NTA JEE Main 10th April 2019 Shift 2

If the tangent to the curve $$y = \frac{x}{x^2 - 3}$$, $$x \in R$$, $$x \neq \pm\sqrt{3}$$, at a point $$(\alpha, \beta) \neq (0, 0)$$ on it is parallel to the line $$2x + 6y - 11 = 0$$, then:

A spherical iron ball of radius 10 cm is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm$$^3$$/min. When the thickness of the ice is 5 cm, then the rate at which the thickness (in cm/min) of the ice decreases, is:

If $$\int x^5 e^{-x^2} dx = g(x)e^{-x^2} + c$$, where c is a constant of integration, then g(-1) is equal to

The integral $$\int_{\pi/3}^{\pi/3} \sec^{2/3}x \cdot \text{cosec}^{4/3}x \, dx$$ is equal to

The area (in sq. units) of the region bounded by the curves $$y = 2^x$$ and $$y = x + 1$$, in the first quadrant is

Let $$y = yx$$ be the solution of the differential equation, $$\frac{dy}{dx} + y\tan x = 2x + x^2\tan x$$, $$x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$, such that $$y(0) = 1$$. Then

The distance of the point having position vector $$-\hat{i} + 2\hat{j} + 6\hat{k}$$ from the straight line passing through the point (2, 3, -4) and parallel to the vector, $$6\hat{i} + 3\hat{j} - 4\hat{k}$$ is

If the plane $$2x - y + 2z + 3 = 0$$ has the distances $$\frac{1}{3}$$ and $$\frac{2}{3}$$ units from the planes $$4x - 2y + 4z + \lambda = 0$$ and $$2x - y + 2z + \mu = 0$$, respectively, then the maximum value of $$\lambda + \mu$$ is equal to:

A perpendicular is drawn from a point on the line $$\frac{x-1}{2} = \frac{y+1}{-1} = \frac{z}{1}$$ to the plane $$x + y + z = 3$$ such that the foot of the perpendicular Q also lies on the plane $$x - y + z = 3$$. Then the coordinates of Q are

Minimum number of times a fair coin must be tossed so that the probability of getting at least one head is more than 99% is: