Trigonometry

Q. If $(1+x{)}^n=C_0+C_{1}x+C_2{x}^2+\dots +C_n{x}^n,$ then
the value of $\underset{0\le r<s\le n}{\sum \sum }(r+s)(C_r+C_s)$ is

1. $n\cdot 2^n$
2. None of these
3. $n^2\cdot 2^{2n}$
4. $n^2\cdot 2^n$

Q. If $(1+x{)}^n=C_0+C_{1}x+\dots ..+C_n{x}^n,$ then the value of $\underset{0\le r<s\le n}{\sum \sum }(C_r+C_s)$ is equal to

1. $(n-1)\cdot 2^n$
2. $2^n$
3. $2^{2n}$
4. $n\cdot 2^n$

Q. ${\sum }_{m=1}^n\left({\sum }_{k=1}^m\left({\sum }_{p=k}^m{}^n{C}_m{.}^m{C}_p{.}^p{C}_k\right)\right)=$

1. $3^n+2^n$
2. $3^n-2^n$
3. $4^n-3^n$
4. $4^n-1$

Q. The incircle of a scalene triangle $ABC$ touches side $AB,BC$ and $AC$ at $P,Q$ and $R$ respectively. If $O$ is the incentre of triangle $ABC,$ then which of the following statements is (are) correct?

$\text{I.}$ Area of $△OBC$ is $\frac{1}{2}\times BC\times \text{radius of incircle}$
$\text{II.}$ $AB+CQ=BQ+AC$
$\text{III.}$ Area of $△ABC$ is $\frac{1}{2}\times \text{perimeter of}△ABC\times \text{radius of incircle}$

1. $\text{I}$ and $\text{III}$
2. $\text{I}$ only
3. $\text{II}$ and $\text{III}$
4. $\text{I},\text{II}$ and $\text{III}$