GATE EE 2022 Question Paper (05-Feb-2022) (Shift 2)

e${}^A$ denotes the exponential of a square matrix A. Suppose λ is an eigenvalue and ν is the corresponding eigen-vector of matrix A.

Consider the following two statements:

Statement 1: e${}^{\lambda }$ is an eigenvalue of e${}^A$.

Statement 2: ν is an eigen-vector of e${}^A$

Which one of the following options is correct?

The frequencies of the stator and rotor currents flowing in a three-phase 8-pole induction motor are 40 Hz and 1 Hz, respectively. The motor speed, in rpm, is _________. (round off to nearest integer)

Let f(x) = ${\int }_0^{\mathrm{x}}{\mathrm{e}}^{\mathrm{t}}(\mathrm{t}-1)(\mathrm{t}-2)\mathrm{dt}$ then f(x) decreases in the interval

A 280 V, separately excited DC motor with armature resistance of 1 Ω and constant field excitation drives a load. The load torque is proportional to the speed. The motor draws a current of 30 A when running at a speed of 1000 rpm. Neglect frictional losses in the motor. The speed, in rpm, at which the motor will run, if an additional resistance of value 10 Ω is connected in series with the armature, is __________. (round off to nearest integer)

The fuel cost functions in rupees/hour for two 600 MW thermal power plants are given by

Plant 1: C₁ = 350 + 6P₁ + 0.004 (P₁)²

Plant 2: C₂ = 450 + aP₂ + 0.003 (P₂)²

where P1 and P2 are power generated by plant 1 and plant 2, respectively, in MW and a is constant. The incremental cost of power (λ) is 8 rupees per MWh. The two thermal power plants together meet the total power demand of 550 MW. The optimal generation of plant 1 and plant 2 in MW, respectively, are