## Saturday, October 23, 2021

### 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

20. The table below displays the results of a survey regarding the number of pets each student in a class has. The average number of pets per student in this class is 2.

What is the value of k for this table?

1) 9
2) 2
3) 8
4) 4

There are 22 + k students in the room. They have a total of 6 + 20 + 10 + 4k = 36 + 4k pets.

According to the problem, (36 + 4k) / (22 + k) = 2

(36 + 4k) / (22 + k) = 2
(36 + 4k) = 2(22 + k)
36 + 4k = 2k + 44
2k = 8
k = 4

You could also solve this by plugging in the values for k to see which one works.

22. How many negative solutions to the equation 2x3 - 4x2 + 3x - 1 = 0 exist

1) 1
2) 2
3) 3
4) 0

If you graph y = 2x3 - 4x2 + 3x - 1, you will see that the curve only has one real root at x = 1. The other two roots are imaginary.

There are no negative roots.

23. A study finds that 80% of the local high school students text while doing homework. Ten students are selected at random from the local high school. Which expression would be part of the process used to determine the probability that, at most, 7 of the 10 students text while doing homework?

1) 10C6 (4/5)6 (1/5)4
2) 10C7 (4/5)10 (1/5)7
3) 10C8 (7/10)10 (3/10)2
4) 10C9 (7/10)9 (3/10)1

Part of the process means that it could be anything from the probability that 0 of 10 through 7 out of 10.

Even if you weren't familiar with how to set up the expression, there are hints in the other choices that allow you to eliminate them:

In Choice (2), the exponents are 10 and 7, but they should add up to 10.

In Choice (3), the exponents are 10 and 2, but they should add up to 10.

In Choice (4), the exponents are 9 and 1, which add up to 10, but the fractions are incorrect. The 80% is 4/5, not 7/10. Likewise, if we wanted at most 7, the first part wouldn't be 10C9 because we don't want 9 out of 10.

Choice (1) shows the probability of exactly 6 out of 10 text while doing homework, which is one part of the process.

24. In which interval of f(x) = cos(x) is the inverse also a function?

1) -π/2 < x < π/2
2) -π/2 < x < π/2
3) 0 < x < π
4) π/2 < x < 3π/2

Answer: 3) 0 < x < π

Between 0 and π, the value of f(x) is decreasing from 1 to -1. In that interval, no value repeats for f(x).

For the other intervals, the curve has either a minimum or maximum value, and the values for f(x) will repeat. You could draw a horizontal line through the graph and hit multiple points.

More to come. Comments and questions welcome.

More Regents problems.

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