GATE EC question library

Consider a system represented by the block diagram shown below. Which of the following signal flow graphs represent(s) this system? Choose the correct option(s).

All the diodes in the circuit given below are ideal. Which of the following plots is/are correct when $V_{I}$ (in Volts) is swept from $-M$ to $M$?

Two fair dice (with faces labeled $1,2,3,4,5$, and $6$) are rolled. Let the random variable $X$ denote the sum of the outcomes obtained. The expectation of $X$ is __________ (rounded off to two decimal places).

Consider the vectors $$\boldsymbol{a}=\left[\begin{array}{l} 1 \\ 1 \end{array}\right], \quad \boldsymbol{b}=\left[\begin{array}{c} 0 \\ 3 \sqrt{2} \end{array}\right]$$ For real-valued scalar variable $x$, the value of $$\min _{x}\|\boldsymbol{a} x-\boldsymbol{b}\|_{2}$$ is $\_\_\_\_\_$ (rounded off to two decimal places). $\|\cdot\|_{2}$ denotes the …

$X$ and $Y$ are Bernoulli random variables taking values in $\{0,1\}$. The joint probability mass function of the random variables is given by: $$\begin{array}{l} P(X=0, Y=0)=0.06 \\ P(X=0, Y=1)=0.14 \\ P(X=1, Y=0)=0.24 \\ P(X=1, Y=1)=0.56 \end{array}$$ The mutual information $I(X ; Y)$ is $\_\_\_\_\_\_\_\_$ (rounded off to two decimal places).

The diode in the circuit shown below is ideal. The input voltage (in Volts) is given by $V_{I}=10 \sin 100 \pi t$, where time $t$ is in seconds. The time duration (in $\mathrm{ms}$, rounded off to two decimal places) for which the diode is forward biased during one period of the input is $\_\_\_\_$

In the circuit shown below, the $\mathrm{AND}$ gate has a propagation delay of $1 \: \mathrm{ ns}$. The edge-triggered flip-flops have a set-up time of $2 \: \mathrm{ ns}$, a hold-time of $0 \: \mathrm{ns}$, and a clock-to-$Q$ delay of $2$ ns. The maximum clock frequency (in $\mathrm{MHz}$, rounded off to the nearest integer) such that there …

An ideal p-n junction germanium diode has a reverse saturation current of $10 \mu \mathrm{~A}$ at $300 \: \mathrm{ K}$. The voltage (in Volts, rounded off to two decimal places) to be applied across the junction to get a forward bias current of $100 \: \mathrm{mA}$ at $300 \: \mathrm{K}$ is $\_\_\_\_\_\_\_$. (Consider the Boltzmann constant …

A $50 \Omega$ lossless transmission line is terminated with a load $\mathrm{Z}_{\mathrm{L}}$ of $(50-\mathrm{j} 75) \Omega$. If the average incident power on the line is $10 \: \mathrm{mW}$, then the average power delivered to the load (in $\mathrm{mW}$, rounded off to one decimal place) is $\_\_\_\_\_\_\_$.

Two resistors are connected in a circuit loop of area $5 \mathrm{~m}^{2}$, as shown in the figure below. The circuit loop is placed on the $x-y$ plane. When a time-varying magnetic flux, with flux-density $B(t)=0.5 t$ (in Tesla), is applied along the positive $z$-axis, the magnitude of current $I$ (in Amperes, rounded off to two decimal places) in …

​​​If ' $\rightarrow$ ' denotes increasing order of intensity, then the meaning of the words [ charm $\rightarrow$ enamor $\rightarrow$ bewitch] is analogous to [bored $\rightarrow \_\_\_\_\_\_ \longrightarrow$ weary] Which one of the given options is appropriate to fill the blank?

​​​​$\text{P, Q, R, S}$, and $\text{T}$ have launched a new startup. Two of them are siblings. The office of the startup has just three rooms. All of them agree that the siblings should not share the same room. If $\text{S}$ and $\text{Q}$ are single children, and the room allocations shown below are acceptable to all. …

​​​​Five years ago, the ratio of Aman's age to his father's age was $1: 4$, and five years from now, the ratio will be $2:5$. What was his father's age when Aman was born?

​​​​For a real number $x>1$, $$\frac{1}{\log _{2} x}+\frac{1}{\log _{3} x}+\frac{1}{\log _{4} x}=1$$ The value of $x$ is

The greatest prime factor of $\left(3^{199}-3^{196}\right)$ is

​Sequence the following sentences $(\mathrm{P}, \mathrm{Q}, \mathrm{R}, \mathrm{S})$ in a coherent passage: $\text{P}$: Shifu's student exclaimed, "Why do you run since the bull is an illusion?" $\text{Q}$: Shifu said, "Surely my running away from the bull is also an illusion." $\mathrm{R}$ : Shifu once proclaimed that all life is illusion. …

​​​​​Four identical cylindrical chalk-sticks, each of radius $r=0.5 \mathrm{~cm}$ and length $l=10 \mathrm{~cm}$, are bound tightly together using a duct tape as shown in the following figure. The width of the duct tape is equal to the length of the chalk-stick. The area (in $\mathrm{cm}^{2}$ ) of the duct tape required to wrap the bundle of …

The bar chart shows the data for the percentage of population falling into different categories based on Body Mass Index $\text{(BMI)}$ in $2003$ and $2023$. Based on the data provided, which one of the following options is INCORRECT?

Examples of mirror and water reflections are shown in the figures below: An object appears as the following image after first reflecting in a mirror and then reflecting on water. The original object is

​​​Two identical sheets $\text{A}$ and $\text{B}$, of dimensions $24 \mathrm{~cm} \times 16 \mathrm{~cm}$, can be folded into half using two distinct operations, $\text{FO1}$ or $\text{FO2}$. In $\text{FO1}$, the axis of folding remains parallel to the initial long edge, and in $\text{FO2}$, the axis of folding remains parallel to the initial …