JEE (Advanced) 2013 Paper-2

Let $$w=\frac{\sqrt{3}+i}{2}$$ and $$p={{w^{n}}:n=1,2,3,...}$$. Further $$H_{1}= \left\{Z\epsilon C : Rez>\frac{1}{2}\right\}$$ and $$H_{2}= \left\{Z\epsilon C : Rez<\frac{-1}{2}\right\}$$ where C is the set of all complex numbers. If $$Z_{1}\epsilon P \cap H_{1}$$, $$Z_{2}\epsilon P \cap H_{2}$$ and O represents the origin, then $$\angle z_{1}Oz_{2}=$$

If $$3^x = 4^{x - 1}$$, then x =

Let $$\omega$$ be a complex cuberoot of unity with $$\omega \neq 1$$ and $$P = [p_{ij}]$$ be a $$n \times n$$ matrix with $$p_{ij} = \omega^{i+j}$$. Then $$P^2 \neq 0$$, when n =

The function $$f(x) = 2 \mid x \mid + \mid x + 2 \mid - \mid \mid x + 2 \mid - 2 \mid x \mid \mid$$ has a local minimum or a local maximum at x =

For a $$x \in R$$ (the set of all real numbers), $$a \neq -1$$,
$$\lim_{n \rightarrow \infty}\frac{(1^a + 2^a + ... + n^a)}{(n + 1)^{a - 1}[(na + 1) + (na + 2) + ... + (na + n)]}= \frac{1}{60}$$Then a =

Circle(s) touching x - axis at a distance 3 from the origin and having an intercept of length $$2\sqrt{7}$$ on y-axis is (are)

Two lines $$L_1 : x = 5, \frac{y}{3 - \alpha} = \frac{z}{-2}$$ and $$L_2 : x = \alpha, \frac{y}{-1} = \frac{z}{2 - \alpha}$$ are coplanar. Then $$\alpha$$ can take value(s)

In a triangle PQR, P is the largest angle and $$\cos P = \frac{1}{3}$$. Further the incircle of the triangle touches the sides PQ, QR and RP at N, L and M respectively, such that the lengths of PN, QL and RM are consecutive even integers. Then possible length(s) of the side(s) of the triangle is (are)

Area of S =

$$\min_{z \in S} \mid 1 - 3i - z \mid =$$