The quadratic equation $${x^2} - 4x + 4 = 0$$ is to be solved numerically, starting with the initial guess $${x_0} = 3.$$ The Newton- Raphson method is applied once to get a new estimate and then the Secant method is applied once using in the initial guess and this new estimate. The estimated value of the root after the application of the Secant method is ________.

The integral $$\,\int_{{x_1}}^{{x_2}} {{x^2}dx\,\,} $$ with $${x_2} > {x_1} > 0$$ is evaluated analytically as well as numerically using a single application of the trapezoidal rule. If $${\rm I}$$ is the exact value of the integral obtained analytically and $$J$$ is the approximate value obtained using the trapezoidal rule, which of the following statements is correct about their relationship?

Consider the following complex function $$f\left( z \right) = {9 \over {\left( {z - 1} \right)\left( {z + 2} \right)}}.$$ Which of the following is ONE of the residues of the above function ?

In a catchment, there are four rain-gauge stations, P, Q, R, and S. Normal annual precipitation values at these stations are 780 mm, 850 mm, 920 mm, and 980 mm, respectively. In the year 2013, stations Q, R, and S, were operative but P was not. Using the normal ratio method, the precipitation at station P for the year 2013 has been estimated as 860 mm. If the observed precipitation at stations Q and R for the year 2013 were 930 mm and 1010 mm, respectively; what was the observed precipitation (in mm) at station S for that year?