NTA JEE Main 12th April 2014 Online
For the following questions answer them individually
NTA JEE Main 12th April 2014 Online - Question 81
Let f, g : R → R be two functions defined by $$f(x) = \begin{cases} x\sin\left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x = 0 \end{cases}$$ and $$g(x) = xf(x)$$. Statement I: f is a continuous function at x = 0. Statement II: g is a differentiable function at x = 0.
NTA JEE Main 12th April 2014 Online - Question 82
Let $$f$$ and $$g$$ be two differentiable functions on R such that $$f'(x) > 0$$ and $$g'(x) < 0$$ for all $$x \in R$$. Then for all x:
NTA JEE Main 12th April 2014 Online - Question 83
The integral $$\int \frac{\sin^2 x \cos^2 x}{(\sin^3 x + \cos^3 x)^2} dx$$ is equal to:
NTA JEE Main 12th April 2014 Online - Question 84
If [ ] denotes the greatest integer function, then the integral $$\int_0^{\pi} [\cos x] dx$$ is equal to:
NTA JEE Main 12th April 2014 Online - Question 85
If for a continuous function f(x), $$\int_{-\pi}^{t} (f(x) + x) dx = \pi^2 - t^2$$, for all $$t \geq -\pi$$, then $$f\left(-\frac{\pi}{3}\right)$$ is equal to:
NTA JEE Main 12th April 2014 Online - Question 86
The general solution of the differential equation, $$\sin 2x\left(\frac{dy}{dx} - \sqrt{\tan x}\right) - y = 0$$, is:
NTA JEE Main 12th April 2014 Online - Question 87
If $$\hat{x}$$, $$\hat{y}$$ and $$\hat{z}$$ are three unit vectors in threedimensional space, then the minimum value of $$|\hat{x} + \hat{y}|^2 + |\hat{y} + \hat{z}|^2 + |\hat{z} + \hat{x}|^2$$
NTA JEE Main 12th April 2014 Online - Question 88
A symmetrical form of the line of intersection of the planes $$x = ay + b$$ and $$z = cy + d$$ is:
NTA JEE Main 12th April 2014 Online - Question 89
If the distance between planes, $$4x - 2y - 4z + 1 = 0$$ and $$4x - 2y - 4z + d = 0$$ is 7, then d is:
NTA JEE Main 12th April 2014 Online - Question 90
A number x is chosen at random from the set {1, 2, 3, 4, ..., 100}. Define the event: A = the chosen number x satisfies $$\frac{(x-10)(x-50)}{(x-30)} \geq 0$$. Then P(A) is:
