This exam was adminstered in January 2024.

More Regents problems.

__January 2024 Algebra 2 Regents__

__January 2024 Algebra 2 Regents__

__Part I __

__Part I__

Each correct answer will receive 2 credits. No partial credit.

*17. Which equation does not represent an identity?
(1) x*

^{2}- y

^{2}= (x + y)(x - y) (2) (x - y)

^{2}= (x - y)(x - y) (3) (x + y)

^{2}= x

^{2}+ 2xy + y

^{2}(4) (x + y)

^{3}= x

^{3}+ 3xy + y

^{3}

**Answer: (4) (x + y) ^{3} = x^{3} + 3xy + y^{2} **

Choice (4) does not show the expansion of a binomial raised to the third power. The exponents for each term should add up to 3, but the middle term is 3xy, with two exponents of 1.

Choice (1) is the Difference of Squares rule. Eliminate Choice (1).

Choice (2) is the definition of squaring a binomial (or squaring anything). Eliminate Choice (2).

Choice (3) is the expansion of a binomial squared: x^{2} + xy + xy + y^{2}. Eliminate Choice (3).

The third line of Pascal's Triangle is 1 3 3 1, and the expansion of (x + y)^{3} is x^{3} + 3x^{2}y + 3xy^{2} + y^{3}.

The correct answer is Choice (4).

*18. Two surveys were conducted to estimate the proportion of teens who use social media at least once per day.*

Based on these results, it was determined that approximately 75% of teens use social media at least once per day. What is the best explanation of the difference in the results between the two surveys?

(1) The smaller sample size of five teens resulted in a smaller margin of error and should provide a more accurate estimate.

(2) The smaller sample size of five teens resulted in a bigger margin of error and should provide a more accurate estimate.

(3) The larger sample size of 50 teens resulted in a smaller margin of error and should provide a more accurate estimate.

(4) The larger sample size of 50 teens resulted in a bigger margin of error and should provide a more accurate estimate.

Based on these results, it was determined that approximately 75% of teens use social media at least once per day. What is the best explanation of the difference in the results between the two surveys?

(1) The smaller sample size of five teens resulted in a smaller margin of error and should provide a more accurate estimate.

(2) The smaller sample size of five teens resulted in a bigger margin of error and should provide a more accurate estimate.

(3) The larger sample size of 50 teens resulted in a smaller margin of error and should provide a more accurate estimate.

(4) The larger sample size of 50 teens resulted in a bigger margin of error and should provide a more accurate estimate.

**Answer: (3) The larger sample size of 50 teens resulted in a smaller margin of
error and should provide a more accurate estimate. **

A larger sample size means that the results will be more accurate and result is a smaller margin of error.

Five students is too small a sample size to be representative of the entire population.

*19. Given f(x) = x*

(1) 1

(2) -1

(3) 3

(4) -3

^{3}- 3 and f^{-1}= ∛(x - 3b), the value of b is(1) 1

(2) -1

(3) 3

(4) -3

**Answer: (2) -1 **

The definition of inverse functions is that f

^{-1}(f(x)) = x. The inverse undoes the function and yields the original input.

You can plug in the choices and see which one works, or you can rewrite the equation.

y = x^{3} - 3

x = y^{3} - 3

x + 3 = y^{3}

∛(x + 3) = y

So if 3 = -3b, then b = -1

The correct Choice is (2).

*20. Robert is buying a car that costs $22,000. After a down payment of $4000, he borrows the remainder from a bank, a six year loan at 6.24% annual interest rate. The following formula can be used to calculate his monthly loan payment.*
R = ((P)(i)) / (1 - (1 + i)

R = monthly payment

P = loan amount

i = monthly interest rate

t = time, in months

Robert’s monthly payment will be

(1) $298.31

(2) $300.36

(3) $307.35

(4) $367.10

^{-t}

R = monthly payment

P = loan amount

i = monthly interest rate

t = time, in months

Robert’s monthly payment will be

(1) $298.31

(2) $300.36

(3) $307.35

(4) $367.10

**Answer: (2) $300.36 **

Make the substitutions and put the expression into your calculator.

Note that the amount borrowed is 22000 - 4000 = 18000, 6.24% is the annual rate, which must be divided by 12, and the number of months is 6 years times 12 months / year, or 72.

R = ( (18000)(0.0624/12) ) / (1 - (1 + 0.0625/12)^{-72}

R = 300.355 or $300.36, which is Choice (2).

*21. Given tan θ = -4/3 where π/2 < θ < π, what is the value of sec θ?*

(1) -5/3

(2) -3/5

(3) 4/5

(4) 5/3

(1) -5/3

(2) -3/5

(3) 4/5

(4) 5/3

**Answer: (1) -5/3 **

Given that tan = sin / cos and sec = 1 / cos, then sec is either 5/3 or -5/3 depending upon the value of θ, which is this question put the point in Quadrant II where cosine is negative.

So the correct answer is Choice (1) -5/3.

*22. To solve the equation 7 / (x + 7) + 4x / (x - 7) = (3x + 7) / (x - 7), Joan’s first step is to multiply both sides by the least common denominator. Which statement is true?*

(1) -14 is an extraneous solution.

(2) 7 and -7 are extraneous solutions.

(3) 7 is extraneous solutions.

(4) There are no extraneous solutions.

(1) -14 is an extraneous solution.

(2) 7 and -7 are extraneous solutions.

(3) 7 is extraneous solutions.

(4) There are no extraneous solutions.

**Answer: (3) 7 is extraneous solutions. **

The equation is not defined for x = 7 or x = -7. If you multiply the equation by the least common demoninator, you need to multiple by (x + 7)(x - 7), which means that there will be two possible extraneous solutions. However, it's possible that either one, or both, are NOT solutions at all.

7 / (x + 7) + 4x / (x - 7) = (3x + 7) / (x - 7)

7 (x - 7) + 4x (x + 7) = (3x + 7)(x + 7)

7x - 49 + 4x^{2} + 28x = 3x^{2} + 28x + 49

x^{2} + 7x - 98 = 0

(x + 14)(x - 7) = 0

x = -14 or x = 7

Only 7 is an extraneous solution.

*23. Beginning July 1, 2019, Michelle deposited $250 into an account that yields 0.15% each month. She continued to make $250 deposits into this account on the first of each month for 3 years. Which expression represents the amount of money that was in the account after her last deposit was made on June 1, 2022?*

(1) 250(1.0015)

(2) 250(1.0015)

(3) (250 - 250(1.0015)

(4) (250 - 250(1.0015)

(1) 250(1.0015)

^{3}(2) 250(1.0015)

^{36}(3) (250 - 250(1.0015)

^{3}) / (1 - 1.0015)(4) (250 - 250(1.0015)

^{36}) / (1 - 1.0015)**Answer: (4) (250 - 250(1.0015) ^{36}) / (1 - 1.0015) **

Choices (1) and (2) are compound interest with no extra deposits. Eliminate Choices (1) and (2).

Choice (3) only has 3 months of interest. Eliminate Choice (3).

Choice (4) is the correct geometric series and the correct choice.

*24. A study of the red tailed hawk population in a given area shows the population, H(t), can be represented by the function H(t) = 50(1.19)*

(1) H(t) = 50(1.015)

(2) H(t) = 50(1.15)

(3) H(t) = 50(1.19)

(4) H(t) = 50(1.19)

^{t}where t represents the number of years since the study began. In terms of the monthly rate of growth, the population can be best approximated by the function(1) H(t) = 50(1.015)

^{12t}(2) H(t) = 50(1.15)

^{t/12}(3) H(t) = 50(1.19)

^{12t}(4) H(t) = 50(1.19)

^{t/12}

**Answer: (1) H(t) = 50(1.015) ^{12t} **

The annual rate needs to be converted to the monthly rate.

50(1.19^{1/12})^{12t} = 50(1.0146...)^{12t}

This is the same as Choice (1), which is the correct answer.

More to come.

Comments and questions welcome.

More Regents problems.

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