Algebra Problems of the Day (Integrated Algebra Regents, June 2011)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones. The Integrated Algebra Regents covered most of the same material as the current Algebra Regents, with a few differences.

More Regents problems.

Integrated Algebra Regents, June 2011

Part I: Each correct answer will receive 2 credits.


6. What is 3√(250) expressed in simplest radical form?

1) 5√(10)
2) 8√(10)
3) 15√(10)
4) 75√(10)

Answer: 3) 15√(10)

It is obvious that 25 is a factor of 250. Is it the largest perfect square that is a factor of 250? Yes, 250 = 25 * 10, and 10 does not have any perfect squares as factors.

An alternative way to look at this is that 250 = 2 * 5 * 5 * 5, which means that you can remove one pair of factors from under the square root symbol.

So 3 √(250) = 3 * √(25) * √(10) = 3 * 5 * √(10) = 15√(10).

7. A survey is being conducted to determine which school board candidate would best serve the Yonkers community. Which group, when randomly surveyed, would likely produce the most bias?

1) 15 employees of the Yonkers school district
2) 25 people driving past Yonkers High School
3) 75 people who enter a Yonkers grocery store
4) 100 people who visit the local Yonkers shopping mall

Answer: 1) 15 employees of the Yonkers school district

Looking for the most bias, Choice (1) has a combination of the fewest people surveyed (and 15 is a very small sample size) and the population is the most limited, oversampling for one specific demographic (school district employees).

8. An 8-foot rope is tied from the top of a pole to a stake in the ground, as shown in the diagram below.

If the rope forms a 57° angle with the ground, what is the height of the pole, to the nearest tenth of a foot?

1) 4.4
2) 6.7
3) 9.5
4) 12.3

Answer: 2) 6.7

You have a right triangle. You know the length of the hypotenuse. You are looking for the side opposite of the known angle. That means that you need to use the sine function.

sin 57 = x / 8

x = 8 * sin 57 = 6.7 ..., which is choice (2).

Choice (1) is the result if you used cosine by mistake.

Choices (3) and (4) both larger than the hypotenuse and can be immediately eliminated. These answers from writing hypotenuse over opposite (or over the adjacent) and solving the proportion incorrectly.

9. How many different ways can five books be arranged on a shelf ?

1) 5
2) 15
3) 25
4) 120

Answer: 4) 120

There are 5! permutations, which is 5! = 5 * 4 * 3 * 2 * 1 = 120.

10. What is the slope of the line passing through the points (-2,4) and (3,6)?

1) -5/2
2) -2/5
3) 2/5
4) 5/2

Answer: 3) 2/5

The x values are increasing and the y values are increasing, so the slope is positive. Eliminate Choices (1) and (2).

You want the change in the rise over the run, which is the change in y over the change in x.

6 - 4 = 2 and 3 - (-2) = 5, so the slope is 2/5, which is Choice (3).

More to come. Comments and questions welcome.

More Regents problems.

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