GATE EC 2023 Question Paper (05-Feb-2023) (Shift 2)

A random variable X, distributed normally as 𝑁(0, 1), undergoes the transformation Y = h(X), given in the figure. The form of the probability density function of 𝑌 is

(In the options given below, 𝑎, 𝑏, 𝑐 are non-zero constants and 𝑔(𝑦) is piece-wise continuous function)

The value of the line integral ${\int }_{\mathrm{P}}^{\mathrm{Q}}({\mathrm{z}}^2\mathrm{dx}+3{\mathrm{y}}^2\mathrm{dy}+2xzdz)$ along the straight line joining the points 𝑃 (1, 1, 2) and 𝑄 (2, 3, 1) is

Let 𝒙 be an 𝑛 × 1 real column vector with length $\mathrm{l}=\sqrt{{\mathrm{x}}^{\mathrm{T}}\mathrm{x}}$. The trace of the matrix 𝑃 = 𝒙𝒙${}^{\text{T}}$ is

The $\frac{{\mathrm{V}}_{\mathrm{OUT}}}{{\mathrm{V}}_{\mathrm{IN}}}$ of the circuit shown below is

In the circuit shown below, D₁ and D₂ are silicon diodes with cut-in voltage of 0.7 V. V${}_{IN}$ and V${}_{OUT}$ are input and output voltages in volts. The transfer characteristic is