This exam was adminstered in August 2024.
More Regents problems.
August 2024 Algebra 2 Regents
Part I
Each correct answer will receive 2 credits. No partial credit.
17. Given q(x) = 2log(x) and r(x) = (x - 2)3 - 4, what is a solution of
q(x) = r(x) to the nearest tenth?
(1) 1.1
(2) 3.7
(3) 3.9
(4) 4.3
Answer: (2) 3.7
This is a MULTIPLE-CHOICE question. Do not attempt to solve it algebraically. It will NOT be a fun time.
You can graph both equations on your calculator and look for the point where they intersect. (There's actually a function on the calculator that will help you find it if you know how to use it.)
You can also find the values of q(1.1), r(1.1), q(3.7), r(3.7), etc to find the ones that are the same.
Let's do this just for giggles:
q(1.1) = 0.08, r(1.1) = -4.7
q(3.7) = 1.1, r(3.7) = .91
q(3.9) = 1.2, r(3.9) = 2.9
q(4.3) = 1.7, r(4.3) = 8.2
You will notice that when x = 3.7, the values of q(x) and r(x) are the closest. They differ because of rounding. (I did a little experimenting so that you wouldn't have to, and found that x = 3.725 is pretty close but still not exact.)
Choice (2) is the correct answer.
How would I solve this algebraically? I wouldn't. Simple as that.
If you would like to, please place it in the comments.
18. The volume of a cardboard box can be modeled by V(x), which is the product of the length, width, and height, x. If the length can be represented by L(x) = 18 - 2x and the width can be represented by W(x) = 18 - 2x, then which function represents V(x)?
(1) V(x) = 4x2 - 72x + 324
(2) V(x) = 4x3 - 72x2 + 324x
(3) V(x) = -3x + 36
(4) V(x) = 4x3 + 324x
Answer: (2) V(x) = 4x3 - 72x2 + 324x
It should be "obvious" that since we have to multiply three terms with the variable x in them that the final formula will contain x3 in it.
V = (18 - 2x)(18 - 2x)(x)
= (324 - 36x - 36x + 4x2)(x)
= 324x - 72x2 + 4x3
Choice (2) is the correct choice.
Choice (4) is missing the middle term. Choice (1) is the Area of the base. Choice (3) is nonsense because it just added the three expressions.
19. The expression 8x/2 * 8x/3 is equivalent to
(1) 6√(85x)
(2) 64(5x/6)
(3) 5√(82x)
(4) 64(x2/6)
Answer: (1) 6√(85x)
Keep the base and add the exponents x/2 + x/3.
The common denominator is 6, so 3x/6 + 2x/6 = 5x/6, and the expression will be 85x/6, which is not one of the listed choices.
However, the one-sixth power is the same as taking the sixth root of the expressions.
Therefore, 85x/6 is the same as (85x)(1/6 which is the same as 6√(85x). This is Choice (1).
20. If θ is an angle in standard position whose terminal side passes through the point (-3,-4), which statement is true?
(1) sec θ > 0 and tan θ > 0
(2) sec θ < 0 and tan θ < 0
(3) sec θ > 0 and tan θ < 0
(4) sec θ < 0 and tan θ > 0
Answer: (4) sec θ < 0 and tan θ > 0
Point (-3,-4) is in Quadrant III where sin is negative, cos is negative, and tan is positive. Eliminate Choices (2) and (3).
Sec = 1/cos and has the same sign and cos, which is negative in Quadrant III. So the correct answer is Choice (4).
21. What is the value of y for the system shown below?
2x + 3y - z = -3
6x - y + 4z = 3
(1) -27
(2) 6
(3) 3
(4) -3
Answer: (3) 3
Solve the system of equations by eliminating the x term first since 2 and 3 are factors of 6.
3x + 4y - 5z = -27
2x + 3y - z = -3
6x - y + 4z = 3
6x + 8y - 10z = -54
6x + 9y - 3z = -9
6x - y + 4z = 3
First 2 equations: y + 7z = 45
Last 2 equations: 10y - 7y = -12
Combined: 11y = 33
y = 3
Choice (3) is the correct answer.
22. The number of employees who work nights and weekends at a department store is summarized in the table below.
Let N represent the event “works nights” and let W represent the event “works weekends.” Based on the table, are N and W independent events?
(1) Yes, because P(N) • P(W) = P(N ∩ W).
(2) Yes, because P(N) • P(W) =/= P(N ∩ W).
(3) No, because P(N) • P(W) = P(N ∩ W).
(4) No, because P(N) • P(W) =/= P(N ∩ W).
Answer: (1) Yes, because P(N) • P(W) = P(N ∩ W).
The answer is either Yes because they are equal or No because they are not equal. Multiply the probabilities.
There are a total of 120 employees. P(N) = 20/120 and P(W) = 48/120, and (20/120)(48/120) = (1/6)(2/5) = 1/15.
P(N ∩ W) = 8/120 = 1/15.
So P(N) • P(W) = P(N ∩ W).
The correct answer is Choice (1).
23. Which expression is equivalent to x8 - y8?
(1) (x - y)8
(2) (x2 + y2)2(x2 - y2)2
(3) (x4 + y4)(x2 + y2)(x + y)(x - y)
(4) (x + y)4(x - y)4
Answer: (3) (x4 + y4)(x2 + y2)(x + y)(x - y)
The given expression is a Difference of Perfect Squares and canb be factored into two conjugates. One of those factors will still be a Difference of Perfect Squares and can be factored again. And then again.
x8 - y8
(x4 + y4)(x4 - y4)
(x4 + y4)(x2 + y2)(x2 - y2)
(x4 + y4)(x2 + y2)(x + y)(x - y)
The correct answer is Choice (3).
If you factor Choice (1), you will find that it is equivalent to Choice (4), so both could be eliminated.
24. A research assistant receives a first year salary of $90,000 and a 2% annual raise throughout the first ten years of employment. In total, how much money will be earned over the first ten years, to the nearest dollar?
(1) $91,837
(2) $109,709
(3) $877,917
(4) $985,475
Answer: (4) $985,475
Using the formula for a Geometric Series: Sn = (a1 - a1rn) / (1 - r), we can calculate the answer.
S10 = (90000 - 90000*1.0210) / (1 - 1.02) = 985474.89, which is Choice (4).
End of Part I. How did you do?
Comments and questions welcome.
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