JEE (Advanced) 2012 Paper-1

The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each persongets at least one ball is

Let $$f(x) = \begin{cases}x^2 \mid \cos \frac{\pi}{x} \mid & x \neq 0\\0, & x = 0\end{cases}, x \in R$$, then f is

The function $$f: [0, 3] \rightarrow [1, 29]$$, defined by $$f(x) = 2x^3 - 15x^2 + 36x + 1$$, is

If $$\lim_{x \rightarrow \infty}\left(\frac{x^2 + x + 1}{x + 1} - ax - b\right) = 4$$, then

Let z be a complex number suchthat the imaginary part of z is nonzero and $$a = z^2 + z + 1$$ is real. Then a cannot take the value

The ellipse $$E_1 : \frac{x^2}{9} + \frac{y^2}{4} = 1$$ is inscribed in a rectangle R whosesides are parallel to the coordinate axes. Another ellipse $$E_2$$ passing through the point(0, 4) circumscribes the rectangle R. The eccentricity of the ellipse $$E_2$$ is

Let $$P = [a_{ij}]$$ be a $$3 \times 3$$ matrix and let $$Q = [b_{ij}]$$, where $$b_{ij} = 2^{i + j}a_{ij}$$ for $$1 \leq i, j \leq 3$$. If the determinant of P, is 2, then the determinant of the matrix Q is

The integral $$\int \frac{\sec^2 x}{(\sec x + \tan x)^{\frac{9}{2}}} dx$$ equals (for some arbitrary constant K)

The point P is the intersection of the straight line joining the points Q(2,3,5) and R(1, -1, 4) with the plane 5x - 4y - z = 1. If S is the foot of the perpendicular drawn from the point 7(2, 1, 4) to QR, then the length of the line segment PS is

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line 4x - 5y = 20 to the circle $$x^2 + y^2 = 9$$ is