JEE (Advanced) 2011 Paper-1

The probability of the drawn ball from $$U_2$$ being white is

Given that the drawn ball from $$U_2$$ is white, the probability that head appeared on the coin is

Let $$a_1, a_2, a_3, ….., a_{100}$$ be an arithmetic progression with $$a_1 = 3$$ and $$S_p = \sum_{i=1}^{p}a_i, 1 \leq p \leq 100$$. For any integer n with $$1 \leq n \leq 20$$, let $$m = 5n$$. If $$\frac{S_m}{S_n}$$ does not depend on n, then $$a_2$$ is

Consider the parabola $$y^2 = 8x$$. Let $$\triangle _1$$ be the area of the triangle formed by the end points of its latus rectum and the point $$P\left(\frac{1}{2}, 2\right)$$ on the parabola, and $$\triangle_2$$ be the area of the triangle formed by drawing tangents at P and at the end points of the latus rectum. Then $$\frac{\triangle_1}{\triangle_2}$$ is

The positive integer value of $$n > 3$$ satisfying the equation $$\frac{1}{\sin\left(\frac{\pi}{n}\right)} = \frac{1}{\sin\left(\frac{2 \pi}{n}\right)} + \frac{1}{\sin\left(\frac{3 \pi}{n}\right)}$$ is

Let $$f(\theta) = \sin\left(\tan^{-1}\left(\frac{\sin \theta}{\sqrt{\cos 2\theta}}\right)\right)$$, where $$-\frac{\pi}{4} < \theta < \frac{\pi}{4}$$. Then the value of $$\frac{d}{d(\tan \theta)}(f(\theta))$$ is

If z is any complex number satisfying $$\mid z - 3 - 2i \mid \leq 2$$, then the minimum value of $$\mid 2z - 6 + 5i \mid$$ is

The minimum value of the sum of real numbers $$a^{-5}, a^{-4}, 3a^{-3}, 1, a^{8}$$ and $$a^{10}$$ with a > 0 is

Let $$f : [1, \infty) \rightarrow [2, \infty)$$ be a differentiable function such that $$f(1) = 2$$. If $$6 \int_{1}^{x} f(t) dt = 3xf(x) - x^3$$ forall $$x \geq 1$$, then the value of f(2) is