NTA JEE Main 2nd April 2017 Offline

The normal to the curve $$y(x-2)(x-3) = x + 6$$ at the point where the curve intersects the $$y$$-axis passes through the point:

Let $$I_{n} = \int \tan^{n}x \, dx$$ ($$n > 1$$). If $$I_{4} + I_{6} = a\tan^{5}x + bx^{5} + c$$, then the ordered pair $$(a, b)$$, is equal to

The integral $$\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \frac{dx}{1 + \cos x}$$ is equal to

The area (in sq. units) of the region $$\{(x, y) : x \geq 0, x + y \leq 3, x^{2} \leq 4y \text{ and } y \leq 1 + \sqrt{x}\}$$ is

If $$(2 + \sin x)\frac{dy}{dx} + (y+1)\cos x = 0$$ and $$y(0) = 1$$, then $$y\left(\frac{\pi}{2}\right)$$ is equal to

Given, $$\vec{a} = 2\hat{i} + \hat{j} - 2\hat{k}$$ and $$\vec{b} = \hat{i} + \hat{j}$$. Let $$\vec{c}$$ be a vector such that $$|\vec{c} - \vec{a}| = 3$$, $$|\vec{a} \times \vec{b} \times \vec{c}| = 3$$ and the angle between $$\vec{c}$$ and $$\vec{a} \times \vec{b}$$ be 30°. Then $$\vec{a} \cdot \vec{c}$$ is equal to:

If the image of the point $$P(1, -2, 3)$$ in the plane, $$2x + 3y - 4z + 22 = 0$$ measured parallel to the line, $$\frac{x}{1} = \frac{y}{4} = \frac{z}{5}$$ is $$Q$$, then $$PQ$$ is equal to:

The distance of the point $$(1, 3, -7)$$ from the plane passing through the point $$(1, -1, -1)$$, having normal perpendicular to both the lines $$\frac{x-1}{1} = \frac{y+2}{-2} = \frac{z-4}{3}$$ and $$\frac{x-2}{2} = \frac{y+1}{-1} = \frac{z+7}{-1}$$, is:

For three events, $$A$$, $$B$$ and $$C$$, $$P$$(Exactly one of $$A$$ or $$B$$ occurs) = $$P$$(Exactly one of $$B$$ or $$C$$ occurs) = $$P$$(Exactly one of $$C$$ or $$A$$ occurs) = $$\frac{1}{4}$$ and $$P$$(All the three events occur simultaneously) = $$\frac{1}{16}$$. Then the probability that at least one of the events occurs, is:

If two different numbers are taken from the set $$\{0, 1, 2, 3, \ldots, 10\}$$; then the probability that their sum as well as absolute difference are both multiple of 4, is: