JEE (Advanced) 2011 Paper-1

Let $$\alpha$$ and $$\beta$$ be the roots of $$x^2 - 6x - 2 = 0$$, with $$\alpha > \beta$$. If $$a_n = \alpha^n - \beta^n$$ for $$n \geq 1$$, then the value of $$\frac{a_{10} - 2a_8}{2a_9}$$ is

A straight line L through the point (3, -2) is inclined at an angle $$60^\circ$$ to the line $$\sqrt{3} x + y = 1$$. If L also intersects the x-axis, then the equation of L is

Let $$P = \left\{\theta : \sin \theta - \cos \theta = \sqrt{2} \cos \theta\right\}$$ and $$Q = \left\{\theta : \sin \theta + \cos \theta = \sqrt{2} \sin \theta\right\}$$ be two sets. Then

The vector(s) which is/are coplanar with vectors $$\hat{i} + \hat{j} + 2\hat{k}$$ and $$\hat{i} + 2\hat{j} + \hat{k}$$, and perpendicular to the vector $$\hat{i} + \hat{j} + \hat{k}$$ is/are

Let $$f : R \rightarrow R$$ be a function such that
$$f(x + y) = f(x) + f(y), \forall x, y \in R$$
If f(x) is differentiable at x = 0, Then

Let M and N be two $$3 \times 3$$ non-singular skew-symmetric matrices such that $$MN = NM$$. If $$P^T$$ denotes the transpose of P, then $$M^2N^2(M^TN)^{-1}(MN^{-1})^T$$ is equal to

Let the eccentricity of the hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ be reciprocal to that of the ellipse $$x^2 + 4y^2 = 4$$. If the hyperbola passes through a focus of the ellipse, then

If the point P(a, b, c), with reference to (E), lies on the plane 2x + y + z = 1, then the value of 7a + b + c is

Let $$\omega$$ be a solution of $$x^3 - 1 = 0$$ with $$Im(\omega) > 0$$. If a = 2 with b and c satisfying (E), then the value of $$\frac{3}{\omega^a} + \frac{1}{\omega^b} + \frac{3}{\omega^c}$$ is equal to

Let b = 6, with a and c satisfying (E). If $$\alpha$$ and $$\beta$$ are the roots of the quadratic equation $$ax^2 + bx + c = 0$$, then $$\sum_{n=0}^{\infty}\left(\frac{1}{\alpha} + \frac{1}{\beta}\right)^n$$ is