JEE (Advanced) 2016 Paper-1

Let m be the smallest positive integer such that the coefficient of $$x^2$$ in the expansion of $$(1 + x)^2 + (1 + x)^3 + ....... + (1 + x)^{49} + (1 + mx)^{50}$$ is $$(3n + 1)^{51}C_3$$ for some positive integer n. Then the value of n is

The total number of distinct $$x \in [0, 1]$$ for which $$\int_{0}^{x} \frac{t^2}{1 + t^4}dt = 2x - 1$$ is

Let $$\alpha, \beta \in R$$ be such that $$ \lim_{x \rightarrow 0}\frac{x^2 \sin (\beta x)}{\alpha x - \sin x} = 1$$. Then $$6(\alpha + \beta)$$ equals

Let $$z = \frac{-1 + \sqrt{3}i}{2}$$, where $$i = \sqrt{-1}$$, and $$r, s \in \left\{1, 2, 3\right\}$$. Let $$P = \begin{bmatrix}(-z)^r & z^{2s} \\z^{2s} & z^r \end{bmatrix}$$ and I be the identity matrix of order 2. Then the total number of ordered pairs $$(r, s)$$ for which $$P^2 = -I$$ is