NTA JEE Main 5th September 2020 Shift 1

The number of words, with or without meaning, that can be formed by taking 4 letters at a time from the letters of the word 'SYLLABUS' such that two letters are distinct and two letters are alike, is

The natural number $$m$$, for which the coefficient of $$x$$ in the binomial expansion of $$\left(x^m + \frac{1}{x^2}\right)^{22}$$ is 1540, is

If the line, $$2x - y + 3 = 0$$ is at a distance $$\frac{1}{\sqrt{5}}$$ and $$\frac{2}{\sqrt{5}}$$ from the lines $$4x - 2y + \alpha = 0$$ and $$6x - 3y + \beta = 0$$ respectively, then the sum of all possible values of $$\alpha$$ and $$\beta$$ is __________.

Let $$f(x) = x \cdot \left[\frac{x}{2}\right]$$, for $$-10 < x < 10$$, where $$[t]$$ denotes the greatest integer function. Then the number of points of discontinuity of $$f(x)$$ is equal to

Four fair dice are thrown independently 27 times. Then the expected number of times, at least two dice show up a three or a five, is