CSIR NET Maths previous year question papers

$\lim(n\to\infty) \int_{0}^{1}\int_{0}^{1}\cdots\int_{0}^{1} (x_{1}^{2} + \cdots + x_{n}^{2})/(x_{1} + \cdots + x_{n})$ dx$_{1} \dots$ dx$_{n}$ equals

In an examination question paper, all questions are 'True' or 'False' type. These are arranged in such a way that three-fourth of times a question with answer 'True' is followed by a question with answer 'True'. Also two-third of times a question with answer 'False' is followed by a question with answer 'False'. If the question paper has 100 questions, the approximate probability that the correct answer of the 100-th question is 'True', is