This exam was adminstered in June 2022.
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June 2022
Part I
Each correct answer will receive 2 credits. No partial credit.
17. The expression (m - 3)2 is equivalent to
(1) m2 + 9
(2) m2 - 9
(3) m2 - 6m + 9
(4) m2 - 6m - 9
Answer: (3) m2 - 6m + 9
The expression (m - 3)2 is (m - 3)(m - 3). If you "FOIL" it (I hate that term), you will get m2 - 3m - 3m + 9, which is Choice (3).
Choice (2) is the product of conjugates (m + 3)(m - 3).
Choice (1) has no roots and Choice (4) has irrational roots.
18. Mrs. Rossano asked her students to explain why (3,-4) is a solution
to 2y + 3x = 1. Three student responses are given below.
Andrea:
“When the equation is graphed on a calculator, the point can be found
within its table.”
Bill:
“Substituting x = 3 and y = -4 into the equation makes it true.”
Christine:
“The graph of the line passes through the point (3,-4).”
Which students are correct?
(1) Andrea and Bill, only
(2) Bill and Christine, only
(3) Andrea and Christine, only
(4) Andrea, Bill, and Christine
Answer: (4) Andrea, Bill, and Christine
The whole point of tables of values and graphs and points on the line is that they are all solutions to an equation. This means that if you substitute the coordinates/values into the equation, you will get a true statement.
All three of the above statements are correct, so the correct response is Choice (4).
19. Four quadratic functions are shown below.
(1) The maximum of f(x) is less than the maximum of j(x).
(2) The maximum of g(x) is less than the maximum of h(x).
(3) The maximum of f(x) equals the maximum of g(x).
(4) The maximum of h(x) equals the maximum of j(x).
Answer: (3) The maximum of f(x) equals the maximum of g(x).
The maximums of f(x), g(x) and h(x) are "obvious" in that you can tell by looking (if you know what you're looking for). For j(x), you may want to graph it or find the axis of symmetry.
The maximum of f(x) is 5, when x = -1. It's in the table.
The maximum of g(x) is 5, when x = 4. The function is in vertex form.
The maximum of h(x) is 4, when x = 1. It's on the graph.
The maximum of j(x) occurs when x = -1/(2(1/2) = -1. It is -1/2(-1)2 + (-1) + 4 = 2.5.
The maximum of f(x) is not less than j(x). The maximum of g(x) is not less than h(x). The maximums of f(x) and g(x) ARE equal, so this is the correct choice. The maximums of h(x) and j(x) are not equal.
20. An example of a sixth-degree polynomial with a leading coefficient of
seven and a constant term of four is
(1) 6x7 - x5 + 2x + 4
(2) 4 + x + 7x6 - 3x2
(3) 7x4 + 6 + x2
(4) 5x + 4x6 + 7
Answer: (2) 4 + x + 7x6 - 3x2
The leading coefficient is the number in front of the variable with the largest exponent. When the polynomial is written in standard form, the term with the largest exponent goes first.
Choice (1) has a leading coefficient of 6 and a constant of 4.
Choice (2) has a leading coefficient of 7 and a constant of 4. This is the correct answer.
Choice (3) has a leading coefficient of 7 and a constant of 6.
Choice (4) has a leading coefficient of 4 and a constant of 7.
21. In the equation A = P(1 + r)t, A is the total amount, P is the principal amount, r is the annual interest rate, and t is the time in years. Which statement correctly relates information regarding the annual interest rate for each given equation?
(1) For A = P(1.025)t, the principal amount of money is increasing at a 25% interest rate.
(2) For A = P(1.0052)t, the principal amount of money is increasing at a 52% interest rate.
(3) For A = P(0.86)t, the principal amount of money is increasing at a 14% interest rate.
(4) For A = P(0.68)t, the principal amount of money is increasing at a 68% interest rate.
Answer: (3) For A = P(0.86)t, the principal amount of money is increasing at a 14% interest rate.
You want to find the value of r and if it is positive or negative, and then turn r into a percentage by multiplying by 100%.
Choice (1) shows an increase at a rate of 2.5%, not 25%.
Choice (2) shows an increase at a rate of 0.52%, not 52%.
Choice (3) shows an decrease with 86% remaining, so the rate is 100 - 86 = 14%. This is the correct answer.
Choice (4) shows 68% of the money remaining, meaning that there was a decrease of 32%.
22. It takes Tim 4.5 hours to run 50 kilometers. Which expression will allow him to change this rate to minutes per mile?
Answer: (1) [See image]
I overheard some students complain about this question because they didn't know how to convert miles and kilometers. But they didn't need to know how. It's given in the choices. The problem is that many students don't learn about simplifying units. Too often, units are stripped off the question and tacked on again when there's an answer.
Units can be multiplied, divided and "cancelled" like any other factor can.
Look at the image below, which shows which factors can be cancelled. Which one results in miles per minute?
Choice (1) shows minutes per mile. Choice (1) is the correct answer.
Choice (2) shows a mess -- (miles minutes) per hour2.
Choice (3) shows miles per minute.
Choice (4) shows another mess -- (miles minutes) per km2.
23. When the equation (x - 1) / 2 - a / 4 = (3a) / 4 is solved for x in terms of a, the solution is
(1) (3a)/2 + 1
(2) a + 1
(3) (4a + 1) / 2
(4) 2a + 1
Answer: (4) 2a + 1
To isolate x, use inverse opereations.
(x - 1) / 2 - a / 4 = (3a) / 4
(x - 1) / 2 = (3a) / 4 + a / 4
(x - 1) / 2 = (4a) / 4
(x - 1) / 2 = a
x - 1 = 2a
x = 2a + 1
24. If a sequence is defined recursively as a1 = -3 and an = -3an-1 - 2, then a4 is
(1) -107
(2) -95
(3) 55
(4) 67
Answer: (4) 67
Each number in the sequence is -3 times the previous number minus 2.
a1 = -3
a2 = -3(-3) - 2 = 7
a3 = -3(7) - 2 = -23
a4 = -3(-23) - 2 = 67
End of Part I. How did you do?
Comments and questions welcome.
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