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.


11. Which verbal expression is represented by 2(x + 4)?

(1) twice the sum of a number and four
(2) the sum of two times a number and four
(3) two times the difference of a number and four
(4) twice the product of a number and four

Answer: (1) twice the sum of a number and four

Twice means you are multiplying something by two. Sum means that you are adding two things. Twice the sum means that you have to add something first before multiplying it.

Choice (2) would be 2x + 4.

Choice (3) would be 2(x - 4).

Choice (4) would be 2(4x).

12. How many cubes with 5-inch sides will completely fill a cube that is 10 inches on a side?

(1) 50
(2) 25
(3) 8
(4) 4

Answer: (3) 8

You can fit two smaller cubes side by side and front to back. That means you can have a layer of four cubes. You also have a second layer of four cubes on top of that for a total of 8.

Mathematically, (2)³ = 8.



13. A school newspaper will survey students about the quality of the school’s lunch program. Which method will create the least biased results?

(1) Twenty-five vegetarians are randomly surveyed.
(2) Twenty-five students are randomly chosen from each grade level.
(3) Students who dislike the school’s lunch program are chosen to complete the survey.
(4) A booth is set up in the cafeteria for the students to voluntarily complete the survey.

Answer: (2) Twenty-five students are randomly chosen from each grade level.

You want to choose students at random from across the entire student population.

Choice (1) specifically targets vegetarians, who would likely have a bias toward the school lunch program.

Choice (3) is selecting people who dislike the menu in a survey about the quality of the menu, and that is a bias.

Choice (4) is self-selecting, which is biased because people with stronger feelings are more likely to take part in it.

14. The vertex of the parabola y = x² + 8x + 10 lies in Quadrant

(1) I
(2) II
(3) III
(4) IV

Answer: (3) III

You can quickly graph it and see that it is in Quadrant III.

Or you could find that the Axis of Symmetry is x = -b/(2a) = -8/2 = -4. And when x = -4, y = (-4)² + 8(-4) + 10 = -6. The vertex is (-4, -6), which is in Quadrant III.



15.In the figure below, ABCD is a square and semicircle O has a radius of 6.

What is the area of the figure?

(1) 36 + 6π
(2) 36 + 18π
(3) 144 + 18π
(4) 144 + 36π

Answer: (3) 144 + 18π

The area of the semicircle is 1/2 π r², which is 1/2 π (6)² = 1/2 π (36) = 18π. At this point, you can eliminate (1) and (4).

The area of the square is s² and the side is equal to the diamater of the semicircle, which is twice the radius. So A = (12)² = 144.






More to come. Comments and questions welcome.

More Regents problems.