NTA JEE Main 12th January 2019 Shift 2

If the function f given by $$f(x) = x^3 - 3(a-2)x^2 + 3ax + 7$$, for some $$a \in R$$ is increasing in (0, 1] and decreasing in [1, 5), then a root of the equation, $$\frac{f(x) - 14}{(x-1)^2} = 0$$, $$(x \neq 1)$$ is:

The integral $$\int \frac{3x^{13} + 2x^{11}}{(2x^4 + 3x^2 + 1)^4} dx$$, is equal to

The integral $$\int_1^e \left\{\left(\frac{x}{e}\right)^{2x} - \left(\frac{e}{x}\right)^x\right\} \log_e x \, dx$$ is equal to

$$\lim_{n \to \infty} \left(\frac{n}{n^2 + 1^2} + \frac{n}{n^2 + 2^2} + \frac{n}{n^2 + 3^2} + \ldots + \frac{1}{5n^2}\right)$$ is equal to

If a curve passes through the point (1, -2) and has slope of the tangent at any point (x, y) on it as $$\frac{x^2 - 2y}{x}$$, then the curve also passes through the point

Let $$\vec{a}$$, $$\vec{b}$$ and $$\vec{c}$$ be three unit vectors, out of which vectors $$\vec{b}$$ and $$\vec{c}$$ are non-parallel. If $$\alpha$$ and $$\beta$$ are the angles which vector $$\vec{a}$$ makes with vectors $$\vec{b}$$ and $$\vec{c}$$ respectively and $$\vec{a} \times (\vec{b} \times \vec{c}) = \frac{1}{2}\vec{b}$$, then $$|\alpha - \beta|$$ is equal to:

If an angle between the line, $$\frac{x+1}{2} = \frac{y-2}{1} = \frac{z-3}{-2}$$ and the plane, $$x - 2y - kz = 3$$ is $$\cos^{-1}\left(\frac{2\sqrt{2}}{3}\right)$$, then a value of k is

Let S be the set of all real values of $$\lambda$$ such that a plane passing through the points $$(-\lambda^2, 1, 1)$$, $$(1, -\lambda^2, 1)$$ and $$(1, 1, -\lambda^2)$$ also passes through the point $$(-1, -1, 1)$$. Then S is equal to:

In a class of 60 students, 40 opted for NCC, 30 opted for NSS and 20 opted for both NCC and NSS. If one of these students is selected at random, then the probability that the student selected has opted neither for NCC nor for NSS is:

In a game, a man wins Rs. 100 if he gets 5 or 6 on a throw of a fair die and loses Rs. 50 for getting any other number on the die. If he decides to throw the die either till he gets a five or a six or to a maximum of three throws, then his expected gain/loss (in rupees) is: