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

Let $${E_1}$$ and $${E_2}$$ be two ellipses whose centres are at the origin. The major axes of $${E_1}$$ and $${E_2}$$ lie along the $$x$$-axis and the $$y$$-axis, respectively. Let $$S$$ be the circle $${x^2} + {\left( {y - 1} \right)^2} = 2$$. The straight line $$x+y=3$$ touches the curves $$S$$, $${E_1}$$ and $${E_2}$$ at $$P, Q$$ and $$R$$ respectively. Suppose that $$PQ = PR = {{2\sqrt 2 } \over 3}$$. If $${e_1}$$ and $${e_2}$$ are the eccentricities of $${E_1}$$ and $${E_2}$$, respectively, then the correct expression(s) is (are)

Consider the hyperbola $$H:{x^2} - {y^2} = 1$$ and a circle $$S$$ with center $$N\left( {{x_2},0} \right)$$. Suppose that $$H$$ and $$S$$ touch each other at a point $$P\left( {{x_1},{y_1}} \right)$$ with $${{x_1} > 1}$$ and $${{y_1} > 0}$$. The common tangent to $$H$$ and $$S$$ at $$P$$ intersects the $$x$$-axis at point $$M$$. If $$(l, m)$$ is the centroid of the triangle $$PMN$$, then the correct expressions(s) is(are)