If the tangent to the curve y = x³ at the point P(t, t³) meets the curve again at Q, then the ordinate of the point which divides PQ internally in the ratio 1 : 2 is :

If $${e^{\left( {{{\cos }^2}x + {{\cos }^4}x + {{\cos }^6}x + ...\infty } \right){{\log }_e}2}}$$ satisfies the equation t² - 9t + 8 = 0, then the value of
$${{2\sin x} \over {\sin x + \sqrt 3 \cos x}}\left( {0 < x < {\pi \over 2}} \right)$$ is :

An ordinary dice is rolled for a certain number of times. If the probability of getting an odd number 2 times is equal to the probability of getting an even number 3 times, then the probability of getting an odd number for odd number of times is :

The locus of the mid-point of the line segment joining the focus of the parabola y² = 4ax to a moving point of the parabola, is another parabola whose directrix is :