NTA JEE Main 2nd September 2020 Shift 2

For a positive integer $$n$$, $$\left(1 + \frac{1}{x}\right)^n$$ is expanded in increasing powers of $$x$$. If three consecutive coefficients in this expansion are in the ratio, 2 : 5 : 12, then $$n$$ is equal to ___________.

If the variance of the terms in an increasing A.P. $$b_1, b_2, b_3, \ldots, b_{11}$$ is 90 then the common difference of this A.P. is ___________.

If $$y = \sum_{k=1}^{6} k\cos^{-1}\left\{\frac{3}{5}\cos kx - \frac{4}{5}\sin kx\right\}$$ then $$\frac{dy}{dx}$$ at $$x = 0$$ is ___________.

Let $$[t]$$ denote the greatest integer less than or equal to $$t$$. Then the value of $$\int_1^2 |2x - [3x]| \; dx$$ is ___________.

Let the position vectors of points 'A' and 'B' be $$\hat{i} + \hat{j} + \hat{k}$$ and $$2\hat{i} + \hat{j} + 3\hat{k}$$, respectively. A point 'P' divides the line segment AB internally in the ratio $$\lambda : 1$$ $$(\lambda > 0)$$. If O is the origin and $$\vec{OB} \cdot \vec{OP} - 3|\vec{OA} \times \vec{OP}|^2 = 6$$ then $$\lambda$$ is equal to ___________.