JEE (Advanced) 2015 Paper-1

The number of distinct solutions of the equation
$$\frac{5}{4} \cos^2 2x + \cos^4 x + \sin^4x + \cos^6 x + \sin^6 x = 2$$
in the interval $$[0, 2\pi]$$ is

Let the curve C be the mirror image of the parabola $$y^2 = 4x$$ with respect to the line x + y + 4 = 0. If A and B are the points of intersection of C with the line y = -5, then the distance between A and B is

The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least 0.96, is

Let n be the numberof ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let m be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that exactly four girls stand consecutively in the queue. Then the value of $$\frac{m}{n}$$ is

If the normals of the parabola $$y^2 = 4x$$ drawn at the end points of its latus rectum are tangents to the circle $$(x - 3)^2 + (y + 2)^2 = r^2$$, then the value of $$r^2$$ is

Let $$f : R \rightarrow R$$ be a function defined by $$f(x) = \begin{cases}[x], & x \leq 2\\0 & x > 2\end{cases}$$,
where [x] is the greatest integer less than or equal to x. If $$I = \int_{-1}^{2} \frac {xf(x^2)}{2 + f(x + 1)} dx,$$ then the value of (4I - 1) is

A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of $$V mm^3$$, has a 2 mm thick solid wall and is open at the top. The bottom of the container is a solid circular disc of thickness 2 mm and is of radius equal to the outer radius of the container. If the volume of the material used to make the container is minimum when the inner radius of the container is 10 mm, then the value of $$\frac{V}{250 \pi}$$ is

Let $$F(x) = \int_{x}^{x^2 + \frac{\pi}{6}} 2 \cos^2t dt$$ for all $$x \in R$$ and $$f : \left[0, \frac{1}{2}\right] \rightarrow [0, \infty)$$ be a continuous function. For $$a \in \left[0, \frac{1}{2}\right]$$, if $$F' (a) + 2 $$ is the area of the region bounded by x = 0, y = 0, y = f(x) and x = a, then f(0) is

Let X and Y be two arbitrary, $$3 \times 3$$, non-zero, skew-symmetric matrices and Z be an arbitrary $$3 \times 3$$, non-zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?

Which of the following values of $$\alpha$$ satisfy the equation