Free Math Questions Calculator Not Permitted Exam 3

Question 1

The mean number of students per classroom, y, at Central High School can be estimated using the equation y=0.8636 x + 27.227 , where x represents the number of years since 2004 and x less or equal than 10. Which of the following statements is the best interpretation of the number 0.8636 in the context of this problem?

Accepted Answer

The estimated yearly increase in the mean number of students per classroom

Answer

Choice D is correct. When an equation is written in the form y equals m x plus b comma the coefficient of the x-term (in this case 0.8636) is the slope. The slope of a linear equation gives the amount that the mean number of students per classroom (represented by y) changes per year (represented by x).

Choice A is not the correct answer. This answer may result from a misunderstanding of slope and y-intercept. The y-intercept of the equation represents the estimated mean number of students per classroom in 2004.

Choice B is not the correct answer. This answer may result from a misunderstanding of the limitations of the model. Students may see that x less or equal than 10 and erroneously use this statement to determine that the model finds the mean number of students in 2014.

Choice C is not the correct answer. This answer may result from a misunderstanding of slope. The student recognizes that slope models the rate of change, but may think that a slope of less than 1 represents a decreasing function.

Question 2

What is the value of a + 7 ?

Accepted Answer

10

Answer

This question is incomplete as the value of the variable 'a' is not provided. To find the value of the expression 'a + 7', a numerical value for 'a' must be known. If, for example, 'a' were equal to 3, then substituting this value into the expression would result in 3 + 7 = 10, matching the correct answer.

Question 3

Which of the following is equal to (14 - 2i) ( 7 + 12i ) ?

Accepted Answer

122 + 154i

Answer

Choice D is correct. Applying the distributive property to multiply the binomials yields the expression 98 plus 168 i minus 14 i minus 24 i squared. The note in the stem of the question reminds students that i equals square root of minus 1 end root comma therefore i squared equals minus 1. Substituting this value into the expression gives the student 98 plus 168 i minus 14 i minus left parenthesis minus 24 right parenthesis comma and combining like terms results in 122 plus 154 i.

Choice A is not the correct answer. This answer may result from a combination of errors. The student may not correctly distribute when multiplying the binomials, multiplying only the first terms together and the second terms together. The student may also think that i squared equals 1.

Choice B is not the correct answer. This answer may result from a misunderstanding of how to multiply binomials. The distributive property is not applied correctly and the student may only multiply the first two terms together and the last two terms together.

Choice C is not the correct answer. This answer results from misapplying the statement i equals square root of minus 1 end root.

Question 4

What is the value of 3 x plus 2 y ?

Accepted Answer

24

Answer

This question is incomplete because the numerical values for the variables 'x' and 'y' are not provided. To evaluate the algebraic expression '3x + 2y', specific values for both 'x' and 'y' are required. For instance, if x=4 and y=6, then 3(4) + 2(6) = 12 + 12 = 24, which would yield the correct answer.

Question 5

In the system of linear equations above, a is a constant. If the system has no solution, what is the value of a ?

Accepted Answer

6

Answer

Choice C is correct. If the system of equations has no solution, the graphs of the equations in the xy-plane are parallel lines. To be parallel, the lines must have the same slope, and this will be true if the expression a x minus 3 y is a multiple of the expression 1 half x minus 1 fourth y. Since minus 3 y equals 12 open parentheses minus 1 fourth y close parentheses comma the expression a x minus 3 y would have to be 12 times the expression 1 half x minus 1 fourth y. This means a x equals 12 open parentheses 1 half x close parentheses comma so a equals 6. The resulting system is 1 half x minus 1 fourth y equals 5 and 6 x minus 3 y equals 20 comma which is equivalent to 6 x minus 3 y equals 60 and 6 x minus 3 y equals 20 comma which has no solution.

Choice A is not the correct answer. This may result from the misconception that if each equation in a system has the same x-coefficient, the system cannot have a solution. But if a equals 1 half comma subtracting the two equations eliminates x and produces a solution to the system.

Choice B is not the correct answer. This may result from trying to make the second equation in the system a multiple of the first by looking at the ratio of the constants on the right sides, 20 over 5 comma and wrongly concluding that the second equation must be 4 times the first, which would give a equals 4 open parentheses 1 half close parentheses comma or a equals 2. But the two equations in a system are multiples only if the system has infinitely many solutions, not if the system has no solution. 

Choice D is not the correct answer. The student may have found the factor, 12, that multiplies the left side of the first equation to yield the left side of the second, but then neglected to find a equals 12 open parentheses 1 half close parentheses comma or a equals 6.

Math Practice Test

Math Practice Test

Free Math Questions Calculator Not Permitted Exam 3