Q109 / 155 · WBJEE 2020 · Mathematics

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

  • A
    10
  • Correct answer
    B
    11
  • C
    12
  • D
    13

Solution

Hint: Total students was gave wrong answer to exactly i - questions $=2^{n-i}-2^{n-(i+1)}$ Total wrong answer given \(\sum_{i=0}^{n-1} i \times \left(2^{n-i}-2^{n-(i+1)}\right)+n \times 1\) $\Rightarrow 2^{n-1}+\ldots \ldots 1=2047$ $\Rightarrow 2^{n}=2048$ $\Rightarrow \mathrm{n}=11$

Hence, the correct answer is option: 'B'.

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