GATE EC question library

Match the logic gates in Column A with their equivalents in Column B.

For the output $\text{F}$ to be $1$ in the logic circuit shown, the input combination should be

If $e^{y}=x^{\frac{1}{x}}$, then $y$ has a

The Boolean function realized by the logic circuit shown is

The Taylor series expansion of $\dfrac{\sin x}{x-\pi}$ at $x=\pi$ is given by

The eigen values of the following matrix are $$\left[\begin{array}{ccc} -1 & 3 & 5 \\ -3 & -1 & 6 \\ 0 & 0 & 3 \end{array}\right]$$

What are the minimum number of $2$-to-$1$ multiplexers required to generate a $2$-input $\text{AND}$ gate and a $2$-input $\text{Ex-OR}$ gate?

Refer to the NAND and NOR latches shown in the figure. The inputs $\left(P_{1}, P_{2}\right)$ for both the latches are first made $(0,1)$ and then, after a few seconds, made $(1,1)$. The corresponding stable outputs $\left(Q_{1}, Q_{2}\right)$ are

All the four entries of the $2 \times 2$ matrix $\mathbf{P}=\left[\begin{array}{ll}p_{11} & p_{12} \\ p_{21} & p_{22}\end{array}\right]$ are nonzero, and one of its eigenvalues is zero. Which of the following statements is true?

The system of linear equations $$\begin{array}{l} 4 x+2 y=7 \\ 2 x+y=6 \end{array}$$ has

For real values of $x$, the minimum value of the function $f(x)=\exp (x)+\exp (-x)$ is

Which of the following functions would have only odd powers of $x$ in its Taylor series expansion about the point $x=0?$

For a Hertz dipole antenna, the half power beam width $\text{(HPBW)}$ in the $\text{E}$-plane is

Consider the amplitude modulated (AM) signal $A_{c} \cos \omega_{c} t+2 \cos \omega_{m} t \cos \omega_{c} t$. For demodulating the signal using envelope detector, the minimum value of $A_{c}$ should be

Consider the matrix $\mathbf{P}=\left[\begin{array}{cc}0 & 1 \\ -2 & -3\end{array}\right]$. The value of $\mathbf{e}^{\mathbf{P}}$ is

In the Taylor series expansion of $\exp (x)+\sin (x)$ about the point $x=\pi$, the coefficient of $(x-\pi)^{2}$ is

Four messages band limited to $\text{W, W, 2W}$ and $\text{3W}$ respectively are to be multiplexed using $\text{Time Division Multiplexing (TDM)}.$ The minimum bandwidth required for transmission of this $\text{TDM}$ signal is

Consider the frequency modulated signal $10 \cos \left[2 \pi \times 10^{5} t+5 \sin (2 \pi \times 1500 t)+7.5 \sin (2 \pi \times 1000 t)\right]$ with carrier frequency of $10^{5} \mathrm{~Hz}$. The modulation index is

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. If the bits $0$ and $1$ are transmitted using bipolar pulses, the minimum bandwidth required for distortion free transmission is

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. Assuming the signal to be uniformly distributed between its peak to peak value, the signal to noise ratio at the quantizer output is