NTA JEE Main 9th January 2019 Shift 1

The maximum volume in cu.m of the right circular cone having slant height 3 m is:

If $$\theta$$ denotes the acute angle between the curves, $$y = 10 - x^2$$ and $$y = 2 + x^2$$ at a point of their intersection, then $$\tan\theta$$ is equal to:

For $$x^2 \neq n\pi + 1$$, $$n \in N$$ (the set of natural numbers), the integral $$\int x\sqrt{\frac{2\sin(x^2-1) - \sin 2(x^2-1)}{2\sin(x^2-1) + \sin 2(x^2-1)}} \; dx$$ is equal to (where c is a constant of integration):

The value of $$\int_0^{\pi} \cos x^3 \; dx$$ is:

The area (in sq. units) bounded by the parabola $$y = x^2 - 1$$, the tangent at the point (2, 3) to it and the $$y$$-axis is:

If $$y = y(x)$$ is the solution of the differential equation, $$x\frac{dy}{dx} + 2y = x^2$$ satisfying $$y(1) = 1$$, then $$y\left(\frac{1}{2}\right)$$ is equal to:

Let $$\vec{a} = \hat{i} - \hat{j}$$, $$\vec{b} = \hat{i} + \hat{j} + \hat{k}$$ and $$\vec{c}$$ be a vector such that $$\vec{a} \times \vec{c} + \vec{b} = \vec{0}$$ and $$\vec{a} \cdot \vec{c} = 4$$, then $$|\vec{c}|^2$$ is equal to:

The plane through the intersection of the planes $$x + y + z = 1$$ and $$2x + 3y - z + 4 = 0$$ and parallel to $$y$$-axis also passes through the point:

The equation of the line passing through $$(-4, 3, 1)$$, parallel to the plane $$x + 2y - z - 5 = 0$$ and intersecting the line $$\frac{x+1}{-3} = \frac{y-3}{2} = \frac{z-2}{-1}$$ is:

Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Let X denote the random variable of number of aces obtained in the two drawn cards. Then $$P(X = 1) + P(X = 2)$$ equals: