Algebra 2 Problems of the Day (Algebra 2/Trigonometry Regents, June 2013)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Algebra 2/Trigonometry Regents, June 2013

Part I: Each correct answer will receive 2 credits.


25. Expressed with a rational denominator and in simplest form,

x / (x - SQRT(x) ) is:

1) ( x² + x SQRT(x) ) / ( x² - x )
2) - SQRT(x)
3) ( x + SQRT(x) ) / (1 - x)
4) ( x + SQRT(x) ) / (x - 1)

Answer: 4) ( x + SQRT(x) ) / (x - 1)

To rationalize the denominator, multiply by the conjugate. Note that it says simplest form, so after you multiply, you might be able to simplify.

x / ( x - SQRT(x) ) * ( x + SQRT(x) ) / ( x + SQRT(x) )
= x ( x + SQRT(x) ) / ( ( x - SQRT(x) ) ( x + SQRT(x) ) )
= x ( x + SQRT(x) ) / ( x² - x )
= x ( x + SQRT(x) ) / ( x (x - 1) )
= ( x + SQRT(x) ) / (x - 1)

26. What is the common ratio of the sequence

1/64 a⁵b³, -3/32 a³b⁴, 9/16 ab⁵, ... ?

1) -3b/(2 a²)
2) -6b/(a²)
3) -3a²/b
4) -6a²/b

Answer: 2) -6b/(a²)

Look at what is happening in each of the terms:

The numerator is being multiplied by -3, and the denominator is divided by 2, which means it is multipled by 1/2. So each term is multiplied by -3 / (1/2), which is -6.

Each a term has its exponent reduced by 2, so that's a^-2 or 1/a².

Each b term has its exponent increased by 1, so that's b¹ or just b.

That makes the common ratio -6b / a²

27. In triangle KLM, KL = 20, LM = 13, and m∠K = 40. The measure of ∠M

1) must be between 0° and 90°
2) must equal 90°
3) must be between 90° and 180°
4) is ambiguous

Answer: 1) must be between 0° and 90°

Angle M is opposite side KL. Angle K is opposite LM.

THe Law of Sines says that sin M / 20 = sin 40 / 13

So sin M = 20 * sin 40 / 13 = 0.9889

M = 81.45 degrees

According to the Regents, the answer is (4) ambiguous, but I have to assume that's based on just looking at the size of the angles. Angle M MUST BE bigger than Angle K. Without doing any work, you don't know if it is over 90, 90 or under 90. HOWEVER, with the Law of Sines, we have enough information to calculate the answer. We KNOW FOR A FACT that it is about 81 degrees.

Whether the Regents ever corrected this mistake, I have no idea.

If your teacher is using these old questions for test prep and marks you incorrect, bring this to his or her attention, politely.

End of Part I

More to come. Comments and questions welcome.

More Regents problems.

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