JEE (Advanced) 2011 Paper-2

Let P(6, 3) be a point on the hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$. If the normal at the point P intersects the x-axis at (9, 0), then the eccentricity of the hyperbola is

A value of b for which the equations
$$x^2 + bx - 1 = 0$$
$$x^2 + x + b = 0$$,
have one root in common is

Let $$\omega \neq 1$$ be a cube root of unity and S bethe set of all non-singular matrices of the form
$$\begin{bmatrix}1 & a & b\\\omega & 1 & c\\\omega^2 & \omega & 1\end{bmatrix}$$,
where each of a, b, and c is either $$\omega$$ or $$\omega^2$$. Then the number of distinct matrices in the set S is

The circle passing through the point (-1,0) and touching the y-axis at (0, 2) also passes through the point

If
$$\lim_{x \rightarrow 0} \left[1 + x \ln (1 + b^2)\right]^{\frac{1}{x}} = 2b \sin \theta, b > 0 $$ and $$\theta \in (-\pi, \pi]$$, then the value of $$\theta$$ is

Let $$f:[-1, 2] \rightarrow [0, \infty)$$ be a continuous function such that $$f(x) = f(1 - x)$$ for all $$x \in [-1, 2]$$. Let $$R_1 = \int_{-1}^{2} x f(x)dx$$ and $$R_2$$ be the area of the region bounded by y = f(x), x = -1, x = 2, and the x-axis. Then

Let $$f(x) = x^2$$ and $$g(x) = \sin x$$ for all $$x \in R$$. Then the set of all x satisfying $$(f\circ g \circ g \circ f)(x) = (g\circ g \circ f)(x)$$, where $$(f \circ g)(x) = f(g(x))$$, is

Let (x, y) be any point on the parabola $$y^2 = 4x$$. Let P be the point that divides the line segment from (0, 0) to (x, y) in the ratio 1:3. Then the locus of P is

If
$$f(x) = \begin{cases}-x-\frac{\pi}{2}, & x \leq -\frac{\pi}{2}\\-\cos x, & -\frac{\pi}{2}<x \leq 0\\x-1, & 0 < x \leq 1\\\ln x, & x > 1\end{cases}$$, then

Let E and F be two independent events. The probability that exactly one of them occurs is $$\frac{11}{25}$$ and the probability of none of them occurring is $$\frac{2}{25}$$. If P(T) denotes the probability of occurrence of the event T , then