Algebra Problems of the Day (Integrated Algebra Regents, January 2013)

While I'm waiting for new Regents exams to come along, I'm revisiting some of the older NY Regents exams.

More Regents problems.

Administered January 2013

Part I: Each correct answer will receive 2 credits.


6. The expression 100n² - 1 is equivalent to

(1) (10n + 1)(10n - 1)
(2) (10n - 1)(10n - 1)
(3) (50n + 1)(50n - 1)
(4) (50n - 1)(50n - 1)

Answer: (1) (10n + 1)(10n - 1)

The Difference of Squares Rules: when you have two terms which are perfect squares with a subtraction operation, then the expression factors into two conjugates where each term is a square root of one of the terms in the original expression. "Conjugates" means that the two binomials will be the same, except that one will have addition and the other will have subtraction. The result of this is when you multiply two conjugates, the "inner" and "outer" terms will create a zero pair and go away, leaving only two terms in the quadratic expression.

7. In right triangle ABC shown below, what is the value of cos A?

(1) 12/20
(2) 16/20
(3) 20/12
(4) 20/16

Answer: (2) 16/20

Cosine can never be greater than 1, so Choices (3) and (4) are eliminated immediately.

Since the Cosine is the side adjacent to the angle divided by the hypotenuse (the longest side, which makes the fraction less than 1), the ratio is 16/20.



8. A bag contains five green gumdrops and six red gumdrops. If Kim pulls a green gumdrop out of the bag and eats it, what is the probability that the next gumdrop she pulls out will be red?

(1) 5/11
(2) 5/10
(3) 6/11
(4) 6/10

Answer: (4) 6/10

There had been 11 gumdrops in the bag. One green was removed and not replaced, (at least we hope it wasn't replaced since it was eaten). That leaves 10 gumdrops in the bag, 6 of which are red.

The probability of taking out a red one next is therefore 6/10.

Choice (1) is the probability of green if the first green had been replaced.

Choice (2) is the probability of red if a red gumdrop had been removed and not replaced first.

Choice (3) is the probability of red if the first gumdrop had been replaced.

9. Which graph represents a function?

(see image)

Answer: (3)

A function can only have one y value for any given x value. That means that a function must past the Vertical Line Test -- if a Vertical Line can be drawn so that it passes through two points on the line, then the line cannot be a function.

Choice (1) is a circle which fails the Vertical Line Test.

Choice (2) is an actual Vertical Line, so it obviously fails the test.

Choice (4) shows every point from Quadrant II reflected in Quadrant III, so it fails the test.



10. The current population of a town is 10,000. If the population, P, increases by 20% each year, which equation could be used to find the population after t years?

(1) P =10,000(0.2)^t
(2) P =10,000(0.8)^t
(3) P =10,000(1.2)^t
(4) P =10,000(1.8)^t

Answer: (3) P =10,000(1.2)^t

If the population is increasing, that is growth, which requires a rate of greater than 1, so Choices (1) and (2) are eliminated because they show decay.

If it is growing an additional 20%, then next year there will be 120% of the current population, and 120% is 1.2 as a decimal.

Choice (4) is just there to match up with Choice (2).






More to come. Comments and questions welcome.

More Regents problems.