JEE (Advanced) 2015 Paper-2

For any integer k, let $$\alpha_k = \cos \left(\frac{k\pi}{7}\right) + i \sin \left(\frac{k\pi}{7}\right),$$ where $$i = \sqrt -1.$$ The value of the expression $$\frac{\sum_{k = 1}^{12}|\alpha_{k + 1} - \alpha_k|}{\sum_{k = 1}^{3}|\alpha_{4k - 1} - \alpha_{4k - 2}|}$$ is

Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is 6: 11 and the seventh term lies in between 130 and 140, then the common difference of this A.P. is

The coefficient of $$x^9$$ in the expansion of $$(1 + x)(1 + x^2)(1 + x^3)...(1 + x^{100})$$ is

Suppose that the foci of the ellipse $$\frac{x^2}{9} + \frac{y^2}{5} = 1$$ are $$(f_1, 0)$$ and $$(f_2, 0)$$ where $$f_1 > 0$$ and $$f_3 < 0.$$ Let $$P_1$$ and $$P_2$$ be two parabolas with a common vertex at (0, 0) and with foci at $$(f_1, 0)$$ and $$(2f_2, 0),$$ respectively. Let $$T_1$$ be a tangent to $$P_1$$ which passes through $$(2f_2, 0)$$ and $$T_2$$ be a tangent to $$P_2$$ which passes through $$(f_1, 0)$$. If $$m_1$$ is the slope of $$T_1$$ and $$m_2$$ is the slope of $$T_2,$$ then the value of $$\left(\frac{1}{m_1^2} + m_2^2\right)$$ is

Let m and n be two positive integers greater than 1. If
$$\lim_{\alpha \rightarrow 0} \left(\frac{e^{\cos(\alpha^n)} - e}{\alpha^m}\right) = -\left(\frac{e}{2}\right)$$ then the value of $$\frac{m}{n}$$ is

If
$$\alpha = \int_{0}^{1} \left(e^{9x + 3 \tan^{-1}x}\right)\left(\frac{12 + 9x^2}{1 + x^2}\right) dx$$
where $$\tan^{-1}x$$ takes only principal values, then the value of $$\left(\log_e |1 + \alpha| - \frac{3\pi}{4}\right)$$ is

Let $$f : R \rightarrow R$$ be a continuous odd function, which vanishes exactly at one point and $$f(1) = \frac{1}{2}$$. Suppose that $$F(x) = \int_{-1}^{x} f(t) dt$$ for all $$x \in [-1, 2]$$ and $$G(x) = \int_{-1}^{x} t |f(f(t))| dt$$ for all $$x \in [-1, 2].$$ If $$\lim_{x \rightarrow 1} \frac {F(x)}{G(x)} = \frac{1}{14},$$ then the value of $$f \left(\frac{1}{2}\right)$$ is

Suppose that $$\overrightarrow{p}, \overrightarrow{q}$$ and $$\overrightarrow{r}$$ are three non-coplanar vectors in $$R^3$$. Let the components of a vector $$\overrightarrow{s}$$ along $$\overrightarrow{p}, \overrightarrow{q}$$ and $$\overrightarrow{r}$$ be 4, 3 and 5, respectively. If the components of this vector $$\overrightarrow{s}$$ along $$(-\overrightarrow{p} + \overrightarrow{q} + \overrightarrow{r}), (\overrightarrow{p} - \overrightarrow{q} + \overrightarrow{r})$$ and $$(-\overrightarrow{p} - \overrightarrow{q} + \overrightarrow{r})$$ are x, y and z, respectively, then the value of 2x + y + z is

Let S be the set of all non-zero real numbers $$\alpha$$ such that the quadratic equation $$ax^2 - x + a = 0$$ has two distinct real roots $$x_1$$ and $$x_2$$ satisfying the inequality $$|x_1 - x_2| < 1.$$ Which of the following intervals is(are) a subset(s) of S?

If $$\alpha = 3 \sin^{-1} \left(\frac{6}{11}\right)$$ and $$\beta = 3 \cos^{-1} \left(\frac{4}{9}\right) where the inverse trigonometric functions take only the principal values, then the correct option(s) is(are)