NTA JEE Main 9th April 2013 Online

Statement-1: The equation $$x\log x = 2 - x$$ is satisfied by at least one value of $$x$$ lying between 1 and 2.
Statement-2: The function $$f(x) = x\log x$$ is an increasing function in $$[1, 2]$$ and $$g(x)=2-x$$ is a decreasing function in $$[1, 2]$$ and the graphs represented by these functions intersect at a point in $$[1, 2]$$.

If the surface area of a sphere of radius $$r$$ is increasing uniformly at the rate 8 cm$$^2$$/s, then the rate of change of its volume is:

If $$\int \frac{dx}{x+x^7} = p(x)$$ then, $$\int \frac{x^6}{x+x^7}dx$$ is equal to:

If $$x = \int_0^y \frac{dt}{\sqrt{1+t^2}}$$, then $$\frac{d^2y}{dx^2}$$ is equal to :

The area bounded by the curve $$y = \ln(x)$$ and the lines $$y = 0$$, $$y = \ln(3)$$ and $$x = 0$$ is equal to:

Let $$\vec{a} = 2\hat{i} - \hat{j} + \hat{k}$$, $$\vec{b} = \hat{i} + 2\hat{j} - \hat{k}$$ and $$\vec{c} = \hat{i} + \hat{j} - 2\hat{k}$$ be three vectors. A vector of the type $$\vec{b} + \lambda\vec{c}$$ for some scalar $$\lambda$$, whose projection on $$\vec{a}$$ is of magnitude $$\sqrt{\frac{2}{3}}$$ is :

The vector $$(\hat{i} \times \vec{a} \cdot \vec{b})\hat{i} + (\hat{j} \times \vec{a} \cdot \vec{b})\hat{j} + (\hat{k} \times \vec{a} \cdot \vec{b})\hat{k}$$ is equal to:

A vector $$\vec{n}$$ is inclined to the $$x$$-axis at 45°, to the $$y$$-axis at 60° and at an acute angle to the $$z$$-axis. If $$\vec{n}$$ is a normal to a plane passing through the point $$(\sqrt{2}, -1, 1)$$ then the equation of the plane is :

If the lines $$\frac{x+1}{2} = \frac{y-1}{1} = \frac{z+1}{3}$$ and $$\frac{x+2}{2} = \frac{y-k}{3} = \frac{z}{4}$$ are coplanar, then the value of $$k$$ is :

The probability of a man hitting a target is $$\frac{2}{5}$$. He fires at the target $$k$$ times ($$k$$, a given number). Then the minimum $$k$$, so that the probability of hitting the target at least once is more than $$\frac{7}{10}$$, is :