NTA JEE Main 11th January 2019 Shift 1

If $$x \log_e(\log_e x) - x^2 + y^2 = 4$$ $$(y > 0)$$, then $$\frac{dy}{dx}$$ at $$x = e$$ is equal to:

The maximum value of the function $$f(x) = 3x^3 - 18x^2 + 27x - 40$$ on the set $$S = \{x \in R : x^2 + 30 \le 11x\}$$ is:

If $$\int \frac{\sqrt{1-x^2}}{x^4} dx = A(x)\left(\sqrt{1-x^2}\right)^m + C$$, for a suitable chosen integer m and a function $$A(x)$$, where C is a constant of integration, then $$(A(x))^m$$ equals:

The value of the integral $$\int_{-2}^{2} \frac{\sin^2 x}{[\frac{x}{\pi}] + \frac{1}{2}} dx$$ (where $$[x]$$ denotes the greatest integer less than or equal to x) is

The area (in sq. units) of the region bounded by the curve $$x^2 = 4y$$ and the straight line $$x = 4y - 2$$ is:

If $$y(x)$$ is the solution of the differential equation $$\frac{dy}{dx} + \left(\frac{2x+1}{x}\right)y = e^{-2x}$$, $$x \gt 0$$, where $$y(1) = \frac{1}{2}e^{-2}$$, then:

Let $$\vec{a} = \hat{i} + 2\hat{j} + 4\hat{k}$$, $$\vec{b} = \hat{i} + \lambda\hat{j} + 4\hat{k}$$ and $$\vec{c} = 2\hat{i} + 4\hat{j} + (\lambda^2 - 1)\hat{k}$$ be coplanar vectors. Then the non-zero vector $$\vec{a} \times \vec{c}$$ is:

The plane containing the line $$\frac{x-3}{2} = \frac{y+2}{-1} = \frac{z-1}{3}$$ and also containing its projection on the plane $$2x + 3y - z = 5$$, contains which one of the following points?

The direction ratios of normal to the plane through the points (0,-1,0) and (0,0,1) and making an angle $$\frac{\pi}{4}$$ with the plane $$y - z + 5 = 0$$ are: 2,-1,1; $$2, \sqrt{2} - \sqrt{2}$$; $$\sqrt{2}, 1, -1$$; $$2\sqrt{3}, 1, -1$$

Two integers are selected at random from the set $$\{1, 2, \ldots, 11\}$$. Given that the sum of selected numbers is even, the conditional probability that both the numbers are even is: