Shortcut to find an answer to such question in an exam.
Newton-Raphson method is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function. It is one example of a root-finding algorithm.
Here f(x) = x3-3x-5
Just put the options one by one in f(x) and check whether we are getting "0.000..."(correct up to 3 decimal places) as the answer.
op A: x= 2.222 ie. f(x) = (2.222)3-3(2.222)-5 = -0.695354... // op A is not correct
op B: x= 2.275 ie. f(x) = (2.275)3-3(2.275)-5 = -0.050453... // op B is not correct
op C: x= 2.279 ie. f(x) = (2.279)3-3(2.279)-5 = -(0.000)236361... // op C is correct root of given equation (upto 3 decimal places..
So Op C is the answer.