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 II: Each correct answer will receive 2 credits. Partial credit is available.


28. Determine the sum and the product of the roots of the equation 12x² + x - 6 = 0.

Answer:

You do not need to find what the roots are to get the sum and product of the roots.

The product of the roots is given by the formula c / a. The sum is given by the formula -b / a.

The sum is -1 / 12 = -1/12

The product is -6 / 12 = -1/2

If you wanted to find the roots:

12x² + x - 6 = 0, a = 12, b = 1, c = -6.
Find the factors of ac that add up to b.
The factors of -72 that add up to 1 are 9 and -8.
12x² + 9x - 8x - 6 = 0
3x(4x + 3) - 2(4x + 3) = 0
(3x - 2)(4x + 3) = 0
3x - 2 = 0 or 4x + 3 = 0
3x = 2 or 4x = -3
x = 2/3 or x = -3/4.

The product of the roots is (2/3)(-3/4) = -6/12 = -1/2

The sum of the roots is 2/3 - 3/4 = 8/12 - 9/12 = -1/12.

Make sure these are labeled.

29. Solve algebraically for x:

log₂₇ (2x - 1) = 4/3

Answer:

Rewrite the equation using exponents: 27^(4/3) = 2x - 1
3⁴ = 2x - 1
81 = 2x - 1
82 = 2x
x = 41

30. Find the number of possible different 10-letter arrangements using the letters of the word “STATISTICS.”

Answer:

The number of 10-letter arrangements of 10 letters is 10! (that is, 10 factorial). However, there are duplicate letters that need to be accounted for because of the repetition: 3 Ss, 3 Ts, and 2 Is.

10!
3! * 3! * 2!

= 10 * 9 * 8 * 7 *6 * 5 * 4 * 3 * 2 * 1
3 * 2 * 1 * 3 * 2 * 1 * 3 * 2 * 1

= 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
3 * 2 * 1 * 3 * 2 * 1 * 2 * 1

= 10 * 7! = 10 * 5040 = 50,400

You could have entered the equation into your calculator at any point after writing the factorial notation, if your calculator handles it.

I simplified to 7! just because I knew that 7! = 5040, which is basically where I have the factorials memorized to. For 8!, I usually do 8 * 5040 in my head. The numbers aren't so bad.

31. Express the product of cos 30° and sin 45° in simplest radical form.

Answer:

cos 30° = SQRT(3) / 2 and sin 45° = SQRT (2) / 2

The product of SQRT (3) / 2 * SQRT (2) / 2 = SQRT (6) / 4. This cannot be reduced.

Looking at the answer key, they also accepted SQRT (3) / (2 SQRT (2)).

More to come. Comments and questions welcome.

More Regents problems.

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