GATE EE 2017 Question Paper (11-Feb-2017) (Shift 1)

A function f (x) is defined as f≤ft( x ) = ≤ft\{ {*20c e^x,&x < 1\\ x + ax^2+bx,&x ≥ 1 } ., where x ϵ R. Which one of the following statements is TRUE?

Consider the differential equation $\left(t^2-81\right)\frac{dy}{dt}+5ty=\sin \left(t\right)$ with y(1) = 2π. There exists a unique solution for this differential equation when t belongs to the interval

Consider the line integral $I=\underset{C}{\int }\left(x^2+i{y}^2\right)dz$, where z = x + iy. The line C is shown in figure below.



The value of I is

Two passive two-port networks are connected in cascade as shown in figure. A voltage source is connected at port 1.



Given V₁ = A₁V₂ + B₁I₂

I₁ = C₁V₂ + D₁I₂

V₂ = A₂V₃ + B₂I₃

I₂ = C₂V₃ + D₂I₃

A₁, B₁, C₁, D₁, A₂, B₂, C₂, and D₂ are the generalized circuit constants. If the Thevenin equivalent circuit at port 3 consists of a voltage source V${}_T$ and an impedance Z${}_T$ connected in series, then

Let a causal LTI system be characterized by the following differential equation, with initial rest condition

$\frac{d^{2}y}{d{t}^2}+7\frac{dy}{dt}+10y\left(t\right)=4x\left(t\right)+5\frac{dx\left(t\right)}{dt}$

where x(t) and y(t) are the input and output respectively. The impulse response of the system is (u(t) is the unit step function)