In the following differential equation, the numerically obtained value of , at , is __________ (Round off to 2 decimal places).

The per-unit power output of a salient-pole generator which is connected to an infinite bus, is given by the expression, where d is the load angle. Newton-Raphson method is used to calculate the value of for. If the initial guess is 30°, then its value (in degree) at the end of the first iteration is

Only one of the real roots of  lies in the interval  and bisection method is used to find its value. For achieving an accuracy of 0.001, the required minimum number of iterations is ___________ .        

Let  and . The value of  equals _________. (Give the answer up to three decimal places)

Let x and y be integers satisfying the following equations

The value of (x + y) is ________ .

The function  is to be solved using Newton-Raphson method. If the initial value of  is taken as 1.0, then the absolute error observed at  iteration is ________.        

When the Newton-Raphson method is applied to solve the equation, the solution at the end of the first iteration with the initial guess value as is

Solution of the variables  and  for the following equations is to be obtained by employing the Newton-Raphson iterative method.        

Equation (i)         

Equation (ii)        

Assuming the initial values  and , the Jacobian matrix is

Let .  The iterative steps for the solution using Newton-Raphson's method is given by        

Equation is required to be solved using Newton's method with an initial guess . Then, after one step of Newton’s method, estimate  of the solution will be given by

A differential equation  has to be solved using trapezoidal rule of integration with a step size h = 0.01s. Function u(t) indicates a unit step function.  If , then value of x at t = 0.01s will be given by:

The differential equation is discretized using Euler's numerical integration method with a time step ΔT > 0.  What is the maximum permissible value of ΔT to ensure stability of the solution of the corresponding discrete time equation?

The value of  computed using Simpson’s rule with a step size of h=0.25 is:

A system governed by the differential equation  is initially at rest. Determine the value of y at t=0.1s using the trapezoidal rule of integration with a step size of 0.05s

The number of positive real roots of the equation  is ________________