Question 93

# In a certain examination paper, there are n questions. For j = 1,2 …n, there are $$2^{n-j}$$ students who answered j or more questions wrongly. If the total number of wrong answers is 4095, then the value of n is

Solution

Let there only be 2 questions.

Thus there are $$2^{2-1}$$ = 2 students who have done 1 or more questions wrongly, 2$$^{2-2}$$ = 1 students who have done all 2 questions wrongly .

Thus total number of wrong answers = 2 + 1 = 3= $$2^n - 1$$.

Now let there be 3 questions. Then j = 1,2,3

Number of students answering 1 or more questions incorrectly = 4

Number of students answering 2 or more questions incorrectly = 2

Number of students answering 3 or more questions incorrectly = 1

Total number of incorrect answers = 1(3)+(2-1)*2+(4-2)*1 = 7 = $$2^3-1$$

According to the question , the total number of wrong answers = 4095 = $$2^{12} - 1$$.

Hence Option A.

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