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In a certain test, there are n questions. In this test $2^{n-i}$ students gave wrong answers to at least i questions, where i $=1,2, \ldots \ldots \mathrm{n}$. If the total number of wrong answers given is 2047 , then $\mathrm{n}$ is equal to
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Verified Answer
The correct answers are:
11
Hint:
Total students was gave wrong answer to exactly i - questions $=2^{n-i}-2^{n-(i+1)}$
Total wrong answer given \(\sum_{i=0}^{n-1} i \times \left(2^{n-i}-2^{n-(i+1)}\right)+n \times 1\)
$\Rightarrow 2^{n-1}+\ldots \ldots 1=2047$
$\Rightarrow 2^{n}=2048$
$\Rightarrow \mathrm{n}=11$
Total students was gave wrong answer to exactly i - questions $=2^{n-i}-2^{n-(i+1)}$
Total wrong answer given \(\sum_{i=0}^{n-1} i \times \left(2^{n-i}-2^{n-(i+1)}\right)+n \times 1\)
$\Rightarrow 2^{n-1}+\ldots \ldots 1=2047$
$\Rightarrow 2^{n}=2048$
$\Rightarrow \mathrm{n}=11$
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