This problem requires us to determine the number of multiple choice questions (MCQs) in a test, given the total number of questions, total marks, and the marks awarded for each type of question.

Setting Up the Equations for the Test

Let's define the variables:

• Let $m$ represent the number of multiple choice questions.
• Let $e$ represent the number of essay questions.

We are given the following information:

• The total number of questions is 20.
• Each MCQ is worth 3 marks.
• Each essay question is worth 11 marks.
• The total marks for the test is 100.

From this information, we can set up two equations:

1. Equation for the total number of questions:

   $$m + e = 20$$

2. Equation for the total marks:

   $$3m + 11e = 100$$

Solving for the Number of Multiple Choice Questions

We can use the first equation to express $e$ in terms of $m$:

$$e = 20 - m$$

Now, substitute this expression for $e$ into the second equation:

$$3m + 11(20 - m) = 100$$

Distribute the 11:

$$3m + 220 - 11m = 100$$

Combine the terms involving $m$:

$$(3 - 11)m + 220 = 100$$

$$-8m + 220 = 100$$

Subtract 220 from both sides of the equation:

$$-8m = 100 - 220$$

$$-8m = -120$$

Divide both sides by -8 to solve for $m$:

$$m = \frac{-120}{-8}$$

$$m = 15$$

Verification

To ensure our answer is correct, let's find the number of essay questions and check if the total marks match.

Number of essay questions, $e = 20 - m = 20 - 15 = 5$.

Total marks = (Number of MCQs × Marks per MCQ) + (Number of Essay Questions × Marks per Essay Question)

Total marks = ($$15 \times 3$$) + ($$5 \times 11$$)

Total marks = $$45 + 55$$

Total marks = $$100$$

The calculated total marks match the given total marks, confirming our result.

Conclusion

Therefore, the exam has 15 multiple choice questions.