Let PQ be a focal chord of length 6.25 units of the parabola y2 = 4x. If O is the vertex of the parabola, then 10 times the area (in sq. units) of $$\Delta$$POQ is equal to ___________.
Consider a triangle ABC whose vertices are A(0, $$\alpha$$, $$\alpha$$), B($$\alpha$$, 0, $$\alpha$$) and C($$\alpha$$, $$\alpha$$, 0), $$\alpha$$ > 0. Let D be a point moving on the line x + z $$-$$ 3 = 0 = y and G be the centroid of $$\Delta$$ABC. If the minimum length of GD is $$\sqrt {{{57} \over 2}} $$, then $$\alpha$$ is equal to ____________.
The probability distribution of X is :
X | 0 | 1 | 2 | 3 |
---|---|---|---|---|
P(X) | $${{1 - d} \over 4}$$ | $${{1 + 2d} \over 4}$$ | $${{1 - 4d} \over 4}$$ | $${{1 + 3d} \over 4}$$ |
For the minimum possible value of d, sixty times the mean of X is equal to _______________.
Let $${S_1} = \{ x \in [0,12\pi ]:{\sin ^5}x + {\cos ^5}x = 1\} $$
and $${S_2} = \{ x \in [0,8\pi ]:{\sin ^7}x + {\cos ^7}x = 1\} $$
Then $$n({S_1}) - n({S_2})$$ is equal to ______________.