GATE EC 2025 Question Paper (15-Feb-2025) (Shift 2)

Consider the matrix $A$ below:
$$A=\left[\begin{array}{cccc}2& 3& 4& 5\\ 0& 6& 7& 8\\ 0& 0& \alpha & \beta \\ 0& 0& 0& \gamma \end{array}\right]$$
For which of the following combinations of $\alpha$, $\beta$ and $\gamma$, is the rank of $A$ at least three?

1. (i) $\alpha =0$ and $\beta =\gamma \ne 0$
2. (ii) $\alpha =\beta =\gamma =0$
3. (iii) $\beta =\gamma =0$ and $\alpha \ne 0$
4. (iv) $\alpha =\beta =\gamma \ne 0$

Consider the following series:
(i) ${\sum }_{n=1}^{\infty }\frac{1}{\sqrt{n}}$
(ii) ${\sum }_{n=1}^{\infty }\frac{1}{n(n+1)}$
(iii) ${\sum }_{n=1}^{\infty }\frac{1}{n!}$

A pot contains two red balls and two blue balls. Two balls are drawn from this pot randomly without replacement.
What is the probability that the two balls drawn have different colours?

Consider a frequency-modulated (FM) signal

$f(t)=A_c\cos (2\pi f_{c}t+3\sin (2\pi f_{1}t)+4\sin (6\pi f_{1}t))$,

where $A_c$ and $f_c$ are, respectively, the amplitude and frequency (in Hz) of the carrier waveform. The frequency $f_1$ is in Hz, and assume that $f_c>100f_1$. The peak frequency deviation of the FM signal in Hz is ________.

Consider an additive white Gaussian noise (AWGN) channel with bandwidth W and noise power spectral density $\frac{N_0}{2}$. Let $P_{av}$ denote the average transmit power constraint. Which one of the following plots illustrates the dependence of the channel capacity C on the bandwidth W (keeping $P_{av}$ and $N_0$ fixed)?