NTA JEE Main 3rd September 2020 Shift 1

If $$\left(\frac{1+i}{1-i}\right)^{\frac{m}{2}} = \left(\frac{1+i}{i-1}\right)^{\frac{n}{3}} = 1$$, $$(m, n \in N)$$ then the greatest common divisor of the least values of m and n is

The value of $$0.16^{\log_{2.5}\left(\frac{1}{3} + \frac{1}{3^2} + \frac{1}{3^3} + \ldots \infty\right)}$$ is __________

The diameter of the circle, whose centre lies on the line $$x + y = 2$$ in the first quadrant and which touches both the lines $$x = 3$$ and $$y = 2$$ is __________

If $$\lim_{x \to 0}\left\{\frac{1}{x^8}\left(1 - \cos\frac{x^2}{2} - \cos\frac{x^2}{4} + \cos\frac{x^2}{2}\cos\frac{x^2}{4}\right)\right\} = 2^{-k}$$ then the value of k is

Let $$A = \begin{bmatrix} x & 1 \\ 1 & 0 \end{bmatrix}$$, $$x \in R$$ and $$A^4 = [a_{ij}]$$. If $$a_{11} = 109$$, then $$a_{22}$$ is equal to __________