GATE EE 2019 Question Paper (09-Feb-2019) (Shift 2)

Consider a 2 × 2 matrix M = ≤ft[ {*20c v_1&v_2 } ], where, v₁ and v₂ are the column vectors. Suppose M^ - 1 = ≤ft[ {*20c u_1^T\\ u_2^T } ], where u${}^T$ ₁ and u${}^T$ ₂ are the row vectors. Consider the following statements.

Statement: u${}^T$ ₁v₁ = 1 and u${}^T$ ₂v₂ = 1

Statement: u${}^T$ ₁v₂ = 0 and u${}^T$ ₂v₁ = 0

Which of the following options is correct?

The closed loop line integral _≤ft| z | = 5^ {z^3 + z^2 + 8}z + 2dz evaluated counter-clockwise, is

A periodic function f(t), with a period of 2π, is represented as its Fourier series,

$f\left(t\right)=a_0+{\mathrm{\Sigma }}_{n=1}^{\infty }{a}_n\cos nt+{\mathrm{\Sigma }}_{n=1}^{\infty }{b}_n\sin nt$

If

f≤ft( t ) = \{ *20c At,&0 ≤ t ≤ π \\ 0,&π < t < 2π ,

the Fourier series coefficients a₁ and b₁ of f(t) are

The asymptotic Bode magnitude plot of a minimum phase transfer function G(s) is shown below.



Consider the following two statements.

Statement I: Transfer function G(s) has three poles and one zero.

Statement II: At very high frequency (ω→∞), the phase angle $\angle G\left(j\omega \right)=-\frac{3\pi }{2}$.

Which one of the following options is correct?

The transfer function of a phase lead compensator is given by

$D\left(s\right)=\frac{3\left(s+\frac{1}{3T}\right)}{\left(s+\frac{1}{T}\right)}$

The frequency (in rad/sec), at which ∠D(jω) is maximum, is