NTA JEE Main 6th April 2014 Offline
For the following questions answer them individually
NTA JEE Main 6th April 2014 Offline - Question 81
If $$x = -1$$ and $$x = 2$$ are extreme points of $$f(x) = \alpha \log|x| + \beta x^2 + x$$, then:
NTA JEE Main 6th April 2014 Offline - Question 82
The slope of the line touching both the parabolas $$y^2 = 4x$$ and $$x^2 = -32y$$ is:
NTA JEE Main 6th April 2014 Offline - Question 83
The integral $$\int\left(1 + x - \frac{1}{x}\right)e^{x+\frac{1}{x}} dx$$ is equal to:
NTA JEE Main 6th April 2014 Offline - Question 84
The integral $$\int_0^{\pi} \sqrt{1 + 4\sin^2\frac{x}{2} - 4\sin\frac{x}{2}} \, dx$$ equals:
NTA JEE Main 6th April 2014 Offline - Question 85
The area (in sq. unit) of the region described by $$A = \{(x, y) : x^2 + y^2 \leq 1$$ and $$y^2 \leq 1 - x\}$$ is:
NTA JEE Main 6th April 2014 Offline - Question 86
Let the population of rabbits surviving at a time $$t$$ be governed by the differential equation $$\frac{dp(t)}{dt} = \frac{1}{2}\{p(t) - 400\}$$. If $$p(0) = 100$$, then $$p(t)$$ equals:
NTA JEE Main 6th April 2014 Offline - Question 87
If $$\begin{bmatrix} \vec{a} \times \vec{b} & \vec{b} \times \vec{c} & \vec{c} \times \vec{a} \end{bmatrix} = \lambda \begin{bmatrix} \vec{a} & \vec{b} & \vec{c} \end{bmatrix}^2$$, then $$\lambda$$ is equal to:
NTA JEE Main 6th April 2014 Offline - Question 88
The image of the line $$\frac{x-1}{3} = \frac{y-3}{1} = \frac{z-4}{-5}$$ in the plane $$2x - y + z + 3 = 0$$ is the line:
NTA JEE Main 6th April 2014 Offline - Question 89
The angle between the lines whose direction cosines satisfy the equations $$l + m + n = 0$$ and $$l^2 = m^2 + n^2$$ is:
NTA JEE Main 6th April 2014 Offline - Question 90
Let $$A$$ and $$B$$ be two events such that $$P\left(\overline{A \cup B}\right) = \frac{1}{6}$$, $$P(A \cap B) = \frac{1}{4}$$ and $$P(\bar{A}) = \frac{1}{4}$$, where $$\bar{A}$$ stands for the complement of the event $$A$$. Then the events $$A$$ and $$B$$ are:
