Differentiation...

The number of rectangles that can be obtained by joining four of the twelve vertices of a $$12$$ sided regular polygon is

Three vertices are chosen randomly from the seven vertices of a regular $$7$$ -sided polygon. The probability that they form the vertices of an isosceles triangle is

Let f and g be differentiable function such that $${f}'\left ( x \right )=2g\left ( x \right )$$ and $${g}'\left ( x \right )=-f\left ( x \right )$$, and let $$T\left ( x \right )=\left ( f\left ( x \right ) \right )^{2}-\left ( g\left ( x \right ) \right )^{2}$$. Then $${T}'\left ( x \right )$$ is equal to

A college offers $$7$$ courses in the morning and $$5$$ courses in the evening. Find the number of ways a student can select exactly one course either in the morning or in the evening.

The derivative of $$\displaystyle (\tan x)^{x}$$ is equal to-

The sides of a quadrilateral are all positive integers and three of them are $$5, 10, 20.$$ How many possible value are there for the fourth side?

If $$ { _{  }^{ n }{ C } }_{ 4 },{ _{  }^{ n }{ C } }_{ 5 }$$ and $$ { _{  }^{ n }{ C } }_{ 6 }$$ are in AP, then $$n$$ is

If j, k, and n are consecutive integers such that $$0 < j < k < n$$ and the units (ones) digit of the product jn is 9, what is the units digit of k ? 

Two classrooms A and B having capacity of $$25$$ and $$(n-25)$$ seats respectively. $$A_n$$ denotes the number of possible seating arrangements of room $$'A'$$, when 'n' students are to be seated in these rooms, starting from room $$'A'$$ which is to be filled up to its capacity. If $$A_n-A_{n-1}=25!(^{49}C_{25})$$ then 'n' equals: