AIEEE 2004 (Maths) Mock Test

The coefficient of $x^n$ in expansion of $(1+x)(1-x{)}^n$ is

If $S_n={\sum }_{r=0}^n\frac{1}{{}^n{C}_r}$ and $t_n={\sum }_{r=0}^n\frac{r}{{}^n{C}_r},$ then $\frac{t_n}{S_n}$ is equal to

Let $T_r$ be the $r$ th term of an A.P. whose first term is $a$ and common difference is $d.$ If for some positive integers $m,n,m\ne n,T_m=\frac{1}{n}$ and $T_n=\frac{1}{m},$ then $a-d$ equals

The sum of the first $n$ terms of the series $1^2+2\cdot 2^2+3^2+2\cdot 4^2+5^2+2\cdot 6^2+...$ is $\frac{n(n+1{)}^2}{2}$ when $n$ is even. When $n$ is odd the sum is

The sum of series $\frac{1}{2!}+\frac{1}{4!}+\frac{1}{6!}+...$ is