NTA JEE Main 25th February 2021 Shift 2

$$\operatorname{cosec}\left[2\cot^{-1}(5) + \cos^{-1}\left(\frac{4}{5}\right)\right]$$ is equal to:

A function $$f(x)$$ is given by $$f(x) = \frac{5^x}{5^x + 5}$$, then the sum of the series $$f\left(\frac{1}{20}\right) + f\left(\frac{2}{20}\right) + f\left(\frac{3}{20}\right) + \ldots + f\left(\frac{39}{20}\right)$$ is equal to:

Let $$x$$ denote the total number of one-one functions from a set $$A$$ with 3 elements to a set $$B$$ with 5 elements and $$y$$ denote the total number of one-one functions from the set $$A$$ to the set $$A \times B$$. Then:

The shortest distance between the line $$x - y = 1$$ and the curve $$x^2 = 2y$$ is:

The integral $$\int \frac{e^{3\log_e 2x} + 5e^{2\log_e 2x}}{e^{4\log_e x} + 5e^{3\log_e x} - 7e^{2\log_e x}} dx$$, $$x > 0$$, is equal to (where $$c$$ is a constant of integration)

If $$I_n = \int_{\pi/4}^{\pi/2} \cot^n x \, dx$$, then

$$\lim_{n \to \infty} \left[\frac{1}{n} + \frac{n}{(n+1)^2} + \frac{n}{(n+2)^2} + \ldots + \frac{n}{(2n-1)^2}\right]$$ is equal to

A plane passes through the points $$A(1, 2, 3)$$, $$B(2, 3, 1)$$ and $$C(2, 4, 2)$$. If $$O$$ is the origin and $$P$$ is $$(2, -1, 1)$$, then the projection of $$\vec{OP}$$ on this plane is of length:

In a group of 400 people, 160 are smokers and non-vegetarian; 100 are smokers and vegetarian and the remaining 140 are non-smokers and vegetarian. Their chances of getting a particular chest disorder are 35%, 20% and 10% respectively. A person is chosen from the group at random and is found to be suffering from the chest disorder. The probability that the selected person is a smoker and non-vegetarian is:

Let $$A$$ be a set of all 4-digit natural numbers whose exactly one digit is 7. Then the probability that a randomly chosen element of $$A$$ leaves remainder 2 when divided by 5 is: