A debate club consists of 6 girls and 4 boys. A team of 4 members is to be select from this club including the selection of a captain (from among these 4 members ) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is

Let $$m$$ be the smallest positive integer such that the coefficient of $${x^2}$$ in the expansion of $${\left( {1 + x} \right)^2} + {\left( {1 + x} \right)^3} + ........ + {\left( {1 + x} \right)^{49}} + {\left( {1 + mx} \right)^{50}}\,\,$$ is $$\left( {3n + 1} \right)\,{}^{51}{C_3}$$ for some positive integer $$n$$. Then the value of $$n$$ is

Let RS be the diameter of the circle $${x^2}\, + \,{y^2} = 1$$, where S is the point (1, 0). Let P be a variable point (other than R and S) on the circle and tangents to the circle at S and P meet at the point Q. The normal to the circle at P intersects a line drawn through Q parallel to RS at point E. Then the locus of E passes through the point (s)

The circle $${C_1}:{x^2} + {y^2} = 3,$$ with centre at $$O$$, intersects the parabola $${x^2} = 2y$$ at the point $$P$$ in the first quadrant, Let the tangent to the circle $${C_1}$$, at $$P$$ touches other two circles $${C_2}$$ and $${C_3}$$ at $${R_2}$$ and $${R_3}$$, respectively. Suppose $${C_2}$$ and $${C_3}$$ have equal radil $${2\sqrt 3 }$$ and centres $${Q_2}$$ and $${Q_3}$$, respectively. If $${Q_2}$$ and $${Q_3}$$ lie on the $$y$$-axis, then