Answer :
A test is to be conducted with certain types of questions and each type of question weighs certain number of points.
A test would consist of two types of questions. These two types will be assigned variables that will denote the number of questions respectively as follows:
[tex]\begin{gathered} \text{True and False: x} \\ \text{MCQS : y} \end{gathered}[/tex]We are given that the entire test will consits of 20 questions. We can express the total number of questions on the test in terms of number of True and False questions ( x ) and number of MCQS ( y ) as follows:
[tex]\begin{gathered} \text{Total number of Questions = True and False + MCQS} \\ \textcolor{#FF7968}{20}\text{\textcolor{#FF7968}{ = x + y }}\textcolor{#FF7968}{\ldots Eq1} \end{gathered}[/tex]Further information is given to us in the questions regarding the number of points aloted to each type. The total weightage of each type of question on the test can be expressed as a product of ( number of each type * point weight of each type ).
The point weights for each type of questions are:
[tex]\begin{gathered} \text{True and False ( x ) : 3 points each} \\ \text{MCQs ( y ) : 11 points each} \end{gathered}[/tex]The total weights of each types of questions are:
[tex]\begin{gathered} \text{True and False ( points ) = 3}\cdot x \\ \text{MCQS ( points ) = 11}\cdot x \end{gathered}[/tex]We are given that the entire test is worth ( 100 points ). We express the total number of points of the test in terms of total weight of each type of question as follows:
[tex]\begin{gathered} test\text{ points = True and False ( points ) + MCQS ( points )} \\ \textcolor{#FF7968}{100}\text{\textcolor{#FF7968}{ = 3}}\textcolor{#FF7968}{\cdot x}\text{\textcolor{#FF7968}{ + 11}}\textcolor{#FF7968}{\cdot y\ldots}\text{\textcolor{#FF7968}{ Eq2}} \end{gathered}[/tex]We have two equations that express the total number of questions ( Eq 1 ) and total points ( Eq2 ) of the test in terms of number of True and False questions ( x ) and number of MCQs on the test ( y ).
[tex]\begin{gathered} \textcolor{#FF7968}{x}\text{\textcolor{#FF7968}{ + y = 20 }}\textcolor{#FF7968}{\ldots Eq1} \\ \textcolor{#FF7968}{3x}\text{\textcolor{#FF7968}{ + 11y = 100 }}\textcolor{#FF7968}{\ldots}\text{\textcolor{#FF7968}{ Eq2}} \end{gathered}[/tex]We will solve the above two equations simultaneously using Elimination method.
Step1: Multiply Eq1 with ( -3 )
[tex]\begin{gathered} -3\cdot\text{ ( x + y ) = -3}\cdot20 \\ \textcolor{#FF7968}{-3x}\text{\textcolor{#FF7968}{ - 3y = -60 }}\textcolor{#FF7968}{\ldots}\text{\textcolor{#FF7968}{ Eq3}} \end{gathered}[/tex]Step2: Add Eq 3 into Eq 2
[tex]\begin{gathered} -3x\text{ - 3y = -60 } \\ 3x\text{ + 11y = 100} \\ =========== \\ 8y\text{ = 40 } \\ \textcolor{#FF7968}{y}\text{\textcolor{#FF7968}{ = 5}} \\ =========== \end{gathered}[/tex]Step3: Back susbtitue the value of ( y ) into ( Eq1 )
[tex]\begin{gathered} x\text{ + ( 5 ) = 20 } \\ \textcolor{#FF7968}{x}\text{\textcolor{#FF7968}{ = 15 }} \end{gathered}[/tex]Therefore, the number of each type of questions that must be put on the test should be.
[tex]\begin{gathered} \text{\textcolor{#FF7968}{True and False ( x ) = 15}} \\ \text{\textcolor{#FF7968}{MCQs ( y ) = 5}} \end{gathered}[/tex]