NTA JEE Main 15th April 2018 Shift 1

If $$x^2 + y^2 + \sin y = 4$$, then the value of $$\frac{d^2y}{dx^2}$$ at the point (-2, 0) is:

If a right circular cone having maximum volume, is inscribed in a sphere of radius 3 cm, then the curved surface area (in cm$$^2$$) of this cone is:

If $$f\left(\frac{x-4}{x+2}\right) = 2x + 1$$, $$(x \in R - \{1, -2\})$$, then $$\int f(x)dx$$ is equal to (where C is a constant of integration):

The value of the integral $$\int_{-\pi/2}^{\pi/2} \sin^4 x\left(1 + \log\left(\frac{2+\sin x}{2-\sin x}\right)\right)dx$$ is:

The area (in sq. units) of the region $$\{x \in R : x \geq 0, y \geq 0, y \geq x - 2$$ and $$y \leq \sqrt{x}\}$$, is:

Let $$y = y(x)$$ be the solution of the differential equation $$\frac{dy}{dx} + 2y = f(x)$$, where $$f(x) = \begin{cases} 1, & x \in [0, 1] \\ 0, & \text{otherwise} \end{cases}$$. If $$y(0) = 0$$, then $$y\left(\frac{3}{2}\right)$$ is:

If $$\vec{a}$$, $$\vec{b}$$, and $$\vec{c}$$ are unit vectors such that $$\vec{a} + 2\vec{b} + 2\vec{c} = \vec{0}$$, then $$|\vec{a} \times \vec{c}|$$ is equal to:

A variable plane passes through a fixed point (3, 2, 1) and meets x, y and z axes at A, B and C respectively. A plane is drawn parallel to yz-plane through A, a second plane is drawn parallel zx plane through B and a third plane is drawn parallel to xy-plane through C. Then the locus of the point of intersection of these three planes, is:

An angle between the plane, $$x + y + z = 5$$ and the line of intersection of the planes, $$3x + 4y + z - 1 = 0$$ and $$5x + 8y + 2z + 14 = 0$$, is:

A box 'A' contains 2 white, 3 red and 2 black balls. Another box 'B' contains 4 white, 2 red and 3 black balls. If two balls are drawn at random, without replacement, from a randomly selected box and one ball turns out to be white while the other ball turns out to be red, then the probability that both balls are drawn from box 'B' is: