This exam was adminstered in June 2022.
More Regents problems.
June 2022 Algebra 2 Regents
Part I
Each correct answer will receive 2 credits. No partial credit.
1. For all positive values of x, which expression is equivalent to x3/4?
(1) ∜(x3)
(2) ∛(x4)
(3) (x3)4
(4) 3(x4)
Answer: (1) ∜(x3)
The 4 in the denominator of the fraction in the exponent refers to taking the fourth root of the expression. The 3 in the numberator is the third power. You can rewrite 3/4 as 3 * (1/4), which says third power and the fourth root. This is Choice (1).
Choice (2) inverts the fraction. It is the same as x4/3. Eliminate Choice (2).
Choice (3) is equivalent to x12. Eliminate Choice (3).
Choice (4) is just flat out wrong. You'd have to make multiple errors to arrive at it.
2. Mrs. Favata's statistics class wants to conduct a survey to see how students feel about changing the school
mascot's name. Which plan is the best process for gathering an appropriate sample?
(1) Survey students in a random sample of senior homerooms.
(2) Survey every tenth student entering art classes in the school.
(3) Survey every fourth student entering the cafeteria during each lunch period.
(4) Survey all members of the school's varsity sports teams.
Answer: (3) Survey every fourth student entering the cafeteria during each lunch period.
If you want the least biased, and the most random, sample, then the cafeteria is the best choice. Everyone (or just about everyone) goes there, so sampling every fourth student would be a good sample.
In Choice (1), sampling only seniors ignore 3/4 of the student body. Worse, seniors are graduating, so they may have less interest in a new mascot the following year.
In Choice (2), only art students are being asked. None of the student who do not take art are being surveyed.
In Choice (4), the survey is limited to sports teams, who would likely have stronger opinions about the mascot than the non-sports students who are not being surveyed.
3. Given x =/= 3, the expression (2x3 + 7x2 - 3x - 25) / (x + 3) is equivalent to
(1) 2x2 + x - 6 - 7/(x + 3)
(2) 2x2 + 13x - 36 + 83/(x + 3)
(3) 2x2 + x - 13
(4) x2 + 4x - 15 + 20/(x + 3)
Answer: (1) 2x2 + x - 6 - 7/(x + 3)
Since 3 is not a factor of 25, the polynomial canno be divided by (x + 3) evenly, so eliminate Choice (1). Additionally, since 2x3 divided by x is 2x2, not x2, Choice (4) can be eliminated.
If you multiply (x + 3)(2x), you get 2x2 + 6x2. If you subtract that from 2x3 + 7x2, you get x2. So the first term in the quotient is 2x2 and the second term will be x, not 13x. The correct response is Choice (1).
4. In a group of 40 people, 20 have brown hair, 22 have blue eyes, and 15 have both brown hair and blue eyes. How
many people have neither brown hair nor blue eyes?
(1) 0
(2) 13
(3) 27
(4) 32
Answer: (2) 13
If you add the number of peope with brown hair to the number of people with blue eyes, you have 20 + 22 = 42 people. However, you counted 15 of them twice, so subtract 15.
42 - 15 = 27 people have either brown hair or blue eyes. So the number that have neither is 40 - 27 = 13, which is Choice (2).
You can see the trap they set by using 27 as one of the incorrect responses.
5. Consider the function y = h(x), defined by the graph below.
Which equation could be used to represent the graph shown below?
(1) y = h(x) − 2
(2) y = h(x − 2)
(3) y = −h(x)
(4) y = h(−x)
Answer: (3) y = −h(x)
In the first graph, all the y values are positive. In the second graph, the line has been reflected over the x-axis and all the corresponding values are negative.
All of the values of h(x) have been multiplied by -1, which is Choice (3).
In Choices (1), (2) and (4), the curve would still be heading upward. Choice (1) would lower it 2 units. Choice (2) would shift it two units to the right. Choice (4) would reflect it over the y-axis so that it started at infinity and declined toward 0.
6. For the polynomial p(x), if p(3) = 0, it can be concluded that
(1) x + 3 is a factor of p(x)
(2) x − 3 is a factor of p(x)
(3) when p(x) is divided by 3, the remainder is zero
(4) when p(x) is divided by −3, the remainder is zero
Answer: (2) x − 3 is a factor of p(x)
If (x - 3) is a factor of p(x) then (3 - 3) = 0 and the polynomial will have a product of 0.
Choice (1) is a simple misconception confusion the zeroes of a function.
Choices (3) and (4) suggest that either 3 or -3 are factors of p(x), but neither would affect any specific value of x, such as x = 3.
7. The solution to the equation 5ex + 2 = 7 is
(1) -2 + ln(7/5)
(2) (ln 7/ln 5) - 2
(3) -3/5
(4) -2 + ln(2)
Answer: (1) -2 + ln(7/5)
Working backward, we get
5ex + 2 = 7
ex + 2 = 7/5
ln(ex + 2) = ln(7/5)
x + 2 = ln(7/5)
x = ln(7/5) - 2 = -2 + ln(7/5)
Choice (1) is the correct answer.
Note that ln(7/5) = ln(7) - ln(5). However, it is NOT TRUE that ln(7/5) = ln(7)/ln(5). Choice (2) is a trap.
8. Consider the system of equations below?
−x − 3y + 2z = 0
2x − 4y + z = 10
What is the solution to the given system of equations?
(1) (1,1,2)
(2) (3,−1,0)
(3) (5,−1,2)
(4) (3,5,8)
Answer: (2) (3,−1,0)
Substitution is probably the quickest method of solving the problem.
x = 1, y = 1, x = 2
1 + 2(1) - 2 = 1 + 2 - 2 = 1, check
-(1) - 3(1) + 2(2) = -1 - 3 + 4 = 0, check
2(1) - 4(1) + 2 = 2 - 4 + 2 =/= 10. Eliminate Choice (1).
x = 3, y = -1, x = 0
3 + 2(-1) - 0 = 3 - 2 - 0 = 1, check
-(3) - 3(-1) + 2(0) = -3 + 3 + 0 = 0, check
2(3) - 4(-1) + 0 = 6 + 4 + 0 = 10, check.
Choice (2) is the solution.
More to come.
Comments and questions welcome.
More Regents problems.
No comments:
Post a Comment