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

A 50 kVA transformer has an efficiency of 95% at full load and unity power factor. Assume the core losses are negligible. The efficiency of the transformer at 75% of the full load and 0.8 power factor is ______________ % (rounded off to one decimal place).

A three-phase squirrel-cage induction motor has a starting torque of 100% of the full load torque and a maximum torque of 300% of the full load torque. Neglecting the stator impedance, the slip at the maximum torque is ______________ % (rounded off to two decimal places).

Two magnetically coupled coils, when connected in series-aiding configuration, have a total inductance of 500 mH. When connected in series-opposing configuration, the coils have a total inductance of 300 mH. If the self-inductance of both the coils are equal, then the coupling coefficient is ______________ (rounded off to two decimal places).

The solution of an ordinary differential equation $y^{\prime \prime \prime }+3y^{\prime \prime }+3y^{\prime }+y=30e^{-t}$ is
$y(t)=(c_0+c_{1}t-c_2{t}^2+c_3{t}^3)e^{-t}$
Given $y(0)=3$, $y^{\prime }(0)=-3$ and $y^{\prime \prime }(0)=-47$, the value of $(c_0+c_1+c_2+c_3)$ is ______________ (rounded off to nearest integer).
Note: $y^{\prime \prime \prime }=d^{3}y/d{t}^3$, $y^{\prime \prime }=d^{2}y/d{t}^2$, $y^{\prime }=dy/dt$ and $c_0,c_1,c_2,c_3$ are constants.

A random variable $X$ has a probability density function
$f_X(x)=\{\begin{array}{ll}{e}^{-x},& x\ge 0\\ 0,& \text{otherwise}\end{array}$
The probability of $X>2$ is ______________ (rounded off to three decimal places).