JEE (Advanced) 2016 Paper-2

Area of the region $$\left\{(x, y) \epsilon R^2 : y \geq \sqrt{\mid x + 3 \mid}, 5y \leq x + 9 \leq 15\right\}$$ is equal to

Let P be the image of the point (3,1,7) with respect to the plane $$x - y + z = 3$$. Then the equation of the plane passing through P and containing the straight line $$\frac{x}{1} = \frac{y}{2} = \frac{z}{1}$$ is

Let $$f(x) = \lim_{n \rightarrow \infty}\left(\frac{n^n(x + n)\left(x + \frac{n}{2}\right).........\left(x + \frac{n}{n}\right)}{n!(x^2 + n^2)\left(x^2 + \frac{n^2}{4}\right).........\left(x^2 + \frac{n^2}{n^2}\right)}\right)^{\frac{x}{n}}$$, for all x > 0. Then

Let $$a, b \epsilon R$$ and $$f: R \rightarrow R$$ be bedefined by $$f(x) = a \cos (\mid x^3 - x \mid) + b \mid x \mid \sin(\mid x^3 + x \mid)$$. Then f is

Let $$f : R \rightarrow (0, \infty)$$ and $$g : R \rightarrow R$$ be be twice differentiable functions such that $$f''$$ and $$g''$$ are continuous functions on R. Suppose $$f'(2) = g(2) = 0, f''(2) \neq 0$$ and $$g'(2) \neq 0$$. If $$\lim_{x \rightarrow 2}\frac{f(x)g(x)}{f'(x)g'(x)} = 1$$, then

Let $$f:\left[-\frac{1}{2}, 2\right] \rightarrow R$$ and $$g:\left[-\frac{1}{2}, 2\right] \rightarrow R$$ be functions defined by $$f(x) = [x^2 - 3]$$ and $$g(x) = \mid x \mid f(x) + \mid 4x - 7 \mid f(x)$$, where $$[y]$$ denotes the greatest integer less than or equal to y for $$y \epsilon R$$. Then

Let $$a, b \epsilon R$$ and $$a^2 + b^2 \neq 0$$. Suppose $$S = \left\{z \epsilon C : z = \frac{1}{a + ibt}, t \epsilon R, t \neq 0\right\}$$, where $$i = \sqrt{-1}$$. If $$z = x + iy$$ and $$z \epsilon S$$, then (x, y) lies on

Let P be the point on the parabola $$y^2 = 4x$$ which is at the shortest distance from the center S of the circle $$x^2 + y^2 - 4x - 16y + 64 = 0$$. Let Q be the pomt on the circle dividing the line segment SP internally. Then

Let $$a, \lambda, \mu \epsilon R$$. Consider the system of linear equations
$$ax + 2y = \lambda$$
$$3x - 2y = \mu$$
Which of the following statement(s) is(are) correct?

Let $$\hat{u} = u_1\hat{i} + u_2\hat{j} + u_3\hat{k}$$ be a unit vector in $$R^3$$ and $$\hat{\omega} = \frac{1}{\sqrt{6}}(\hat{i} + \hat{j} + 2 \hat{k})$$. Given that there exists a vector $$\overrightarrow{v}$$ in $$R^3$$ such that $$\mid \hat{u} \times \overrightarrow{v} \mid = 1$$ and $$\overrightarrow{\omega} . (\hat{u} \times \overrightarrow{v}) = 1$$. Which of the following statement(s) is(are) correct?