This exam was adminstered in January 2025.
More Regents problems.
January 2025 Algebra 2 Regents
Part I
Each correct answer will receive 2 credits. No partial credit.
17. If 4(105x-2) = 12, then x equals
(1) 2.3/5
(2) 1/3( log12/log40 + 5)
(3) (log(3) + 2) / 5
(4) 1/5 (log12/log4 + 2)
Answer: (3) (log(3) + 2) / 5
The rules for logarithms are derived from the rules of exponents. Remember that if you have difficulty. Use inverse operations to find x.
4(105x-2) = 12
105x-2) = 3
log103 = 5x - 2
log(3) + 2 = 5x
(log(3) + 2) / 5 = x
This is Choice (3).
18. A random sample of 152 students was surveyed on a particular day
about how they got to school. The survey results are summarized in
the table below.
Which statement is best supported by the data?
(1) The probability of being late given that a student walked is greater
than the probability that a student walked given that the student
was late.
(2) The probability of being late given that a student walked is less
than the probability that a student walked given that the student
was late.
(3) The probability of being late given that a student walked is equal
to the probability that a student walked given that the student was
late.
(4) The probability of being late given that a student walked cannot be
determined.
Answer: (1) The probability of being late given that a student walked is greater than the probability that a student walked given that the student was late.
Write in the subtotals and the grand total to make the calculations easier:
In Choice (1), the probability of being late given that a student walked is 4/22. The probability that a student walked given that the student was late is 4/30. The first probability is greater than the second, so this is the correct answer.
Choice (2) says the first should be less, but it is not. Choice (3) says that the two are equal, but they are not. Choice (4) says the probability cannot be determined, but it can be. (We just did it!)
The correct answer is Choice (1).
19. If f(x) = 3√(x) + 4, then f-1(x) equals
(1) 3√(x - 4)
(2) (x - 4)3
(3) x3 + 1/4
(4) -3√(x) - 4
Answer: (2) (x - 4)3
Substitute the x with a y, then set the expression equal to x. Finally, rewrite the equation to solve for y.
x = 3√(y) + 4
x - 4 = 3√(y)
(x - 4)3 = y
The correct answer is Choice (1).
20. Given the equation S(x) = 1.7sin(bx) + 12, where the period of S(x)
is 12, what is the value of b?
(1) π/6
(2) 24π
(3) π/12
(4) 6π
Answer: (1) π/6
The period of a sine function is 2π/b. We want to find b. The "+ 12" in the function is meaningless to this problem because it has nothing to do with the period, which is also 12.
2π/b = 12
2π/12 = b
π/6 = b
Choice (1) is the correct answer.
21. Jin solved the equation √(4 - x) = x + 8 by squaring both sides. What extraneous solution did he find?
(1) -5
(2) -12
(3) 3
(4) 4
Answer: (2) -12
The extraneous value would be one that wouldn't make a valid equation. For examples, if a "solution" makes the expression x + 8 into a negative number, then that solution is extraneous because the value of the square root cannot be negative. The only possible answer from the four choices is (2) -12.
If you wanted to solve it to check (or if there had been more possibilities):
√(4 - x) = x + 8
4 - x = (x + 8)2
4 - x = x2 + 16x + 64
0 = x2 + 17x + 60
0 = (x + 12)(x + 5)
x + 12 = 0 or x + 5 = 0
x = -12 or x = -5
The solution x = -5 is valid because √(4 - (-5) = (-5) + 8 is a true statement becasue 3 = 3.
However, √(4 - (-12)) = (-12) + 8 is not a true statement because 4 =/= -4.
The correct answer is Choice (2).
22.The expression (x2 + y2)2 is not equivalent to
(1) (x2 - y2)2 + (2xy)2
(2) (x + y)4 + 2(xy)2
(3) x2(x2+2y2) + (y2)2
(4) (2x2 + y2)2 - (3x4 + 2x2y2)
Answer: (2) (x + y)4 + 2(xy)2
If you want a calculator "shortcut", replace x with a value such as 5 and y with a value such as 7 (note that 5 and 7 are mutually prime) and calculate all five expressions to see which doesn't match.
On first look, my "educated guess" would be Choice (2), but let's check all of them:
(x2 + y2)2 = x4 + 2x2y2 + y4
In Choice (1) (x2 - y2)2 + (2xy)2
= x4 - 2x2y2 + y4 + (4xy)2
= x4 + 2x2y2 + y4
In Choice (2), (x + y)4 is NOT equal to x4 + y4.
(x + y)4 + 2(xy)2
= x4 + 4x3y + 6x2y2 + 4xy3 + y4 + 4x2y2
This is totally different from the original expression. Choice (2) is the correct answer.
In Choice (3) x2(x2+2y2) + (y2)2
= x4 + 2x2y2 + y4
In Choice (4) (2x2 + y2)2 - (3x4 + 2x2y2)
= 4x4 + 4x2y2 + y4 - 3x4 - 2x2y2
= x4 + 2x2y2 + y4
23. The height of a running trail is modeled by the quartic function
y = f(x) shown below, where x is the distance in miles from the start of
the trail and y is the height in feet relative to sea level.
If this trail has a minimum height of 16 feet below sea level, which
function(s) could represent a running trail whose minimum height is
half of the minimum height of the original trail?
(1) I, only
(2) II, only
(3) I and III
(4) II and III
Answer: (4) II and III
Half of the minimum height would be 8 feet below sea level instead of 16 feet. That mean that the minimum is either raised by 1/2 or raised by 8 feet.
In function I, putting 1/2 in front of x inside the parentheses, the function has changed the period but hasn't changed either the baseline or the amplitude. Function I is no good, so you can eliminate Choices (1) and (3).
Function II must be true because it's in both of the remaining choices. Adding 8 to the y value will raise the minimum from -16 to -8.
Function III shrinks the y value by 1/2, so -16 will become -8. Function III fits the description.
Choice (4) is the correct answer.
24. The crew aboard a small fishing boat caught 350 pounds of fish on Monday. From that Monday through the end of the week on Friday, the weight of the fish caught increased 15% per day. The total weight, in pounds, of fish caught is approximately
(1) 411
(2) 612
(3) 1748
(4) 2360
Answer: (4) 2360
Use the formula for the Geometric Series to find the sum:
S5 = (350 - 350(1.15)5) / (1 - 1.15)
P = 2359.833...
The correct answer is Choice (4).
End of Part I. Comments and questions welcome.
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