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

Part I: Each correct answer will receive 2 credits.


6. Which expression is equivalent to (9x²y⁶)^(-1/2)

1) 1/(3xy³)
2) 3xy³
3) 3/(xy³)
4) xy³/3

Answer: 1) 1/(3xy³)

The exponent (-1/2) is the product of (-1)(1/2). The (-1) tells you to take the reciprocal, and the (1/2) tells you to take the square root. When taking the square root, apply it to the coefficient the normal way and halve all the exponents on the variables.

(9x²y⁶)^(-1/2)
= (9x²y⁶)^((-1)(1/2)
= (3xy³)^(-1)
= 1/(3xy³)

7. In a certain high school, a survey revealed the mean amount of bottled water consumed by students each day was 153 bottles with a standard deviation of 22 bottles. Assuming the survey represented a normal distribution, what is the range of the number of bottled waters that approximately 68.2% of the students drink?

1) 131-164
2) 131-175
3) 142-164
4) 142-175

Answer: 2) 131-175

The 68.2% of the population refers to one standard deviation below and above the mean.

153 - 22 = 131 and 153 + 22 = 175.

8. What is the fourth term in the binomial expansion (x - 2)⁸?

1) 448x⁵
2) 448x⁴
3) -448x⁵
4) -448x⁴

Answer: 3) -448x⁵

Refer to the Binomial Theorem in the back of the booklet.

The first term will have x⁸ and each term after will have the exponent reduced by 1. So the 2nd will be x⁷, the 3rd will be x⁶ and the 4th will be x⁵. Eliminate Choices (2) and (4).

The first term will have (-2)⁰ -- that is, it's not there because it only has factors of x. The second term with have (-2)¹, then (-2)², then (-2)³. This is (-8), which means that the coefficient will be negative. The answer must be Choice (3).

The last part of the term is _nC₃. If you use your calculator, or if you can doodle that far along Pascal's triangle, you will see that

₈C₃ = 8 * 7 * 6 / (3 * 2 * 1) = 56

Multiply (-8)(56) = -448

9. Which value of k satisfies the equation 8^3k+4 = 4^2k-1?

1) -1
2) 9/4
3) -2
4) -14/5

Answer: 4) -14/5

8 is 2³ and 4 is 2². You can rewrite each expression as a power of 2 and then compare the exponents.

8^3k+4 = (2³)^{3k + 4} = 2^{9k + 12}

4^2k-1 = (2²)^{2k - 1} = 2^{4k - 2}

9k + 12 = 4k - 2
5k = -14
k = -14/5

10. There are eight people in a tennis club. Which expression can be used to find the number of different ways they can place first, second, and third in a tournament?

1) ₈P₃
2) ₈C₃
3) ₈P₅
4) ₈C₅

Answer: 1) ₈P₃

Since the order matters, you want a Permutation not a Combination. Eliminate Choices (2) and (4). Also, you should have noticed that (2) and (4) have the SAME value.

There are 8 possibilities for first, then 7 for second and then 6 for third. That is 8P3.

More to come. Comments and questions welcome.

More Regents problems.

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