JEE (Advanced) 2008 Paper-1

Consider the two curves
$$C_1 : y^2 = 4x$$
$$C_2 : x^2 + y^2 - 6x + 1 = 0$$
Then,

If $$0 < x < 1$$, then
$$\sqrt{1 + x^2}[\left\{x \cos (\cot^{-1} x) + \sin(\cot^{-1} x)\right\}^2 - 1]^{\frac{1}{2}}=$$

The edges of a parallelopiped are of unit length andare parallel to non-coplanar unit vectors $$\hat{a}, \hat{b}, \hat{c}$$ such that
$$\hat{a} . \hat{b} = \hat{b} . \hat{c} = \hat{c} . \hat{a} = \frac{1}{2}$$.
Then, the volume of the parallelopiped is

Let a and b be non-zero real numbers. Then, the equation
$$(ax^2 + by^2 + c)(x^2 - 5xy + 6y^2) = 0$$
represents

Let
$$g(x) = \frac{(x - 1)^n}{\log \cos^m (x - 1)};0 < x < 2$$, m and n are integers, $$m \neq 0, n > 0$$, and let p be the left hand derivative of $$\mid x - 1 \mid$$ at x=1.
If $$\lim_{x \rightarrow 1+}g(x) = p$$, then

The total number of local maxima and local minima of the function
$$f(x) = \begin{cases}(2 + x)^3, & -3<x \leq -1\\x^{\frac{2}{3}} & -1 < x < 2\end{cases}$$ is

A straight line through the vertex P of a triangle PQR intersects the side QR at the point S and the circumcircle of the triangle PQR at the point T. If S is not the centre of the circumcircle, then

Let $$P(x_1, y_1)$$ and $$Q(x_2, y_2), y_1 < 0, y_2 < 0$$, be the end of the latus rectum of the ellipse $$x^2 + 4y^2 = 4$$. The equations of parabolas with latus rectum PQ are

Let
$$S_n = \sum_{k=1}^{n}\frac{n}{n^2 + kn + k^2}$$ and $$T_n = \sum_{k=0}^{n-1}\frac{n}{n^2 + kn + k^2}$$, for $$n = 1, 2, 3, ..., $$ Then,

Let f(x) be a non-constant twice differentiable function defined on $$(-\infty, \infty)$$ such that f(x) = f(1 - x) and $$f'\left(\frac{1}{4}\right) = 0$$. Then,