NTA JEE Main 19th April 2014 Online

If the volume of a spherical ball is increasing at the rate of $$4\pi$$ cc/sec then the rate of increase of its radius (in cm/sec), when the volume is $$288\pi$$ cc is:

If non-zero real numbers b and c are such that $$\min f(x) > \max g(x)$$, where $$f(x) = x^2 + 2bx + 2c^2$$ and $$g(x) = -x^2 - 2cx + b^2$$, $$(x \in R)$$; then $$\left|\frac{c}{b}\right|$$ lies in the interval:

If m is a non-zero number and $$\int \frac{x^{5m-1}+2x^{4m-1}}{(x^{2m}+x^m+1)^3} dx = f(x) + c$$, then $$f(x)$$ is equal to:

Let, the function F be defined as $$F(x) = \int_1^x \frac{e^t}{t} dt$$, $$x > 0$$, then the value of the integral $$\int_1^x \frac{e^t}{t+a} dt$$, where a > 0, is:

The area of the region (in square units) above the x-axis bounded by the curve $$y = \tan x$$, $$0 \leq x \leq \frac{\pi}{2}$$ and the tangent to the curve at $$x = \frac{\pi}{4}$$ is:

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

If $$\vec{x} = 3\hat{i} - 6\hat{j} - \hat{k}$$, $$\vec{y} = \hat{i} + 4\hat{j} - 3\hat{k}$$ and $$\vec{z} = 3\hat{i} - 4\hat{j} - 12\hat{k}$$, then the magnitude of the projection of $$\vec{x} \times \vec{y}$$ on $$\vec{z}$$ is:

If the angle between the line $$2(x+1) = y = z + 4$$ and the plane $$2x - y + \sqrt{\lambda}z + 4 = 0$$ is $$\frac{\pi}{6}$$, then the value of $$\lambda$$ is:

Equation of the line of the shortest distance between the lines $$\frac{x}{1} = \frac{y}{-1} = \frac{z}{1}$$ and $$\frac{x-1}{0} = \frac{y+1}{-2} = \frac{z}{1}$$ is:

Let A and E be any two events with positive probabilities.
Statement I: $$P(E/A) \geq P(A/E)P(E)$$.
Statement II: $$P(A/E) \geq P(A \cap E)$$.