Question: In a certain test there are $ n $ questions. In the test $ {2^{n-i}} $ students gave wrong answers to at least $ i $ questions, where $ i=1,\ 2,\ ……n $ . If the total number of wrong answers given is 2047, then $ n $ is equal to
Options:
A) 10
B) 11
C) 12
D) 13
Show Answer
Answer:
Correct Answer: B
Solution:
- Since the number of students giving wrong answers to at least $ i $ question $ (i=1,\ 2,……..,n)={2^{n-i}} $ . The number of students answering exactly $ i\ (1\le i\le -1) $ questions wrongly = {the number of students answering at least $ i $ questions wrongly, $ i=1,\ 2,\ ………,n)} $ - {the number of students answering at least $ (i+1) $ questions wrongly $ (2\le i+1\le n)} $ $ ={2^{n-i}}-{2^{n-(i+1)}}(1\le i\le n-1) $ . Now, the number of students answering all the $ n $ questions wrongly $ ={2^{n-n}}=2^{0} $ . Thus the total number of wrong answers $ =1({2^{n-1}}-{2^{n-2}}+2({2^{n-2}}-{2^{n-3}})+3({2^{n-3}}-{2^{n-4}}) $ $ +……….+(n-1)(2^{1}-2^{0})+n(2^{0}) $ $ ={2^{n-1}}+{2^{n-2}}+{2^{n-3}}+………+2^{0} $ = $ 2^{n}-1 $ $ (\because \ Its\ a\ G\text{.P}\text{.}) $
$ \therefore $ As given $ 2^{n}-1=2047\Rightarrow 2^{n}=2048=2^{11} $
$ \Rightarrow n=11 $
BETA
