JEE (Advanced) 2011 Paper-2

Let L be anormal to the parabola $$y^2 = 4x$$. If L passes through the point (9, 6), then L is given by

Let $$f : (0, 1) \rightarrow R$$ be defined by
$$f(x) = \frac{b - x}{1 - bx}$$, where b is a constant such that 0 < b < 1. Then

Let $$\omega = e^{\frac{i \pi}{3}}$$ and a, b, c, x, y, z be non-zero complex numbers such that
$$a + b + c = x$$
$$a + b \omega + c \omega^2 = y$$
$$a + b \omega^2 + c \omega = z$$.
Ten the value of $$\frac{\mid x \mid^2 + \mid y \mid^2 + \mid z \mid^2}{\mid a \mid^2 + \mid b \mid^2 + \mid c \mid^2}$$ is

The number of distinct real roots of $$x^4 - 4x^3 + 12 x^2 + x - 1 = 0$$ is

Let $$y'(x) + y(x)g'(x) = g(x)g'(x), y(0) = 0, x \in R$$, where $$f'(x)$$ denotes $$\frac{d f(x)}{dx}$$ and $$g(x)$$ is a given non-constant differentiable function on R with g(0) = g(2) = 0. Then the value of y(2) is

Let M be a $$3 \times 3$$ matrix satisfying
$$M\begin{bmatrix}0\\1\\0 \end{bmatrix} = \begin{bmatrix}-1\\2\\3 \end{bmatrix}, M\begin{bmatrix}1\\-1\\0 \end{bmatrix} = \begin{bmatrix}1\\1\\-1 \end{bmatrix}$$, and $$M\begin{bmatrix}1\\1\\1 \end{bmatrix} = \begin{bmatrix}0\\0\\12 \end{bmatrix}$$. Then the sum of the diagonal entries of M is

Let $$\overrightarrow{a} = -\hat{i} - \hat{k}, \overrightarrow{b} = -\hat{i} + \hat{k}$$ and $$\overrightarrow{c} = \hat{i} + 2\hat{j} + 3\hat{k}$$ be three given vectors. If $$\overrightarrow{r}$$ is a vector such that $$\overrightarrow{r} \times \overrightarrow{b} = \overrightarrow{c} \times \overrightarrow{b}$$ and $$\overrightarrow{r} . \overrightarrow{a} = 0$$, then the value of $$\overrightarrow{r} . \overrightarrow{b}$$ is

The straight line 2x - 3y = 1 divides the circular region $$x^2 + y^2 \leq 6$$ into two parts. If
$$S = \left\{\left(2, \frac{3}{4}\right), \left(\frac{5}{2}, \frac{3}{4}\right), \left(\frac{1}{4}, -\frac{1}{4}\right), \left(\frac{1}{8}, \frac{1}{4}\right)\right\}$$,
then the number of point(s) in S lying inside the smaller part is

Match the statements given in Column I with the values given in Column II

Match the statements given in Column I with the intervals/union of intervals given in Column II