Algebra 2 Problems of the Day (Algebra 2 Regents, June 2012)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Algebra 2/Trigonometry Regents, June 2012

Part I: Each correct answer will receive 2 credits.


1. What is the product of (2/5 x - 3/4 y²) and (2/5 x + 3/4 y²)?

1) 4/25 x² - 9/16 y⁴
2) 4/25 x - 9/16 y²
3) 2/25 x² - 3/4 y⁴
4) 4/5 x

Answer: 1) 4/25 x² - 9/16 y⁴

The multiplication of two conjugates (the same binomial except one has a plus and the other a minus) will have a product that is the Difference of Two Perfect Squares.

In general: (ax + by) * (ax - by) = a²x² - b²y².

Replace a with 2/5 and b with 3/4 and you get:

(2/5 x + 3/4 y) * (2/5 x - 3/4 y) = (2/5)²x² - (3/4)²y²
= 4/25 x² - 9/16 y²

2. What is the domain of the function shown below?

1) -1 < x < 6
2) -1 < y < 6
3) 2 < x < 5
4) 2 < y < 5

Answer: 1) -1 < x < 6

Domain refers to the x values, so eliminate Choices (2) and (4) which refer to y.

The only values of x that have points are from -1 through 6, inclusive. That is Choice (1).

Note that Choice (4) is the range of the function.

3. What is the solution set for 2cosθ - 1 = 0 in the interval 0° < θ < 360°

1) {30°, 150°}
2) {60°, 120°}
3) {30°, 330°}
4) {60°, 300°}

Answer: 4) {60°, 300°}

Solve for cosθ, and then find the values of θ between 0 and 360 that will give the correct answer.

2cosθ - 1 = 0
2cosθ = 1
cosθ = 1/2

The values on the Unit Circle when cosθ = 1/2 is at 60 degrees above and below the x-axis. However, since the stated interval was 0 to 360, we want 60 and 300, not 60 and -60.

-5 - -3 = -2 and 2 - 6 = -4

Point B is (-5 - 2, 2 - 4), or (-7, -2)

4. The expression ∛(64a¹⁶) is equivalent to

1) 8a⁴
2) 8a⁸
3) 4a⁵ ∛(a)
4) 4a⁵ ∛(a⁵)

Answer: 3) length of sides

The cube root of 64 is 4 because 4³ = 64.

To find the cube root of a¹⁶, break it into a⁵ * a⁵ * a⁵ * a¹. So the cube root is a⁵ * ∛(a)

So the final answer is 4 * a⁵ * ∛(a)

A dilation will retain the orientation and the shape of the original. Since the shape is similar, the size angles will be the same and lines that were parallel will remain parallel.

5. Which summation represents 5 + 7 + 9 + 11 + ... + 43?

Answer: 2) (see image)

The summations are like little loops (FOR/NEXT, or DO UNTIL, etc, for programmers)

In Choice (1), all the integers from 5 to 43 are being added, but we only want the odd numbers. Eliminate Choice (1).

In Choice (2), the first number is 2(1) + 3 = 5 and the last is 2(20) + 3 = 43. The sequence is counting by 2s. This is the solution.

In Choice (3), the first number is 2(4) - 3 = 5 but the last is 2(24) - 3 = 45, not 43. Had the top number been 23 instead of 24, this would have worked. Eliminate Choice (3).

In Choice (4), the sequence is increasing by 3 and not 2. Eliminate Choice (4).

Once you see that the series is increasing by 2, Choices (1) and (4) should have been eliminated immediately.

More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction! You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below. Preorder the softcover or ebook at Amazon. Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides. Available in softcover or ebook at Amazon. If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.