GATE EC question library

A speech signal, band limited to $4 \; \mathrm{kHz}$ and peak voltage varying between $+5 \mathrm{V}$ and $-5 \mathrm{V}$, is sampled at the Nyquist rate. Each sample is quantized and represented by $8$ bits. The number of quantization levels required to reduce the quantization noise by a factor of $4$ would be

The following plot shows a function $y$ which varies linearly with $x$. The value of the integral $I=\displaystyle{}\int_1^2 y d x$ is

For $|x| \ll 1$, $\operatorname{coth}(x)$ can be approximated as

$\displaystyle{}\lim _{\theta \rightarrow 0} \frac{\sin (\theta / 2)}{\theta}$ is

Which one of the following functions is strictly bounded?

For the function $e^{-x}$, the linear approximation around $x=2$ is

$\mathrm{X}=01110$ and $\mathrm{Y}=11001$ are two $5$-bit binary numbers represented in two's complement format. The sum of $\mathrm{X}$ and $\mathrm{Y}$ represented in two's complement format using $6$ bits is

It is given that $X_{1}, X_{2}, \cdots ,X_{M}$ are $M$ non-zero, orthogonal vectors. The dimension of the vector space spanned by the $2 M$ vectors $X_{1}, X_{2}, \cdots, X_{M},-X_{1},-X_{2}, \cdots,-X_{M}$ is

Consider the function $f(x)=x^{2}-x-2$. The maximum value of $f(x)$ in the closed interval $[-4,4]$ is

The following binary values were applied to the $\mathrm{X}$ and $\mathrm{Y}$ inputs of the NAND latch shown in the figure in the sequence indicated below: $$\mathrm{X}=0, \mathrm{Y}=1 ; \quad \mathrm{X}=0, \mathrm{Y}=0 ; \quad \mathrm{X}=1, \mathrm{Y}=1 \text{.}$$ The corresponding stable $\mathrm{P}, \mathrm{Q}$ outputs will be

In delta modulation, the slope overload distortion can be reduced by

In the following scheme, if the spectrum $M(f)$ of $m(t)$ is as shown, then the spectrum $Y(f)$ of $y(t)$ will be

The rank of the matrix $\left[\begin{array}{ccc}1 & 1 & 1 \\ 1 & -1 & 0 \\ 1 & 1 & 1\end{array}\right]$ is

$\nabla \times \nabla \times \mathrm{P},$ where $\mathrm{P}$ is a vector, is equal to

$\displaystyle{}\iint(\nabla \times \mathrm{P}) \bullet \mathrm{ds},$ where $\mathrm{P}$ is a vector, is equal to

A probability density function is of the form $\qquad p(x)=\mathrm{Ke}^{-\alpha|x|}, x \in(-\infty, \infty)$ The value of $\text{K}$ is

A solution for the differential equation $\qquad \dot{x}(t)+2 x(t)=\delta(t)$ with initial condition $x(0-)=0$ is

A low-pass filter having a frequency response $\mathrm{H}(\mathrm{j} \omega)=\mathrm{A}(\omega) \;e^{j{\phi(\omega)}}$ does not produce any phase distortion, if

The values of voltage $\left(\mathrm{V}_{\mathrm{D}}\right)$ across a tunnel-diode corresponding to peak and valley currents are $\text{V}_{\text{P}}$ and $\text{V}_{\text{V}}$ respectively. The range of tunnel-diode voltage $\text{V}_{\text{D}}$ for which the slope of its $\text{I}-\text{V}_{\text{D}}$ characteristics is negative would be

The concentration of minority carriers in an extrinsic semiconductor under equilibrium is