Wednesday, November 03, 2021

Algebra 2 Problems of the Day (Algebra 2/Trigonometry Regents, January 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, January 2012

Part I: Each correct answer will receive 2 credits.

11. When x-1 + 1 is divided by x + 1, the quotient equals

1) 1
2) 1/x
3) x
4) -1/x

Answer: 2) 1/x

You could multiply each response by (x + 1), which would immediately eliminate Choices (1) and (3).

Choice (2) is (1/x)(x + 1) = 1 + 1/x = 1 + x-1, which is what we'll looking for.

Choice (4) is (-1/x)(x + 1) = -1 - 1/x = -1 - x-1, which is probably there for people who confuse negative exponents with negative numbers.

(x-1 + 1)(x / x) = (1 + x) / x

Divide (x + 1) / x by (x + 1) and the result is 1 / x, which is Choice (2).

12. If the amount of time students work in any given week is normally distributed with a mean of 10 hours per week and a standard deviation of 2 hours, what is the probability a student works between 8 and 11 hours per week?

1) 34.1%
2) 38.2%
3) 53.2%
4) 68.2%

Answer: 3) 53.2%

Look at the image below, which can be found in the back of the booklet. Students working for 8-10 hours is one standard deviation below the mean. There would be 15.0% and 19.1% of the population. Students working 10-11 hours would fall in one-half of a standard deviation above the mean. That's another 19.1%.

So 15.0 + 19.1 + 19.1 = 53.2

13. What is the conjugate of 1/2 + 3/2i?

1) -1/2 + 3/2i
2) 1/2 - 3/2i
3) 3/2 + 1/2i
4) -1/2 - 3/2i

Answer: 2) 1/2 - 3/2i

Like with the Difference of Squares Rule, conjugates are two binomials that are the same except that one has a plus separating them and the other has a minus. When multiplied, this causing the "Inner" and "Outer" product to sum up to a zero pair (that is, they cancel).

With Complex Numbers there is the additional note that real term comes first and the imaginary term comes second, so that the sign to be changed is in between the Real and Imaginary, and not the other way around.

So the conjugate of 1/2 + 3/2i is 1/2 - 3/2i. It's that simple. No need to overthink it.

Choice (1) flips the wrong sign.

Choice (4) flips both signs.

Choice (3) flips the numbers about. Don't do that.

14. Given angle A in Quadrant I with sin A = 12/13 and angle B in Quadrant II with cos B = -3/5, what is the value of cos (A - B)?

1) 33/65
2) -33/65
3) 63/65
4) -63/65

Answer: 1) 33/65

First find A and B, so you know the value of A - B.

A = sin-1(12/13) = 67.38 degrees

B = cos-1(-3/5) = 126.87 degrees

A - B = -59.49, so the angle with be in Quadrant IV, and the cosine will be positive. Also, it is very close to 60 degrees, and the cosine of 60 is 1/2, so the answer should be very close to one-half.

cos(-59.49) = 0.522. Looking at the four choices, the answer has to be Choice (1).

(0.522)(65) = 33.97, and we can assume the difference comes from rounding errors made during the previous calculations.

Note that the problem used a 3-4-5 right triangle and a 5-12-13 right triangle. I've used 3-4-5 enough to know that the angles are approximately 37-53-90. I'm less familiar with the angles of 5-12-13, but they are approximately 23-67-90. Knowing this could've gotten you 67 and 127 quicker, and then -60.

15. Which expression represents the third term in the expansion of (2x4 - y)3?

1) -y3
2) -6x4y2
3) 6x4y2
4) 2x4y2

Answer: 3) 6x4y2

First off, eliminate Choice (1). It's just silly.

The first term will have x(4)(3) and each term after will go down by four. The first term with have y0 and each term will count up by 1. This means that the third term will have x4y2, which is in three of the four choices.

The coefficient of the third term will be 3C2(2)1(-1)2 = 6.

More to come. Comments and questions welcome.

More Regents problems.

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