Algebra Problems of the Day (Integrated Algebra Regents, August 2010)

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, August 2010

Part I: Each correct answer will receive 2 credits.


6. The quotient of (9.2 × 10⁶) and (2.3 × 10²) expressed in scientific notation is

1) 4,000
2) 40,000
3) 4 X 10³
4) 4 X 10⁴

Answer: 4) 4 X 10⁴

Put the expression (9.2 × 10⁶) / (2.3 × 10²) into your calculator, and you will get the result of 40,000. (Note: If you got 400,000,000, then you multiplied by 10² instead of dividing. You didn't use parentheses.)

However, 40,000 is NOT in scientific notation. It is equivalent to 4 X 10⁴, which is Choice (4).

7. In a recent town election, 1,860 people voted for either candidate A or candidate B for the position of supervisor. If candidate A received 55% of the votes, how many votes did candidate B receive?

1) 186
2) 837
3) 1,023
4) 1,805

Answer: 2) 837

If 55% voted for A and the rest voted for B, then 45% of the 1,860 people voted for B.

1860(.45) = 837, which is Choice (2).

8. WWhich expression is equivalent to 121 − x²?

1) (x - 11)(x - 11)
2) (x + 11)(x - 11)
3) (11 - x)(11 + x)
4) (11 - x)(11 - x)

Answer: 3) (11 - x)(11 + x)

The expression is the Difference Bewteen Two Perfect Squares, which can be factored into two conjugates, which is the square root of each of the perfect squares, in order.

Choices (1) and (4) are not conjuagates. These will each have a middle term of -22x.

Choice (2) has a product of x² - 121 instead of 121 - x². Eliminate this.

Choice (3) is the correct answer.

9. Given:

U = {1, 2, 3, 4, 5, 6, 7, 8}
B = {2, 3, 5, 6}
Set B is a subset of set U. What is the complement of set B?

1) { }
2) {2, 3, 5, 6}
3) {1, 4, 7, 8}
4) {1, 2, 3, 4, 5, 6, 7, 8}

Answer: 3) {1, 4, 7, 8}

The complement of B is every element of U that is not in B.

Removing 2, 3, 5, and 6 from U leaves 1, 4, 7, 8. This is Choice (3).

Choice (1) is the Empty (Null) Set. Choice (2) is Set B, not the complement. Choice (4) is U, the Universal Set.

10. Which graph can be used to find the solution of the following system of equations?

y = x² + 2x + 3
2y − 2x = 10

Answer: 1) [SEE IMAGE]

You can enter the first equation (the quadratic equation) into your graphing calculator. The second equation (the linear equation) has to be rewrtten to start with y = to use your calculator.

Without using the calculator, the middle term of the quadratic is "+ 2x" and the leading coefficient is positive 1, so the Axis of Symmetry is x = -1. This eliminates Choices (3) and (4). The linear equation will have a positive slope (-A/B), which eliminates Choice (2).

Choice (1) is correct.

If you needed to rewrite the linear equation:

2y − 2x = 10
2y = 2x + 10
y = x + 5

The correct linear equation is shown in (1) and (3).

More to come. Comments and questions welcome.

More Regents problems.

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