$$\,\,P\,\,\,\left( {0,3} \right),\,\,Q\,\,\,\left( {0.5,4} \right),\,\,$$ and $$\,\,R\,\,\,\left( {1,5} \right)\,\,\,$$ are three points on the curve defined by $$\,\,f\left( x \right),\,\,$$ Numerical integration is carried out using both Trapezoidal rule and Simpson's rule within limits $$x=0$$ and $$x=1$$ for the curve. The difference between the two results will be

Solve the equation $$x = 10\,\cos \,\left( x \right)$$ using the Newton-Raphson method. The initial guess is $$x = {\pi \over 4}.$$ The value of the predicted root after the first iteration, up to second decimal, is _____________.

Gauss-Seidel method is used to solve the following equations (as per the given order). $$${x_1} + 2{x_2} + 3{x_3} = 5$$$ $$$2{x_1} + 3{x_2} + {x_3} = 1$$$ $$$\,3{x_1} + 2{x_2} + {x_3} = 3$$$
Assuming initial guess as $${x_1} = {x_2} = {x_3} = 0,$$ the value of $${x_3}$$ after the first iteration is __________.

The error in numerically computing the integral $$\,\int\limits_0^\pi {\left( {\sin \,x + \cos \,x} \right)dx\,\,\,} $$ using the trapezoidal rule with three intervals of equal length between $$0$$ and $$\pi $$ is _________.