NTA JEE Main 6th September 2020 Shift 2

The number of words (with or without meaning) that can be formed from all the letters of the word 'LETTER' in which vowels never come together is_____.

Consider the data on x taking the values $$0, 2, 4, 8, \ldots, 2^n$$ with frequencies $$^nC_0, ^nC_1, ^nC_2, \ldots, ^nC_n$$ respectively. If the mean of this data is $$\frac{728}{2^n}$$, then n is equal to_______.

The sum of distinct values of $$\lambda$$ for which the system of equations:
$$(\lambda - 1)x + (3\lambda + 1)y + 2\lambda z = 0$$
$$(\lambda - 1)x + (4\lambda - 2)y + (\lambda + 3)z = 0$$
$$2x + (3\lambda + 1)y + 3(\lambda - 1)z = 0$$
Has non-zero solutions, is_______.

Suppose that a function $$f : \mathbb{R} \to \mathbb{R}$$ satisfies $$f(x+y) = f(x)f(y)$$ for all $$x, y \in \mathbb{R}$$ and $$f(1) = 3$$. If $$\sum_{i=1}^{n} f(i) = 363$$, then $$n$$ is equal to_____.

If $$\vec{x}$$ and $$\vec{y}$$ be two non-zero vectors such that $$|\vec{x} + \vec{y}| = |\vec{x}|$$ and $$2\vec{x} + \lambda\vec{y}$$ is perpendicular to $$\vec{y}$$, then the value of $$\lambda$$ is_____.