Algebra Problems of the Day (Integrated Algebra Regents, January 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, January 2011

Part I: Each correct answer will receive 2 credits.


6. What is the value of x in the equation 2(x − 4) = 4(2x + 1)?

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

Answer: 1) -2

You xan try each of the choices or you can solve the problem.

2(x − 4) = 4(2x + 1)

x - 4 = 2(2x + 1)

x - 4 = 4x + 2

-6 = 3x

-2 = x

That is Choice (1).

7. WThe rectangle shown below has a diagonal of 18.4 cm and a width of 7 cm.
To the nearest centimeter, what is the length, x, of the rectangle?

1) 11
2) 17
3) 20
4) 25

Answer: 2) 17

READ THE CHOICES!!! NOW READ THEM AGAIN!!!!!

Only one choice makes ANY sense, so it must be the answer.

We know that x CANNOT be 11 because 7 - 11 - 18.4 does NOT make a triangle. The two legs only add up to 18, which is less than 18.4. Eliminate Choice (1).

We also know that x CANNOT be greater than or equal to the hypotenuse of the right triangle, so it cannot be 20 or 25. Eliminate (3) and (4).

That leaves Choice (2) as the answer.

If you didn't see that, you had to do Pythagorean Theorem:

x² + 7² = 18.4²

x² = 289.56

x² = 289.56

x = 17.016...

8. When a³ − 4a is factored completely, the result is

1) (a - 2)(a + 2)
2) a(a - 2)(a + 2)
3) a²(a - 4)
4) a(a - 2)²

Answer: 2) a(a - 2)(a + 2)

Eliminate Choice (1) because it's only quadratic and not cubic. You can eitehr factor the expression or multiply the choices.

a(a² - 4)

a(a - 2)(a + 2)

This is Choice (2).

9. Which ratio represents sin x in the right triangle shown below.

1) 28/53
2) 28/45
3) 45/53
4) 53/28

Answer: 1) 28/53

The sine ratio is the length of the side opposite the angle, which is 28, over the length of the hypotenuse, which is 53.

So the sin x = 28/53.

Choice (2) shows tan x, which is the opposite over the adjacent side.

Choice (3) shows cos x, which is the adjacent over the hypotenuse.

Choice (4) shows the cosecant ratio, which is the reciprocal of the sine. It is useful to know sometimes, but not here.

10. What is the value of the expression (a³ + b⁰)² when a = −2 and b = 4?

1) 64
2) 49
3) -49
4) -64

Answer: 2) 49

The last thing that happens when you do Order of Operations, after substituting, is SQUARING whatever is in the parentheses. That means that the answer cannot be negative. Eliminate Choice (3) and (4).

Also notice that b has an exponent of 0, so it doesn't matter what b is, b⁰ = 1.

So (-2)³ = -8, and -8 + 1 = -7. (-7)² = 49. Which is Choice (2).

More to come. Comments and questions welcome.

More Regents problems.

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