Question

There are \[30\] questions in a multiple choice test. A student gets \[1\] mark for each unattempted question, \[0\] mark for each wrong answer and \[4\] marks for each correct answer. A student answered \[x\] questions correctly and scored \[60\]. Then the number of possible values of \[x\] is
\[\begin{align}
  & \left( A \right)15 \\
 & \left( B \right)10 \\
 & \left( C \right)6 \\
 & \left( D \right)5 \\
\end{align}\]

Answer 1

Hint: We are going to take the two possible cases of the values of \[x\], where \[x\] is minimum and maximum , and we get \[15\] and \[10\] as the values of \[x\] . and then, take the values to be \[6\] and \[5\] and calculate the total marks.

Formulae Used:
For finding the maximum value of \[x\], consider rest questions be unattempted,
\[x\times 4=\]total marks
And for minimum, let the rest be wrong
\[x\times 4+\left( 30-x \right)\times 1=\]total marks

Complete step by step solution:
For \[x\] correct options, the marks that the student gets for the correct answers:
\[x\times 4\]
As the total number of the questions is \[30\], therefore, the number of rest of the answers is:
\[30-x\]
So, if the rest questions are wrong, the marks that the student gets for is \[0\] as given
In this case,
\[\begin{align}
  & x\times 4=60 \\
 & \Rightarrow x=15 \\
\end{align}\]
Now, if the rest questions are unanswered,
\[\begin{align}
  & x\times 4+\left( 30-x \right)=60 \\
 & \Rightarrow 4x+30-x=60 \\
 & \Rightarrow 3x=30 \\
 & \Rightarrow x=10 \\
\end{align}\]
Now considering the other two options,
If the student answers \[6\] questions correctly, then, the rest \[24\] questions are left, now if they are unattempted, then the total marks
\[6\times 4+24\times 1=24+24=48\ne 60\]
And if the student answers \[5\] questions correctly, then the rest \[25\] questions are left, now if they are unattempted, then the total marks
\[5\times 4+25\times 1=20+25=45\ne 60\]
Therefore, only possible values of \[x\] are \[15\] and \[10\].

Note: Finding the maximum and minimum values of \[x\] only tells us that the two options are correct, it is important to check whether other options are correct or not. This is found by calculating the total marks for both the cases which gives that only two options are correct.