JEE (Advanced) 2009 Paper-2

The maximum value of the function $$f(x) = 2x^3 - 15x^2 + 36x - 48$$ on the set $$A = \left\{x \mid x^2 + 20 \leq 9x \right\}$$ is

Let (x, y, z) be points with integer coordinates satisfying the system of homogeneous equations:
$$3x - y - z = 0$$
$$-3x + z = 0$$
$$-3x + 2y + z = 0$$.
Then the number of such points for which $$x^2 + y^2 + z^2 \leq 100$$ is

Let ABC and ABC' be two non-congruent triangles with sides $$AB = 4, AC = AC' = 2\sqrt{2}$$ and angle $$B = 30^\circ$$. The absolute value ofthe difference between the areas of these triangles is

Let p(x) be a polynomial of degree 4 having extremumat x = 1, 2 and
$$\lim_{x \rightarrow 0}\left(1 + \frac{p(x)}{x^2}\right) = 2$$.
Then the value of p(2) is

Let $$f : R \rightarrow R$$ be a continuous function which satisfies
$$f(x) = \int_{0}^{x}f(t)dt$$.
Then the value of $$f(\ln 5)$$ is

The centres of two circles $$C_1$$ and $$C_2$$ each of unit radius are at a distance of 6 units from each other. Let P be the mid point of the line segment joining the centres of $$C_1$$ and $$C_2$$ and C be a circle touching circles $$C_1$$ and $$C_2$$ externally. If a common tangent to $$C_1$$ and C passing through Pis also a common tangent to $$C_2$$ and C, then the radius ofthe circle C is

The smallest value of k, for which both the roots of the equation
$$x^2 - 8kx + 16(k^2 - k + 1) = 0$$
are real, distinct and have values at least 4, is

If the function $$f(x) = x^3 + e^{\frac{x}{2}}$$ and $$g(x) = f^{-1}(x)$$, then the value of $$g'(1)$$ is

The mass M shown in the figure oscillates in simple harmonic motion with amplitude A. The amplitude of the point P is

A piece of wire is bent in the shape of a parabola $$y = kx^2$$ (y-axis vertical) with a bead of mass m onit. The bead canslide on the wire withoutfriction. It stays at the lowest point of the parabola when the wire is at rest. The wire is now accelerated parallel to the x-axis with a constant acceleration a. The distance of the new equilibrium position of the bead, where the bead can stay at rest with respect to the wire, from the y-axis is