Question

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

• A. $10$
• B. $11$
• C. $12$
• D. $13$

Answer 1

The correct option is B

$11$

Explanation for correct option:

Finding the value of $\mathbf{n}$:

Let's write the form of,

$$\begin{gathered} 2^0=2^1-1 \\ {2}^0+2^1=2^2-1 \\ {2}^0+2^1+2^2=2^3-1 \end{gathered}$$

Given that, the sum of exponents of $2$ is one less than the higher power.

According to the question, the total wrong answers given in the question are $2047$.

Since,

$2^{11}=2048$

Similarly, we can also write,

$2^{11}=2047+1$

$2047=2^{11}–1$

Therefore, $n=11$.

Hence, the correct option is B.