NTA JEE Main 10th April 2019 Shift 1

Let $$f(x) = e^x - x$$ and $$g(x) = x^2 - x$$, $$\forall$$ x ∈ R. Then the set of all x ∈ R, where the function $$h(x) = (f \circ g)(x)$$ is increasing, is:

If $$\int \frac{dx}{(x^2 - 2x + 10)^2} = A\left(\tan^{-1}\left(\frac{x-1}{3}\right) + \frac{f(x)}{x^2 - 2x + 10}\right) + C$$, then (where C is a constant of integration)

The value of $$\int_0^{2\pi} [\sin 2x(1 + \cos 3x)]dx$$, where [t] denotes the greatest integer function is

$$\lim_{n \to \infty}\left(\frac{(n+1)^{1/3}}{n^{4/3}} + \frac{(n+2)^{1/3}}{n^{4/3}} + \ldots + \frac{(2n)^{1/3}}{n^{4/3}}\right)$$ is equal to

The region represented by $$|x - y| \leq 2$$ and $$|x + y| \leq 2$$ is bounded by a

If $$y = y(x)$$ is the solution of the differential equation $$\frac{dy}{dx} = (\tan x - y)\sec^2 x$$, $$x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$, such that $$y(0) = 0$$, then $$y\left(-\frac{\pi}{4}\right)$$ is equal to:

Let A(3, 0, -1), B(2, 10, 6) and C(1, 2, 1) be the vertices of a triangle and M be the mid-point of AC. If G divides BM in the ratio, 2 : 1, then $$\cos(\angle GOA)$$ (O being the origin) is equal to

If the length of the perpendicular from the point $$(\beta, 0, \beta)$$, ($$\beta \neq 0$$) to the line, $$\frac{x}{1} = \frac{y-1}{0} = \frac{z+1}{-1}$$ is $$\sqrt{\frac{3}{2}}$$, then $$\beta$$ is equal to

If Q(0, -1, -3) is the image of the point P in the plane $$3x - y + 4z = 2$$ and R is the point (3, -1, -2), then the area (in sq. units) of $$\triangle PQR$$ is

Assume that each born child is equally likely to be a boy or girl. If two families have two children each, then the conditional probability that all children are girls given that at least two are girls is: