NTA JEE Main 10th April 2015 Online

If Rolle's theorem holds for the function $$f(x) = 2x^3 + bx^2 + cx$$, $$x \in [-1, 1]$$ at the point $$x = \frac{1}{2}$$, then $$2b + c$$ is equal to

The distance from the origin, of the normal to the curve, $$x = 2\cos t + 2t\sin t$$, $$y = 2\sin t - 2t\cos t$$ at $$t = \frac{\pi}{4}$$, is:

The integral $$\int \frac{dx}{(x+1)^{3/4}(x-2)^{5/4}}$$, is equal to

For $$x > 0$$, let $$f(x) = \int_1^x \frac{\log t}{1-t} dt$$. Then $$f(x) + f\left(\frac{1}{x}\right)$$ is equal to

The area (in square units) of the region bounded by the curves $$y + 2x^2 = 0$$ and $$y + 3x^2 = 1$$, is equal to

If $$y(x)$$ is the solution of the differential equation $$(x + 2)\frac{dy}{dx} = x^2 + 4x - 9$$, $$x \neq -2$$ and $$y(0) = 0$$, then $$y(-4)$$ is equal to

Let $$\vec{a}$$ and $$\vec{b}$$ be two unit vectors such that $$|\vec{a} + \vec{b}| = \sqrt{3}$$. If $$\vec{c} = \vec{a} + 2\vec{b} + (\vec{a} \times \vec{b})$$, then $$2|\vec{c}|$$ is equal to:

If the points $$(1, 1, \lambda)$$ and $$(-3, 0, 1)$$, are equidistant from the plane, $$3x + 4y - 12z + 13 = 0$$, then $$\lambda$$ satisfies the equation:

If the shortest distance between the line $$\frac{x-1}{\alpha} = \frac{y+1}{-1} = \frac{z}{1}$$, $$(\alpha \neq -1)$$, and $$x + y + z + 1 = 0 = 2x - y + z + 3$$ is $$\frac{1}{\sqrt{3}}$$, then value of $$\alpha$$ is:

Let X be a set containing 10 elements and P(X) be its power set. If A and B are picked up at random from P(X), with replacement, then the probability that A and B have equal number of elements is: