GATE IN 2024 Question Paper (04-Feb-2024) (Shift 2)

The current $i(t)$ drawn by a circuit is given as
$i(t)=4+30\cos (t)\text{–}20\sin (t)+15\cos (3t)\text{–}10\sin (3t)\text{ A}$
The root-mean-square value of $i(t)$ is ________ A (rounded off to one decimal place).

A linear potentiometer ($0-10\text{ k}\Omega$) is used to measure the water level as shown in the figure. The resistance between A and C varies linearly from $0$ to $10\text{ k}\Omega$ for a change in water level from $0$ to $20\text{ cm}$. The sensor is excited using a DC voltage source, $V_s=10\text{ V}$ with an internal resistance, $R_s=200\Omega$. If $V_{out}=5\text{ V}$, the water level is ________ cm (rounded off to one decimal place).

The switch in the following figure has been closed for a long time ($t<0$). It is opened at $t=0$ seconds. The value of $d{v}_c/dt$ at $t=0^{+}$ is ________ V/s (rounded off to nearest integer).

Consider a system given by the following first order differential equation:
$\frac{dy}{dt}=y+2t-t^2$
where, $y(0)=1$ and $0\le t<\infty$. Using a step size $h=0.1$ for the improved Euler method, the value of $y(t)$ at $t=0.1$ is ________ (rounded off to two decimal places).

Indian Premier League has divided the sixteen cricket teams into two equal pools: Pool-A and Pool-B. Four teams of Pool-A have blue logo jerseys while the rest four have red logo jerseys. Five teams of Pool-B have blue logo jerseys while the rest three have red logo jerseys.
If one team from each pool reaches the final, the probability that one team has a blue logo jersey and another has a red logo jersey is ________ (rounded off to one decimal place).