2^{\mathrm{T}}$

The eigenvalues of A are _______

Consider a system of linear equations Ax = b, where

$A={\left[\begin{array}{ccc}1& -\sqrt{2}& 3\\ -1& \sqrt{2}& -3\end{array}\right]}_{,}$$B=\left[\begin{array}{c}1\\ 3\end{array}\right]{}_{.}$

This system of equations admits _________.

For the circuit shown, the locus of the impedance Z(jω) is plotted as ω increases from zero to infinity. The values of R₁ and R₂ are: