Free Math Questions Calculator Not Permitted Exam 3 — Questions and Answers

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?

The estimated mean number of students per classroom in 2004
The estimated mean number of students per classroom in 2014
The estimated yearly decrease in the mean number of students per classroom
The estimated yearly increase in the mean number of students per classroom (Correct answer)

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

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).\r\n\r\nChoice 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.\r\n\r\nChoice 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.\r\n\r\nChoice 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 ?

10 (Correct answer)
12
7
51

Correct answer: 10

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 ) ?

74
122
74 + 154i
122 + 154i (Correct answer)

Correct answer: 122 + 154i

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.\r\n\r\nChoice 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.\r\n\r\nChoice 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.\r\n\r\nChoice 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 ?

24 (Correct answer)
34
54

Correct answer: 24

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 ?

2
6 (Correct answer)
12

Correct answer: 6

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.\r\n\r\nChoice 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.\r\n\r\nChoice 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. \r\n\r\nChoice 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.

Question 6: Anise needs to complete a printing job using both of the printers in her office. One of the printers is twice as fast as the other, and together the printers can complete the job in 5 hours. The equation above represents the situation described. Which of the following describes what the expression 1 over x represents in this equation?

The time, in hours, that it takes the slower printer to complete the printing job alone
The portion of the job that the slower printer would complete in one hour (Correct answer)
The portion of the job that the faster printer would complete in two hours
The portion of the job that the faster printer would complete in two hours

Correct answer: The portion of the job that the slower printer would complete in one hour

Choice B is correct. From the description given, 1 fifth is the portion of the job that the two printers, working together, can complete in one hour, and each term in the sum on the left side is the part of this 1 fifth of the job that one of the printers contributes. Since one of the printers is twice as fast as the other, 2 over x describes the portion of the job that the faster printer is able to complete in one hour and 1 over x describes the portion of the job that the slower printer is able to complete in one hour.\r\n\r\nChoice A is not the correct answer. The student may have not seen that in this context, the rates (that is, the work completed in a fixed time) of the printers can be added to get the combined rate, but the times it takes each printer to complete the job cannot be added to get the time for both printers working together, since the time for printers working together is less than, not greater than, the times for each printer alone. Hence the terms in the sum cannot refer to hours worked. In fact, the time it would take the slower printer to complete the whole job is x hours.\r\n\r\nChoice C is not the correct answer. The student may have seen that 1 over x is the smaller term in the sum, wrongly concluded that the smaller term must apply to the faster printer, and then assumed the 2 in the numerator of the second term implies the equation describes work completed in 2 hours. In fact, the portion of the job that the faster printer could complete in 2 hours is open parentheses 2 close parentheses open parentheses 2 over x close parentheses equals 4 over x.\r\n\r\nChoice D is not the correct answer. The student may have correctly seen that the value 1 fifth on the right side refers to the portion of the job completed, but not seen that in this context, the rates (that is, the work completed in a fixed time) of the printers can be added to get the combined rate, but the times it takes each printer to complete the job cannot be added to get the time for both printers working together. Hence the terms in the sum cannot refer to hours worked. In fact, the time it takes the slower printer to complete 1 fifth of the job is x over 5 hours.

Question 7: The equation of a circle in the xy-plane is shown above. What is the diameter of the circle?

26 (Correct answer)
144
12

Correct answer: 26

Completing the square yields the equation open parentheses x minus 3 close parentheses squared plus open parentheses y plus 4 close parentheses squared equals 169 comma the standard form of an equation of the circle. Understanding this form results in the equation r squared equals 169 comma which when solved for r gives the value of the radius as 13. Diameter is twice the value of the radius; therefore, the diameter is 26.

Question 8: What is one possible solution to the equation?

5 or 7 (Correct answer)
12
24
36

Correct answer: 5 or 7

Students should look for the best solution methods for solving rational equations before they begin. Looking for structure and common denominators will prove very useful at the onset, and will help prevent complex computations that do not lead to a solution.\r\n \r\nIn this problem, multiplying both sides of the equation by the common denominator open parentheses x plus 1 close parentheses open parentheses x minus 1 close parentheses yields 24 open parentheses x minus 1 close parentheses minus 12 left parenthesis x plus 1 right parenthesis equals left parenthesis x plus 1 right parenthesis left parenthesis x minus 1 right parenthesis.\r\n \r\nMultiplication and simplification then yields 12 x minus 36 equals x squared minus 1 comma or x squared minus 12 x plus 35 equals 0. Factoring the quadratic gives open parentheses x minus 5 close parentheses open parentheses x minus 7 close parentheses equals 0 comma so the solutions occur at x equals 5 and x equals 7 comma both of which should be checked in the original equation to ensure that they are not extraneous. In this case, both values are solutions.

Free Math Questions Calculator Not Permitted Exam 3

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?