Question

In a certain test, there are $n$ questions. In this test $2^{n-i}$ students gave wrong answers to at least $i$ questions; where $i = 1,2............. n-1, n$. If the total number of wrong answers given is $2047$, then $n$ is equal to

• A. 10
• B. 11
• C. 12
• D. 13

Answer 1

Answer: (B)

11

Answer 2

The no. of students answering exactly $i\left(1 \le i \le n-1\right)$ questions wrongly is $2^{n-i}-2^{n-i-1}$. The no. of students answering all n questions wrongly is $2^{\circ}$. Thus, the total number of wrong answer is $1\left(2^{n-1}-2^{n-2}\right)+2\left(2^{n-2}-2^{n-3}\right)+\ldots\left(n-1\right)\left(2^{1}-2^{\circ}\right)+n\left(2^{\circ}\right)$ $\Rightarrow 2^{n-1}+2^{n-2}+2^{n-3} +.........+2+1=2^{n}-1$ Thus $2^{n}-1=2047$ $\Rightarrow2^{n}=2048=2^{11}$ $\therefore n=11$