In a certain  test , there are n  questions , in the test $2^{n-i}$ students  gave wrong  answers to  atleast  i questions , where  i=1,2,3,…..,n . if  the  total  number  of  wrong  answers  given is  2047, then  the  value of  n is

The number  of  students  answering  exactly $i=(1\le i\le n-1)$ questions wrongly is $2^{n-i}-2^{n-i-1}$. The number  of  students  answering  all n  questions  wrongly  is 2047. Thus, the total  number  of  wrong  answer is $1(2^{n-1}-2^{n-2})+2(2^{n-2}-2^{n-3})+\mathrm{....}+(n-1)(2^1-2^0)+n\left(2^0\right)$

$$\begin{gathered} =2^{n-2}+2^{n-2}+2^{n-3}+\mathrm{.....}+2^0 \\ =2^n-1 \end{gathered}$$

Thus, $=2^n-1=2047$

$$\begin{gathered} \Rightarrow 2^n=2048=2^{11} \\ \Rightarrow n=11 \end{gathered}$$