Let a and b be two unit vectors such that a . b = 0. For some x, y$$ \in $$R, let $$\overrightarrow c = x\overrightarrow a + y\overrightarrow b + \overrightarrow a \times \overrightarrow b $$. If | $$\overrightarrow c $$| = 2 and the vector c is inclined at the same angle $$\alpha $$ to both a and b, then the value of $$8{\cos ^2}\alpha $$ is ..............

Let a, b, c three non-zero real numbers such that the equation $$\sqrt 3 a\cos x + 2b\sin x = c,x \in \left[ { - {\pi \over 2},{\pi \over 2}} \right]$$, has two distinct real roots $$\alpha $$ and $$\beta $$ with $$\alpha + \beta = {\pi \over 3}$$. Then, the value of $${b \over a}$$ is ............

A farmer F₁ has a land in the shape of a triangle with vertices at P(0, 0), Q(1, 1) and R(2, 0). From this land, a neighbouring farmer F₂ takes away the region which lies between the sides PQ and a curve of the form y = xⁿ (n > 1). If the area of the region taken away by the farmer F₂ is exactly 30% of the area of $$\Delta $$PQR, then the value of n is .................

Let S be the circle in the XY-plane defined the equation x² + y² = 4.

Let E₁E₂ and F₁F₂ be the chords of S passing through the point P₀ (1, 1) and parallel to the X-axis and the Y-axis, respectively. Let G₁G₂ be the chord of S passing through P₀ and having slope$$-$$1. Let the tangents to S at E₁ and E₂ meet at E₃, then tangents to S at F₁ and F₂ meet at F₃, and the tangents to S at G₁ and G₂ meet at G₃. Then, the points E₃, F₃ and G₃ lie on the curve