Numerical Method

Using Newton-Raphson method, a root correct to 3 decimal places of x^{3}-3 x-5=0

What is the sum to infinity of the series,
3+6 x^{2}+9 x^{4}+12 x^{6}+\ldots \text { given }|x| \lt 1

The velocity v (in kilometer/minute) of a motorbike which starts from rest, is given at fixed intervals of time t (in minutes) as follows:

The approximate distance (in kilometers) rounded to two places of decimals covered in 20 minutes using Simpson's 1/3^{rd} rule is _______________.

The secant method is used to find the root of an equation f(x)=0. It is started from two distinct estimates x_{a} and x_{b} for the root. It is an iterative procedure involving linear interpolation to a root. The iteration stops if f(x_{b}) is very small and then x_{b} is the solution. The procedure is given below. Observe that there is an expression which is missing and is marked by ?. Which is the suitable expression that is to be put in place of ? so that it follows all steps of the secant method?

With respect to the numerical evaluation of the definite integral,K=\int_{a}^{b}x^{2}dx , where a and b are given, which of the following statements is/are TRUE?
(I) The value of K obtained using the trapezoidal rule is always greater than or equal to the exact value of the definite integral.
(II) The value of K obtained using the Simpson's rule is always equal to the exact value of the definite integral.